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BOUGHT WITH THE INCOME
FROM THE
SAGE ENDOWMENT FUND
THE GIFT OF
Henrg ^> Sa^e
1891
/?i3fc7i-^- \tlV/ii/o<l.
DnSTMfMATICS IWSARY
DIFFEEENTIAL AND INTEGEAL
CALCULUS
By D. A. MURRAY, Ph.D.
Pbofesbob of Applibd Matuehatics in McGill Untvebsity.
INTRODUCTORY COURSE IN DIFFERENTIAL EQUA-
TIONS, FOR Students in Classical and Enoinkeb-
ING Colleges. Pp. xvi + 236.
A FIRST COURSE IN INFINITESIMAL CALCULUS.
Pp. xvii + 439.
DIFFERENTIAL AND INTEGRAL CALCULUS. Pp.
xvlii + 491.
PLANE TRIGONOMETRY, roK Colleoes and Second-
ABT Schools. With a Protractor. Pp. xiii + 212.
SPHERICAL TEIGONOMETRY, foe Colleges and
Secondaet Schools. Pp. x + 114.
PLANE AND SPHERICAL TEIGONOMETRY. In One
Volume. With a Protractor. Pp. 349.
PLANE AND SPHERICAL TRIGONOMETRY AND
TABLES. In One Volume. Pp. 448.
PLANE TRIGONOMETRY AND TABLES. In One Vol-
ume. With a Protractor. Pp. 324.
LOGARITHMIC AND TRIGONOMETRIC TABLES. Five-
place AND FOUE-PLACE. Pp. 99.
NEW YORK: LONGMANS, GREEN, & CO.
DIFFERENTIAL AND INTEGRAL
CALCULUS
BY
DANIEL A. MURRAY, Ph.D.
Fbofessos of Applied Mathematics in McGill
Ukivebsitt
LONGMANS, GREEN, AND CO.
91 AND 93 FIFTH AVENUE, NEW YORK
LONDON, BOMBAY, AND CALCUTTA
1908
COPTBIOHT, liOS, BT
LONGMANS, GEEEN, AND CO.
All rights reserved.
^"■%
Kfotiniiol) ^tim
J. S. Gushing Co. — Berwick & Smith Co.
Norwood, Mass., U.S.A.
PEEFACE.
The topics in this book are arranged for primary courses in
calculus in which the formal division into differential calculus
and integral calculus is deemed necessary. The book is mainly
made up of matter from my Infinitesimal Calculus. Changes,
however, have been made in the treatment of several topics, and
some additional matter has been introduced, in particular that
relating to indeterminate forms, solid geometry, and motion. The
articles on motion have been written in the belief that familiarity
with the notions of velocity and acceleration, as treated by the
calculus, is a great advantage to students who have to take
mechanics.
Part of the preface of my Infinitesimal Calculus applies equally
well to this book. Its purpose is to provide an introductory
course for those who are entering upon the study of calculus
either to prepare themselves for elementary work in applied
science or to gratify and develop their interest in mathematics.
Little more has been discussed than what may be regarded as the
essentials of a primary course. An attempt is made to describe
and emphasise the fundamental principles of the subject in such
a way that, as much as may reasonably be expected, they may
be clearly understood, firmly grasped, and intelligently applied
by young students. There has also been kept in view the devel-
opment in them of the ability to read mathematics and to prose-
cute its study by themselves.
With regard to simplicity and clearness in the exposition of
the subject, it may be said that the aim has been to write a book
that will be found helpful by those who begin the study of
calculus without the guidance and aid of a teacher. For these
students more especially, throughout the work suggestions and
remarks are made concerning the order in which the various
V
VI PREFACE.
topics may be studied, the relative importance of the various
topics in a first study of calculus, the articles that must be
thoroughly mastered, and the articles that may advantageously
be omitted or lightly passed over at its first reading, and so on.
The notion of anti-differentiation is presented simultaneously
with the notion of differentiation, and exercises thereon appear
early in the text ; but when integration is formally taken up the
idea of integration as a process of summation is considered before
the idea of integration as a process which is the inverse of
differentiation. There is considerable difference of opinion as to
the propriety or the advantage of this order. The decision to
follow it here has been made mainly for the reason that students
appear — at least so it seems to me, but other teachers may have
a different experience — to understand more clearly and vividly
the relation of integration to many practical problems when the
summation idea is put in the forefront. In teaching, the one
order can be taken as readily as the other.
In several technical schools the time assigned to calculus is
not sufficient for a fair study of Taylor's theorem. What may
be regarded as the irreducible requisite for a slight working
acquaintance with Taylor's and Maclaurin's series is indicated
at the beginuing of Chapter XV., and may be taken at an early
stage in the course.
An explanation of hyperbolic functions can be made more
naturally and more fully, perhaps, in a course in calculus than
in any other course in elementary mathematics. For this reason,
and also because students will meet them in their later work and
reading, a note on these functions appears in the latter part of
the book.
Owing to the pressure of other subjects the time allotted to
mathematics in quite a number of technical schools is rather
brief. Where this is the case, and where there is a lack of
maturity in the students, it is better not to try to cover too
much ground, but to lay stress on fundamental principles, to
drill in the elementary processes, and to train in making simple
applications. Thus this book, small as it may be regarded even
for a short course, contains more matter than can be thoroughly
studied in the few months allotted to calculus in colleges and
PREFACE. vii
technical schools where such conditions exist. Several topics,
however (for example, the investigation of series), which in some
cases are not studied by technical students owing to lack of time,
are very important, particularly for those who take a first course
in the calculus as an introduction to a more extended study of
the subject and as part of the preparation necessary for more
advanced work in mathematics. For the sake of these students
more especially, but not exclusively on their account, many definite
references for collateral reading or inspection are given throughout
the text.
It is hoped that these references will add to the helpfulness
of the book. With but very few exceptions those are chosen
which are easily accessible to all college students. Some of the
references will aid the learner by presenting an idea of the text
in the words of another ; but the larger number of them are
intended to direct students to places where they will either re-
ceive fuller information or be impressed with some of the impor-
tant modern ideas of mathematics. Turning up such references
as these will increase the mathematical interest of the student
and widen his outlook. It will also help to train the pupils in
the use of mathematical literature, and, by arousing and exercis-
ing their critical faculties, will greatly benefit those who may
intend to teach mathematics in the secondary schools. Of course
the lists of references are not exhaustive, and, while care has
been taken in making them, it is to be expected that several
othet equally serviceable lists can be arranged. It is intended
that these lists shall be revised and supplemented by those who
may use the book.
Not many examples involving a technical knowledge of engi-
neering, physics, or chemistry have been inserted. Few young
students understand examples of this kind without considerable
explanation, and thus it seems better to refer the pupils to the
more specialised text-books dealing with calculus (for instance,
those of Perry, Young and Linebarger, and Mellor), which contain
many examples of a technical character.
For learners who can afford but a minimum of time for this
study the essential articles of a short course are indicated after
the table of contents.
Vlll PREFACE.
I take this opportunity of thanking Mr. T. Eidler Davies,
Lecturer in Mathematics at McGill University, for his kind
assistance in the revision of the proof sheets.
D. A. MURKAY.
Jolt 6, 1908.
CONTENTS.
DIFFERENTIAL CALCULUS.
ART.
2.
3.
4.
5.
6.
CHAPTER I.
Introductory Problems.
Speed of a moving train
To determine the speed of a falling body
To determine the slope of a tangent
To determine the area of a plane figure .
To find a function when its rate of change is known
To find the equation of a curve when its slope is known
Elementary notions used in infinitesimal calculus .
PAGE
2
2
6
10
11
11
11
CHAPTER II.
ALGEBRAit; NOTIOTSS WHICH ABB FREQUENTLY USED IN THir CaLCULUS.
8. Variables ... 13
9. Functions 14
10. ConsUnts 16
11. Classification of functions 16
12. Notation 18
13. Graphical representation of functions of one variable ... 19
14. Limits 20
15. Notation 23
15 a. Continuous variation. Interval of variation .... 24
16. Continuous functions. Discontinuous functions .... 25
CHAPTER m.
Infinitesimals, Derivatives, Differentials, Anti-derivatives, and
Anti-differentials.
18. Infinitesimals, infinite numbers, finite numbers
19. Orders of magnitude. Orders of infinitesimals.
20. Changes in the variable and the function
21. Comparison of these corresponding changes
ix
. 28
Orders of infinites 29
. 30
. 31
X CONTENTS.
AET. FACE
22. The derivative of a function of one variable 32
23. Notation 35
24. Tiie geometrical meaning and representation of the derivative of a
function . . . . 37
25. The physical meaning of the derivative of a function ... 39
26. General meaning of the derivative : the derivative is a rate . . 40
27. Differentials ... 42
27 a. Anti-derivatives and anti-difierentials 45
CHAPTER IV.
DiFFEKENTIATION OF THE ORDINARY FUNCTIONS.
General BesuUs in Differentiation.
29. The derivative of the sum of a function and a constant, say
(t>(x) + c 46
30. The derivative of the product of a constant and a function, say
c<f,(x) 48
31. The derivative of the sum of a finite number of functions . . 49
32. The derivative of the product of two or more functions ... 50
33. The derivative of the quotient of two functions .... 62
34. The derivative of a function of a function 64
35. The derivative of one variable with respect to another when both
are functions of a third variable .... .55
36. Differentiation of inverse functions . . ... 66
Differentiation of Particular Functions.
A. Algebraic Functions.
37. Differentiation of a" . . 56
B. Logarithmic and Exponential Functions.
38. Note. To find lim„^ (l + ^Y 61
39-41. Difierentiatioii of log„ u, a", w . . . . . 62-^
C. Trigonometric Functions.
42-48. Differentiation of sin u, cos u, tan u, cot u, sec u, esc u, vers u 66-71
D. Inverse Triyonometric Functions.
49-66. Differentiation of sin"'^, cos"'m, tan-'«, c^t"' «, sec"'«,
csc-^M, vers-'tt 71-76
56. Differentiation of implicit functions : two variables ... 76
CONTENTS. XI
CHAPTER V.
Some Geometrical, Physical, and Analytical Applications.
Geometric Derivatives and Differentials.
AET. PAGE
59. Slope of a curve at any point : rectangular coordinates . . .79
60. Angles at which two curves intersect 81
61. Equations of the tangent and normal drawn at a point on a curve . 88
62. Lengths of tangent, subtangent, normal, and subnormal ; rectangu-
lar coordinates 84
63. Slope of a curve at any point : polar coordinates ... 87
64. Lengths of tangent, subtangent, normal, and subnormal : polar
coordinates 88
65. Applications involving rates ... . . 90
66. Small errors and corrections ; relative error 92
66 a. Applications to algebra . 93
67. Geometric derivatives and differentials . . 95-102
CHAPTER VI.
Successive Differentiation.
68. Successive derivatives 103
69. The nth derivative of some particular functions .... 108
70. Successive differentials 109
71. Successive derivatives of y with respect to x when both are func-
tions of a third variable 109
72. Leibnitz's theorem 110
73. Application of differentiation to elimination Ill
CHAPTER VII.
Further Analytical and Geometrical Applications.
74. Increasing and decreasing functions 113
75. Maximum and minimum values of a function. Critical points on
the graph, and critical values of the variable .... 114
76. Inspection of the critical values of the variable for maximum or
minimum values of the function 117
77. Practical problems in maxima and minima ... 121
78. Points of inflexion : rectangular coordinates . ... 126
CHAPTER VIII.
Differentiation OF Functions of Several Variables.
79. Partial derivatives. Notation . . ... . 128
80. Successive partial derivatives 131
XU . CONTENTS.
AKT. PAGE
81. Total rate of variation of a function of two or more variables . 132
82. Total differential 134
83. Approximate value of small errors 136
84. Differentiation of implicit functions ; two variables . . . 137
85. Condition that an expression of the form Fdx + Qdy be a total
differential 138
86. Illustrations: partial differentials, total differentials, partial de-
rivatives. Illustration A 139
87. Illustration B . . . . .... 140
88. Illustration C 141
CHAPTER IX.
Change of Variable.
89. Change of variable 143
90. Interchange of the dependent and independent variables . . 143
91. Change of the dependent variable 144
92. Change of the independent variable 145
93. Dependent and independent variables both expressed in terms of a
single variable 146
CHAPTER X.
Concavity and Convexity. Contact and Curvature. Evoluteb
AND Involutes.
94. Concavity and convexity : rectangular coordinates . . 148
95. Order of contact ... . . 149
96. Osculating circle . 152
97. The notion of curvature 153
98. Total curvature. Average curvature. Curvature at a point . 154
99. The curvature of a circle 155
100. To find the curvature at any point of a curve : rectangular coordi-
nates 155
101. The circle of curvature at any point of a curve .... 156
102. The radius of curvature : polar coordinates . ... 159
103. Evolute of a curve 160
104. Properties of the evolute 161
105. Involutes of a curve 164
CHAPTER XI.
Rolle's Theorem. Theorems of Mean Value. Approximate
Solution of Equations.
107. Rolle's theorem . . 166
108. Theorem of mean value 169
CONTENTS. xiii
109. Approximate solution of equations 171
110. Theorem of mean value derived from RoUe's theorem . . . 174
111. Another form of the theorem of mean value 176
112. Second theorem of mean value 176
113. Extended theorem of mean value 177
CHAPTER XII.
Indeterminate Forms.
114. Indeterminate forms ....
115. Classification of indeterminate forms
116. Generalized theorem of mean value
117. Evaluation of functions when they take the form -
118. Evaluation of functions when they take the form -^
119. Evaluation of other indeterminate forms
180
181
182
183
186
187
CHAPTER XIII.
Special Topics relating to Corves.
120. Family of curves. Envelope of a family of curves . . 190
121. Locus of ultimate intersections of the curves of a family . 191
122. Theorem 193
123. To find the envelope of a family of curves having one parameter . 194
124. Envelope of a family of curves having two parametei's . . .197
125. Kectilinear asymptotes 199
126. Asymptotes parallel to the axes 201
127. Oblique asymptotes ... ... 203
128. Rectilinear asymptotes : polar coordinates . . 205
129. Singular points 206
130. Multiple points 206
131. To find multiple points, cusps, and isolated points . . . 209
132. Curve tracing 211
133. Note supplementary to Art. 127 212
CHAPTER XIV.
Applications to Motion.
134. Speed, displacement, velocity
136. To find velocity of a point moving on a curve
136. Composition of displacements
137. Resolution of a displacement into components
138. Composition and resolution of velocities
139. Component velocities of a point moving on a curve 220
214
216
216
218
219
xiv CONTENTS.
ABT. PAGE
140. Acceleration 223
141. Acceleration : particular cases 224
CHAPTEB XV.
Infinite Series.
142. Infinite series : definitions, notation
143. Questions concerning infinite series
144. Study of infinite series
145. Definitions. Algebraic properties of infinite series
146. Tests for convergence
147. Differentiation of infinite series term by term
148. Examples in the differentiation of series
230
231
233
234
237
240
240
CHAPTER XVI.
Taylor's Theorem.
150. Derivation of Taylor's theorem 242
161. Another form of Taylor's theorem 246
152. Maclaurin's theorem and series . 247
153. Relations between the circular functions and exponential functions 250
154. Another method of deriving Taylor's and Maclaurin's series . 252
155. Application of Taylor's theorem to the determination of condi-
tions for maxima and minima . . . . . 254
156. Application of Taylor's theorem to the deduction of a theorem on
the contact of curves 255
167. Applications of Taylor's theorem in elementary algebra . . 256
CHAPTER XVII.
Applications to Surfaces and Twisted CnRVBS.
158. Introductory .....
159. Tangent line to a twisted curve
160. Equation of plane normal to a skew curve
161. Tangent lines and tangent plane to a surface
162. Normal line to a surface . . . .
257
259
. 260
262
. 264
163. Equations of tangent line and normal plane to a skew curve . 266
CONTENTS. XV
INTEGRAL CALCULUS.
CHAPTER XVm.
Integration.
ART. Piai
164. Integration and integral defined. Notation 269
166. Examples of the summation of infinitesimals .... 271
166. Integration as summation. The definite integral . . 276
167. Integration as the inverse of differentiation. The indefinite integral
Constant of integration. Particular integrals .... 281
168. Geometric or graphical representation of definite integrals
Properties of definite integrals 284
169. Greometrlc or graphical representation of indefinite integrals
Geometric meaning of the constant of integration .... 287
170. Integral curves 289
171. Summary ... 290
CHAPTER XIX.
Elementary Inteoralb.
173. Elementary integrals 293
174. General theorems in integration 294
175. Integration aided by substitution 296
176. Integration by parts 298
177. Further elementary integrals 301
178. Integration of /(a) da; vf hen /(a;) is a rational fraction . 305
179. Integration of a total differential 309
CHAPTER XX.
Simple Geometrical Applications of Integration.
181. Areas of curves : Cartesian coordinates 313
182. Volumes of solids of revolution 320
183. Derivation of the equations of curves 324
CHAPTER XXI.
Integration of Irrational and Trigonometric Functions.
Integration of Irrational Functions.
185. The reciprocal substitution . . 327
186. Difierential expressions involving Va + bx 328
xvi CONTENTS.
187. A. Expressions of form F{x, y/x?- + aa; + Xi)dx. B. Eipressions
of form F{x, y/ - x'^ -{■ ax + b)dx . . .... 329
188. To find f !»■(« + bx')Pdx 332
Integration of Trigonometric Functions.
189. Algetraic transformations 336
190. Integrals reducible to i F{u)du, in which u is one of the trigo-
nometric ratios 337
191. Integration aided by multiple angles 338
192. Reduction formulas 339
CHAPTER XXII.
Approximate Integration. Mechanical Integration.
193. Approximate integration of definite integrals .... 344
194. Trapezoidal rule for measuring areas and evaluating definite inte-
grals ... . . 344
195. Parabolic rule for measuring areas and evaluating definite integrals 346
196. Mechanical devices for integration 348
CHAPTER XXIII.
Integration of Infinite Series.
197. Integration of infinite series term by term ... . 350
198. Expansions obtained by integration of known series . . . 360
199. Approximate integration by means of series . . . 353
CHAPTER XXIV.
Successive Integration. Multiple Integrals. Applications.
201. Successive integration : one variable. Applications
202. Successive integration : several variables
203. Finding areas : rectangular coordinates .
204. Finding volumes : rectangular coordinates
205. Finding volumes : polar coordinates
356
357
359
360
363
CHAPTER XXV.
Further Geometrical Applications of Integration.
207. Volumes of solids of known cross-section . . . 366
208. Areas : polar coordinates 367
209. Lengths of curves : rectangular coordinates . . . . 370
CONTENTS. xvii
ART. PAGK
210. Lengths of curves : polar coordiuates 373
211. Areas of surfaces of revolution 374
212. Areas of surfaces 2 =/(x, y) . . 378
213. Mean values ... .... 380
214. Note to Art. 104 384
CHAPTER XXVI.
Note on Centre of Mass and Moment or Inertia.
215. Mass, density, centre of mass . . ... 385
216. Moment of inertia. Radius of gyration .... 390
CHAPTER XXVII.
Differential Equations.
217. Definitions. Classifications. Solutions 394
218. Constants of integration. General solutions. Particular solutions 395
Equations of the First Order.
219. Equations of the form /(i)da;+ Ji'(2^)djf = 0 . . . .395
220. Homogeneous equations . . . . . 396
221. Exact diSereiitial equations. Integrating factors .... 396
222. The linear equation 397
223. Equations not of the first degree in tlie derivative :
The form x =f{y, p) ; the form y =f{x, p) ; Clairaut's equation 398
224. Singular solutions 400
225. Orthogonal trajectories 401
Equations of the Second and Higher Orders.
226. Linear equations with constant coeflicients. Homogeneous linear
equations 406
227. Special equations of the second order :
dx-i -•''-2'-' ■ J[dx^' dx' ) '■' \dx^' dx' ^1
APPENDIX.
Note A. Hyperbolic functions 413
Note B. Intrinsic equations 423
Note C. Length of a curve in space 427
Collection of Examples 431-450
XVlll
CONTENTS.
Integrals fob Retiew Exercises and for Refebence
Figures
Answers ...
Index . .
PAGE
451-458
459-464
. 465
. 486
SHORT COURSE
FOR STUDENTS HAVING A MINIMUM OF TIME
(The Roman numerals refer to chapters, the Arabic to articles.)
I. ; II. ; in. ; IV. ; V. 57-65 ; VI. 68-70 ; VII. ; VIII. 79-84, 86 ; IX. (if
time permits) ; X. ; XI. 108, 109 ; XIII. (part) ; XV. ; XVI. 149,
154, Exs. 150-152; XVIII.; XIX.; XX.; XXI. 184-186, (188-192,
if time permite) ; XXII. ; XXIII. ; XXIV. ; XXV. 206-211, 213.
Recommended for students looking forward to engineering and to
courses in mechanics : XIV., XXVI.
DIFFERENTIAL CALCULUS
CHAPTER I.
INTRODUCTORY PROBLEMS.
3, The infinitesimal calculus is one of the most powerful mathe-
matical instruments ever invented.* Many practical problems can
be solved by its means with wonderful ease and rapidity. Even
a slight acquaintance with. the calculus is very helpful in the study
of many other subjects, for example, geometry, astronomy, physics,
and engineering; and the fullest knowledge possible about the
calculus is necessary for advance in these subjects. Some of
the higher branches of mathematics consist largely of special
investigations in the infinitesimal calculus and extensions of its
principles, methods, and applications.f
In this book the fundamental notions and principles of the
calculus are, to a certain extent, explained, and applications are
made to the solution of some simple practical problems. As a
preliminary to the study there is in this chapter a discussion of
a few problems. This discussion introduces in an informal way
the notions and principles and methods which are at the founda-
tion of the infinitesimal calculus, and also provides material which
serves to illustrate a few of the articles that follow.^
* The calculus is divided into two parts, the differential calculus and the
integral calculus. Concerning its invention see Art. 164, note.
t The word " infinitesimal " serves to distinguish the subject from other
branches of mathematics, such as the calculus of finite difierences, the cal-
culus of variations, the calculus of quaternions, etc.
J An important fact in the history of the calculus is that the problems in
Arts. 3-6 were the occasion of the invention and development of some divi-
sions of the subject.
1
2 DIFFERENTIAL CALCULUS. [Ch. I.
Note. A knowledge of the meaning of the term speed or rate of motion is
presupposed in the following two articles. If a body moves through equal
distances in equal times, it is said to have uniform speed. The average speed
of a body during the time that it is moving through a certain distance, is the
uniform speed at which a body will pass over that distance in that time.
For instance, if a bicyclist wheels 36 miles in 3 hours, his average speed is
12 miles per hour ; if a body moves through 45 feet in 5 minutes, its average
speed is 9 feet per minute. The number which indicates the average speed
of a body while it is moving through a certain distance, is the ratio of the
number of units of length in the distance to the number of units of time spent
during the motion. In other words, the measure of the speed is the ratio of
the measure of the distance to the measure of the corresponding time. Thus,
in the instances above, 12 = 36 : 3, 9 = 45 : 5.
Any reader of this book knows what is meant by the statements that a
train is running at a particular instant at the rate of 30 miles an hour, and
that at another instant, some minutes later say, it is running at the rate of 40
miles an hour. This notion, viz. the speed of a moving body at a par-
ticular instant, will be developed further by the examples that follow.
2. Speed of a moving train. Suppose that a person is standing
by a railway and wishes to ascertain the speed at which a train
is going by him. A way to determine this speed approximately
would be to find the distance passed over in five seconds by the
train, or by a definite mark on the train, say a vertical line. (The
place where the observer stands may be at one end of, or upon,
the measured distance.) If the observer knew the distance passed
over in three seconds, he would get the speed more accurately ;
yet more accurately, if he knew the distance passed over in one
second; more accurately still, if he knew the distance passed
over in half a second ; and so on. The point to be noted and
emphasised in this illustration is this : the less the time and the
corresponding distance that can be observed, the more nearly will
the observer obtain the actual speed of the train just at the
moment when it is passing him.
3. To determine the speed of a falling body. Let a body fall
vertically from rest. It is known that in I seconds from the
time of starting, the body passes through ^gt^ feet. (Here g
denotes a number whose approximate value is 32.2.) That is, if
s denotes the number of feet through which the body falls in t
seconds, s = ^gt\
2,3.]
INTBODUCTOBY PBOBLEMS.
As the body descends its speed is continually changing and grow-
ing greater; but at any particular instant it has some definite
speed. Let it be required to find the speed after it has been
falling for : (a) 4 seconds ; (6) t^ seconds.
(a) To find speed after the body has been falling from rest for 4 seconds.
A method of getting an approximate value of this speed is as follows. Find
the distance through which the body would fall in 4 seconds ; then find the
distance through which it would fall in a little more than 4 seconds. There-
from deduce the average value of the speed from the end of the fourth second
to the last instant (Note, Art. 1). This average speed may be taken as an
approximate value of the speed at the end of the fourth second. The smaller
the interval of time which is taken after the fourth second, the more nearly
will the average speed for the interval be equal to the actual speed just at
the end of the fourth second. This is also apparent from the following
calculations :
a
i .
o «
r
o
Length of fall,
in feet.
Increase In time
after 4 seconds.
(In seconds.)
Correapondinp
increase in
distance,
in feet.
Average speed during increased
time, in feet per second.
4.
8?
4.1
8.405 3
.1
.40.5?
4.05 5f or
130.41
4.01
8.04005 flr
.01
.04005 J/
4.005 3
128.961
4.001
8. 0040005 jr
.001
.0040005 (/
4.0005?
128.8161
4.0001
8.000400005 ff
.0001
.000400005^
4.00005 g
128.80161
4 + h
(8+4A + JA2)j,
h
(4A + i/j2)^
e-iv
128,8 + 16.1 X ft
It is evident that the less the increase given to the 4 seconds, the more
nearly does the average speed during this additional time approach to 128.8
feet per second. The last line of the table shows that, no matter how short
a time h may be, the average speed during this time has a definite value,
namely (128.8 + 16.1 x h) feet per second. The number in brackets becomes
more and more nearly equal to 128.8 when h is made smaller and smaller ; the
difference between it and 128.8 can be made as small as one pleases, merely
by decreasing h, and will become still less when h is further diminished.
Since the number (128.8 + 16.1 x h) behaves in this way, the speed of the
falling body at the end of the fourth second is manifestly 128.8 feet per
second.
4 DIFFERENTIAL CALCULUS. [Ch. I.
(6) To find the speed after the body has been falling for t\ seconds. Let
St denote the distance in feet through which the body has fallen in the ti
seconds. It is known that si = i gti'. (1)
Let A«i (read " delta «i") denote any increment given to ti, and Asi denote
the corresponding increment of si.
Note 1. Here AJi does not mean A x ^i. The symbol A is used with a
quantity to denote any difference^ change, or increment, positive or negative
(i.e. any increase or decrease), in the quantity. Thus Ax and Ay denote
" increment of a," "increment of y," "difference in x," " difference in y."
Then si + Asi = J g{ti + A«i)^. (2)
Hence, by (1) and (2), Asi = gt^. • ACi + i g{^t{)K
.■.^=gtx + \g.^t,. (3)
Ati
Here ^ Is the average speed for the time A«i and the corresponding
^h , .„ Asi
distance Asi. Now the smaller A«i Is taken, the more nearly will ^
approximate to the actual speed which the falling body has at the end of
the «ith second. But when A«i is taken smaller and smaller (in other words,
when Ati approache. nearer and nearer to zero), the second member of equa-
tion (3) approaches nearer and nearer to gt\. Equation (8) also shows that
— - can be made to differ as little as one pleases from gti, merely by taking
Ail
AJi small enough. Hence it is reasonable to conclude that at the end of the
tith second
the speed of the falling body = gti feet per second. (4)
Here ti may be any value of (. So it is usual to express conclusion (4)
thus : the speed of a body that has been falling for t seconds is gt feet per
second. This result (speed = gt feet per second) is a general one, and can
be applied to special cases. Thus at the end of the fourth second the speed
is jr X 4 or 128.8 feet per second, as found in (a) ; at the end of 10 seconds
the speed is \0g or 322 feet per second.
The two principal points to be noted in tills lllnstratlon are :
(1) No matter what the value of A*i may be, or how small Afj
As
may be, the quantity — ■' has a definite value, namely, gh + ^g- At, ;
(2) When A<i is taken smaller and smaller, — * gets nearer and
Ati
nearer to gr<, ; and the difference between them can be made as
small as one pleases by giving Ai, a definite small value; this
difference remains less than the assigned value when Atx further
decreases.
4-]
INTRODUCTORY PROBLEMS.
Note 2. The definite small value refeiTed to in (2) can be easily found.
For example, suppose that |^ is to differ from gh by not more than * say (*
being any small quantity, as a millionth, or a million-millionth).
Then
-^^-9h<k. But ^-5,«, = Jj,.4«iby(3).
ig • Ati ^k; accordingly Ati <
2k
g'
Note 3. It should be observed, as shown by equation (3) , that the value of
-^ depends upon the values of both «i and A«i. On the other hand, the
value to which ^^^ tends to become equal as Mi decreases, depends (see (4))
upon «i alone. The quantity A«i is any increment whatever of <i, but it does
not depend upon the value of ti.
4. To determine the slope of the tangent to the parabola y = 3?:
(a) at the point whose abscissa is 2 ; (b) at the point whose abscissa
is Xi.
(a) Let VOQ, Fig. 1, be the
parabola y = 3^, and P be the
point whose abscissa is 2.
Draw the secant J^Q. If PQ
turns about P until Q coin-
cides with P, then PQ will
take the position FT and be-
come the tangent at P. The
angle QPR will then become the angle BPT.
Note 1. This conception of a tangent to a curve has probably been
already employed by the student in finding the equations of tangents to circles,
parabolas, ellipses, and hyperbolas. The process generally followed in the
analytic treatment of the conic sections is as follows : The equation of the
secant PQ is found subject to the condition that P and Q are on the curve ;
then Q is supposed to move along the curve until it reaches P. The resulting
form of the equation of the secant is the equation of the tangent at P. The
calculus method (now to be shown) of finding tangents to curves is preferred
by some teachers of analytic geometry ; e.g. see A. L. Candy, Analytic
Oeometry, Chap. V.
Draw the ordinates LP and MQ ; draw PE parallel to OX.
Let PR be denoted by Ax, and RQ by Ay. Then the slope of
the secant PQ is ^. /^For tan RPQ = ^.\
Ax \ PR J
FlQ. 1.
DIFFERENTIAL CALCULUS.
[Ch. I.
The following table shows the value of — ^ for various values
of Ax.
Value of 05.
Corresponding
value of y.
(Increase over (B).
Ay
(Increase over y).
Corresponding
value of '^■
Ax
2.
4.
_
_
2.1
4.41
.1
.41
4.1
2.01
4.0401
.01
.0401
4.01
2.001
4.004001
.001
.004001
4.001
2.0001
4.00040001
.0001
.00040001
4.0001
i-^h
4 + 4 ft + ft2
h
4A + ft2
4 + A
It is apparent from this table that the less Ax is, the more nearly does
-^ approach the value 4. The last line shows that, no matter how small Ax
Ax .
(or h) may be, -^ has a definite value, namely i + h. This number becomes
Ax
more and more nearly equal to 4 when h is made less and less ; the difference
between it and 4 can be made as small as one pleases, merely by decreasing h
to a certain definite value, and will continue to be as small or smaller when
h is further diminished. Because the number i + h behaves in this way,
it is evident that -^ will reach the value 4 when Ax decreases to zero.
Ax
Accordingly the slope of the tangent PT is 4 ; and hence angle TPR or
PWL is 76° 57' 49".
(V) To determine the slope of the tangent at the point whose
abscissa is Xi.
Let (Fig. 1) P be the point (x,, j/i). Draw the secant PQ, and the
ordinates PL and QM ; draw PB parallel to OX Let PB, the difference
between the abscissas of P and Q, be denoted by Axi, and let BQ, the
difierence between the ordinates of P and Q, be denoted by Ayi. Then
tangent QPB=^ = ^.
* PB Axi
If Q be moved along the curve toward P, the secant PQ will approach
the position of PT, the tangent at P; at last, when Q reaches P, the secant
PQ becomes the tangent PT. As Q approaches P, Axi becomes less and
less, and when Q reaches P, Axi becomes zero. Conversely, as Axi decreases,
PQ approaches the position PT. Accordingly, the slope of the tangent PT
can be determined by finding what the slope of the secant PQ, namely ^,
approaches when Axi approaches aero. ^i
4.J INTRODUCTORY PROBLEMS. 7
Hence, on subtraction, Ayi = 2 ki . Azi + (Aaii)*. (1)
.:^ = 2xi + Axi. (2)
This equation shows that — ^ approaches nearer to 2 xi when Axi decreases.
. Axi
It also shows that -^ can be made to differ as little as one pleases from 2 Xj,
merely by taking Axi small enough, and that this difference will become
smaller when Axi is further diminished. (For instance, if it is desired that
— ^ — 2 xi be less than any positive small quantity, say e, it is only necessary
to take Axi less than «.) Accordingly,
the slope of FT (the tangent at P) = 2 xi. (3)
The two principal points to be noted in tliis illnstration are :
(1) No matter what the value of Axi may be, or how small Aic,
may be, the quantity —^ has a definite value, namely 2 Xi + Aa^j.
' Am
(2) When Ax^ decreases, the quantity -~ approaches the
A?/ '
value 2*1; the difference between ~ and 2x, can be made as
AX]
small as any number that may be assigned, by giving Axi a
definite small value ; this difference remains less than the
assigned value when Axj further decreases.
Note 1. The value of --^, as shown by Equation (2), depends upon the
values of both xi and Axi. On the other hand, the value to which — ^
Axi
tends to become equal as Axi decreases, depends (Equation (3)) upon xi
alone. The value of Axi does not depend upon the value of xi ; for Q
(Fig. 1) may be taken anywhere on the curve.
Note 2. The method used in getting result (3) does not depend upon
the particular value of xi- The result is perfectly general, and may be
expressed thus : " the slope of the curve y = x^ is 2 x." 'J'his general result
can be used for finding the slope at particular points on the curve. For
instance, it xi = 2, the slope is 4, as found in (a) ; if Xi = — 1, the slope
is — 2, and accordingly, the angle made by the tangent with the x-axis is
116° 34'. (It is advisable to make a figure showing this.)
Note 3. In the infinitesimal calculus, as well as in other branches of
mathematics, it is very important for the student always to have a clear
8 DIFFERENTIAL CALCULUS. [Ch. I.
understanding of the meaning of the operations which he performs with
numbers, and to interpret rightly the numerical results obtained by these oper-
ations. Thus, if it is stated that 6 men work 5 days at 2 dollars per day each,
the numbers 6, 5, and 2 are treated by the operation called multiplication,
and the number 60 is obtained. The calculator then applies, or interprets,
this numerical result as meaning, not 60 men, or 60 days, buo that the men
have earned 60 dollars. In the curve above, y = x^. This does not mean
that at any point on the curve the ordinate is equal to the square on the
abscissa, i.e. a length is equal to an area. By y = x^ it is meant that the
number of units of length in any ordinate is equal to the square of the num-
ber of units of length in the corresponding abscissa. Again, the- result in
Equation (3) does not mean that the slope of FT is twice OL. The result
means that the number which is the value of the trigonometric tangent of
the angle TPR is twice the number of units of length in OL.
Many persons who can perform operations of the calculus easily and
accurately, cannot correctly or confidently interpret the results of these
operations in concrete practical problems in geometry, physics, and engi-
neering. Thus, some engineers who have had a fairly extended course in
calculus discard it when possible, and solve practical problems by much
longer and more laborious methods. Such a misfortune will not happen to
those who early get into the habit of giving careful thought to finding out the
real meaning of the operations and results of the calculus. They will not
only "understand the theory," but they can use the calculus as a tool with
ease and skill.
Note 4. In Fig. 1 let a point Qi be taken on the curve to the left of P,
and draw the secant QiP. (The drawing for this note is left to the student.)
It is obvious from the figure that the same tangent FT is obtained, whether
the secant QiF revolves until Qj reaches P, or QF revolves until Q reaches
P. This may also be deduced algebraically. Let the coordinates of Qi he
xi — Axi, t/i — Ayi. [Here the Axi and Ayi are not necessarily the same in
amount as the Axi and Ayi in (6).] Draw the ordinate QiMi. Then
yi(=LF)=Xi%
yi - Ayi (= MiQi) = {xi - Aa;i)2.
Whence, it follows that -^ = 2 aii — Axi.
Accordingly, when Axi approaches zero, -^ approaches the value 2 Xi.
Note 5. Thoughtful beginners in calculus are frequently, and not un-
naturally, troubled by the consideration that when Ati (Art. 3 6) is diminished
to zero, —J has the form - ; and likewise, when Axi (Art. 4 6) becomes
zero, -r^ becomes ^. It is true that ^ is indeterminate in form ; and, if
4] INTRODUCTORY PROBLEMS. 9
it is presented without any information being given concerning the whence
and ttie wherefore of its appearance, a value for it cannot be determined.
In the oases in Arts. 3, 4, however, there is given information which makes
it possible to tell the meaning of the quantity - that appears at the final stage
of each of these problems. In these cases one knows how the quantities
Asi , Ayi
aF ^ Ar" '"'* ^'^^"^^''9 ■when Ati and Azi respectively are approaching
zero ; and by means of this knowledge he can confidently and accurately
state what these ratios will become when Ati and Aa;i actually reach zero.*
Note 6. Moreover, it should be carefully noted that at the final stages
in the solution of the problems in Arts. 3 and 4, -— ^ is not regarded as a
Ati
fraction composed of two quantities, Asi and Ati, hut as a single quantity,
namely the speed after ti seconds ; likewise that — =-i is then not regarded
AXi
as a fraction at all, but as a single quantity, namely the slope of the tangent
at P.
Note 7. The student should not be satisfied until he clearly perceives,
and understands, that the method employed in solving the problems in
Arts. 3 and 4 is not a tentative one, but is general and sure, and that the
results obtained are not indefinite or approximate, but are certain and exact.
EXAMPLES.
1. Assnming the result in Art. 4 (6), namely, that the slope of the tangent
at a point (xi, j/i) on the curve y = x^ is 2 Xi, find the slope and the angle
made vrith the x-axis by the tangent at each of the points whose abscissas are
.5, 0, 1, 1.5, 2, 2.5, 3, 4, -2, -3, - i, -f - f .
2. In the curve in Ex. 1 find the cobrdinates of the points the tangents at
which make angles of 20», 30°, 45°, 60°, 85°, 115", 145°, 160°, 170°, respec-
tively, with the x-azis.
Ay
3. Draw figures of the following curves. Find the value of — at any
Aw
point (x, y) in the case of each curve ; then find what — is approaching
when Ax approaches zero :
(a) x2 + y2 = i6; (6) j, = x2 + x-|-l; (c) y = 3?;
(d) f = ix; (e) 9x2+162/2 = 144; (/) 9x2 - 16y« = 144 ;
{g) y^=4px; (h) bH^ + aV = a'^V' ; (0 b^x^ - aY = a-f)'-
* The mathematical phraseology and notation employed to express these
ideas is given in Chapter II.
10
DIFFEUENTIAL CALCULUS.
[Ch. I.
fSuGGESTiON. In (a), (a; + Aa)^ + {V + ^vT = 16. It can then be de-
'- tiy 2 1 + Aa; "I
ducedthat^ = -2y:p^-J
Compare the results found In {g), (h), and (i), with those found in
analytic geometry.
4. Using the results obtained in Ex. 3, find the slopes and the angles made
with the X-axis by the tangents in the following cases :
(a) The curve in Ex. 3 (a), at the points whose abscissas are
4, 2, 1, 0, -1.6, -3.5.
(6) The curve in Ex. 3 (c), at the points whose abscissas are
-3, -2,-1, 0, 1.5, 2.5.
(c) The curve in Ex. 3 (d), at the points whose abscissas are
0, 1, 2, 3, 6, 8.
(d) The curve in Ex. 3 (e), at the points whose abscissas are
0, 1, 2, 4, - .5, - 1.5.
(e) The curve in Ex. 3 (/), at the points whose abscissas are
4, 8, 10, -5, -7.
6. Using the results obtained in Ex. 3, find the points on the curve in
Ex. 3 (a) the tangents at which make angles 40° and 136° with the x-axis.
6. Do as in Ex. 5 for the curves whose equations are given in Ex. 3 (c),
(d), (e), and (/).
1. Do some of the examples in Art. 62. Make careful drawings in each
case.
5. To detennine the area of a plane figure. A plane area, say
ABCD, may be supposed to be divided into an exceedingly great
number of exceedingly small rect-
angles. It will be seen later
that the limit of the sum of these
rectangles when they are taken
smaller and smaller, is the area.
The calculus furnishes a way to
find this limit. Even at this
„ „ stage in the study of the calculus
Fig. 2. ° , •'
the student can get some useful
ideas concerning this problem by making a brief inspection of
Art. 165, Exs. («), (6), (c). [Art. 14 discusses the term " limit."]
yffTT>\.
/ ^ fc. ij
tt- IIII-S.sJ'
. / -rrrrrrr.
A/ _ .._ : ^'v
y. 'k
J.'. ._ :: ^
. . _ 1
/ :; ::_ -
t _ _ _: :: i
t.__ — J
\"":-X;'^ ^^;" — /
NttUX
6-7.] INTRODUCTORY PROBLEMS. 11
6. (a) To find a function when its rate of change at any (every)
moment is known, or, in more general terms, when its law of change
is known. In Art. 3 (6) a particular example has been given of
this general problem, viz. to determine the rate of change of a func-
tion at any moment. The calculus not only provides a method of
solving this general problem, but also provides a method of solving
the inverse problem which is stated above.
(6) To find the equation of a curve when its slope at any (every)
point is known. In Art. 4 (b) a particular example has been given
of this general problem, viz. to determine the slope of a curve at
any point on it. The calculus not only provides a method of
solving this problem, but it also provides a method of solving the
inverse problem which has just been stated. Problem (6) is a
special case of problem (a), for the slope at a point on a curve
really shows " the law of change " existing between the ordinate
and the abscissa of the point (see Art. 26).
A brief inspection of Arts. 24-26, 167,169, at this time, will repay
the beginner.
Note. Differential calcnlns and integral calculus. The subject of
infinitesimal calculus is frequently divided into two parts ; namely , differential
calculus and integral calculus. This division is merely a formal division ;
though oftentimes convenient, it is by no means necessary. Examples of the
kind given in Arts. 2-4 formally belong to " the differential calculus," and
those described in Arts. 5, 6, to " the integral calculus."
7. Elementary notions used in infinitesimal calculus. The prob-
lems used in Arts. 2-4 put in evidence some notions and methods,
the consideration and development of which constitute an impor-
tant part of infinitesimal calculus. These notions are :
(1) The notion of varying quantities which may approach as
near to zero as one pleases, such as At^ and Ax, in the last stages
of the solution of the problems in Arts. 3 and 4.
(2) The notion of a varying quantity, such as —J in Art. 3
for ^ in Art. 4 \ which approaches a fixed number when Af,
(or Aa^i) varies and decreases towards zero, and approaches in such
a way that the diiference between the varying quantity and the
fixed number can be made to become, and remain, as small as one
pleases, merely by decreasing Af, (or Ax^).
12 DIFFERENTIAL CALCULUS. [Ch. I.
The infinitesimal calculus gives mathematical definiteness and
exactness to these notions, and a convenient notation has been
invented for dealing with them. From these notions, with the
help of this notation, it has developed methods and obtained
results which are of great service in such widely separated fields
of study as geometry, astronomy, physics, mechanics, geology,
chemistry, and political economy.
A review of certain notions of algebra is not only highly advan-
tageous but absolutely necessary for a satisfactory understanding
of the calculus and for good progress in its study. Accordingly,
Chapter II. is devoted to the consideration of the notions of a
variable, a function, a limit, and continuity.
Note. Reference for collateral reading. Perry, Calculua for Engi-
neers, Preface, and Arts. 1-18.
CHAPTER II.
ALGEBRAIC NOTIONS WHICH ARE FREQUENTLY
USED IN THE CALCULUS.
8. Variables. When in the course of an investigation a quan-
tity can take different values, the quantity is called a variable
quantity, or, briefly, a variable. For instance, in the example in
Art. 3, the distance through which the body falls and its speed
both vary from moment to moment, and, accordingly, are said to
be variables. Again, if the x in the expression a^ + 3 be allowed
to take various values, then x is said to be a variable, and ar" + 3
is likewise a variable. If a steamer is going from New York to
Liverpool, its distance from either port is a variable.
In general a variable can take an unlimited number of values.
Note 1. ITninbers. The values of a variable are indicated by numbers.
In preceding matliematical work various kinds of numbers have been met ;
such as 2, 7, |, V2, v^, ir^ 3.14159 •••, logioS = .90309 •••, e = 2.71828 ■-.,
V— 5, 3 V— 1, 4+3 V— 1. The student Is supposed to be acquainted
with the divisions of numbers into real and imaginary, integral and frac-
tional, rational and irrational, positive and negative. In general in this
book real numbers only aroused.
Graphical representation of real numbers. Draw a straight line LM,
L C Q A D BO it
-i 1 VT' 3^/10
Fig. 3.
which is supposed to be unlimited in length both to the right and to the
left. Choose any point O, and take any distance OA for unit length. Also
let it be arranged for convenience (as has been done in trigonometry and
analytic geometry) that positive numbers be measured from 0 towards M,
and negative numbers from 0 towards L. Then the point A represents the
number 1 ; if OB = 3 OA, B represents the number 3 ; if OC = J OA, C
represents the number — J. If OD is the length of a diagonal of a square
whose side is OA, then OD = v'2, and D represents the number y/i ; if OG
be the length of a diagonal of a rectangle who.se sides are OA and OB, then
00 = -v/lO, and O represents the number VlO. It is a topic for a more ad-
vanced course than this to show that all real numbers can be represented on
13
14 DIFFERENTIAL CALCULUS. [Ch. II.
the unlimited line LM, that to each point on LM there corresponds (on the
scale OA = 1) a definite real number, and that to each real number there
corresponds a definite point on the line.
Absolate ralae of a number. The value of a number without regard to
sign is called its absolute value. Thus the absolute values of the numbers
1, — 2, ^, — ^ are 1, 2, \, \. The absolute value of a number x is denoted
by the symbol |x|.
Note 2. Infinite numbers. Sometimes the value of a variable " be-
comes unlimited in magnitude," i.e. "increases beyond all bounds." The
variable is then said to become infinite in magnitude, and its value is then
called infinity. If the unlimited value is positive, it is denoted by the
symbol + oo; if it is negative, it is denoted by the symbol — oo. For ex-
ample, if X be an angle, as x increases from 45° to 90°, tana; increases from
+ 1 to -I- 00 ; and as x decreases from 135° to 90," tanx decreases from — 1
to — xi. •
The symbol oo does not denote a definite number in the same vyay as 2,
say, denotes a number ; the symbol oo merely means that the measure of the
variable concerned is unlimitedly great, or, in other words, is beyond all
bounds.*
9. Functions. When two variables are so related that the value
of one of them depends upon the value of the other, each is said to be
a function of the other.
For example, the area of a circle depends upon the length of its radius,
and so the area is said to be a function of the radius. To a definite value of
the radius, e. g. 2 inches, there corresponds a definite value of the area, viz.
IT X 2^ inches, i.e. 12.57 sq. in.
Another example : the length of the side of ^ square depends upon the
area of the square, and so the side is said to be a function of the area. To a
definite value of the area, say 9 sq. in., there corresponds a definite side, viz.,
a side 3 inches in length.
The idea of- a function is sometimes expressed thus : When
two variables are so related that to any arbitrarily assigned definite
value of one of them there corresponds a definite value (or set of
definite values') of the other, the second variable is said to be a
function of the first.'f
* For further notes on numbers, and especially for references for reading,
see Infinitesimal Calculus, Art. 8. Additional references are Pierpont,
Theory of Functions of Seal Variables, Chaps. I., II. ; Veblen-Lennes, Infini-
tesimal Analysis, Chaps. I., II., and the references given on pages 10, 11, 19.
t See Veblen-Lennes, Infinitesimal Analysis, Chap. III. (and its historical
note on page 44).
9.] FUNCTIONS. 15
For example, suppose y = x'^ + 2x — 5. (1)
When the value 3 is assigned to x, y must take the corresponding value
3^ + 2 X 3 — 5, i.e. 10 ; when a; is — 2, y must be — 6. In these cases y is
said to be a function of x ; also x is called the independent variable and y is
called the dependent variable.
On the other hand vfhen the value 30 is assigned to y, x must have the
corresponding values 5 and — 7. (These values are obtained by substituting
30 for y in (1), and then solving for x.) When y is 115, x must be 10 or
— 12. In these cases x is said to be a function of y ; also y is called the
independent variable, and x is called the dependent variable.
Ex. Given that a^-/-6x-82/-7 = 0: (2)
(a) assign values to x and find the corresponding values of y\
(b) assign values to y and find the corresponding values of x.
Independent variable; dependent variable. The variable which
can take arbitrarily assigned values is usually termed the inde-
pendent variable; the other variable, whose values must then be
determined in order that they may correspond to these assigned
values, is usually termed the dependent variable. It is evident
that if the second definition above be followed, " function " and
" dependent variable " are synonymous terms.
One-rained functions. Many-valned functions. When a function
has only one value corresponding to each value of the independent
variable, the function is called a one-valued function ; when it has
two values it is called a two-valued function. If a function has several
values corresponding to each value of the independent variable,
it is called a multiple-valued function, or a many-valued function.
For example: In (1), y is a one-valued function of x, and a is a two-
valued function ofy. 1i y = x^, yisa, single- valued function of x ; ity= Vx,
y is a two-valued function of x.
If 2/ = sin X, y is a one-valued function of x.
liy = sin-i x, i.e. (using another notation) if y = arc sinx,* y is a many-
valued function of x.
Inverse functions. If y is a function of x, then, on the other
hand, a; is a function of y. The second function x is called the
inverse function of the first function y. That is, if
y=fi^), (3)
then x = 4,(y), (4)
* See Plane Trigonometry, Arts. 17, 88.
16 DIFFERENTIAL CALCULUS. [Ch. II.
in which <^(y) denotes an expression in y which is obtained by
solving equation (3) for x.
E.g. in (1), y = a;^ + 2 x - 5.
On solving for x, there is obtained the inverse function,
X = - 1 ± y/y + 6.
Again, if y = a', the inverse function is x = logo y ; if y = sin x, the inverse
function is x = sin-' y; or as it is frequently written x = arc sin y.
Fnnctlons of two variables. Fnnctions of more than two variables.
The value of a function may depend upon the values assigned to
two or more other variables. In such a case the first variable is
said to be a function of the other two variables.
E.g. It e = x^ + y"^ + 18, z is said to be a function of x and y ;
if o = «2 + w^ + «2 + 5, r is a function of u, w, and t.
10. Constants. A quantity whose value never changes through-
out an investigation is called a constant.
If a constant remains the same in all investigations, it is called
an absointe constant.
Thus 2, .33, IT, are absolute constants.
A quantity which has a fixed value in one investigation and
another fixed value in another investigation is called an arbitrary
constant.
Thus let the equations of a straight line, (x, y) denoting any
point on the line, be
y = mx -\- b and x cos a + y sin a =p.
Here m and h, a and p, are arbitrary constants. For any partic-
ular line a and p have fixed particular values, and so also have
m and b.
11. Classification of Functions.
A. Explicit and implicit fanctions. When a function is expressed
directly in terms of the dependent variable, like y in equation (1),
Art. 9, the function is said to be an explicit function. When
the function is not so expressed, as in equation (2), Art. 9, it is
said to be an implicit function. If relation (2), Art. 9, were solved
for y, then y would be expressed as an explicit function of x ; thus
y=±{x-Z)-L
10, 11.] CLASSIFICATION OF FUNCTIONS. 17
On solving the same relation for x, the variable x is expressed
as an explicit function of y; thus
x=±(y + A) + 3.
B. Algebraic and transcendental functions. Functions may also
be classified according to the operations involved in the relation
connecting a function and its dependent variable (or variables).
When the relation involves only a finite number of terms, and
the variables are affected only by the operations of addition, sub-
traction, multiplication, division, raising of powers, and extraction
of roots, the function is said to be algebraic; in all other cases
it is said to be transcendental. Thus 2icP + 3x — 7, -\/x + - , are
X
algebraic functions of x ; sin x, tan (x + a), cos"' x, I', e^', log x,
log 3 a;, are transcendental functions of x. The elementary tran-
scendental functions are the trigonometric, anti-trigonometric, ex-
ponential, and logarithmic. Examples of these have just been
given.
C. Rational and irrational fanctions. Algebraic functions are
subdivided into rational functions and irrational functions. Ex-
pressions involving x which consist of a finite number of terms
of the form ax", in which a is a constant and n a positive integer,
e.g. 3a;<-2af' + 4x-f 5,
are called rational integral functions of x.
When these expressions have more than two terms they are
also called polynomials in x.
If an expression in x, in which x has positive integral expo-
nents only, and which has a finite number of terms, includes
division by a rational integral function of x,
x-1
x'-\-5 x-\-2
'■^- 3^+7"^^ 3^+9 + ^^-2'
it is called a rational fractional function of x.
Eational integral functions and rational fractional functions
are included together in the term rational functions.
18 DIFFERENTIAL CALCULUS. [Ch. IL
An expression which involves root extraction of terms involv-
ing X is called an in-cUional function of x ;
e.g. Vx, Va^ + 3a; + 6-f-9a;-2.
D. Continnons and discontinnous functions. A discussion on this
exceedingly important classification of functions is contained in
Art. 16.
12. Notation. In general discussions variables are usually
denoted by the last letters of the alphabet, x, y, z, u, v, •••, and
constants by the first letters, a, b, c, ■■•.
The mere fact that a quantity is a function of a single variable,
X, say, is indicated by writing the function in one of the forms
f(x), F(x), <l>(x), •••,fi(x),fi{x), •■: If one of these occurs alone,
it is read "a function of x" or "some function of x"; if several
are together, they are read " the /-function of x," " the 2?'-function
of X," "the phi-function of x," ■■-. The letter y is often used to
denote a function of x.
The fact that a quantity is a function of several variables,
X, y, z, •••, say, is indicated by denoting the quantity by means of
some one of the symbols, f(x, y), <t>(x, y), F(x, y, z), ^(x, y, z, u), ■■■.
These are read " the /-function of x and y," " the phi-function of
X and y," " the i?'-f unction of x, y, and z," etc.
Sometimes the exact relation between the function and the
dependent variable (or variables) is stated ; as, for example,
f{x) = !j? + Zx-l,OTy = a? + Sx — l;F{x,y)=:2e' + 7e' + xy-l.
In such cases the /-function of any other number is obtained by
substituting this number for x in f{x), and the jP-function of any
two numbers is obtained by substituting them for x and y respec-
tively in F(x, y). Thus
/(z) = z2-f3«-7,/(4) = 42-f 3 •4-7 = 21;
F(t,z) = 2e' + 7e' + tz-l,F(2,3)=2e^ + 7e? + 5.
In a way the phrases "expressiou containing x" and "function of x"
may be regarded as synonymous. In finding the value of an explicit func-
tion corresponding to a particular value of the variable, the expression in-
volving the variable is treated simply as a pattern form in which to substitute
the value of the variable.
12, 13.] GRAPHICAL REPUESENTATION OF FUNCTIONS. 19
EXAMPLES.
1. Calciilate/(2) and/(.l)when/(x) = 3v^ + -+7a;2 + 2. Write /(y),
/(m),/(8mx). =^
2. Calculate /(2, 3), /(- 2, 1), and /(-I, -1) when f(x, y)-
3x^ + izy + Ty^-13x + 2y-n. Write f{u, v), /(sin x, 2).
3. Calculate z as a function of x when j/ =f(x) = "*" and z =f(y).
4. Given that f(x) = x^ + 2 and j;'(a;) = 4 + v^, calculate f[F(x')] and
8. UJlx, y) = ax^ + bxy + cy^, write /(t/, x), /(x, x), and/(?/, y).
6. If 2/= /Cx)=5^-±-^, show that X =/(!/).
ex— a
2 X 1
7. If y = <f,(x) = - -, show that x = <t>(y), and that x = <f>\x), in
ox ii
which ^'^(i) is used to denote ^[^(x)].
8. If /(i) = |±i, show that /2(x)=x, P(x) = x, /8(x)=x, etc., in
which /2(x) is used to denote/[/(x)],/3(x) to denote /{/[/(x)]}, etc.
9. If /(x) = ^^^ , show that /('^)-/Cy) = »-y .
Note. Notation for inverse functions. The student is already familiar
with the trigonometric functions and their inverse functions, and with the
notation employed ; thus, y = tanx, and x = tan-* y. In general if ^ is a
function of x, say y =:f(x), then i is a function of y. The latter is often
expressed thus : x =/"' (y). For instance, it y = log x, x = log-i (y). This
notation was explained in England first by J. F. W. Herschell in 1813, and at
an earlier date in Germany by an analyst named Burmann. See Herschell,
A Collection of Examples of the Application of the Calculus of Finite
Differences (Cambridge, 1820), page 5, note.
13. Graphical representation of functions of one variable. This
topic is discussed in algebra and in analytic geometry.
For instance, if y = 7 x + 5, (1)
the line whose equation is (1) is the graph of the function y in (1).
If a:^ + 2^ = 25, (2)
the circle whose equation is (2) is the graph of the function y in
(2). Important properties of a function can sometimes be in-
20 DIFFERENTIAL CALCULUS. [Ch. II.
ferred or deduced from an inspection of its graph.* Illustrations
of this will appear in later articles.
14. Limits. The notion that varying quantities may have fixed
limiting values is very important and should be clearly understood
when the study of the calculus is entered upon.
Limit of a variable. When a variable y, say, on taking successive
values approaches nearer and nearer to a constant value a, in such
a way thai the absolute value of the difference between y and a be-
comes and remains less than any preassigned positive quantity, the
constant a is said to be the limit of the variable y, and y is said to
approach the limit a,
EXAMPLES.
1. The area of a regular polygon inscribed in a circle varies when the
number of its sides is increased. Also, this area then approaches nearer
and nearer to the area of the circle. Further, the difference between the
area of the circle and the area of the polygon with the increasing number of
sides can be made less than any quantity that may be arbitrarily assigned,
simply by increasing the number of the sides. Moreover, this difference re-
mains less than the arbitrarily assigned quantity, when the number of sides
is still further increased.
This is mathematically expressed thus :
" The limit of the area of a regular polygon inscribed in a circle, when
the number of sides is increasing beyond all bounds, is the area of the circle ; "
and also expressed thus :
" The area of the polygon approaches the area of the circle as a limit when
the number of its sides is increasing beyond all bounds."
(In this case the varying polygonal area is always less than its limit, the
area of the circle.)
2. Discuss the case of the area of the regular circumscribing polygon when
the number of its sides is continually increasing.
(In this case the varying polygonal area is always greater than its limit.)
3. Discuss the cases of the lengths of the varying perimeters of the poly-
gons in Exs. 1, 2.
4. The number — , in which n is a positive integer, decreases as n in-
creases, and its value approaches nearer and nearer to zero when n is increased.
• Not every function can be represented by a curve ; see Infinitesimal
Calculus, page 20, footnote.
11.] LIMITS. 21
Also, — can be made to difier from zero by as small a positive number as
may be assigned, simply by increasing n ; and the difEerence between — and
2"
zero continues to remain less than the assigned number when n is still farther
increased.
Accordingly, — approaches zero as limit, when n becomes unlimitedly
great. In other words :
the limiting value of — , for n increasing beyond all bounds, is zero.
5. Let Sn denote the sum of n terms of the geometric series
The first term is 1 ; the sum of the first two terms is 1 j ; the sum of the first
three terms is 1| ; the sum of the first four terms is 1}^ ; and so on. It thus
seems to be the case that the more terms are taken, the nearer is their sum
to 2. This is clearly evident on writing the sum of n terms ; for
i - 1 2-1
Accordingly (see Ex. 4), S„ approaches 2 as limit when n increases be-
yond all bounds ;
in other words :
the limiting value of the series 1 + i + i H — , the number of whose terms is
unlimited, is 2.
N.B. The following trigonometric examples of limits are important, and
loill be employed in later articles. Proofs of 6, 7, 8, are given in text-books
on trigonometry.
6. (a) When an angle 8 is approaching 0° the limiting value of sin 0 is 0.
(6) When angle 6 is approaching 90° the limit of sin $ is 1.
(c) When angle 6 is approaching 0° the limit of cosfl is 1.
(d) When angle 6 is approaching 90° the limit of cosO is 0.
(e) When angle 0 is approaching 0° the limit of tan 6 is 0.
(/) When angle $ is approaching 90° tan 6 becomes unlimitedly great.
7. Show that, 0 being the number of radians in the angle, the limiting
value of the fraction ?1I1_, when 0 is approaching zero, is unity.
0
In Fig. 4, angle AOP = 0 radians ; QBR is a circular arc described about
O as centre with radius r ; QMS is a chord drawn at right angles to OA,
22
DIFFERENTIAL CALCULUS.
[Ch. II.
and accordingly is bisected hj OAat M; QT and BT are tangents drawn
at Q and B, which must meet at some point T on OA.
FiQ. 4.
By trigonometry, MQ = r sine, a,Tc QB — rB, QT=rta.ne (1)
By geometry, chord QB < arc QBB < broken line QTB ;
i.e. 2MQ<2a.TcBQ<2QT.
.•.,from(l), 2rsinfl<2rfl<2rtane.
sin«<«<tane. (2j
0 1
.•., on division by sin fl, 1 < < (3)
sin e cos 9
Now let 0 approach zero.
From the fact in Ex. 6 (c) , the limit of is then 1 .
^ ^ cos«
Accordingly, since by relation (3), the value of lies between 1 and
8in0
a
a number which is approaching 1 as its limit, the limit of must also be
Bind
1. Hence, the limit of , when e is approaching zero, is 1.*
&
8. Show that the limiting value of — — is 1 when 6 approaches zero.
$
[Suggestion. Divide the quantities in relation 2, Ex. 7, by tan 9.]
0.2 /jS
9. Show that the limit of ~ , when x approaches a, is 2 a.
X — a
10. Show that the limit of the sum 2 — 1 + ^
increases beyond all bounds, is |.
to n terms, as n
* For another proof see Plane Trigonometry, pages 143, 144.
15.] NOTATION. 23
11. In Ex. (a), Art. 4, -^ varies with Ax, and approaches 4 as Ax
Ax
approaches zero. By decreasing Ax the difference between =2 and 4 can be
Ax
made less than any positive number that may be assigned, and will remain
less than this number when Ax continues to decrease. That is, the limit of
—2, as Aa; approaches zero, is 4.
Ax
Show that in Ex. (6), Art. 4, the limit of — , as Aa: approaches zero, is 2 x.
Ax
Note 1. In each of these cases -^ finally reaches its limit. In Ex-. 10
Ax
the variable sum can never reach its limit.
A?
12. In Ex. (6), Art. 3, -^ varies with At, and approaches gt as At
At
approaches zero. By decreasing M the difference between — and gt can be
At
made less than any positive number that may be assigned, and will remain
less than this number when At continues to decrease. Accordingly, the limit
As
of — , as At approaches zero, is gt.
As
In Ex. (a), Art. 3, the limit of — , as At approaches zero, is 128.8.
At
As
In each of these cases — can reach its limit.
At
Another form of the definition of a limit. In the following defini-
tion, which is longer than the preceding one, the circumstances
under which the dependent variable approaches a limit are expli-
citly expressed.
Definition of a limit. Let there be a function of a variable, and
let the variable approach a particular value. If, at the same time
as the variable approaches the particular vahie, the function also
approaches a fixed constant in such a way that the absolute value
of the difference betioeen the function and the constant may be made
less than any positive number that may be assigned; and if, more-
over, this difference continues to remain less than the assigned num-
ber when the variable approaches still nearer to the particular value
chosen for it; then the constant is the limit of the function when the
variable approaches the particular value.
Ex. Eead Exs. 1-12, with this definition in mind.
15. Notation. The limit of a variable quantity, and the con-
dition under which this limit is approached, are expressed by
24 DIFFEBENTIAL CALCULUS. [Oh. IL
means of a certain mathematical shorthand. Thus the last sen-
tence in Ex. 5, Art. 14, is expressed :
Lim„^(H-i-fi +••■)= 2.
The results found in Ex. 11 are expressed :
Lim^^--^ = 4; Lim,^^-^ = 2a;.
Aa; Ax
The result found in Ex. 6 {h) is expressed :
Lim»^ir sin 6 = 1.
The symbol = is placed between a variable and a constant in
order to indicate that the variable approaches the constant as a
limit. Thus 6 = - above, means that 6 approaches ^ as a limit.
Note. The symbol = is used to indicate an approach, to equality. The
symbol = is used by many instead of = to indicate the same idea. Various
other notations are also employed.
Ex. Express the results in Exs. 1-12 in the mathematical manner of
writing.
IS a. Continuous variation. Interval of variation. When a vari-
able number, x say, takes in succession in the order of their mag-
nitudes all values from a number a to a number b, x is said to
vary continuously from a to b. The set of numbers from a to 6
constitute what is called the interval from a to b, and this interval
is denoted by [a, 6] or by (a, 6).* »
The notion of a variable that varies continuously through an
interval [a, 6] may be described graphically.
V A p B
a X b
Fig. 5.t
On this line let the distances be measured from 0, OA = a, and
OB = b. The point A thus corresponds to the number a, and the
* This symbol should not be confounded with a similar symbol which has
an altogether different meaning, the symbol denoting a point in analytic
geometry.
t The point 0 may happen to be between A and B or may be to the right
of 5.
16.] CONTINUOUS FUNCTIONS. 25
point B to the number b. Let P be any point on the segment
AB, and x its corresponding number. " Then as the point P
moves along the line from A to B, it passes in succession through
all the points from AtoB; and thus its corresponding number x
takes for its successive values all numbers, in the order of their
magnitudes, from a to b.
16. Continuous functions. Discontinuous functions. A function
f{x) in said to be continaoas for the value x = c it it satisfies both the
following conditions :
(1) Its value is finite when x = c, i.e. /(c) is finite ;
(2) The difference /(c + h) —f(c) approaches zero as the abso-
lute value of h approaches zero.
If, in the case of- a function f(x), either of the conditions (1) and
(2) is not fulfilled when x has a particular value, say x = c, then the
function f(x) is said to be discontinnons for the value x = c, or, more
briefly, discontinuous at c.
A function f{x) is said to vary contlnnonsly from a to &, or to be
continnons in the interval (a, 6),* when it is continuous for every
value of X between a and b.
The last definition may be written more fully on making use of
the first :
A function f(x) is said to be a continnons function of x for all
values of x from x = a to x = b, if it satisfies the following
conditions :
(1) Its value is finite for all values of x between a and b;
(2) Any two numbers between a and b (say c and c-\-h) being
taken, the difference/(c + ^)— /(c) approaches zero as the abso-
lute value of h approaches zero.
Note 1. Condition (2) may be roughly expressed in the following way,
which helps to bring out its practical meaning :
The change made in f{x) is exceedingly small when an exceedingly small
change is made in x, while the value of x lies between a and 6. Or, in other
words, the value of /(x) does not take a sudden jump of either a finite or an
unlimited amount when x changes by only an exceedingly small amount at
any value between a and 6.
• See Art. 15 a.
26 DIFFERENTIAL CALCULUS. [Ch. II.
EXAMPLES.
1. Let f{x) =22 + 3 a; -7.
This function is finite for all finite values of x ; accordingly, f{x) satisfies
condition (1) for any finite values of a and 6.
Let xi and xi + ft be any finite values of x. Then
/(xi) = Xi2 + 3xi-7,
and /(xi + ft) = (xi + ft)2 + 3(xi + ft) - 7.
Hence, the difference /(xi + ft) - /(xi) = ft(2 Xi + ft + 3).
This difference approaches zero when ft approaches zero ; acpordingly,
/(x) satisfies condition (2).
Since x'' + 3x — 7 thus satisfies conditions (1) and (2), it is continuous for
all finite values of x.
This example may be made more concrete by giving x\ a value, 3 say.
Then /(3 + ft) - /(3) = 9 ft + fts,
which approaches zero when ft approaches zero.
. •. /(x) is continuous for a; = 3.
2. Show that the function is continuous for values of x from — 4 to
X— 1
+ J, and for values of x from | to 5.
8. Show that the function, /(x) = , is discontinuous when x = 1.
X — 1
Give X the value 1 + ft.
1 1
Then/(l + ft)=-
(l+ft)-l ft
The value of /(I + ft) evidently increases beyond all bounds when ft ap-
proaches zero. Thus /(x) does not satisfy condition (1) when x = 1 ; and,
accordingly, is discontinuous for the value x = 1.
Note 1. Further examination shows that when x is passing through the
value 1, is going through an unlimitedly great change in value.
X — 1
"When X is a little less than 1, say .99999, then — ^ = ^ =—1000000.
x-1 .99999-1
"When a: is a little more than 1, say 1.000001 , then ^— = ^ = + 1.000000.
■' x-1 1.000001-1
The' difference between the values of x here is 1.000001— .99999, i.e.
.000002 ; the difference between the corresponding values of the function is
1000000 -(- 1000000), i.e. 2000000.
16.] CONTINUOUS FUNCTIONS. 27
In general :
When a; is a little less than 1, say 1 — A, in which A is a very small number,
then _!_ = 1 ^_1.
z-1 {\-h)-l h'
when x is a little more than 1, say 1 + A,
then _L_ = I = 1.
x-1 (l + ;i)_l h
Accordingly, /(1 + /i)_/(1_a)=?.
h
The smaller h is made, the greater this difference becomes ; and it in-
creases beyond all bounds when h approaches zero. Thus — - — experiences
X — 1
an unlimited change in value when x passes through the value 1.
4. Show that the function tan x is discontinuous when x= — - Also show
2
that when x passes through the value - , tan x takes an unlimitedly great
change in value.
Note 2. Some functions experience finite changes in value when the
variable passes through particular values.
For example :
1 1
the function f{x) = 2 (4^-3- 1)-- (4^-3 + 1)
changes its value from — 2 to + 2 (i. e. by the amount 4) when x is passing
through the value 3.*
Note 3. References for collateral reading on Limits and Continuous
and Discontinuous Functions. Several are given in Infinitesimal Calculus,
p. 29 ; to these add Pierpont, Theory of Functions of Real Variables, Vol. I. ,
Chap. VI., VII., Veblen-Lennes, Infinitesimal Analysis, Chaps. IV., V.
• See Infinitesimal Calculus, pages 26-29, Exs. 3, 6, Notes 5, 6, 9.
CHAPTER III.
INFINITESIMALS, DERIVATIVES, DIFFERENTIALS,
ANTI-DERIVATIVES, AND ANTI-DIFFERENTIALS.
17. In this chapter some of the principal terms used in the
calculus are defined and discussed, and one of the main problems
of the calculus is described. In the first study of the calculus
it is better, perhaps, not to read all this chapter very closely,
but after a cursory reading of it to proceed to Chapter IV., and,
while working the examples in that chapter, to re-read carefully
the articles of this chapter. These articles can also be reviewed
most profitably when the special problems to which they are
applied are taken up. Articles 22, 23, however, should be care-
fully studied before Chapter IV. is begun.
18. Infinitesimals, infinite numbers, finite numbers. An infini-
tesimal is a variable which has zero for its limit. (See definition
of a limit. Art. 14.) That is, if a denote an infinitesimal,
a = 0, or limit a = 0.
For instance, in Ex. (a), Art. 4, when PR is approaching zero it
is an infinitesimal. So also, at the same time, are angle QPT
and the triangle PQR. Again, when angle 6 is an infinitesimal
sin d and tan 0 are infinitesimal ; cos 6 is an infinitesimal when 6
is approaching ^ ; when n is increasing beyond all bounds 1 -=- 2"
is an infinitesimal.
Note. The infinitesimal of the calculus is not the same as the infinitesimal
of ordinary speech. The latter is popularly defined as " an exceedingly small
quantity," and is usually understood to have a fixed value. The infinitesimal
of the calculus, on the other hand, is a variable which approaches zero in a
particular way.
28
17-19.] INFINITESIMALS. 29
The following statements are in accordance -with, or follow
directly from, the definitions of a limit and an infinitesimal.
(1) The difference between a variable and its limit is an
infinitesimal. That is, on denoting the variable by x and the
limit by a,
if limit x = a, i.e. if x = a,
then « = a + a, in which o = 0.
(2) If the difference between a constant and a variable is an
infinitesimal, then the constant is the limit of the variable. In
"J ""J —
x = a-\
-«,
in which
« = 0,
then
x = a,
i.e.
limit
x = a.
This principle has been employed in the exercises in Arts. 3, 4.
It is evident that the reciprocal of an infinitesimal approaches
a number which is greater than any number that can be named,
namely, an infinite number. Accordingly, an infinite nnmber may
be defined as the reciprocal of an infinitesimal. Numbers which
are neither infinitesimal nor infinite are called finite nnmbers.
19. Orders of magnitude. Orders of infinitesimals. Orders of
infinites. Let m and n each denote a number which may be
finite, infinite, or infinitesimal. When the limiting value of the
ratio — is a finite number, m and n are said to be finite with
n
respect to each otiier and to be of the same order of magnitude;
when the ratio — either has the limit zero or is beyond all bounds,
n
m and n are said to be of different orders of magnitude.
For instance, 1,897,000,000 and .000001 are of the same order of magni-
tude. Tan 90° and tan 45° are of different orders of magnitude. Logx
and X are of different orders of magnitude when x is an infinite number.
This is shown in Art. 118, Ex. 1.
That infinitesimals may be of different orders of magnitude is
shown by the following illustration.
30
DIFFERENTIAL CALCULUS.
[Ch. III.
Suppose that the edge BL of the cube in Fig. 6 is divided into any number
of parts, and that each part, as Bb, becomes infinitesimal. Through each
point of division, as 6, let planes be passed at right angles to BL. The
cube is thereby divided into an infinite number of
infinitesimal slices like Bd. Now suppose that the
edge BA is divided like BL into parts like Bf which
become infinitesimal, and let a plane be passed
through each point of division / at right angles to BA.
The slice Bd is thereby divided into an infinite num-
ber of infinitesimal parallelopipeds like Ck. Finally
suppose that the edge BC is divided into parts which
become infinitesimal like Bg, and that through each
point of division, as g, a plane is passed at right
angles to BC. Then Ck is thereby divided into an
infinite number of infinitesimal parallelopipeds like
DL
kg. Since the limiting value of each of the ratios
Ck
is infinite,
Bd
Bd' Ck' kg'
the parallelopipeds DL, Bd, Ck, kg, are all of different orders of magnitude.
This illustration also serves to show that infinites may be of
different orders of magnitude.
Each of the three ratios, — , --, — , is an infinite number.
kg kg kg
But the
ratio of the first to the second, viz. ,
DL
i.e. is an infinite num-
Bd
Bd
kg kg '
ber; accordingly the first and second ratios are of different orders of magni-
tude. Similarly it can be shown that the second and third ratios are of
different orders of magnitude.
Note. On infinitesimals see Infinitesimal Calculus, pages 32-38, espe-
cially the Beferences, page 38.
20. Changes or increments in the variable and the function.
A. Change in the variable. Suppose that
y=f(^),
and that x has a particular value, say Xi. Then y has a particu-
lar value, viz. ?/i =/(x').
Now suppose that x changes from x^ by a certain amount, which
may be denoted by Ax.
This symbol Ax — which is read 'delta-x' (see Art. 3, Note 1) —
means simply a change or difference made in the value of x.
This change, which may be either an increase or a decrease, is
often called
the increment of the variable x.
20,21.] CHANGE IN THE FUNCTION. 31
Increment of x (i.e. Aa;) = («e!« value of x) — (the old value of x).
E.g. if X changes from the value 4 to 4.2,
its increment = 4.2 — 4, or .2 ; i.e. Ax = .2.
If X changes from the value 4 to 3.6,
its increment = 3.6 — 4, or — .4 ; i.e. Ax = — .4.
B, Chan^ in the fanction. When a variable x changes, its
function y changes and, accordingly, has an increment. This incre-
ment is denoted by Ay,
Thus Ay = (new value of y) — (old value of y).
E.g. let y = 5x^-ix + b.
ltx = i, then !/ = 3 X 42 - 4 X 4 + 5 = 37. (a)
Let X receive an increment Ax = .2.
Then y receives an increment Aj/, and the new value of y, viz.,
y + Ay = 3 x (4.2)2 _ 4 x (4.2)+ 6 = 41.12. (6)
.-. On subtraction in (a) and (6), Ay = 41.12 - 37 = 4.12. (c)
In general : if y =/(»), (1)
and X receives an increment Ax,
then y also receives an increment Ay.
Then (1) becomes y + Ay =f(x + Ax), (2)
and, thus, from (1) and (2), Ay =f{x + Ax) —f(x). (3)
In accordance with the use of the symbol A, the second mem-
ber of (3) may be written AfC^)-
EXAMPLES.
1. Given y = x^ — Sx + 4, calculate the corresponding increment of y, i.e.
Ay, Vfhen :
(a) X = 5 and Aa; = .3 ; (6) x = 3 and Ax =- .2.
2. Given s = S2t^ + 17 « - 5, calculate the corresponding increment of s,
i.e. As, when :
(o) t = S and At = .1 ; (6) < = 6 and At = .1 ; (f) « = 8 and At = .3.
3. Given r = sin ff, find the increment of r, i.e. Ar, when :
(a) e = 37°, A« = 20' ; (6) » = 216°, AS = 1°.
4. Given r = cos 6, find the increment of r, when :
(a) e = 37°, M = 20' ; (6) » = 216°, AS = 1°.
8. See tables of results on pages 3, 6, for examples on increments.
21. Comparison of the corresponding changes (or increments) made
in a function and the variable. These increments are compared by
forming the ratio, increment of the function
increment of the variable
32 DIFFERENTIAL CALCULUS. Ch. III.
That is, if the function is denoted by /(«), . (1)
by forming the ratio f{x + ^^-f{x-)^ ^2)
The fraction expressed by the form (2) is called the
difference-qaotient of the function.
EXAMPLES.
1. In the example worked in Art. 20, B, in whicli
Aa: = .2, and the corresponding ^y = 4.12,
^=il^ = 20.6.
Aa; .2
2. See last columns of tables, pages 3, 6, for examples of comparison of
increments.
A;/
3. Calculate the difference-quotients -^ m Ex. 1, Art. 20.
As
4. Calculate the difference-quotients — in Ex. 2, Art. 20.
6. Calculate the difference-quotients — in Exs. 3, 4, Art. 20.
22. The derivative of a function of one variable. Suppose that
the function fix)
denotes a continuous function of x. Let x receive an increment
Ax ; then the function becomes
fix + Aa-). (a)
Hence the corresponding increment of the function is
f{x + i,x)-f{x). (b)
This may be written A [/(a;)].
The ratio of this increment of the function to the increment of
the variable is f(x + Aa) -f(x) . A[/(x)] ,s
Ax ' *-^- Ax ^"^
The limit of this ratio when Ax approaches zero, i.e.
.. f(x + Ax)-f(x) .. A/(x) ,,,
is called the derived fanction of fix) with respect to x; or the
deriratlTe (or the derlvate) of /(x) with respect to x; or the
«-derlTatiTe ot fix). It is also called the differential coefficient
of fix), a name which is explained in Art. 27.
22.] DIFFERENTIATION. 33
If y also be used to denote the function, that is, if
y =/W,
then if x receive an increment Ax, y will receive a corresponding increment
(positive or negative), which may be denoted by Ay, i.e.
y + Ay=f{x^- Ax).
Hence Ay =f{x + Ax) - f(x) ;
and . Ay^/(x + Ax)-/W.
Aa; Ax
.-. lim,.,o^ = lim,,^^^(' + ^^)-/W- (/)
Ax Ax
The process of finding the derivative of a function is called
differentiation. This process is a perfectly general one, as indi-
cated in steps (a), (6), (c), and (d). It may be described in
words, thus:
(1) Give the independent variable an increment ;
(2) Find the corresponding increment of the function ;
(3) Write the ratio of the increment of the function to the
increment of the variable.
(4) Find the limit of this ratio as the increment of the variable
approaches zero.
For a slightly different description of the process of difierentiation, see
Note 4.
Note 1. To differentiate a function (i.e. to find its derivative) is one
of the three main problems of the inQnitesimal calculus, and is the main
problem of that branch which is called " the differential calculus."
Note 2. The other two main problems of the infinitesimal calculus (see
Arts. 27 a, 164) are the main problems of that branch called " the integral
calculus." It may be said here that while the differential calculus solves the
problem, " when the function is given, to find the derivative," on the other
band the integral calculus solves as one of its two main problems the inverse
problem, namely, " when the derivative is given, to find the function."
EXAMPLES.
1. Find the derivative of x' with respect to x.
Here f(x) = x». (See Fig., p.462.)
Let x receive an increment Ax ;
then /(x + Ax) = (x + Ax)^ = x^ + Z x^Ax + 3 x(Ax)2 + (Ax)».
34 DIFFERENTIAL CALCULUS. [Ch. IIL
.-. fix + Aa;) - /(x) = 3 x^Aa; + 3 x(Ax)2 + (Ax)*.
_./(x + Ax)-/(x)^3^; + SxAx + (Ax)2.
Ax
Ax
If y be used to denote the function, thus y = x', then the first members of
these equations will be successively, y, y + Ay, Ay, -^, lim^i^o — •
Ax Ax
Note 3. It should be observed that the expression (c) depends both on
the value of x and the value of Ax, and, in general, contains terms that
vanish with Ax, as exemplified in Ex. 1. (This is shown clearly in Art. 160.)
On the other hand, the value of the derivative depends on the value which
X has when it receives the increment, and on that alone. Tor this reason, the
derivative of a function is often called the derived fiinction. For instance,
in Ex. 1, if X = 2, the value of the derivative is 12 ; if x = 6, the value of
the derivative is 108. Compare Exs. in Arts. 3, i. (It is probably now
apparent to the beginner that the process used in the problems in Arts. 3, 4,
was nothing more or less than differentiation.)
Note 4. Sometimes Ax is called the difference of the variable x, (6) is
called the corresponding difference of the function, and (c) is called the
difference-quotient of the function. The process of differentiation may then
be described, thus : (1)' Make a difference in the independent variable ;
(2) Calculate the corresponding difference made in the function ; (.3) Write
the ratio of the difference in the function to the difference in the variable ;
(4) Determine the limiting value of this ratio when the difference in the
variable approaches zero as a limit.
8. Find the derivatives, with respect to x, of x, 2x, 3x, ox, x"^, 1 x^,
11 x\ bx\ xs, 5 x', 13 3?, and cx^.
Ans. 1, 2, 3, a, 2x, 14 x, 22 x, 2 6x, 3x2, 15x2, 39x2, ^cx\
3. Calculate the values of these functions and the values of their
derivatives, when x = 1, x = 2, x = 3.
4. Find the derivatives, with respect to x, of : (a) x' + 2, x" — 7,
x2 + A: ; (6) x' + 7, x' - 9, x^ + c.
6. Differentiate x*, x^ + 4 x - 5, -, ? - 3 x + 2 x^, with respect to x.
XX o
6. Find the derivatives, with respect to t, of 3 e^, 4 (3 _ 8 « + -■
3 7
7. Differentiate j/«, -y^ -%y — L^ with respect to y.
4 y
8. Show that, if n is a positive integer, the derivative of a!» with respect
to X, is nx^-^.
Note 5. The result in Ex. 8. as will be seen later, is true for all con-
stant values of n.
23.] NOTATION. 35
9. Assuming the result in Ex. 8, apply it to solve Exs. 4-7.
Note 6. In order that a function may be differentiable (i.e. have a de-
rivative), it must be continuous ; all continuous functions, however, are not
differentiable. For remarks on this topic, see Echols, Calculus, Art. 30.
For an example of a continuous function which has nowhere a determinate
derivative, see Echols, Calculus, Appendix, Note 1, or Harkness and Morley,
Theory of Functions, § 65 ; also Pierpont, Functions, Vol. I., Arts. 367-371.
23. Notation. There are various ways of indicating the derivar
tive of a function of a single variable. (In what follows, the
independent variable is denoted by x. In the case of other
variables the symbols are similar to those now to be described
for functions of x.)
(a) This symbol is often used to denote (d) Art. 22, viz.
/'(as). A
Thus the derivatives (or derived functions) of F(x), <^(y), f{t),
fi(z), with respect to x, y, t, and z, respectively, are denoted by
Fix), <f>'{y), f'{t), fi{z). These are sometimes read " the J'^-prime
function of x," etc.
(b) If y is used to denote the function of x (see Art. 22), the
derivative of y with respect to x is frequently indicated by the
symbol y,^ ^
This is often read "y-prime"; but it is better to say "deriva-
tive of y."
(c) The a>derivative of f{x) is also indicated by the symbol
The brackets in D are usually omitted, and the symbol is written
dx
E
Symbols C, D, and E should be read "the as-derivative of /(x)."
(ri) When y denotes the function, the derivative (see Equation
(/) Art. 22) is sometimes denoted by
36 DIFFERENTIAL CALCULUS. [Ch. III.
The brackets in F and Q are usually omitted, and the symbol
for the derivative is written
^. H
ax
This should be read for a while at least by beginners, "the
derivative of y with respect to x," or more briefly " the x-derivative
ofy." (Other phrases, e.g. " dy by da;," are common, but, unfortu-
nately, are misleading.)
(e) In case (d) the operation of differentiation, and also its
result, namely, the derivative, are alike indicated by the symbol
Dy. I
(/) Sometimes the independent variable x is shown in the
symbol, thus j)^y^ j
Note 1. Mathematics deals with various notions, and it discusses these
notions in a language of its own. In the study of any branch of mathe-
matics, the student has first to clearly understand its fundamental notions,
and then to learn the peculiar shorthand language, made up of signs and
symbols and phrases, which has been in part invented, and in part adapted,
by mathematicians. A striking instance of the great importance of mere
notation is seen in arithmetic. To-day a young pupil can easily perform
arithmetical operations which would have taxed the powers of the great
Greek mathematicians. The one enjoys the advantage of the convenient
Arabic notation* for numerals, the other was hampered by the clumsy
notation of the Greeks.
Note 2. Symbols A and B, and also I and J, have this important quality,
namely, they tend to make manifest the fact that the derivative is a single
quantity. It is not the ratio of two things, but Is the limiting value of a
variable ratio. Symbols C and F have the quality that they indicate, in a way,
the process (Art. 22) by which the derivative is obtained. The symbol —
before a function indicates that the operation of differentiation with respect
to X is to be performed on the function ; it also serves to indicate the result
of the operation. The symbols D and X)i,t in /and J, are simply abbrevia-
tions for the symbol —
dx
• This should really be called the Hindoo notation ; for the Arabs obtained
it from the Hindoos. See Cajori, History of Mathematics.
t The symbol D^y is due to Louis Arbogaste (1759-1803), professor of
mathematics at Strasburg. The symbol ^ was devised by Leibnitz, and
the symbol /', by Lagrange (1736-1813). '^^
24.]
REPRESENTATION OF THE DERIVATIVE.
^7
Note 3. Beginners in the calculus are liable to be misled by the symbols
D, E, 6, and H, especially by H. The symbol ^ does not denote a fraction ;
dx
it does not mean "the ratio of a quantity dy to a quantity dx." Such quan-
tities are not in existence at the stage when ^ is obtained. It should be
dx
thoroughly realized, and never forgotten, that ^ is short for — (w), and
dx dx
that both these symbols are merely abbreviations for lim_^ijj — ^gge Eq. f/")
Art. 22). Some one has remarked that the dy and ax in -^ are merely " the
dx
ghosts of departed quantities " ; but perhaps this is claiming too much for
them.
24. The geometrical meaning and representation of the derivative
of a function. Let f{x) denote a function, and let the geometrical
representation of the function, namely the curve
be drawn.
y =/(«'),
(1)
Fig. 7.
Let P(xi, yi) and Q(xi + Ax^, yj + Ay,) be two points on the
curve. Draw the secant LPQ. Then
Now let secant LQ revolve about P until Q reaches P. Then
the secant LP takes the position of the tangent TP, and the
angle PLX becomes PTX ; then, also, Aa^ reaches zero.
Hence
taiiXrP=lim^,^
AXi
(2)
38 DIFFERENTIAL CALCULUS. [Ch. III.
Now P (xj, yi) is any point on the curve ; hence, on letting
(x, y), according to the usual custom, denote any point on the
curve, and <^ denote the angle made with the a>axis by the
tangent at {x, y),
Aw
tan 4, = lim^^-^ • (3)
The first member of (3) is the slope of the tangent at any point
(«, y) on the curve y =/(x), and the second member is the
derivative of either member of (1). Hence ^, i.e. /'(a;), is the
slope of the tangent at any point {x, y) on the carve y = /(x).
This principle has already been applied in the exercises in
Art. 4.
Cnnre of slopes. If the graph of /'(») be drawn, that is, the
curve y=f'(x), it is called the curve of slopes of the curve
y =f(x). It is also called the derived curve, and sometimes the
differential curve of y =f{x). For instance, the curve of slopes
of the curve y = 3^ is the line y = 2x. The curve of slopes is
the geometrical representative of the derivative of the function ;
the measure of any of its ordinates is the same as the slope of
y = f{x) for the same value of x.
Ex. Sketch the graphs of the functions in Exs., Art. 22. Write the
equations of these graphs. Give the equations of their curves of slopes, and
sketch these curves. (Use the same axes for a curve and its curve of slopes.)
Note 1. Produce BQ (Fig. 7) to meet TP in S, produce PR to R', and
draw B'Q'S' parallel to RQ to meet the curve in Q' and TP in S'. Then
dx ■^ ^ ^ PR PR'
Now, If A*i = PJ?, g = ||; and if Axi = PR', g = f|^- Also,
,. -HO dy
and likewise, limps-^ -=^ = ^ •
Note 2. Hereafter, in general investigations like the above, the symbol x
will be used instead of x\ to denote any particular value of x ; and similarly
in the case of other variables.
25.] MEANING OF THE DERIVATIVE. 39
25. The physical meaning of the derivative of a function. Sup-
pose that the value of» a function, say s, depends upon time ;
i.e. suppose ^^^^^^
After an interval of time A«, the function receives an incre-
ment As; and , . -/, , .^s
.: As =f(t + M) -f(t).
. As_f(t + At)-f(t)^ ,j
"At At ^ ^
As
Since As is the change in the function during the time A*, —
At
is the average rate of change of the function during that time.
As At decreases, the average rate of change becomes more nearly
equal to the rate of change at the time t, and can be made to
differ from it by as little as one pleases, merely by decreasing At.
Hence the second member of (2) is the actual rate of change
at the time t. In words : The derivative of a function with respect
to the time is the rate of change of the function.
* cis
If s denotes a varying distance along a straight line, then —
denotes the rate of change of this distance, i.e. o velocity.
(Por discussions on speed and velocity see text-books on Kine-
matics and Dynamics, and Mechanics.)
^«
Ex. Show that if s = J gf, then — = gt, (See Art. 3 6.)
dt
Note. Newton called the calculus the Method of Fluxions. Variable
quantities were called by him fluents or flowing quantities, and the rate of
flow, i.e. the rate of increase of a variable, he called the fluxion of the
fluent. Thus, if s and x are variable, — and — are their fluxions. Newton
dt dt
indicated these fluxions thus : », x. This notation was adopted in England
and held complete sway there until early in the last century, and the other
notation, that of Leibnitz, prevailed on the continent. At last the continental
notation was accepted in England. " The British began to deplore the very
small progress that science was making in England as compared with its
racing progress on the continent. In 1813 the ' Analytical Society ' was
formed at Cambridge. This was a small club established by George Peacock,
40 DIFFERENTIAL CALCULUS. [Ch. III.
John Herschel, Charles Babbage, and a few other Cambridge students, to
promote, as it was humorously expressed, the principles of pure ' D-ism,'
thai is, the Leibnitzian notation in the calculus against those of ' dot-age,'
or of the Newtonian notation. The struggle ended in the introduction into
Cambridge of the notation -^, to the exclusion of the fluxional notation y.
dx
This was a great step in advance, not on account of any great superiority of
the Leibnitzian over the Newtonian notation, but because the adoption of the
former opened up to English students the vast storehouses of continental
discoveries. Sir William Thomson, Tait, and some other modern writers
find it frequently convenient to use both notations." — Cajori, Histury of
Mathematics, page 283.
26. General meaning of the derivative : the derivative is a rate.
When a variable changes, a function of the variable also changes.
A comparison of the change in the function with the causa] change
in the variable will determine the rate of change of the function
with respect to the variable. The limit of the result of this com-
parison, as the change in the variable approaches zero, evidently
gives this rate. But this limit has been defined as the derivative
of the function with respect to the variable. Accordingly (see
Art. 22, Note 1), the main object of the differential calculus may be
said to be the determination of the rate of change of the function
with respect to its argument.
Note 1. Tlie rate of change of the function with respect to the variable
may also be shown in a manner that explicitly involves the notion of time.
In the case of the function y, when y =f(x), let it be supposed that x receives
a change Arc in a certain finite time At. Accordingly y will receive a change
Ay in the same time M. Then, from the equation preceding (e), Art. 22,
Ay _ f{x + Ax)-f{x) ^ f(x + Ax)-f(x) Ax , ,
At At Ax ' At' ^ ^
Assume that Aa; =?fc 0 when At ^ 0. When At approaches zero, Ax also
approaches zero. On letting A* approach zero, and writing the consequent
limits of the three fractions in (a), there is obtained
^ ■" dy
^=f<{x) ^ ; i.e. ^ = ^ . ^. (1) Whence, ^ = |^ • (2)
dt ' dt dt dx dt dx dx ^ ■'
Result (2) can also be derived directly from
Ay_At_
Aa;~Ax
At
(6)
261 DIFFERENTIALS. 41
(Here it is assumed tliat Ax =^ 0, when A« =^ 0.) When M approaches
zero, Ax approaches zero. On letting At approach zero, and writing the con-
sequent limits of the three fractions in (6), relation (2) is obtained, and
from it relation (I) follows.
Ex. Express relations (1) and (2) in words.
Thus the derivative of a function with respect to a variable may be regarded
as the ratio of the rate of change of the function to the rate of change of the
variable.
NoTK 2. References for collateral reading. McMahon and Snyder,
Dtf. CaL, Arts. 88, 89 ; Lamb, Calculus, Art. 33 ; Gibson, Calculus, Arts.
31-37, 51.
EXAMPLES.
1. A square plate of metal is expanding under the action of heat, and
its side is increasing at a uniform rate of .01 inch per hour; what is the
rate of increase of the area of the plate at the moment when the side is 16
inches long ? At what rate is the area increasing 10 hours later ?
Let X denote the side of the square and A denote i ts area. Then A = x^.
■w„~ AA _ AA Ax „ . „ dA dA dx dA „ z^, ■ ,_
Now — - = ——.-—; whence, —- = —-•--. .•.-— = 2 a: x .01 sq. inches
At Ax At dt dx dt dt
per hour = .02 x sq. inches per hour. Accordingly, at the moment when the
side is 16 inches, the area of the plate is increasing at the rate of .32 sq. inches
per hour. Ten hours later the side is 16.1 inches ; the area of the plate is
then increasing at the rate of .322 sq. inches per hour. The area of the
square is increasing in square inches 2x times as fast as the side is increasing
in linear inches.
2. In the case of a circular plate expanding under the action of heat,
the area is increasing at any instant how many times as fast as the radius ?
If when the radius is 8 inches it is increasing .03 Inches per second, at what
rate is the area increasing ? At what rate is the area increasing when the
radius is 16 inches long ?
8. The area of an equilateral triangle is expanding how many times as
fast as each of its sides ? At what rate is the area increasing when each
side is 15 inches long and increasing at the rate of 2 inches a second ? At
what rate is the area increasing when each side is 30 inches long and increas-
ing at the rate of 2 inches a second ?
4. The volume of a spherical soap bubble is increasing how many times as
fast as its radius ? At what rate (cubic inches per second) is the volume in-
creasing when the radius is half an inch and increasing at the rate of 3 inches
per second 1 At what rate is the volume increasing when the radius is an inch ?
6. A man 5 ft. 10 in. high walks directly away from an electric light 16
feet high at the rate of 3^ miles per hour. How fast does the end of his
shadow move along the pavement 1
42 DIFFERENTIAL CALCULUS. [Ch. III.
27. Differentials, (a) Differential of a Tariable.
Let an independent variable x have a change Ax.
This difference Aa; in x is often called
' the differenticd ofx';
and it is then customary to denote it by the symbol
dx. (1)
(6) Differential of a function.
Let/(a;) denote any differentiable function.
Its derivative (Art. 23) is denoted hyf{x).
The product of the derivative of a function f(x) and the differen-
tial of the independent variable, viz.
f'(x)dx (2)
is ccdled the differential off(x).
In the same fashion as the differential of a variable x is denoted
by dx, the differentials of any other variables w, v, w, y, •■-, are
denoted by du, dv, dw, dy, •••.
Now let y denote the function f{x) ; i.e.
2/ = /(«)•
On taking the derivatives, -^ = f{x). (3)
dx
Then, by the definitions and notation above,
dy=f'{x)dx; (4)
i.e. dy = -^ ■ dx. (5)
doc
The defining equations (4) and (5) may be expressed in words :
The differential of a function y of an independent variable x is
equal to the derivative of the function multiplied by the differential
of the variable, the latter differential being merely a change (or dif-
ference) made in the variable.
ST.] DIFFERENTIALS. 43
The letter d is used as the symbol for the differential.
E.g. the differential of /(«) is written d/(x).
Thus, by definition (6),
df(x) = f{x)dx.
Illustration :
If
y = 3?,
then
dx
.■.dy = ^' dx = 3a?dx.
dx
If
a: = 4, and do; = .01,
dy = 3 xi'x. 01.
= .48.
The actual change made in y when x changes from 4 to 4.01 is
(4.01/ -4^ = .481201.
It will be found that, as in this case, the differential of a func-
tion corresponding to an assigned differential of the variable is
not in general the same as the change in the function ; it is, how-
ever, approximately equal to this change.
Note 1. The differential dx of an independent variable x may be any
arbitrary change, usually small, or it may be an infinitesimal. In the exam-
ples in this article the differentials have arbitrarily assigned or determinable
values ; in the examples in the integral calculus the differentials employed
are usually infinitesimals.
Note 2. It is highlj important to notice that in Equations (3) and (4),
dy and dx are used in altogether different ways.* In (3) and (5), — is used
as a symbol for lim^j-a) — ; and it denotes the definite limiting value of a
Ax
difference-quotient. In (4) and in (5) on the extreme right dx is not zero
(although it may happen to be, and usually is, a small quantity), t and the
dy is such that the ratio dy .dx is equal to f'{z). For instance, In Fig. 7,
• In one respect this double use of dx and dy is unfortunate ; for it tends
to confuse beginners in calculus. Other notation is also used.
t Later on many examples will be found in which this dx is an infinitesimal.
44 DIFFERENTIAL CALCULUS. [Ch. III.
^y- of Equation (2) is tan SPB. As to Equations (4), (5), if dx = PR, then dy
dx
= BS, and if dx= PR', then dy = R'S'. This shows that dy, in (4), is the
increment of the ordinate of the tangent corresponding to an increment dx
of the abscissa. The corresponding increment of the ordinate of the curve
y=f{x) [i.e. the increment of the function /(a;)] in some cases can be
found exactly by means of the equation of tlie curve, and in some cases can
be found, in general only approximately, by means of a very important
theorem in the calculus, namely, Taylor's Theorem (see Chap. XVI.).
Instances of the former are given below ; instances of the latter are given
in Art. 150.
Note 2. It should be clearly understood that, according to the preceding
remarks, cancellation of the dx's in (5) is impossible.
N.B. For geometric illustrations of derivatives and differentials see
Art. 67.
EXAMPLES.
1. In the case of a falling body s = \gt^ (see Art. 3) ; on denoting, as
usual, the differential of the time by dt, ds, the corresponding differential of
the distance is [Ex., Art. 3 (6)] gtdt ; i.e. ds = gldt. The actual change in s
corresponding to the change dt in the time is [see Eq. (2), Art. 3 (6)]
gtdt + lgidty.
2. In the curve y = x'^, dy = 'ixdx. The actual change in y corresponding
to the change da; in K is 2 X da; + (<fa;)2. (See Eq. (1), Art.4.) Thus if a; = 10
and dx = .001, dj) = 2 x 10 x .001 = .02. The actual change in the ordinate of
the curve from a: = 10 to a; = 10 + .001 is (10.001)2 _ 102^ fg .020001. This
change may also be calculated as stated above, viz. 2 x 10 x .001 + (.001)^. The
dy = .02 is the change in the ordinate of the tangent at a; = 10 from x = 10 to
X = 10.001 (see Note 1). (The student should use a figure with this example.)
3. Write the differentials of the functions in the Exs. in Art. 22.
4. Given that y = 7? — ix'^, find dy when a; = 4 and dx = .1. Then find
the change made in y when x changes from 4 to 4.1.
6. Given that 2/ = 2x3 + 7x2-9x + 5, find dy when x = 5 and dx = .2.
Then find the change made in y when i changes from 5 to 5.2.
Note 3. It is evident from these examples that the differential of a
function is an approximation to the change in the function caused by
a differential change in the variable ; and that the smaller the differential
of the variable, the closer is the approximation. When the differential varies
and approaches zero it becomes an infinitesimal.
Ex. Calculate the differentials of the areas in Ex. 2, Art. 26, when the
differential of the radius is .1 inch.
Ex. Calculate the differentials of the areas of the triangles in Ex. 3,
Art. 26, when the differential of the side is .1 inch.
27a.] ANTl-BERIVATIVES AND ANTI-DIFFEEENTIALS. 45
Note 4. It may be remarked here that in problems involving the use
of the diflerential calculus derivatives more frequently occur, and in prob-
lems in integral calculus difierentials (viz. infinitesimal differentials) are
more in evidence.
Note 5. References for collateral reading. Gibson, Calculus, % 60 ;
Lamb, Calculus, Arts. 67, 58.
27 a. Anti-derivatives and anti-difierentials. In Arts. 22 and 27
the derivative and the differential of a function have been defined,
and a general method of deducing them from the function has
been described. With respect to the derivative and the differen-
tial the function is called an anti-derivative and an anti-differential
respectively. Thus, if the function is a?, the axierivative and the
aj-differential are 2 x and 2 xdx respectively ; on the other hand,
3? is said to be an anti-derivative of 2 a; and an anti-differential of
2 xdx. To find the anti-derivatives and the anti-differentials of a
given expression is one of the two main problems of the integral
calculus. (See Art. 22, Notes 1, 2, and Arts. 164, 166, 167.)
Note. Reference for collateral reading. Perry, Calculus for Engi-
neers, Arts. 12-24, 28, 36.
CHAPTER IV.
DIFFERENTIATION OF THE ORDINARY FUNCTIONS.
28. In this chapter the derivatives of the ordinary functions of
elementary mathematics are obtained by the fundamental and
general method described in Art. 22. Since these derivatives are
frequently employed, a ready knowledge of them will prevent
stumbling and thus make the subsequent work in calculus much
simpler and easier; just as a ready command of the sums and
products of a few numbers facilitates arithmetical work. Accord-
ingly these derivatives should be tabulated by the student and
memorized.
N.B. The beginner is earnestly recommended to try to derive these results
for himself. For a synopsis of the chapter see Table of Contents.
GENERAL RESULTS IN DIFFERENTIATION.
29. The deriyative of the sum of a function and a constant, namely,
<|)(X) + c.
Put y = <f>(x) + c.
Let X receive an increment Aa; consequently y receives an
increment, Ay say. That is,
y + Ay = <l>(x + Ax) + c.
.: Ay = <l>(x+ Ax) + c— [<^(a;) + c]
= <}>(x + Ax) — <f>{x).
. ^ _ «/> (a: + Ax) — <^ (a;)
Ax Ax
46
29.] DIFFERENTIATION OF FUNCTIONS.
Let Ax approach zero as a limit ; then
Ax Ax
47
t.e.
(1)
Hence, if constant terms appear in a function, they may be neg-
lected when the function is differentiated.
If u be used to denote </> (x), result (1) can be expressed :
;I<»^«>=S-
(2)
Cob. 1. It follows from (1) that the deriyatire of a constant is
zero. This may also be derived thus : If y = c a constant, then
Ax
-.0 for all
y + Ay = c; and, accordingly, Ay = 0. Hence,
values of Ax; hence, -^, i.e. — (c), is zero.
dx dx
CoK. 2. If two functions differ by a constant, they have the
same derivative.
From (2) and Art. 27, d{u + c) = du.
Note 1. In geometry y = c is the equation of a straight line parallel to the
axis of X and at a distance c from it. The slope of this line is zero ; this is In
accord with Cor. 1.
Note 2. The curves y = (p(x) + c, in which c is an arbitrary constant
(Art. 10), can be obtained by moving the curve y = <t>{x) in a direction
parallel to the y-axis. The result (1) shows that for the same value of the
abscissa, the slope =^ is the same for all the curves. See Figs. 8, 9, below.
dx
Fio. 8.
Fig. 9.
48 DIFFERENTIAL CALCULUS. [Ch. IV.
Note 3. The converse of Cor. 1 is also true; namely, if the- derivative of
a quantity is zero, the quantity is a constant.
Ex. Show this geometrically. (See Art. 24.)
Note 4. The converse of Cor. 2 i."! also true ; namely, if two functions
have the same derivative, the functions differ only by an arbitrary constant.
(By the same derivative is pieant the same expression in the variable and the
fixed constants.) For let <p{x) and F{x) denote the functions, and put
y = 0(x) - Fix).
By hypothesis, Dy = <p'{x) - F'(,x) = 0.
Hence, by Note 3, y = c;
and accordingly, ^(x) = F(x) + c.
Ex. Show this geometrically.
Note 5. If -^ = 4>'(x), then y = it>{x) + c, in which c denotes any con-
dx
Btant. Hence ^(a;) + c is a general expression for all the functions whose
derivatives are <t>'{x). Functions such as <t>{x) + 1, ip{x) — 3, obtained by
giving particular values to c, are particular functions having the same deriva-
tive <p'{x).
Note 6. Notes 4 and 5 come to this : The anti-deriratire of a fnnction
is indefinite, so far as an arbitrary additire constant is concerned.
30. The derivative of the product of a constant and a function, say
Put y = c^(x).
Let X receive an increment Ax; consequently y receives an
increment, Ay say.
That is, y + Ay = c<^(x + Ax).
.-. Ay = c[<^(a; + Ax) — <^(x)].
Ax
>(x-fAx)-<^(x)~|
Aa; J'
.lim^.o^ = lim_ocr^(^±M:^^^1;
Ax Ax
]^
U. ^=c4>'ix);
ax
i.e. ^[c<f.(x)] = c<^'(*)- (1)
ax
so, 31.] DIFFEnENTIATION OF FUNCTIONS. 49
That is, the derivative of the product of a constant and a function
is the product of the constant and the derivative of the function.
If ^(x) be denoted by u, then (1) is written
In particular, ii u = x, — (ex) = c.
dx
From the above and the definition in Art. 27, d[c<^(x)] =
cd[<^(a;)], d(cu) = cdu, d(cx) = cdx.
Ex. See Exs., Art. 22.
3L The derivative of the sum of a finite number of functions, say
+CX) + Fix) + -.
Put y=,i,{x)+F(x) + -.
Then, on giving x an increment Aa; (as in Arts. 29, 30),
y + Ay = <^(a; + Ace) + F(x + Ax) + ....
.-. Ay = 4>{x + Ax) — <^(x) + F{x + Ax) — F(x) -\ .
. Ay _ ^{x + Ax) - <^(x) F{x + Ax) - F{x)
" Ax Aa; Ax
Hence, on letting Ax approach zero,
CCkC
That is, t^e derivative of a sum of a finite number of functions
is the sum of their derivatives.
If the functions be denoted by w, v, w, •••, i.e. if
y = u + v + w-\ ,
the result (1) may be expressed thus :
d^ __ du dv^ , dw , _ __
dx dx dx dx
60 DIFFERENTIAL CALCULUS. [Ch. IV.
From this and Art. 27,
dy = du + dv + dw + •••.
Note 1. The differentiation of the sum of an infinite number of functions
is discussed in Art. 147.
In working the following exercise the result of Ex. 8, Art. 22, may be
used.
Ex. Find the derivatives of
2x8 + 7x2 -10 a; +11, a;2_l7a; + 10, -x2 + 21z-5.
32. The derivative of the product of two functions, say ^{x)F{x.),
Put y = ^{x)F(x).
Then, on giving x an increment Aa;,
y + Ay = <f>(x + \x)F(x + Ax).
.: Ay = 4,(x + Ax)F{x + Ax) — ^{x)F{x).
. Ay _ 4>{^ + Ag) ^(a; + Aa;) - 4>{x)F{x) ,
"'Aa; Ax ' ^'
On letting Ax = 0, the second member approaches the form - •
In order to evaluate this form, introduce ^(x + Ax)F(x) — t^(x -{-
Ax)F(x) in the numerator of this member.* Then, on combining
and arranging terms, (1) becomes
^=A(a;+Ax/^'"+^^)— ^^'^>+Ma;)'^(^+^^)-'^^'^).
Aa; Aa; Ax
Hence, on letting Aa; approach zero,
f^=4>{x)F'(x) + F{x)4.\x). (2)
That is : The derivative of the product of two functions is eqzuil to
the product of the first by the derivative of the second plus the
product of the second by the derivative of the first.
* Equally well, <t>{x) F(x + Ax) — <p{x) F{x + Ax) may be thus introduced.
The student should do this as an exercise.
32.] DIFFERENTIATION OF FUNCTIONS. 51
If the functions be denoted by « and v, that is, if
y = uv,
then (2) may be expressed
^ = u^ + v^. (3)
dx dx dx '
The derivative of the product of any finite number of functions
can be obtained by an extension of (3). For example, if
y = uvw,
then, on regarding vw as a single function,
dx ^ ^dx dx^ ^
du , ( dv , dw\
= vw (- M I W h V )
dx \ dx dxj
du , dv , dw ,,,
= vw \- wu \-uv (4)
da; da; cte
Similarly, if y = uvivz,
dy du , dv , dw , dz ,p.s
-^ = vwz [-UWZ \-uvz \-uvw — (5)
dx dx dx dx dx
In general : In order to find the derivative of a product of several
functions, multiply the derivatim of each function in tui-n by all
the other functions, and add the results.
Note. Another way of obtaining (5) is given in Art. 39 (a).
The differential of the prodnct of two fnnetlons. If
y = uv,
then, from (3) and the definition in Art. 27, it follows that
dy = u — dx + V — dx. (6)
da; da;
But, by Art. 27, —dx = dv, and — dx = du.
dx dx
Hence, (6) may be written
d{uv) = udv + vdu. (7^
52 DIFFERENTIAL CALCULUS. [Ch. IV.
Similarly, if y = uvw,
it follows from (4) that dy = vwdu + wvdv + uvdw.
On division by uvw, this takes the form
d (uvw) _du ,dv ,dw ,q-,
uvw uvw
Ex. 1. Write dy in forms (7) and (8), when y = uvwz.
Ex. 2. Differentiate (x? + l)(x^ -2x + 7} by the above method ; then
expand this product and differentiate, and show that the results are the
same.
Ex. 3. Treat the following functions as indicated in Ex. 2 :
x2(x - l)(a^ + 4), (aa;2 + bx + c)(Zx + m).
Ex. 4. Write the differentials of the functions in Exs. 2, 3.
33. The derivative of the quotient of two functions, say «|>(a3) -i- IX.x).
Put y=^-
F{x)
Then, on proceeding as in Arts. 29-32,
y + i^y = ±i^LtA^.
"^ ^ J'Cx + Ax)
• ^„-'^(^ + Aa;) 4,(x)
" F(x + i^x) F(x)
_'t>(x + Ax)F(x) — ft>(x)F(x + Aa;)
F{x)F(x + Ax)
. Ay _ <^(a; -f Ax)F(x) — ft,{x)F(x + Ax)
' ' Ax F{x)F{x + Ax)Ax
On letting Aa; = 0, the second member approaches the form
In order to evaluate this form, introduce
F(x),j>(x) - F(x)<f>{x)
in the numerator of this member. Then, on combining and
arranging terms, (1) becomes
p^^S<t.(x + Ax)-,l>(x)l _ ^ ^SF(x + Ax)-F{x)l
(1)
0
o'
Ay_
Aa; F(x)F(x + Ax)
33.] DIFFERENTIATION OF FUNCTIONS. 53
Hence, on letting Ax approach zero,
dy _F(x),l>'(x) - 4>(x)F'(x) .^.
da; LF{ic)J ' ^ '
That is : If one function he divided hy another, then the derivative
of the fraction thus formed is equal to the product of the denomi-
nator by the derivative of the numerator minus the product of the
numerator by the derivative of the denominator, all divided by
the square of the denominator.
If the functions be denoted by u and v ; that is, if
then (1) has the form
u
y=v'
dy _ dx dx (2)
dx ~ v^
The differential of tlie quotient of two functions. If y =-, then
from (2) and the definition in Art. 27, ^
v — da; — u—dx ,„,
, dx dx \p)
But, by Art. 27, —dx = du and — da; = dv. Hence (3) may
be written
J,, vdu — udv /,N
^v = ^ (4)
Note. The derivative (1), or (2), can also be obtained by means of Art.
32. For if y = -, then vy = u. Whence v^ + y— = —- From this
V dx dx dx
dy _ldu_ydv^ which reduces to the form in (2) on substituting - for y.
dx V dx V dx V
Ex. 1. Find the derivatives and the differentials of
x» g' + 7 a; - 11
3a:2-7z + 2' x'^ + S' 2x''-9x + 3'
Ex. 2. Calculate the diSerentiala of the functions in Ex. 1 when z = 2
and dx = .1.
54 DIFFERENTIAL CALCULUS. [Ch. IV.
34. The derivative of a function of a function.
Suppose that y = <j>{u),
and that u = F(x),
and that the derivative of y with respect to x is required. (Here
<f>{u) and F(x) are differentiable functions.) The method which
naturally comes first to mind, is to substitute F(x) for u in the
first equation, thus getting y = <l)[F(x)], and then to proceed
according to preceding articles. This method, however, is often
more tedious and diificult than the one now to be shown.
Let x receive an increment Ax ; accordingly, u receives an incre-
ment Am, and y receives an increment Ay. Then
y + ^y = 't>(u + Am).
.-. Ay = <l)(u + Aw) -
- <^(w)-
A?/ <l>(u + Am) -
-<^(m)
" Aa; ~ Aa;
_ <l>(u + Ah) -
-<#•(«)
Am
Am
Aa;
Assume AM=ifcOwhen Aa;=^0. When Aa; approaches zero Ait
approaches zero, and this relation becomes
-^ = — r<^(M)l — ;
dx du^^^ ■'-' dx'
le.pL = ^?l.^. (1)
dx du dx ^ '
Note. It should be clearly understood that the first member of (1) does
not come, and cannot come, from the second member by cancellation of the
du's. Cancellation is not involved at all.
Result (1), which may be expressed more emphatically (Art. 23),
^^^) = £^^)'^^"^' (2)
is an important one and has frequent applications. It may be thus stated :
the derivative of a function with respect to a variable is equal to the product
of the derivative of the function with respect to a second function and the
derivative of the second function with respect to the first named variable.
(Here all the functions concerned are supposed to be diflerentiable.)
34,35.] DIFFERENTIATION OF FUNCTIONS. 65
From (1) and (2) it results that
A^ry) = ^2 , ie^=^. (3)
du^^^ -^(M) du du ^ ^
doc dx
Relations (1) and (2), Note 1, Art. 26, are special applications of (1) [or
(2) and (3)]. The showing of this is left as an exercise for the student.
Ex. 1. Explain why the da's in (1) may not be cancelled.
Ex. 2. Find -^, given that y = u^ and u = x'^ + 1.
dx
Here ^ = 3 u^ ^ = 2a;. .•.^ = eu^x = 6x(x^ + 1)2.
du dx dx
Ex. 3. Find ^ when ?/ = 3 u^ and u = x^-Sx + 7. Verify the result
dx
by the substitution method referred to at the beginning of the article.
Ex. 4. Find — when « = 2 b^ _ 3 » + i and v = efi+l. Verify the
dt
result by the substitution method.
Ex. 6. Show that a function of a function is represented by a curve in
space. (See Echols, Calculus, Appendix, Note 2.)
35. The derivative of one variable with respect to another when
both are functions of a third variable.
Let X = F{t) and y = <^(<).
Now — ^ = — ^ -!- — • Now At, Ax, and Ay reach the limit zero
Ax At At
together. (Assume that A.T=?i=0 when Ay=ifcO.)
Hence, on letting At approach zero,
dy
dy^dt_ „ .
dx ^ ^
dt
This result may also be derived as a special case of result (3),
Art. 34. This is left as an exercise for the student.
Ex. 1. Find ^ when y^St'-Tt + l, and z = 2{» - 13«2 + 11 1.
Here^=6«-7, ^ = 6(2-26« + ll. .-. ^= ^^^^
dt dt etc 6 «2 _ 26 « + 11
Ex. 2. Find ^ when x = 2t^ + nt-l and y = 31^-8^ + 9.
dx
Ex. 3. Find — when « = 7x« - 3 and » = 31= + Ux - 4.
dv
56 DIFFERENTIAL CALCULUS. [Ch. IV.
36. Differentiation of inverse functions. If y is a function of x,
then a; is a function of y ; the second function is said to be the in-
verse function of the first. This is expressed by the following nota-
tion: If y=f(x), then x=f~^(y). Assume that the function /(a:)
and its inverse /"' (y) are continuous and also differentiable.
For cases in which Aa; =^ 0 when Ay=^0 it follows from the
A ?/ ^^ If
equation — • — = 1, since Ax and Ay approach zero together,
Aa; Ay
that^.^ = l.
dx dy
Hence, in such cases,
dy 1
dx dx
dy
DIFFERENTIATION OF PARTICULAR FUNCTIONS.
In the following articles u denotes a continnons function of x,
and differentiation is made with respect to x. The letters a, n, •••,
may denote constants.
N.B. It is advisable for the student to try to obtain the derivatives before
having recourse to the book for help.
A. Algebraic Functions.
37. Differentiation of u".
(a) For n, a positive integer.
Put y = M" ;
i.e. y = uuu ■•■ to n factors.
.-. ^ = M"-'^ + M»-'^ + ••• to n terms (Art. 32)
dx dx ax
dx
In particular, —(x) = 1, and — (x") = raf^K
dx dx
Ex. 1. Give the derivatives with respect to x of
m', 3«*, 7m9, a;8, 3xS 7x^2, 9x» - ITx" + 10* + 40.
36, 37.] DIFFERENTIATION OF FUNCTIONS. 57
Ex. 2. Find the ^-derivative of (2 a: + ly.
Ou denoting tliis function by y, and putting m for 2x + 7, y = m''. Hence
dx dx
Now ^ = 2; hence ^ = 36 u" = 36 (2 x + 7)".
dx dx
The substitution ij for 2 a; + 7 need not be explicitly made. For, if
2/ = (2 a; + 7)18,
then ^ = 18 (2 a: + 7)1'— (2 a; + 7) (Art. 34)
dx dx
= 36 (2 a; + 7)i'.
Ex. 3. Diflerentiate
(5a;2-10)2*, (3i« + 2)i», (4a;2 + 5)8(3x* - 2x+ 7)*
(ft) For n, a negative integer. Let n = — m, and put y = u".
Then y = m"" = —
„" . A(l)_l . A(m~)
.-. ^ = ^ ^ (Art. 33)
~-\du
— mu'^ ' —
= r = ( — m) «<"•'• 3-
= mm" ' —
da;
Ex. 4. Differentiate with respect to x,
u-\ u-\ «-", X-', 3x-6, 17x-i», (x2-3)-«, (3x4 + 7)-6,
3xa-7x» + 2-UA__l..
(c) i'br n, a rational fraction. Let » = -, in which /) and g
are integers.
Put y=u^; then ^ = m'.
On differentiating, g]/*-'^ =pM''->^ •
dx dx
dy_p !/""' (?M _ p M""-' d« _ p ^f-' du _ ^^^,_i dw ^
"da; g y»"' da; g \-i) dx q dx dx
58 DIFFERENTIAL CALCULUS. [Ch. IV.
Ex. S. Find the ^-derivatives of
Vu (i.e. u^), u~^, u^, Vx, x^, V^, VSx^-S,
v/2x2 + 7i_3, VixTl, (3z-7)"^, 3x^-1 x^ + — + ~~~-
(d) i'br n, an incommensurable number. In this case it is also
true that - — (m") =:nM"-' — . This is proved in Art. 39 (6).
Hence, for all constant values of n,
^(««)=nu»-ig. (1)
In particular, if m = x, — (x") = nx"'^.
dx
Ex. 6. Find the ^-derivatives of
M^^ k'^s, 5 x*'', (2x + byi, {Sx^ + Tx- 4)*^.
Ex. 7. Write three functions which have z' for a derivative.
Ex. 8. Do as in Ex. 7 for the functions
x^, i, Vx, Vifi, v^, 6a^---— .
x^ x' y/i
Ex. 9. Show that tAe general form which includes all the fimctions that
have X" for the derivative, is 1- c, in which c is an arbitrary constant.
n + 1
Note 1. The result (1) and the general results, Arts. 29-36, suffice for
the differentiation of any algebraic function.
Note 2. Case (a) can also be treated as follows : Put y = m", and let x
receive an increment Ax ; then u and y receive increments Au and Ay
respectively. Then y + Ay = (u + Au)". On expanding the second member
by the binomial theorem, then calculating Ay and then =^, and finally letting
Ax approach zero, the result will be obtained.
Note .S. It is well to remember that —(x)=:l and — (v^) = — ^•
dx dx 2 Vx
Ex. 10. Do the operations indicated in Note 2.
X'\/t^ 4- 7
Ex. 11. Differentiate "^ • Find the value of the derivative when
x = '2. VWT^
Put y = ^J^^J±.
(.3? + 2)i
37.] DIFFERENTIATION OF FUNCTIONS. 59
(a;! + 2)^-f [x(a:2 + 7)*] - x(x^ + 7)i-f (x^ + 2)*
Then ^ = "^ ^
^ (x= + 2)*
On performing the differentiations indicated in the second member, and
reducing, it is found that
dy_ 4 a:« + 19 x' + 42
"^^ 3(a:2+7)*(22 + 2)i
Hence, when x = 2,
-2 = 1.68, approximately.
dx
Ei. 12. DifEerentiate the following functions with respect to x :
(2x-5)(x2 + llx-3), aa;»+-, i^tZl, ^^^, VTT^, l + 5-^x-7x6,
x" 1 - x2 a + X X*
}^1+^, -^^=, _^, J\±^, (1 + x")", (a + 6x3)S x»(l - X)-,
^ v/a - 6xi' (1 _ ^2)! ^ 1 - X
(a + x) Va — I.
Ex. 13. Find -^ when x^yS + 2x + 3« = 5. Here y is an implicit function
dx
of X. On differentiation of both members with respect to x,
ax dx ax
i.e. 3 x-^3^ + 2 xy' + 2 + 3 ^^ = 0.
dx ax
T^ .,.• dy 2(l + xy»)
From this - - = - „ ;. , o,
dx 3 (1 + xV)
Ex. 14. (a) Find ^ when x and y are connected by the following rela-
dx
tions: y^ + x? -Saxy = 0 ; x< + 2 ax^j/- ay» = 0 ; 7xV+ 2xy»-3iS!/ + 4x2
- 8 y" = 5 ; (a + y)''(62 - y2) + (x + a) V = 0 ; x^ + y' = a" ; aV + 6-x' =
a*6'^. In the last case also obtain =^ directly in terms of x.
dx
(6) In the ellipse 3 1'' + 4 y^ = 7, find the slope at the pointa (1, 1),
(1, -1), (-1,1), (-1, -1).
N.B. The following examples should all be worked by the beginner.
They will serve to test and strengthen his grasp of the fundamental prin-
ciples of the subject, and will give him exercise in making practical applica-
tions of his knowledge. For those who may not succeed in solving them
60 DIFFERENTIAL CALCULUS. [Ch. IV.
after a good endeavour, two examples are worked in the note at the end of
the set.
Ex. IS. A ladder 24 feet long is leaning against a vertical wall. The foot
of the ladder is moved away from the wall, along the horizontal surface of
the ground and in a direction at right angles to the wall, at a uniform rate
of 1 foot per second. Find the rate at which the top of the ladder is descend-
• ing on the wall when the foot is 12 feet from the wall.
Ex. 16. Show that when the top of the ladder is 1 foot from the ground,
the top is moving 575 times as fast as when the foot of the ladder is 1 foot
from the wall.
Ex. 17. Find a curve whose slope at any point (i, y) is 2 x. Find a
general equation that will include the equations of all such curves. Find
the particular curve which passes through the point (1, 2).
Ex. 18. A man standing on a wharf is drawing in the painter of a boat at
the rate of 4 feet a second. If his hands are 6 feet above the bow of the boat,
how fast is the boat moving when it is 8 feet from the wharf ?
Ex. 19. A man 6 feet high walks away at the rate of 4 miles an hour from
a lamp post 10 feet high. At what rate is the end of his shadow increasing
its distance from the post ? At what rate is his shadow lengthening ?
Ex. 20. A tangent to the parabola y^ = 16 a; intersects the z-axis at 45°.
Find the point of contact.
Ex. 21. A ship is 75 miles due east of a second ship. The first sails west
at the rate of 9 miles an hour, the second south at the rate of 12 miles an
hour. How long will they continue to approach each other ? What is the
nearest distance they can get to each other 1
Ex. 22. A vessel is anchored in 10 fathoms of water, and the cable passes
over a sheave in the bowsprit which is 12 feet above the water. If the cable
is hauled in at the rate of a foot a second, how fast is the vessel moving
through the water when there are 20 fathoms of cable out ?
Ex. 23. Sketch the curves y'^ = ix and x^ = 4 y, and find the angles at
which they intersect. (If B denotes the angle between lines whose slopes
are m and n, tan fl = (m — n) -^ (1 + mn) ; see analytic geometry and plane
trigonometry.)
Ex. 24. Sketch the curves xf^ = ix and 3? = %y,
and find the angles at which they intersect.
Note. Examples worked. Ex. 15. Let FT be
the ladder in one of the positions which it takes during
the motion, and let FH be the horizontal projection of
FT. Let FH=x, and HT=y. Then
x' + y» = 576. (1) Fig. 10.
38.] DIFFERENTIATION OF FUNCTIONS. 61
Now X and y are varying with the time ; the time-rate ^ is given, and
le time-rate -^ is requ
dt ^
respect to the time give
dv ^^
the time-rate -2 is required. Differentiation of both members of (1) with
''t*''i
= 0;
lence
dy.
X dx
dt
y dt
In this
case,
dx_
dt '
y-
= 1 foot
per second,
X = 12 feet,
12 V3 feet.
and.
accordingly,
= V242.
- 122 feet =
(2)
- • 1 foot per second = — .577 feet per second.
dt 12 V3
The negative sign indicates that y decreases as x increases. It should be
noticed that the result (2) is general, and that all particular solutions can
be derived from it by substituting in it the particular values of x, y, and — •
dt
Ex. 17. Find a curve whose slope at any point (x, y) is 2x. Find a
general equation that will include the equations of all such curves ; and find
the particular curve which passes through the point (1, 2).
Here ^ = 2 x.
dx
Hence y = x'' + c, (1)
in which c denotes any arbitrary constant. This is the general equation of
all the curves having the slope 2x. .-. y — x^ -{■ 1 ia one of the curves ;
y =x^ — b is another. If the point (1, 2) is on one of the curves (1), then
2 = 1 + c ; whence c = 1, and, accordingly, y = x^ + 1 is the particular curve
passing through (1, 2). As in Ei. 15 it is easier to find first the general solu-
tion ol the problem in question, and therefrom to obtain any particular
solution that may be required. Figure 9 shows some of these curves.
B. Logarithmic and Exponential Functions.
38. Note. To find limmix I 1-)- — J . This limit is required in what
follows.
(o) For m, a positive integer. By the binomial theorem,
,, l,»n-TO — 1 l.m-TO — 1-m — 2 1, /■.,..
m 1-2 m* 1 • 2 • 3 m'
This can be put in the form
(2)
(-:)■
can oe put m me lorm
ifi_n i(i_iVi_2)
fl+l^"=l + l.^A_^ + J — nd} — «d + ....
\ mj 2 1 d I
62 DIFFERENTIAL CALCULUS. [Ch. IV.
On letting m approach infinity, and taking the limits, this becomes *
Iinw»(l+ir=l + l+r7 + 5^+-
\ ml 2 ! 3 !
= 2.718281829 •••. (3)
This constant number is always denoted by the symbol e.
(6) The result (3) is true for all infinitely great numbers, positive and
negative, integral, fractional, and incommensurable. For the proof of (3)
for all kinds of numbers, see Chrystal, Algebra (ed. 1889), Part II., Chap.
XXV., §13, Chap. XXVIII., §§ 1-3; MoMahon and Snyder, Diff. Cal.,
Art. 30, and Appendix, Note B ; Gibson, Calculus, § 48.
Note on e. The transcendental number e frequently presents itself in
investigations in algebra (for instance, as the base of the natural logarithms,
and in the theory of probability), in geometry, and in mechanics. The num-
bers e and ir are perhaps the tv7o most important numbers in mathematics.
They are closely allied, being connected by the very remarkable relation
e*" = — l,t vfhich was discovered by Euler. See references above, and Klein,
Famous Problems (referred to in footnote. Art. 8), pages 55-67.
39. Differentiation of loga u.
Put y = \og„u,
and let x receive an increment \x ; then u and ^ consequently
receive increments Am and Aj/ respectively.
Then y + dty = log„ (u + Am).
.-. A?/ = log„ (m -h Am) — log„ u
=iog.(^^yiog.(i+^y
Ax \ u J Ax
On introducing Am in the second member,
M Alt
u
A.v_l Ml /.. . AiA Am_1i f^,Au\Ai Am
Ax u Am " \^ u J Ax u ° \ u J Ax
* This conclusion is properly reached only after a more rigorous investigar-
tion than is here attempted. (See Arts. 167-171.)
t See Art. 153.
39.] DIFFERENTIATION OF FUNCTIONS. 63
From this, on letting Ax approach zero and remembering that Am
and Ay approach zero with Ax, it follows by Arts. 22, 23, 38, that
dy_l-, du _
dx u ° cZa;'
i.e. 4- (loS« «) = i • logo e ■ ^.
ax u dx
If M = a;, then ~ (logo x) = — . logo e.
ax X
If a = e, then A(iogM) = l^.
dx u dx
If u=x, and a=e, then ^— (log x) = —
Note. When e is the base it is usual not to indicate it in writing the
logarithm.
Ex. 1. Find the derivatives of loga(.3 x^ + 4x — 7), log (_3x^ + 4x — 7),
logio (3 a;2 + 4 a; — 7) . Find the values of these derivatives vf hen re = 3.
Ex. 2. Find the values of the derivatives of log Va:^ + 10, logio Va;' + 10,
when X = 2.
Ex.3. Differentiate the following: log?-^ i^nr^/l+a: i„„l+v^
log (a; + Vx' + a'), log (log x) , x log x.
Ex. 4. Find anti-derivatives of ^^^"^ iix''-l J_
a;2 + 3 a + 5 x' - 7 x - 1 2 x
(a) Logarithmic differentiation. If
y = uvw, (1)
then log y = ]ogu + log v + log w.
r\ j-a i.- 4.- ^dy Idu , Idv , Idw
On differentiation, --^ = -— -H --| — ,
ydx udx vdx wdx
;i^, 10.^1+-, logi±^.
l + X '1-X 1-y/x
whence — = nvw
dx
1 dw 1 dw , J. dw~\ _ /nx
udx vdx wdx J
This result can easily be reduced to the form obtained in
Art. 32. The same method can be used in the case of any finite
number of factors. This method of obtaining result (2) is called
64
DIFFERENTIAL CALCULUS.
[Ch. IV.
the method of logarithmic differentiation. It is frequently more
expeditious than that given in Arts. 32, 33, especially when
several factors are involved.
(I" + 2)*
log !/ = log X + J log (x2 + 7) - i log (3? + 2).
a; 2x
Ex. 5. Find ^ when
dx,
Here,
On differentiation,
1^ = 1 +
ydx X x^ + 7 3 (x2 + 2)
From tliis, on transposing, combining, and reducing,
4 z< + 19 z' + 42
dy.
dx
3 (a;! + 7)^(^2 + 2)*
(a)
Ex. 6. Difierentiate, with respect to x, the following functions
(x + 2y
,^. (x-l){x-2).
^ (a; + l)(a; + 2)'
(c)
V2a: + 6v^a:-6
</{x + 3)*
(4a:-7)J(3« + 5)*
(6) Differentiation of an incommensnrable (constant) power of a
function. This paragraph is supplementary to Art. 37 (d).
Let y = u",
in which n is any constant, commensurable or incommensurabla
Then
log y = n log u.
Note. This deri-
From this
Idy _ndu ^
ydx udx
vation assumes that
^ exists.
dx
and hence
dy
dx
u dx dx
40. Difierentiatioii
1 of
a«.
Put
2/ = o".
Then
log 2/ = M log a.
(See Note i^ljove.)
On differentiation,
lf = loga.^.
ydx
dy 1 du
dx ^ ^ dx'
I.e.
d
du
dx dx
.] DIFFERENTIATION OF FUNCTIONS.
If u = x, then
-^(a*) = a* "log a.
dx
If a = e, then
il«"=-£-
If u = x, and a =
= e, then
/^(e«)=ex;
65
that is, the derivative of e' is itself e*.
Note 1. On the derivation of results in Arts. 39, 40. The derivative
of loga u was deduced by the general and fundamental method, and has
been used in finding the derivative of a". The latter derivative can be
found, however, by the fundamental method, independently of the deriva^
tive of log, M. Moreover, the derivative of loga « can be obtained by means
of the derivative of a". These various methods of finding the derivative
of a" and log„ u are all employed by writers on the calculus. For examples
see Todhunter, Diff. Cal., Arts. 49, 50; Gibson, Calculus, §65, where both
these derivatives are obtained independently of each otlier ; Williamson,
Diff. Cal., Arts. 29, 30; McMahon and Snyder, Diff. Cal., Arts. 30, 31,
where the derivative of the logarithmic function is first obtained and the
derivative of the exponential function is deduced therefrom ; and Lamb,
Calculus, Arts. 35 (Ex. 6), 42, where the derivative of the exponential
function is obtained first and the derivative of the logarithmic function
is deduced therefrom. (See also Echols, Calculus, Art. 33 and foot-note.)
Note 2. On the expansion of e' in a series see Hall and Knight, Higher
Algebra, Art. 220 ; Chrystal, Algebra, Vol. II., Chap. XXVIII., §§ 4, 5; and
other texts. (This expansion is derived by the calculus in Art. 178, Ex. 7.)
Ex. Assuming the expansion for e', show that the derivative of e* is
itself e'.
Note 3. The compound interest law. The function e* "is the only
[mathematical] function known to us whose rate of increase is proportional
to itself ; but there are a great many phenomena in nature which have this
property. Lord Kelvin's way of putting it is that ' they follow the compound
interest law.' " (See Hall and Knight, Higher Algebra, Art. 234, and, in
particular, Perry, Calculus, Art. 97 and Art. 98, Exs. 4, 2.)
Ex. 1. Differentiate, with respect to x, e , 10', 10 , e*^.
Ex. 2. Find the {-derivatives of e^, 10'', e''"*"', lo'^'^'.
Ex. 3. Find the a-derivatives of the following :
e^ — 1 t" + e-' z
Ex. 4. Find anti-derivatives of e''*, xe^, 2 6^+'.
On differentiation, — V = 1 Ji^ log u
66 DIFFERENTIAL CALCULUS. [Ch. IV.
41. Differentiation of w, in which u and v are both functions
of X.
Put y = u\ (1)
Then log y = v log u.
dv
y dx u dx ' ° dx
dy fv du , -, dv\
dx \u dx dxj
le. ^r„») = „«(i'f* + iogM^V (2)
dx \u dx dxj
Note 1. It is tetter not to memorize result (2), but merely to note the
fact that the function in (1) is easily treated by the method of logarithmic
differentiation.
Note 2. The beginner needs to guard against confusing the derivatives
of the functions a", a", and u'.
Ex. 1. Find -4^ when « = af«.
dx "
Here logy = x log x.
On difEerentiation, - -=^ = - + log x ;
y dx X °
whence ^ = x'(l + log i).
Ex. 2. Find the ^-derivatives of
(3x + 7)'\ (3a; + 7)% {(3x+7nX ^x, x-", e«', 1^)', log^.
C. Trigonometkic Functions.
42. Differentiation of sinu.
Put y = sin u.
Then y + Ay = sin (m + Am).
.-. Ay = sin (u + Au) — sin u
= 2 cos [ M + -^ ] sin -^' (Trigonometiy)
41,42.] DIFFERENTIATION OF FUNCTIONS. 67
... ^ = 2cosf« + ^')sin^.J-
Aa; \ 2 J 2 Ax
=-(»+f)
sin —
2 Am
Am Ax
2
Let Ax = 0 ; then also Am = 0, and
Am
lim,!^ 1^ = Iim^„^ cos fu + ^Y lim^„^ __J_ . lim,,^
,• o <^V ^ C^M
I.e. -^ = cos M . 1 —
2
da; dx
^(8inu) = cosug. (1)
In particular, if m = x,
(sinsc) = cosa;. (2)
That is, the rate of change of the sine of an angle with respect
to the angle is equal to the cosine of the angle.
Note 1. Result (2; can also be obtained by geometry. (Ex. Show this.)
See Williamson, Diff. Cal, Art. 28, and other texts.
Note 2. Result (2) shows that as the angle x increases from 0 to — the
rate of increase of the sine is positive, since cos a: is then positive. As x
increases from ^ to x the rate is negative (i.e. the sine decreases), since
o
cos X is then negative. The rate is negative when x increases from t to '-^,
n 2
and the rate is positive 'when x increases from — to 2 ir. This agrees with
what is shown in elementary trigonometry, and it is also apparent on a
glance at the curve y = sin x.
Note 3. Result (2) also shows that if the angle increases at a uniform
rate, the sine increases the faster the nearer the angle is to zero, and
increases more slowly as the angle approaches 90°. This is also apparent
from an inspection of a table of natural sines, or from a glance at. the curve
y — sin X.
Note 4. The derivative of sin u has been found by the general and
fundamental method of differentiation. It is not necessary to use this
68 DIFFERENTIAL CALCULUS. [Cii. IV.
method in finding the derivatives of the remaining trigonometric and anti-
trigonometric functions, for these derivatives can be deduced from that of
the sine.
Ex. 1. Find the awJerivatives of sin 2 u, sin 3 «, sin \u, sin § «, sin y- u.
Ex. 2. Find the anierivatives of sin 2 a;, sin Si, sin J a;, sinSa;^, sin'^Sx,
Bin4a;^ sin' 4 a;.
Ex. 3. Find the derivatives with respect to t of sin 5 1, sin \ {^.
Ex. 4. Find the x-derivatives of ?1IL1?, a; sin 2 x, x'^sai{x + -\-
sin 3 X \ 4 /
Er. 6. At what angles does the curve y = sm.x cross the x-axis ?
Ex. 6. At what points on the curve y = suix is the tangent inclined 30°
to the X-axis.
Ex. 7. Draw the curve y = sin 2 x. At what angles does it cross the
X-axis ?
Ex. 8. Draw the curve y = sin x -!- cos x. Where does it cross the x-axis ?
At what angles does it cross the x-axis ? Where is it parallel to the x-axis ?
Ex.9. Find the x-derivatives of the following: sin nx, sinx", sin"x,
sin(l-|-x2), 8in(nx-l-o), sin(a -1- 6z"), sin'4x, 5^2^, sin(logx), log(sinx),
sin (e») • log x. *
Ex. 10. (a) Find anti-derivatives of
cosx, cos3x, cos(2x-l-5), xcos(x2— 1).
(6) Find anti-differentials of cos2xdi, cos(3z — 7)(tc, x^cosa^fix.
Ex. 11. Calculate d(sinx) when x = 46° and dx = 20', and compare the
result with sin 46° 20' — sin 46°. (Radian measure must be used in the
computation.)
Ex. 12. Compare <i(sin z) when z = 20° and dx = 30', with
sin 20° 30' - sin 20°.
43. Differentiation of cos u.
Put y = cos M.
Then y = sin ( - — m ]•
= -sinM*^;
dx
i.e. -^(cosM) = -sinM— . m
dx dx ^ '
43, 44.] DIFFERENTIATION OF FUNCTIONS. 6.9
In particular, if u = x,
^ (cos 05) = - sin as. (2)
Ex. 1. Obtain derivative (1) by tlie fundamental method.
Ex. 2. Show that result (2) agrees in a general way with what is shown
in trigonometry about the behaviour of the cosine as the angle changes from
0° to 360°. Also inspect the curve y = cos x.
Ex. 3. Find where the curve y = cos x is parallel to the x-axis, and where
its slope is tan 25°.
Ex. 4. Show that the tangents of the curve y = cos x cannot cross the
X-axis at an angle between + 45° and + 135°.
Ex. 6. Find the slope of the tangent to the ellipse x = acoae, y = b sin 6.
(See Art. 35.)
Ex.6. Find the slope of the tangent to the cycloid x — a{e — sme),
y = a(l — cos 8). What angle does this tangent make with the x-axis when
a = 5, and # = - ?
3
Ex.7. Find the x derivatives of the following: cos(2x + 5), cos'Sx,
x^cosx, ~ °"^ ^, cosmxcosnx, xe™", €"co3mx.
1 + cosx
Ex.8. Find anti-differentials of sinxdz, sin^xdx, sin (3 x — 2)(ic,
X sin (x" + i)dx.
Ex. 9. Calculate d cos x when x — 57° and dx = 30', and compare the
result with cos 57° 30' - cos 57°.
44. Differentiation of tan u.
Put y =
tanw.
Then y =
sinu
COSM
COS M — (sin u) — sin u — (cos m)
dx cos'u
__ (cos' M-|-sin^ u) du
cos' u dx
1 du 2 d7i
= sec'w-
i.e.
cos' udx dx'
-^(tan«) = 8ec2«^. (i)
dx dx
70 DIFFERENTIAL CALCULUS. [Ch. IV.
If M = a:, then ^ (tan x) = sec^ x. (2)
Ex. 1. Show the agreement of result (2) with the facts of elementary
trigonometry, and with tlie curve y = tan x.
Ex. 2. Show that the tangents of the curve y = tan x cross the x-axis at
angles varying from + 45° to + 90°.
Ex. 3. State the x-derivatives of tan 2 «, tan 3 «, tan mu, tan nii^, tan 2 x,
tan J X, tan mx, tan 3 x^, tan 4 x', tan tox", tan^ 3 x, tan^ 4 x, tan" mx,
Uu2(|x + 3), log tan |.
Ex. 4. Find anti-diSerentials of sec^ xdx, sec2 2 x dx, sec^ (3 x + a)dx.
Ex. 6. Compute d tan x when x = 20°, dx = 20', and compare the result
with tan 20° 20' - tan 20°.
Ex. 6. When is the differential of tan x infinitely great ?
45. Differentiation of cot u.
Either, substitute S23Ji^ for cot u, and proceed as in Art. 44 ;
sin u
or, substitute tan (90°— m) for cot u, and proceed as in Art. 43;
or, substitute for cot u, and differentiate. It will be
found that *^" "*
^ (cot M) = - cosec2 u^- (1)
dx dx ^ '
li u = x, — — (cot x) = - cosec'* x. (2)
dx ^ '
Ex. Show the general agreement of result (2) with the facts of ele-
mentary trigonometry, and with the curve y = cot x.
46. Differentiation of sec u.
Put y = sec u = -
Then
COSM
dy __ sin u du _ 1 sin u du .
dx cos^ u dx cos u cos u dx '
e. ^ (sec M) = sec M tan M ^. (1)
dx dx ^ '
If a = a;, -^ (sec x) = sec x tan x. (2)
45-49.] DIFFERENTIATION OF FUNCTIONS. 71
47. Differentiation of esc u.
Put 2,= cscM=-J-. Then^ = --^:5^^.
sm M dx sin'' w dx
That is, -^ (cse m) = - csc m cot ?t— ■ (1)
If M = X, -:;-(csc x)=— CSC X cot X. (2")
dx ^ ■'
Note. Or put y = csc u = seel ~— jA, and proceed as in Art. 43.
48. Differentiation of vers u. Put y = vers m = 1 — cos m. Then,
on differentiation, , ,
— (vers u) = sin u —
dx dx
In particular, if m = x,
—- (vers a?) = sin x.
Ex. 1. Find the a:-derivatives of cot (2 x + 3), sec (J a; + 3), csc (3 a; — 7),
vers (6 1 + 2), sec"a;.
Ex. 8. Find the f-derivatives of cot^ (3 « + 1), sec^ (J « — 1), csc-' \{t + 5),
cot (9 £2), sec(7«- 2)2.
Ex. 3. Show that D log (tan x + sec x) = X» log tan (J ir + J x) = sec x.
D. Inverse Teigonometeic Functions.*
49. Differentiation of sin~'(/.
Put y = sin"* u.
Then sin y = u.
On differentiation, cos y-- = — •
dx dx
dy _ 1 du _ 1 du_
dx cosy dx \/l — sin^u dx'
If. = x, -|(ei„-x.) = _^. (2)
• See Murray, Plane Trigonometry, Arts. 17, 88.
72
DIFFERENTIAL CALCULUS.
[Ch. IV.
Note 1. On the ambignity of the deriratlTe of
sin-l as. The result in (2) is ambiguous, since the sign of
the radical may be positive or negative. This ambiguity
is apparent on looking at the curve y = sin-i x, Fig. 11.
Draw the ordinate ABCDE at a; = xi. The tangents
at .8 and D make acute angles with the x-axis, and the
tangents at C and E make obtuse angles with the a-axis.
Hence, at B and D ^ i& positive ; and at C and E -^ \&
dx dx
negative. That is, at B and D — (sin-i a;) = — "*" ;
dx Vl - xi'
and at C and E — (sin"' x) = — — . Thus the sign
dx Vl - a;i2
of — (sin-i x) depends upon the particular value taken of the infinite number
dx
of values of y which satisfy the equation y = sin-i x.
Note 2. If it is understood that there be taken the least positive value of
y satisfying the equation y = sin"' Xi (in which x\ is positive), then the sign
of the derivative is positive. Similar considerations are necessary m (1).
Ex. 1. Show by the graph in Fig. 14, or otherwise, that when z = 1,
dx
(sin-i x) = + CO, and that when x :
Ex. 2. Find the x-derivatives of
1, — (sin-'x) is — 00.
ox
sin-iz», sin-i5-il. sin-i _1^
V2
, sin-'
i + x^ VT
2x
sin-' Vl — x^, Vl — x^ • sin-' x — x, sin-' Vsin x.
Ex. 3. Show that a tangent to the curve y = sin-' x cannot cross the
X-axis at an angle behueen, — 45° and + 45°.
Ex. 4. Find anti-derivatives of
2x
Vl - x^ Vl -x< Vl -x6
50. Differentiation of cos~^u.
Put y = COS"* M.
Then cos y = u.
On differentiation, — sin « -^ = — •
dx dx
dy _ _ 1 du
dx sin y dx
du
Vl — cos^w*^^
50,51.] DIFFERENTIATION OF FUNCTIONS. 73
i.e. 4- (COS-l u)= -^ ^.
ax y/\ _ j^2 dx
d ,„.„_, ^ 1
liu = X, ^ (C08-1)
(/ = cos-1 X where x = — , cross the x-axis.
dx Vl - a;2
Ex. 1. Explain the ambiguity of sign in the derivative of cos"' x by
means of the curve y = cos~i x. Show that if there be taken the least
positive value of y satisfying y = cos"' x, in which x is positive, the sign
of the derivative is negative.
Ex. 2. Determine the angles at which the tangents touching the curve
1
^^'
Ex. 3. Find the x-derivatives of cos"'^ " ~ , cos'^ ~ ^ ,
x2" + 1 1 + x2 a
51. Difierentiation of tan~' u.
Put y = tan"^ u.
Then tan y = u.
On differentiation, sec^ 2/ — = — •
dx dx
■ ^ — "i- du _ 1 du _
dx sec^ ydx 1 + tan^ y dx '
i.e. ^(tan-i«) = -JL_^.
dx 1 + tt2 dac
In particular, if u = x,
Note. The derivative of tan-' x is always positive. This is also evident
on a glance at the curve y = tan-' x.
Ex. 1. Find the x-derivatives of tan-i 2 x, tan-i 2 y, tan-' x^^ tan-i j/».
Ex. 2. Find the t-derivatives of tan-^ 4 «, tan"! «*, tan-i 3 x^.
Ex. 3. Show that the angles made with the x-axis by the tangents to
the curve y = tan-' x are 0°, 45°, and the angles between 0° and 45°.
Ex. 4. Show how to determine the abscissas of the points of ^ = tan-i x,
the tangents at which cross the x-axis at an angle of 30°.
2x '"
Ex. 6. Find the x-derivatives of the following : tan i -, tan"
1 - x2' 1-1- x»'
tan-
^ , Uin-':^^r^'-\ tan-iV-^. ten-'-^^^=^
74 DIFFERENTIAL CALCULUS. [Ch. IV.
Ex. 6. (a) Show that Jtan-'-v/^ cosa;_l_ „. Show, by diflerenti-
Vl + cosa;2
ation, that D i tan-i x + tan-i - J is independent of
Ex. 7. Find anti-differentials of
X.
(fa 2 a: da: z^da:
52. Differentiation of cot~' u. On proceeding in a manner simi-
lar to that in Art. 51, it will be found that
^(COt-lM) = -^-l-^.
If M = a, ^ (cot-i X) = - :ri-5-
' dx 1-1-052
Ex. 1. Show, by means of the curve y = cot-^x, that the derivative of
cot-^ X is always negative.
Ex. 2. Find the x-derivative of cot-' - -f log 'V .
X 'x-Ho
53. Differentiation of sec~' I/.
Put y = sec"' M.
Then sec y = u.
On differentiation, sec y tan y J^ = -r--
^ dy _ 1 du _ 1 du
dx~ sec y tan y da; ~ gee y Vsec^ y — 1 da;'
i.e. #-(sec-iM) = L=.^. (1)
dx MVt*2-l <*« '
If M = a;, then 4-(»f>o-^oe)= — -4=- (2)
Ex. 1. Explain the ambiguity of the result (2). Show that, when x is
positive, the positive value of the radical is taken with the least positive
value of sec-' x.
Ex. 2. Find the x-derivatives of sec-* x^, sec-i -
x^-i-l 2x2-1
x2-l
Ex. 3. Show by differentiation that tan-i ^
see-' "
Va2-
-x*
sec-i ^
= is
independent of x. v'l-x" VI -x^
62-56.] DIFFERENTIATION OF FUNCTIONS. 75
54. Difierentiation of cosec"' u. On proceeding in a manner
similar to that in Art. 53, it will be found that
^(csc-iM) = i ^. (1)
1
du
1
rj dx
a;Vce2-
T
Ifw = x, ^(C8c-ia;) = ^ (2)
Ex. 1. Explain the ambiguity in sign in (2) by means of the graph of
CSC-' u. Show that, when x is positive, the negative value of the radical is
taken with the least positive value of esc-' a.
55. Differentiation of vers"*!/.
Put y = vers"' u.
Then vers y = u.
On differentiation, smy^ = —-
dx dx
■ ^ — 1 <?" _ 1 du
dx sin y dx Vl — eos^ y dx
1 dw.
i.e. 4^ (Ters- J u) = ^ ^. (1)
dx Va u-ui "'"
If M = X, -^ (Ters-i X) = ^ . (2)
dx^ y/iX-xi
2 a:'^
Ex. 1. Find the x-derivative of vers-' -
1 +l2
56. Differentiation of implicit functions : two variables.
N.B. Examples of the difierentiation of implicit functions have been
given in Exs. 13, 14, Art. 37. A preliminary study of these examples will
help to make this article clear.
Let y be an implicit function of x, the function y and the
variable x being connected by a relation
f{x,y) = c. (1)
76 DIFFERENTIAL CALCULUS. [Ch. IV.
If, as sometimes happens, it is impossible or inconvenient to
express y as an explicit function of x, the derivative -^ may be
obtained in the following way :
On taking the ^-derivative of each member of (1), there is
obtained a result of the form
P+Q^ = 0. (2)
ax
From this ^ = -?- (3)
dx Q
Since the x-derivative of f(x, y) is P + Q—, the differential of
, dx
f{x,y) is (Art. 27) Pdx + q'^^dx, i.e. (Art. 27) Pdx+ Qdy.
Ex. 1. Find -^, when xy = c.
ox
Differentiation of tlie members of this equation gives y + x-^ = 0 ; whence
-^ = --. The i-deiivative of xy is y + x-^; accordingly, the differential
dx X ' " dx
of xy is xdy + ydx. [Compare result (7), Art. 32.]
Ex. 2. Write the differentials of the first members of the equations in
Exs. 13, 14, Art. .37.
Ex. 3. Find -y in each of the following cases : (i) i' + y* = a* ;
(ii) x^ + y^ = a^ ; (iii) ^ + fi; = 1 ; ("v) (cos x)y — (sin y)' = 0.
Ex. 4. Write the diSerentials of the first members of the equations in
Ex.3.
Note 1. It should be observed, as illustrated in Equation (2) and the
above examples, that when the differential of f{x, y) is written Pdx + Qdy,
P is the same expression as is obtained by differentiating /(i, y) with respect
to X, and at the same time regarding y as constant or letting y remain
constant, and Q is the same expression as is obtained by differentiating
/(x, y') with respect to y, and at the same time regarding x as constant or
letting X remain constant. Here P is called the partial x-derivative of /(x, ?/),
and Q is called the partial y-derivative of /(x, y). These partial derivatives
are denoted by the symbols ^ ^f ' ^' and -^ ^^' ^^ respectively. With this
5x dy
notation, result (3) may be written
dy_ dx „ dx^^""'^ ^^y
dx df{x,yy d_ ., .
dy 5r^^ ' ^^
56.] DIFFERENTIATION OF FUNCTIONS. 77
Ex. 6. In the exercises above, test the first statement made in this note.
Note 2. Partial derivatives and the difi'erentiation of implicit functions
are discussed further in Chapter VIII.
EXAMPLES.
If.B. It is not advisable for the beginner to work the larger part of
Exs. 1-8 before proceeding to the next chapter. Many of the differentiations
required in these examples are far more difficult than those that are commonly
met in pure and applied mathematics ; but the exercise in working a fair
proportion of them will develop a skill and confidence that will be a great
aid in future work.
Bifierentiate the functions in Exs. 1-4, 6, 7, with respect to x.
1. (i) (2a;-l)(3a; + 4)(a;2 + ll); (ii) (a + a:)(6 + x);
C£±a)r. (y) —^ • (Mi)
{x+b)"' (1 + a;)"' V^^l^i'
(iii) (a+x)'»(6 + x)-; (iv) ii±-^; (v)— ^^; (vi) ^
CYn) -^^- : (viii) _v5±i- ; (jx) Vl + a:' + Vl - x'.
v'l + x2 Va+ Vx Vl + x2 - Vl -x2
(X) ( ^ V: (xi) X (a2 + x^) Va^ - xK
\1+ VI -iV
2. The logarithms of: (i) 7x* + Sx" _ i7a; + 2 ; (ii) -J^lzi^;
(iii) ? ; ^y)Jl+mi. (y)J^^±i.
^ \-V^^zr^^' ^^\i-sinx' ^^Mvrr^-x
8. (i) sin 4x6; (iQcos'Tx; (iii)8ec23x; (iv) tan (8 x + 5) ;
(v) x"logx; (vi) sin^x*; (vii) sinnx-sin»x; (viii) sin (sin x);
(Jx) sin (log nx) ; (x) log (sin nx).
4. (i)logj/|^-|tan-.x; (ii) log ^^IZj _ . ;
(iii) log-^i-±^-itan-ix.
'1 — X 2
6. Showthat i) | ?.:^^^i±.2^ + 1! log (x + Vo^T^) l = VS^T^.
6. (i) tan-i**; (ii) sln-i(cosx); (iii) sin(cos-ix);
') tan-i (n tan x) ; (v) sin-^
(vu) tano»; (vui) e'-\\^^-
(iv) tan-i(ntanx); (v) sin-i^-±-?-55i^ ; (vi) e"' sin" rz;
a + 6cosx
78 DIFFERENTIAL CALCULUS. [Cn. IV.
">(
?r. nn ^^•
U'
(ii) -e-, (iii) r^; (Iv) e^; (v) «(«'); (vi) (2')»^
n) X
8. Find -t=- under each of the following conditions :
(i) 01= + 2 Aa^ + 6y2 + 2 jrx + 2/y + c = 0 ; (ii) (a;2 + ,,2)2 _ a2(-j;2 _ ^,2) =o ;
(iii) »'^ + siny = 0; (iv) sin (a;y) = ma; ; (v) sina;siny + sinxcos j=^;
(vi) e» — e* + x!/ = 0 ; (vii) xs = y ; (viii) ye"* = ax".
dv ^=^
9. Find t;^ in terms of x, when x = e i .
dx
10. Differentiate as follows: (i) 3y^ — ly + 11 with respect to 3y;
(ii) ifi — lit + 1 with respect to < + 2 ; (iii) x with respect to sin x ;
(iv) sin z with respect to cos z ; (v) a; with respect to Vl — x^.
U. (i) Given y = 3u^-7u + 2 and M = 2a;8+3x + 2, find ^ ; (ii) given
(ju '^ ^^
2/ = «• + g2 and s = tan t, find -^ ; (iii) given b = V2gs, s = l gfi, find -j-
da
in two ways ; (iv) u = tan-'(xy), y = e', find -=-•
12. Compute the angle at which the following curves intersect, and sketch
the curves : (i) x'^ — y^ = 9 and xy = i ; (ii) x^ + y^ = 25 and 4 y^ = 9 x ;
(iii) 2/2 = 8(x + 2) and y^ + 4^^; _ i) _ o ; (iy) y = Zz^ -I a,ud y = 2x^
+ 3 ; (v) x2 + y2 = 9 and (x - 4)2 + y2 _ 2 2/ = 15.
13. A point P is moving with uniform speed along a circle of radius a
and centre O ; AB is any diameter, and Q is the foot of the perpendicular
from P on AB. Show that the speed of Q is variable, that at A and B it is
zero, and at O it is equal to the speed of P. (The motion of Q is called
simple harmonic motion.)
SnGGBSTiON : Denote angle AOP hy 6, and OQ by x. Then x = a cos 6 ;
dx
dt
hence ^ = -asin9^.1
dt \
14. Suppose, in Ex. 13, the radius is 18 inches, and P is making 4 revolu-
tions per second : what is the speed of § when AOP is 16°, 30°, 45°, 60°,
76°, 90°, 120°, 150°, respectively?
CHAPTER V.
SOME GEOMETRICAL AND PHYSICAL APPLICATIONS.
GEOMETRIC DERIVATIVES AND DIFFERENTIALS.
57. The variation of functions, the sketching of graphs, and the
determination of maxima and minima, which are discussed in Chapter
VII., can be studied before entering upon this chapter. Por some
reasons it may be preferable to do this.
58. This chapter gives some practical applications of the
preceding principles of the calculus. The applications in Arts.
59-62 are already familiar or obvious. The study of the geometric
derivatives^ and differentials in Art. 67 is not of immediate im-
portance, but •will be found of more interest and value when
Chapters XX., XXV., are taken up. A glance over this article,
however, will serve to make clearer and stronger the notions of
a derivative and a differential.
59. Slope of a curve at any point : rectangular coordinates. By
the sloj^e of a line (rectangular coordinates being used) is meant
the tangent of the angle at which the line crosses the x-axis.
This angle is measured ' counter-clockwise ' from the avaxis to the
line, as explained in trigonometry.
It has been shown in Art. 24 that at any point (x, y) on the curve
y=fix), (1)
or 4.(x,i/)=0. (2)
Hie slope of the tangent is
The slope of the tangent drawn at a point on a curve is commonly
called the slope of the curve at that point.
79
80
DIFFERENTIAL CALCULUS.
[Ch. V.
The slope of the tangent (or the slope of the curve) at a particular
point (a:,, y^ is the number obtained by substituting (x^ y{) in the
expression derived for (3) from (1) or (2). This slope is denoted
by
(4)
When the slope (4) is positive, the tangent crosses the x-axis at
an acute angle ;
When the slope is negative, the tangent crosses the a^axis at an
obtuse angle ;
When the slope is zero, the tangent is parallel to the avaxis ;
When the slope is infinitely great, the tangent is perpendicular
to the X-axis. These facts are illustrated in Fig. 12, in which
the slope is positive at N and P,
negative at L and R,
zero at M and Q,
infinitely great at "Fand S.
Note. Symbol (4) does not mean ' the derivative of y^ with respect to
xi,' which is a meaningless phrase, since Xi and 2/1 are constants.
EXAMPLES.
1. Find the slope of the parabola
4 y = a;2 (1)
at the points {x\, y{), (2, 1), (—3, f) ; and find the angles at which the tan-
gents at the last two points cross the x-axis.
(The student is supposed to draw the figure.)
60.] ANGLES AT WHICH TWO CURVES INTEBSECT. 81
From (1), on differentiation, j^ = n' (2)
This is a general expression, giving the slope of the curve at any point.
From (2), on substitution, the slope at (xi, j/i) (viz., -^ ) = -^.
From (2), on substitution, the slope at (2, 1) = J = 1 ;
accordingly, the tangent drawn at (2, 1) crosses the z-axis at the angle 45°.
— 3
From (2), on substitution, the slope at (— 3, \) = = — 1.5 ;
accordingly, the tangent drawn at ( — 3, |) crosses the a>axis at the angle
123° 41.4'.
2. Find the general expression giving the slope at any point on each of
the curves in Art. 4, Ex. 3.
3. Review the following examples : Ex. in Art. 24 ; Ex. 14 (6) in Art. 37 ;
Exs. 5-8 in Art. 42 ; Exs. 3-6 in Art. 43 ; Ex. 2 in Art. 44 ; Ex. 3 in Art. 49,
Exs. 3, 4, in Art. 51.
4. Plot the following curves ; find the slope of each of them at the points
described, and find the angle at which each of the tangents drawn to the
curves at these points crosses the i-axis : (i) the parabola y^ = 8 a;, where
a: = 2, and where z = 8 ; (it) the parabola a:^ = 8 y, where 2 = 8; {iii) the
circle x^ + y^ = 13 at (2, 3) ; (ic) the circle x' + y- = 18 at (3, 3) ; (t)) the
curve 3 y2 = a:s at (3, 3) ; (»i) the curve 3 y"- = (a; + 1)» at (2, 3) ; (vii) the hy-
perbola x2 _ 2,2 - 20 at (6, 4) ; {viii) the hyperbola xy = 24 at (6, 4).
60. Angles at which two cuires intersect. By the angle (or
angles) at which two curves intersect is meant the angle (or angles)
formed by the tangents drawn to each of them at their point (or points)
of intersection.
By the angles of intersection of a straight line and curve is
meant the angles between the line and the tangents drawn to the
curve at the points of intersection.
The method of finding the angles of intersection of two curves, as illus-
trated in the following examples, may be outlined thus :
1. Find the points of intersection of the curves ;
2. Find the slope of each curve at each of these points ;
thence can be obtained the angles at which the tangents drawn at these
points cross the x-axis.
3. From either the slopes or the angles just described, find the angle
between the tangents at each point of intersection.
82
DIFFERENTIAL CALCULUS.
[Ch. V.
Fig. 13.
EXAMPLES.
1. Find the anglesatwhich the circle
x^ + y^= 72 and the parabola y^ = 6x
intersect. These curves and the tan-
gents concerned are shown in Fig. 13.
On soMng the equations of the
curves simultaneously, the points of
interaection are found : viz. ,
P(6, 6) andif(6-6).
The method of last article applied to each curve at P brings out the
foUovf ing results :
Slope of PTi (i.e. tan Xr,P)= i; whence XTiP = 26°S3.9'.
Slope of PTi (i.e. tan XT^P) = - 1 ; whence XT^P = 135°.
.-. TiPTi = XTiP- XTiP = 135° - 26° 33.9' = 108° 26.1',
and thus, T2PB = 71° 33.9'.
In a similar manner the angle of intersection at B will be found to have
the same value, as is also apparent from the symmetry of the figure.
The angle of intersection may also be found directly from the slopes of
PTi and PT2, for
tan XT^P — tan XT,P
tan TiPTi = tan (XT^P - XTiP) =
-1-i =-3.
1 + tan XTiP ■ tan XTiP
i+C-i xi)
.-. riJTjrr 108° 26.1'.
2. At what angles does the line y = x + 6 inter-
sect the parabola 2y = x^?
The line, parabola, and tangents concerned are
shown in Fig. 14. On solving the equations of
the line and the parabola simultaneously, it is
found that
at P, a: = - 2.6056 ; at Q, at = 4.6056.
Fio. 14.
From 2y = x'', it follows that =^ = x; this is the slope of the parabola
dx
WPQ at any point (x, y).
.: slope of Pr, =-2.6056 ; whence XPiP = 110° 59.8';
slope of QTi = 4.6066 ; whence XT^Q = 77° 46'.
61.] EQUATION OF THE TANGENT. 83
Now, slope of SV = 1 ; whence ^SV = 46°.
.-. SPTi = XTiP- X8V = 65° 59.8';
SQT2 = XTiQ -XSV = 32° 45'.
8. Review Exs. 23, 24, Art.lv, and Ex. 12, Art. 56.
61. Equations of the tangent and the normal drawn at a point on
a curve.
In Fig. 15, Art. 62, P is the point (x,, ^i) on the curve y =/(«) ;
PT is the tangent which touches the curve at P ;
PN, drawn at right angles to PT, is the normal to the curve at P.
The slope of the tangent PT= 'Ml [Art. 59 (4)]. (1)
It is shown in analytic geometry that if the slope of a line is
m, the slope of a line perpendicular to it is Accordingly,
m
the slope of the normal PN= -J • (2)
It is shown in analytic geometry that the equation of a line
which passes through a point (x,, y^ and has a slope m is
y — y^ = m{x — x^.
Accordingly, since PT passes through Pix^, y^ and has the slope
the equation ofthe tangent at {x^, y-^,is y—yi=~ix — x{). (3)
Since PiV passes through P{xi, y{) and has the slope (2),
the equation of the normal at (x^, y{) is y - ^i = - — -i (as — x^ (4)
"■Vl
EXAMPLES.
1. Write the equations of the tangents and normals to the circle and
parabola at P(6, 6) in Fig. 13.
At P, (see Ex. 1, Art. 60), slope of PTi = \.
:. equation of tangent PTi of the parabola is y — 6 = ^(a; - 6) ;
and the equation of the normal to the parabola at P is y — 6 =— 2(a; — 6).
These equations reduce to 2 ;/ — a = 6,
and y + 2 a; = 18, respectively.
2. Find the equations of the tangents and normals drawn to the circle and
parabola at R in Fig. 13.
84
DIFFERENTIAL CALCULUS.
[Ch. V.
3. Write the equations of the tangents to the parabola at P and Q in
Fig. 14 ; also the equations of the nonnals at these points.
Find the lengths of OTi and OT^.
4. Write the equations of the tangents an* normals for each of the curves
and points appearing in Ex. 4, Art. 59.
62. Lengths of tangent, subtangent, normal, and subnormal, for
any point on a curve : rectangular coordinates. Let P be a point
(a;i, ^i) on the curve 2/ =/(«) [or, ^{x, y) = 0].
At P let the tangent PT be drawn ; likewise the normal PN
and the ordinate PM. The length of the line PT, namely, that
part of the tangent which is intercepted between P and the a>axis,
is here termed the length of the tan-
gent. The projection of TP on the
a^axis, namely TM, is called the
subtangent. The length of the line
PK, the part of the normal which
is intercepted between P and the
a^axis, is termed the length of the
normal. The projection of PNon
the a>axis, namely MJt^, is called
the subnormal.
Note 1. The subtangent is measured from the intersection of the tangent
with the X-axis to the foot of the ordinate ; the subnormal is measured from
the foot of the ordinate to the intersection of the normal with the ai-axis.
A subtangent extending to the right from T is positive, and one extending
to the left from T is negative ; a subnormal extending to the right from M is
positive, and one extending to the left from M is negative.
Let angle XTP be denoted by a; then tana = ^'- In the
dxi
triangle TPM: MP^y^; TM= y^ cot a = y^^; .TP= Vi esc a
dyi
the triangle PMN: angle MPN= a ; MN= y, tan MPN= y^^;
. dxi
PN=y^secMPN=y^yjl+f^Y; for, PN=^iMP' + MW
62.] LENGTHS OF TANGENT, ETC. 85
It being understood that y and -^ denote the ordinate and the
(XX
slope of the tangent at any point on the curve, these results may
be written :
sabtangeut = j/ — ^;
ay
subnormal = y — ^;
ax
length of tangent = yyjl+ ij^Yi
length of normal = y-^l + ('^Y-
Note 2. It is better for the student not to use these results as formulas,
but to obtain the lengths of these lines in any case directly from a figure.
EXAMPLES.
N.B. Sketch all the curves and draw all the lines involved in the follovj-
ing examples.
1. In each of the following curves write the equations of the tangent and
the normal, and find the lengths of the subnormal, subtangent, tangent, and
normal, at any point (n, y{), or at the point more particularly described :
(1) Circle s^ + y* = 26 where x=-3; (2) parabola y'^-ix at x = 2;
(3) ellipse b'hfi + aV = a^S'^ > (4) sinusoid y = sin k ; (5) exponential curve
y = eF.
2. Where is the curye y{x — T){x — Z) — x — 1 parallel to the x-axis ?
3. What must o^ be in order that the curves 16x^ + 25y''' = 400 and
49 x^ + dh/^ = 441 intersect at right angles ?
4. In the exponential curve y = be' show that the subtangent is constant
and that the subnormal is — •
a
6. In the semi-cubical parabola 3y''=(x + 1)' show that the subnormal
varies as the square of the subtangent.
6. In the hypocycloid of four cusps, xt + y^ = a^ : (1) Write the equa^
tion of the tangent at (xi, yi) ; (2) show that the part of the tangent inter-
cepted between the axes is of constant length a ; (3) show that the length
of the perpendicular from the origin on the tangent at (x, y) is Vaxy ; (4) if
p, pi be the lengths of the perpendiculars from the origin to the tangent and
normal at any point on the curve, 4p^ + pi" = a^.
86 DIFFERENTIAL CALCULUS. [Ch. V.
7. In the parabola as' + y' = a', write the equation of the tangent at
any point (xi, ^i), and show that the sum of the intercepta made on the axes
by this tangent is constant. Show that this curve touches the axes at (a, 0)
and (0, a).
8. In the cycloid x = a{9 — sin e), y = o(l — cos 8): (1) Calculate the
lengths of the subnormal, subtangent, normal, and tangent at any point
(x, y) ; (2) show that the tangent at any point crosses the y-axis at the angle
-; (3) show that the part of the tangent intercepted between the axes is
otfcosec — 2 a sec-. [See Art. 35.1
2 2 ■•
9. Li the hyperbola xy = c^: (1) Show that for any point (x, y) on
the curve the subnormal is — =^ and the subtangent is — x ; (2) find the
z- and j/-intercept8 of the tangent at any point (xi, yi), and thence deduce a
method of drawing the tangent and normal to the curve at any point on it.
Show that the product of these intercepts is 4 c*.
10. In the semi-cubical parabola ay'^ = x*, show that the length of the
subtangent for any point (x, y) is J x ; thence deduce a way of drawing the
tangent and the normal to the curve at any point on it.
o
11. Show that the parabola %''■ — \y ■ intersects the witch y = — - —
at an angle tan-i 3 ; i.e. 11" 33' 64". *''"'"*
12. Find at what angles the parabola y' = 2 ax cuts the folium of Descartes
x* + y* = 3 axy.
13. In the curve x'»y» = 0"+" show : (1) That the subtangent for any
point varies as the abscissa of the point ; (2) that the portion of the tangent
intercepted between the axes is divided at its point of contact into segments
which are to each other in the constant ratio m -.n; (3) thence, deduce a
method of drawing the tangent and the normal at any point on the curve.
(The curves x^y" = a'"+», obtained by giving various values to m and n, are
called adiabatic curves. Instances of these curves are given in Exs. 9, 10,
and in the parabolas in Exs. 11, 12.)
14. Show that all the curves obtained by giving different values to n in
j-j +(-) =2, touch one another at the point (a, 6). Draw the curves in
which (a, 6) is (4, 7), n = 1, n = 2.
16. Show that the tangents at the points where the parabola ay = x*
meets the folium of Descartes x* + y' = 3 axy are parallel to the x-axis, and
that the tangents at the points where the parabola y' = ax meets the folium
are parallel to the y-axis. Make figures for the curves in which a = 1 and
o = 4.
63.] SLOPE OF A CURVE AT ANT POINT.
63. Slope of a curve at any point : polar coordinates.
be a curve whose equation is
»-=/W, [or ^(r, e)=0], and
P be any point on it having
coordinates r^, 6i, with reference
to the pole 0 and the initial
line OL. Draw OP; then
OP=r^, and angle L0P=6x.
Through P and Q (a neigh-
bouring point on the curve),
draw the chord TPQ, and draw OQ. From P draw PB at right
angles to OQ.
Let angle POQ = A^j, and OQ = ri + Arj ;
then PB = r^ sin A^i, and iZQ = ri + A»-i — rj cos Atfj.
The angle between the radius vector drawn to any point P and
the tangent at P is usually denoted by ip. Since
<i, = lim^g^^ angle RQP,
then, using the general coordinates r, 6, instead of r^, Oj,
HP
taiii/r = lim^j^ —
= lim
r sin Ad
A«=a)
r + Ar — r cos Ad
On replacing cos Ad by its equal, 1—2 sin* ^ Ad, and dividing
numerator and denominator by Ad, this becomes
sin Ad
,. '^ Ad r
tan^ = Iim^,^^ siniAd = rf7'
That is, tan + = r|?. (1)
The angle between the initial line and the tangent at P is
usually denoted by 4>.
88
DIFFERENTIAL CALCULUS.
[Ch. V.
It is apparent from Fig. 17 that
<|> = + + 9.
(2)
Note. Results (1) and (2) are true for all polar curves, whatever the
figure may be. The student is advised to dravf various figures.
64. Lengths of the tangent, normal, subtangent, and subnormal, for
any point on a curve : polar coordinates.
In Fig. 18 0 is the pole and OL is the initial line. At P any
point (ri, ^i), on the curve CR, whose
equation is r=f{d), [or ^{r, fl)=0],
let the tangent PT and the normal
PN be drawn. Produce them to
intersect NT, which is drawn through
O at right angles to the radius vector
OP.
The length of the line PTis termed
the length of the tangent at P; the
projection of PT on NT, namely OT,
is called the polar subtangent for P;
the length of PN is termed the
length of the normal at P; the projec-
tion of PN on NT, namely ON, is called the polar subnormal for P.
Note. In Art. 59 the line used with the tangent and the normal is the
X-axis. Here the line so used is not the initial line, but the line drawn
through the pole at right angles to the radius vector of the point.
In the triangle OPT :
0T= OP tsrn OPT;
Fig. 18.
64.] LENGTHS OF TANGENT, ETC. 89
i.e. (on removing the subscripts from the letters)
polar snbtangent = r tan \{i = r^^;
(If
also, TP= OP sec OPT;
i.e. polar tangent length = r sec ip = ryjl + r^(^\ •
for.- TP=-V0P' + or' =yjr- + »^(f Y= r^l + '^(f Y'l
In the triangle OPN :
angle JVPO = 90 -i^;
OiV^= OP tan xVPO;
I.e. polar subnormal = r cot i/* = ^ ;
also, NP = OP sec NPO ;
polar normal length = r cosec ^ =-^?'« + (^\ •
Or : NP = ^UF+UF = J7+
I.e.
dr
d9
NoTB. In Fig. 18 r increases as 6 increases ; accordingly — is positive,
d8 *''■
and hence the subtangent is positive. Thus when — is positive, the sub-
dr
tangent is measured to the right from an observer at O looking toward P.
When r decreases as 8 increases, and thus — is negative, the subtangent is
dr
measured to the left of the observer looking toward P from 0. The student
is advised to construct figures for the various cases.
EXAMPLES.
N.B. In the following examples make figures, putting o = 4, say. Apply
the general results found in these examples to particular concrete cases, e.^.
0 = 6 and 6 = -, a = 2 and e = — , etc. The angle 8, as used in the equa-
tions of the curves, is expressed in radians.
90 DIFFERENTIAL CALCULUS. [Ch. V.
1. In the following curves calculate the lengths of the subnormal, sub-
tangent, normal, and tangent, at any point (r, ff) : (1) The spiral of
Archimedes r = aO ; (2) the parabolic spiral or lituus r^ = a^B {i.e.
r = aB') ; (3) the hyperbolic spiral (or the reciprocal spiral) rB = a;
(4) the general spiral r = ad". (The preceding spirals are special cases
of this spiral.)
2. From the results in Ex. 1 deduce simple geometrical methods of
drawing tangents and normals to the spirals in (1), (2), (3).
3. Do as in Exs. 1, 2, for the logarithmic spiral r = e**. In this
curve each of the lengths specified varies as the radius vector.
4. (o) In the spiral of Archimedes r = aB, show that tan f = 8. Find
^ and <t> in degrees when angle TOP (Fig. 17) = 40°, and when TOP = 70°.
(6) In the curve r = 4 9, find ^ and <(> when r = 2.
6. (a) In the logarithmic spiral r = ce"', show that ^ is constant.
This spiral accordingly crosses the radii vectores at a constant angle, and
hence is also called the equiangular spiral. (6) Show that the circle is a
special case of the logarithmic spiral, and give the values of </> and a for
this case.
a
6. In the parabola r = asec''-, show that <l> + \l/ = ir. Make a prac-
tical application of this fact to drawing tangents and normals of this curve.
7. In the cardioid r = a(l — cos B), show that <j> =— ^, ^ = -, sub-
6 6 2 2
tangent = 2 a tan - sin^ — Apply one of these facts to drawing the tangent
and normal at a point on the curve.
65. Applications involving rates. Applications of this kind
have already been made in Arts. 26, 37. Rates and differentials
have been discussed in Arts. 25-27. It has been seen, Art. 26,
Eq. (1), that if y =f(x), then
dt ^ 'dt dx dt
In -words, the rate of change of a function of a variable is equal
to the product of the derivative of the function with respect to
the variable and the rate of change of the variable. The following
principles, which are proved in mechanics, will be useful in some
of the examples : (a) If a point is moving at a particular moment
in such a way that its abscissa x is changing at the rate — , and
65.] APPLICATIONS INVOLVING RATES. 91
its ordinate y is changing at the rate -^, and if — denote its rate
dt dt
of motion along its path at that moment, then
(l)'-(l)'-(f^^
J
(6) If a point is moving in a certain direction with a velocity
V, the component of this velocity in a direction inclined at an
angle a to the first direction, is v cos a.
For instance, if a point is moving so that its abscissa is increasing at the
rate 2 feet per second and its ordinate is decreasing at the rate 3 feet per
second, it is moving at the rate v'2'-' + 3^, i.e. Vl3 feet per second. Again,
if a point is moving at the rate of 6 feet per second in a direction inclined
60° to the x-azis, the component of its speed in a direction parallel to the
X-axis is 6 cos 60°, i.e. 3 feet per second, and the component parallel to the
y-axis is 6 cos 30°, i.e. 5.196 feet per second.
EXAMPLES.
II.B. Make figures.
1. If a particle is moving along a parabola y^ =8x at a uniform speed of
4 feet per second, at vrhat rates are its abscissa and its ordinate respectively
increasing as it is passing through the point (x, y) and x has successively the
values 0, 2, 8, 16 ?
2. A particle is moving along a parabola y^ = ix, and, when z = 4, its
ordinate is increasing at the rate of 10 feet per second : find at what rate its
abscissa is then changing, and calculate the speed along the curve at that
time.
3. A particle is moving along the hyperbola xy = 25 with a uniform speed
10 feet per second : calculate the rates at which its distances from the axes
are changing when it is distant 1 unit and 10 units respectively from the
y-axis.
4. A vertical wheel of radius 3 feet is making 25 revolutions per second
about an axis through its centre : calculate the vertical and the horizontal
components of the velocity, (1) of a point 20° above the level of the axis;
(2) of a iwint 65° above the level of the axis.
6. A point is moving along a cubical parabola y = a? : find (1) at what
points the ordinate is increasing 12 times as fast as the abscissa ; (2) at what
points the abscissa is increasing 12 times as fast as the ordinate ; (3) how
many times as fast as the abscissa is the ordinate growing when a; = 10 ?
92 DIFFERENTIAL CALCULUS. [Ch. V.
66. Small errors and corrections : relative error.
If y=f{^), (1)
then by Art. 27 dy =f'{x) ■ dx, (2)
in which dx is an assigned change in x. It has been seen (Note
3, Art. 27) that dy is approxiviately the change in y due to dx.
An important practical application may be made of this principle.
For it follows that if da; be regarded as a small error in the
assigned or measured value of x, then dy is an approximate value
of the consequent error in y.
The ratio ^ or --^ • dx (3)
y A^)
is, approximately, the relative error or the proportional error, i.e.
the ratio of the error in the value to the value itself.
The approximate values of the correction and relative error may also be
deduced from the theorem of mean value. For, if y —f(x), and Ax be an
error m x, then f(x +Ax) — f(x) is the error in y, i.e. the correction that
must be applied to y. Now by (3) Art. 108, on putting a = x and h = Ax,
f(x + Ax)-/(i) =/'(! + 9 . Aar) . Ax.
Hence, on denoting the error in y by Ay,
Ay =/'(x) ■ Ax approximately.
Au fHx')
From this the relative error is, approximately, — = ,; / • Ax. (41
y Ax) ^ '
EXAMPLES.
1. The side a of a square is measured, but there is a possible error
Aa : find approximately the error in the calculated value of the area. Let
A denote the area. Then A = a^ ; whence AA = 2 o • Aa approximately.
2. If the measured length of the side is 100 inches and this be correct
to within a tenth of an inch, find an approximate value of the possible error
in the computed area, and an approximate value of the relative error.
In this case, approximately, Aa = 2 x 100 x .1 = 20 square inches. The
20 1
relative error is, approximately, — — or -!— ; that is, 20 square inches in
1002 600 ^
10,000 square inches, or 1 square inch in 500 square Inches.
66,60a.J APPLICATIONS TO ALGEBRA. 93
3. A cylinder has a height h and a radius r inches ; there is a possible
error Ar inches in »•: find by the calculus an approximate value of the possible
eiTor in the computed volume. If A = 10 inches and the radius is 8 ± .05
inches, calculate approximately the possible error in the computed volume
and the relative error made on taking r = 8 inches.
4. Find approximately the error made in the volume of a sphere by
making an error Ar in the radius r. The radius of a sphere is said to be 20
inches : give approximate values of the errors made in the computed surface
and volume, if there be an error of .1 inch in the length assigned to the radius.
Also calculate the relative errors in the radius, the surface, and the volume,
and compare these relative errors.
6. Two sides of a triangle are 20 inches and 35 inches. Their included
angle is measured and found to be 48° 30'. It is discovered later that there
is an error of 20' in this measurement. Find, by the calculus, approximately
the error in the computed value of the area of the triangle. Compare the
relative errors in the angle and in the area.
6. The exact values of the errors in the computed values in Exs. 1-4
happen to be easily found. Calculate these exact values, and compare with
the approximate values already obtained.
7. (1) Two sides, a, b, of a triangle are measured, and also the included
angle C: show that the approximate amount of the error in the computed
length of the third side c due to a small error AC made in measuring C, is
oftsinC ,Q
y/ai 4 62 _ 2 ab cos G
(2) Calculate the approximate error in the computed value of the third
side in Ex. 5.
66 a. Applications to algebra. Solution of equations having
multiple roots.
The following properties are shown in algebra:
(a) If a is a root of the equation f(x) = 0,
then x — a is a factor of the expression /(a;) ;
and conversely,
if « — a is a factor of the expression f(x),
then a is a root of the equation y(x) = 0.
94 DIFFERENTIAL CALCULUS. [Ch. V.
(6) If a, is an r-f old (or r-tuple) root of the equation f(x) = 0,
then (a; — a)' is a factor of the expression f(x) ;
and conversely,
if (a; — a)' is a factor of the expression f(x),
then a is an r-fold (or r-tuple) root of the equation f{x) = 0.
E.g. the equation a^ — 7x'+16x — 12 = 0
has roots 2, 2, 3.
The equation may be written {x — 2y{x — 3) = 0.
The roots of the equation af - 7 a^ + 16 a; — 12 = 0 are 2, 2, 3 ;
the factors of the expression a^ — 7a^ + 16a; — 12 are (x — 2y, x — 3.
Note. When a number is a root of an equation more than once (^e.g.
the number 2 in the equation above), it is said to be a multiple root of the
equation. If an equation has r roots equal to the same number, the number
is said to be an r-fold or an r-tuple root of the equation.
Theorem A, If f(x) is a rational integral function of x, and
(x — ay is a factor of f{x), then (x — ay-^ is a factor of f'(x).
For, let f(x) = {x — ay<f>{x).
Then f'(x) = r(x — ay '4>(x) + (x — ay<^'{x)
= (a; - a)---' [r^ (a;) + (a; - a)<^'(a;)].
Accordingly, {x — ay~' is a part of the Highest Common Factor
of /(x) and /'(a;).
Also, if (x-a)'-' is a part of the H.C.F. of /(x) and /'(a;),
(x — ay is a factor of /(a;).
From Theorem A and property (6) there follows :
Theorem B. If f(x) is a rational integral function of x, and a is
an r-tuple (or r-fold) root of the equation f(x) — 0, then a is an
(r — lytuple root of the equation f'{x)= 0.
It follows from Theorems A and B that if the equation f(x) = 0
67.]
GEOMETRIC DEBIVATIVES.
95
has multiple roots, they will be revealed on finding the H. C. F.
oifix) and/(x).
Ex. 1. Solve a^ - 2 x2 _ I6x + 36 = 0 (a) by trying for equal roots.
The derived equation is 3 x^ — 4 a; — 16 = 0. (6)
The H. C. F. of the first members of these equations Is x — 3.
Accordingly (x — 3)" is a factor of the first member of (a).
Hence, as found on division by (x — 3)'', (a) may be written
(x-3)S(x + 4)=0;
and thus the roots of («) are 3, 3, — 4.
Ex. 2. Solve the following equations :
(1) 3x3+4x2-x-2 = 0
(2) 4x8 + 16x2 + 21x + 9 = 0
(.3) x4-llx8 + 44x2-76x + 48 = 0
(4) 8x< + 4x» - 62x2 -61x- 15 = 0
(5) x6 + x* - 13 x» - x" + 48 X - 36 = 0.
Ex. 3. Find the condition that x" — px^ + r = 0 may have equal roots.
N.B. It is better to postpone the reading of the larger part of Art. 67
until the topics in it are required, or referred to, in the integral calculus.
67. Geometric derivatives and difierentials.
(a) Derivatiye and differe&tial of an
area : rectangular coordinates. Let PQ
be an arc of the curve y=f(^x). Take
any point on PQ, V{x, y) say, and take
T(x + Ax, y + Ay). Construct the rec-
tangles FjV and TM as shown in Fig. 19.
Draw the ordinate BP, and let the area of
BPVM be denoted by A ; then the area
of M VTN may be denoted by AA.
Now, rectangle FJV"< ilfr7'iV< rectangle JKT;
i.e. y-Ax< AA <(,y + Ay)Ax.
AA
Hence, on division by Ax, y< — <y + Ay.
(1)
96 DIFFERENTIAL CALCULUS. [Ch. V.
On letting Ax approach zero, these quantities (Arts. 18, 22, 23) approach
dA
the values y, =^, y, respectively.
dx
■■^ = V- (2)
That is, the derivative of the area BPVM vfith respect to the abscissa
X of V, is the measure of the ordinate of V. On denoting this measure by y,
result (2) means (Art. 26) that the area BPVM is increasing y times as fast
as the abscissa of V. From (2) it follows by Art. 27 that
dA = v ' dx. (3)
That is, the differential of the area BPVM is the area of a rectangle
whose height is the ordinate MV and whose base is dx, the differential of the
abscissa of V.
Ex. 1. Find the derivative of the area between the a;-axis and the curve
y = x», with respect to the abscissa ; (a) at the point w^hose abscissa is 2 ;
(6) at the point whose abscissa is 4.
(a) !?:^ = !/, (where X = 2,) = 2' = 8. (4)
dx
(b) — =r/, (wherex = 4,) = 48 = 64. (5)
dx
These results mean that, if an ordinate, like VM in the figure, is moving
to the right or left at a certain rate, the area of the figure bounded on one
side by that ordinate is changing, in case (a) at 8 times that rate, and in
case (6) at 64 times that rate.
Ex.2. Find the diHerentials in Ex. 1 (a) and (6), when dx = .1 inch.
Show these differentials on a drawing.
By (3), (4), and (5), in case (a), dA = .8 square inch; in case (6)
dA = 6A square inches.
Note. The area .8 square inch is nearly the actual increase in area
between the curve and the x-axis when the ordinate moves from x = 2 to
X = 2.1 ; and 6.4 square inches is nearly the increase in this area when the
ordinate moves from x = 4 to x = 4. 1 . These increases are calculated in
Ex. 16, Art. 111.
It is evident that the smaller dx is taken, the more nearly will the differen-
tial of the area become equal to the actual increase of the area between the
curve and the x-axis.
Ex. 3. Show that the y-derivative of an area between the curve and the
j/-axis is X. Thence deduce that the ^/-differential of this area is x dy, and make
a figure showing this differential area.
87.] GEOMETRIC BEBIVATIVES. 97
Ex. 4. In the case of the cubical parabola y = x^ find — and — ; then
dx dy
calculate the difierential of the area between this curve and the x-axis at the
point (2, 8) , taking dx = .2. Also calculate the differential of the area between
this curve and the j/-axis at the same point, taking dy = .2. Show these
differentials in a figure.
(6) Deriyative and differential of an area : polar coordinates. Let
FQ be an arc of the curve f(r, B) = 0. On
PQ take any point V(r, e), and take the
point Tr(r + Ar, 6 + AS). About O describe
a circular arc VN intersecting OW in N, and
describe a circular arc WM intersecting OV
in M. Then NW=Ar, and VOW = AS.
Also (PI. Trig., p. 175), area sector VON =
ir^Ae, and area sector 3fO»r= I (r+AryAO.
Draw OP. Let the area of POV be
denoted by A ; then the area of VO W may " p,Q 20
be denoted by AA.
Now, area FO^V< area VO W < area MOW;
i-e. ir^Ae<AA<i(,r + AryA0.
••• i'-^<|^<K'- + Ar)2. /
On letting A0 approach zero, these quantities (Arts. 18, 22, 23) approach
the values ^a
ir^, — , Jr^, respectively.
do
... *4 = 1^2. (1)
Result (1) means that, if the radius vector is revolving at a certain rate,
the area passed over by the radius vector, when its length is r, is increasing
at a rate which is ^ r'' (i.e. the number) times the rate of revolution.
It follows from (1) and Art. 27 that
dA = lrid9. (2)
Ex. 5. Show that in the case of the circle the differential of the area swept
over by a revolving radius is the additional area pa.ssed over.
Ex. 6. In the spiral of Archimedes r = 2 0 find the derivative of the area
swept over by the radius vector, with respect to 0. Calculate the differential
of this area when : (1) e = 30° and d0 = 30' ; (2) r = 2 and d9 = 1°. Make a
figure showing these differentials.
Ex. 7. In the cardioid r = 4(1 — cos 0) find the ^-derivative of the area.
Calculate the differential of the area when : (1) 9=60° and d«=l° ; (2) 0=0 and
do = 2°; (3) S = 330° and dS = 1". Make a figure showing these differentials.
98 DIFFERENTIAL CALCULUS. [Cb. V.
(c) DeriratlTe and differential of tlie len^li of a cure: rectangular
_ coordinates. Let PQ be an arc of the
" /U- ^5 curve y=f{x'). On PQ take any point
^"^^ M{x, y), and take the point N{x + Aa;,
y + Ai/) ; and draw the chord MN. On
denoting the length of the arc PM by s,
the length of the arc MN may be denoted
by As.
When Aa; approaches zero, chord MN
and arc MN approach equality. It can be
y>^^
FiQ. ai. shown rigorously (see Inf. Cal., p. 102) that
,. arc ilfiV ,.„ „ chord JtfJV . ^-.^
liniAi=i=o =limAJtio ; (i.)
Ax Ax
At ,. . V(Ak)2 + (Aj/)2 _
lin..^ ^ = lim.^ --"----"''- =lim..^ aRIJ'
£=Vi-(g)^-
aiJ,^(dyY, (2)
From (2), (3), and Art. 27,
and «*« = Vl+f^\
Ut/j '
<?»/. (6)
Ex. 8. Show that for a given dx and the actual derivative -^ at ilf, the
cb;
second member of (4) gives the length of the Intercept of the tangent,
namely, MT. Show that for a given dx, and using dy to denote the exact
corresponding change in the ordinate, the second members in (4) and (5)
give the length of the chord of the arc, namely, the line MN.
Note. It is shown in Art. I.IT how to find the length of the arc MN
corresponding to an Increment dx in x. The smaller dx is, the more nearly
will MT, arc MN, and chord MN, become equal to one another. See Ex. 6,
Art. 19.
Ex. 9. (1) Calculate the a;-derivative and the {/-derivative of the arc
of the parabola y^ = 4 ax. (2) Find the a;-derivative of the hypocycloid
x^ + y^ = a',
Ex. 10. In the cubical parabola y = i^ calculate the differential of the arc
at the point (2, 8) when : (1) dx = .2 ; (2) dy = .1. Show these differen-
tials in a figure. (The actual increments of the arcs can be computed by
Art. 209.)
67.] GEOMETRIC DEBIYATIVES. 99
{d) DeriratlTe and differential of tlie
length of a cnrre: polar coordinates.
Let PQ be an arc of the curve /(r, 6) = 0.
On PQ take any point V(r, »), and take
W{r + Ar, e + A9). Denote the length of PV
by s ; then the length of VW may be denoted
by As. Draw the chord VW.
Now, as in (c),
,. axcVWI. ds\ ,. chord FW
A« V del "-=" A9 ^.j^ Fig. 22.
About 0 describe a circular arc VM intersecting OW in M, and draw FT
at right angles to O W. Then angle FO ir = A9, and ilf IT = Ar.
.-. TW=OW-OT=r Jr \r -r cos Ad, and Fr=rsiuAe.
.-. chord VW = VC Fr)2 + ( TWy = V()-sin AS)^ + [r(l - cos AS) + Aj-p.
.-. chord FTT ^ j7;iinAgy r . sjujAg . ^j,, ^, Ar-|^.
Atf VV A9 / L JAfl ^ AffJ ^ '^
,. chord VW L.fdry
since, if A» = 0, SllL^ = 1, SBiAl = i, and sin i AS = 0.
A» JA9 ^
Hence, by (1). | = A|-^+(f)'
(3)
On multiplying each member of (2) by — , and then letting M, and con-
Ar
sequently Ar, approach zero, it will be found that
^^^r-JfY+l. (4)
dr ' \drj
From (3), (4), and definition Art. 27,
ds=^r2 + (g)'.de, (5)
and ds = ^lri(^y+l'dr. (6)
Ex. 11. Find the derivative of the arc of the spiral of Archimedes r=ae:
(1) with respect to the angle ; (2) with respect to the radius vector.
Ex. 12. Calculate the diflerential of the arc of the Archimedean spiral
r = 20 when e = 2 radians and de = 1° Make a figure. (The actual incre-
ment of the arc can be computed by Art. 210.)
100
DIFFERENTIAL CALCULUS.
[Ch. V.
(e) DeriratlTe and differential of the Tolnme of a surface of reroln-
tion. Let PQ be an arc of the curve y =/(x). On PQ take any point
L{x, y), and take the point M(^x + Ax,
y + Ay). On letting V denote the volume
obtained by revolving arc PL about OX,
the volume obtained by revolving arc LM
may be denoted by AV. Through L and
Jf*raw the lines shown in the figure.
The volume obtained by revolving arc LM
about the x-axis is greater than the volume
obtained by revolving LG, and is less than the
volume obtained by revolving KM. That is,
ir. UI? .L0< AF<ir. VM'.KM]
ry" . Ax < AV <: TT ■ (y + Ay)^ ■ Ax.
.-. iry^<^<-r-{y + Ayy.
Ax
(1)
On letting Ax approach zero, the three numbers in (1) become
wy^, — , iry^, respectively.
dx
Hence,
OF
dx
= iry*.
From (2) and Art. 27 dV= iry2 . doe.
If PQ had been revolved about the y-axis, then
dr
dy
irae^, and dV=irxi-dy,
(2)
C3)
(4)
Note. According to (3), for a given differential dx the corresponding
difierential of the volume is the volume of a cylinder of radius y and height
dx. The smaller dx is, the more nearly does this volume become equal to the
actual increment, due to dx, in the volume of the solid of revolution.
Ex. 18. Derive the results in (4).
Ex. 14. (1) Find the x-derivative of the volume generated by the revolu-
tion of the parabola y = x'^ about the a;-axis. (2) Find the y-derivative of
the volume generated by the revolution of this curve about the y-axis.
Ex. 16. (1) Calculate the differential of the volume in Ex. 14 (1), taking
dx = .l at the point where a; = 2. (2) Thus also in Ex. 14 (2), taking
dyr= .2 at the point where x = i. (The actual increment in the volume of
the solid due to changes dx and dy can be computed by Art. 182.)
67.]
GEOMETRIC BERIVATIVE8.
101
(/) DeriTatiTe and differential of the area of a enrface of revoln-
tion. Let PQ be an arc of the curve y =f{x). On FQ take any point, say
i(x, y), and take the point M{x + Aa:, y + Ay). Let S denote the area of
the surface generated by revolving arc PL about OX; then the area generated
by revolving arc LM about OX may be de-
noted by AS. There is evidently a straight
line whose length is equal to the length of the
arc LM. Through L and M draw the lines
LM' and ML' parallel to OX and equal in
length to the arc LM. {LM may be supposed
to be a piece of wire, LM' the same piece of
wire when it is stretched out in a horizontal
straight line from L, and ML' the same piece
of wire when it is stretched out in a horizontal
line from M. ) The surface obtained by revolving the arc LM about OX is
greater than the surface obtained by revolving LM' ; for, with the exception
of the point X, each point on LM has a greater ordinate than the corre-
sponding point in the line LM., and consequently a greater radius of swing.
Similarly, the surface obtained by revolving LM is less than the surface
obtained by revolving ML'. That is,
Fig. 24.
2 iry • LM' < surface generated by LM <,2 «" (y + Ay) • L'M;
2 Try ■ arc LM < A A^ < 2 ir (
+ Ay) • arc LM.
arc LM
...2.yHlM<M<2x(y + Ay)
Aa; Ax Ai
(1)
(2)
On letting Az approach zero, the three numbers in (2), by Arts. 20, 22,
23, 67c, take the values
and hence
2iry^, ^, 2iry^, respectively;
dz dx dx
M=2,ry^.
dx dx
(3)
On dividing the members in (1) by Ay, and letting Ay approach zero.
^ = 2.y^.
dy dy
Similarly, if arc PQ revolve about the y-axis.
i^ = 2.x^ (5), and f = 2.x^.
dx dx dy dy
(4)
(6)
From (3), (4), and Art. 67 (c) [(2), (3)],
dS
dx
-'W'+(S)'lf=.-W'HI)" <"
102 DIFFERENTIAL CALCULUS. [Ch. V.
Similarly, in case of revolution about the y-aiis, from (5) and (6),
dS_
—Mil f — ^Rf^■ ™
EesultB(3), (4), (7), show that, for a curve revolTing about the a;-axls,
dS = 2irj/ . d« = 2ir|/^l +(g)^da; = 2ny^jl +I^Ydy; (9)
and (5), (6), (8), show that, for a curve revolving: about the 2/-axl8,
dS = 2nx-d» = 2 rrx^^l+(^Ydx = 2 icxyjl + (^J dy. (10)
Ex. 16. Derive results (5), (6), (8), and (10).
Ex. 17. Find the x-derivative and the ^-derivative of each of the surfaces
described in Ex. 14.
Ex. 18. Calculate the differentials of the surfaces described in Ex. 15.
Make figures showing these differentials. (The actual increments of the
surfaces can be computed by Art. 211.)
Ex. 19. Find — , — , — , ^, for the ellipse h^^ + aY = oPV^- For
ax dx dx dx
a given differential of x, draw figures showing the corresponding differentials
of «, A, V, and x.
ds
Ex.20. Find — for r^=a^cos2e, r=acoae, rrrae'™'", r=a(l+cosfl).
dd
Ex. 21. If (p denote the eccentric angle of the ellipse in Ex. 19, show that
ds
— = oVl — e^ cos''0, e being the eccentricity.
d0
CHAPTER VI.
SUCCESSIVE DIFFERENTIATION.
TT.B. Article 68 contains all that the beginner will find necessary concern-
ing successive diflerentiation for the larger part of the remaining chapters.
Accordingly, the reading of Arts. 69-72 may be deferred until later.
68. Successive derivatives. As observed in many of the pre-
ceding examples, the derivative of a function of x is, in general,
also a function of x. This derivative, which may be called the
first derived function, or the first derivative (pi the function), may
itself be differentiated ; the result is accordingly called the second
derived function, or the second derivative (of the original function).
If the second derivative is differentiated, the result is called the
third derived function, or the third derivative ; and so on. If the
operation of differentiation is performed on a function n times in
succession, the final result is called the nth derived function, or
the nth derivative, of the function.
Ex. If the function is x«, then its first derivative is 4a^; its second
derivative is Mx'^; its third derivative is 24 a; ; its fourth derivative is 24;
its fifth and its succeeding derivatives are all zero.
Notation, (a) If y denote the function of x, then
the first derivative, namely -^(y), is denoted by -^ (Art. 23);
dx ax
the second derivative, namely Y\d)' ^^ '^®"°*®*^ ^^ ^!
the third derivative, namely —-
dx
dx\dx
, is denoted by — ^;
and so on. On this 5)lan of writing,
the nth deriyative is denoted by --^.
rfx"
103
104 DIFFERENTIAL CALCULUS. [Ch. VI.
In this notation the integers 2, 3, ••■, n, are not exponents;
these integers merely indicate the number of times that the
function y is to be differentiated successively with respect to x.
(6) The letter D is frequently used to denote both the operar
tion and the result of the operation indicated by the symbol
— (See Art. 23.) The successive derivatives of v are then
dx ^ ■^ ^
By, D{Dy), D\D{Dy)'], •••; these are respectively denoted by
Dy, Vy, l^y, -, D"y.
Sometimes there is an indication of the variable with respect
to which differentiation is performed ; thus
Dj/, DJy, D^\ ..., D^'y.
Note. Here n is not an exponent ; D"y does not mean (Dy)". (.E.g. see
Exs., p. 108.) D"y is called the derivative of the nth order.
(c) Instead of the symbols shown in (a) and (6), for the succes-
sive derivatives of y, the following are sometimes used, namely,
y', y", y'", -, y'"'-
(d) If the function be denoted by <t>(x), its first, second, third, •••,
and nth derivatives (with respect to x) are generally denoted by
^'{x), <t>"(x), <f>"'(x'), •••, <^'">(x) or </>"(a;), respectively;
also by |,(.),-|^(.),^<,(.), ..,£,,(.).
Note 1. In this book notation (a) is most frequently used. The symbol
D is very convenient, and is especially useful in certain investigations. See
Byerly's Diff. Cat, Lamb's Calculus, Gibson's Calculus (in particular § 67).
For an exposition of simple elementary properties of the symbol D also see
Murray's Differential Equations (edition 1898), Note K, page 208.
Note 2. Instead of the accent notation in (c), the 'dot '-age notation,
y, y, y, —
is sometimes used, particularly in physics and mechanics.
Note 3. Geometrical meaning of -j^ • It has been seen in Arts. 25, 26,
that — , i.e. -j- (y), denotes the rate of change of y, the ordinate of the curve,
68.] SUCCESSIVE DIFFERENTIATION. 105
compared with the rate of change of the abscissa x ; this may be simply
denoted as the a-rate of change of the ordinate. Similarly --4, i.e. -r;-l ^]
dx^ dx\dx/'
dy
IS the rate of change of the slope — of a curve compared with the rate of
change of the abscissa x, or, simply, the avrate of change of the slope.
On a straight line, for instance, the slope is constant, and hence the i-rate
of change of the slope is zero. This is also apparent analytically. For, if
y = mx + c is the equation of the line, then -j- =m, and hence -r^ = 0.
Note 4. Physical meaning of -^' In Art. 25 it has been seen that
if s denotes a varying distance along a straight line, — , i.e. — (s), denotes
dt dt
the rate of change of this distance, i.e. a velocity. Similarly — 5 , i.e. — ( — |
dfl dt\dtj'
denotes the rate of change of this velocity. Rate of change of velocity is
called acceleration. For instance, if a train is going at the rate of 30 miles
an hour, and half an hour later is going at the rate of 40 miles an hour, its
velocity has increased by '10 miles an hour' in half an hour, i.e. as usually
expressed, its acceleration is 10 miles per hour per half an hour. Again, it
is known that it s is the distance through which a body falls from rest
in t seconds, s = ^g^. Hence — := gt ; accordingly, ^-^ = g. That is, the
dt dt^
acceleration of a falling body is ' g feet per second ' per second. (See
text-books on Kinematics, Dynamics, and Mechanics, for a discussion on
acceleration.)
EXAMPLES.
1. Find the second i-derivative of: (i) a;tan->i; (ii) ix^ — 9x +
- — Vx ; (iii) tan i + sec x ; (iv) x'.
X
2. Find Dx*!/, when : (i) y =(x2 + a^) tan-i? ; (ii) y = log(sinx).
a
d*v 1
3. Find ^, when: (i) y=sin-ix; (ii) y = ^ ^ ■
4. Find i)/!/, when : (i) y = x*\ogx; (ii) y = e»^cosx.
6. Find ^, when xy» + 3x + 5y = 0.
dv y^ + S /,^
By Art. 56, ^ = -/ ,■ (1)
•' ' dx 2xy -r o
On difierentiation, ^r-j = —
(2xj, + 5)2t/|-(y2 + 3)(2y + 2x|)
dx^~ (2X2/ + 6)2
106
DIFFERENTIAL CALCULUS.
[Ch. VI.
On substituting the value of -p , and reducing,
(Py _ 2(y2 + 3) (3 ay' + 10 y - 3 x)
dx:^ (2 ij/ + 6)8 ■ •■ -'
6. (i) In the ellipse aV ^ 1,2^2 = a^ft^ calculate D^^y. (ii) Given
y2 + J, = x2, find X»^3y.
Work of part (i) .
Equation of ellipse, aV + f>V = a^^"-
On differentiation, 2a.^y ^ + 2b^ =0.
" dx
Whence
On differentiation in (2),
On substitution from (2), and reduction,
dy _ Vhc
dx ~ d'^y
^y 62
dx2 ~ a'
f dy]
[ y J
dx2- aA dY )'
cPy 62
a262 6*
0)
(2J
whence, by (1),
7. Shovr that the point (J, }) is on the curve log (x + y)= x — y. Show
that at this point ^ = 0, and ^ = J.
dx dx''
dy
d^y .
8. What are the values of ^ and ^^ : (i) at the point (2, 1) on
dx dx'
the ellipse 7 x'' + 10 y'' = 3S ; (ii) at the point (3, 5) on the parabola
y2 = 4 a; + 13 ?
9. Calculate — ^ for the cycloid in Art. 43, Ex. 6. Compute it when
dx^
a = 8 and « = - .
3
X = a(8 — sin 6),
dx
de
' = a(l — cos»).
a(l — cos*), and ?^=asin«.
do
.•.f? = r^^^,byArt.35l = -^^
dx ld9 de J a(l-
sin0
sind
(1 — cos*) 1 — cos 6
2 sin -cos -
2 2
2sin2
0
■ '^o* .7 ■
68] SUCCESSIVE DIFFERENTIATION. 107
dx^ dx \dxj de \dxj dx^ ^ ^^
. d^ _ d^ /■(%\_ d^ l^\ de
dx \dx) de \dx) ' dx
de\
1 a cosec^?
= — =r cosec^ 1^
2 2 a(l — costf) . . „ e . . .e
2 2
When 0 = 8 and e = -, this becomes
(Py _ 1 ^_1
««a:2 32 sin* 30° 2'
10. Verify the following : (i) if )/ = o sin a; + 6 cos x, ^-\-y = 0;
(ii) if u= (sill-' xy\ (1 - a:ii) 0 _ a;|H = 2 ; (ui) if j/ = a cos (log x) +
6sin(logx),x^g + x| + j, = o.
11. Show that if u = 2,2 logy, and y = f(x), ^ = (21ogy + 3)/"^?
,72,, dx^ \dxl
+ 2/(2 1ogj, + l)g-
12. Find -^, in the following cases : w = 4x» + 2x-8, M = 4x' + 4a + 2
(JX
j/ = 4x' + 5x — 4, y=4x3+cx + A;.
13. Given that — ^ = 3 z + 2, find the most general expression for
dx^
— ; then find the most general expression for y.
dx
14. A curve passes through the point (2, 3) and its slope there is 1; at
any point on this curve — - = 2 x; find its equation and sketch the curve.
dx^
16. At any point on a certain curve — ^= 8; the curve passes through
dx^
the origin and touches the line y = x there ; find its equation and sketch the
curve.
16. (1) In the case of simple harmonic motion, Ex. 13 (p. 78), show
that the speed of § is changing at a rate which varies as the distance of Q
from the centre of the circle. (2) What is the acceleration of the velocity
of the boat in Ex. 18, Art. 37 ?
17. In Ex. 14 (p. 78), calculate the rate at which Q is changing its speed
when Q is : (i) at an extremity of the diameter ; (ii) 12 inches from the
centre ; (iii) 6 inches from the centre ; (iv) at the centre.
108 DIFFERENTIAL CALCULUS. [Ch. VI.
18. A body moving vertically has an acceleration or a retardation of
g feet per second due to gravitation, g being a number whose approximate
value is 32.2 : find the most general expression for the distance of the moving
point from a fixed point in its line of motion, after t seconds. Explain the
physical meaning of the constants that are introduced in the course of
integration.
19. A body is projected vertically upwards with an initial velocity of 500
feet per second : find how long it will continue to rise, and what height it
will reach, if the resistance of the air be not taken into account.
20. A rifle ball is fired through a three-inch plank, the resistance of
which causes an unknown constant retardation of its velocity. Its velocity
on entering the plank is 1000 feet a second, and on leaving the plank is
500 feet a second. How long does it take the ball to traverse the plank ?
(Byerly, Problenis in Differential Calculus.)
69. The flth derivative of some particular functions. In a few-
cases the nth derivative of a function can be found. This is
done by differentiating the function a few times in succession,
and thereby being led to see a connection between the successive
derivatives.
EXAMPLES.
1. Let y = x'-
Then Dy = rx''-^ ;
D^ = r(r— l)x'-2;
Dh/ = r(r - t) (r - 2)x'-^.
From this it is evident that
D"y = r(r- l)(r— 2) ••■ (r - n + l)x'-".
Show that D"z'' = n 1
2. Find the nth derivative of the following functions :
(a) e' ; (b) a' ; (c) e" . (d) a"".
3. Show that the nth derivative of sin x is sin I x -t — ^ i .
Suggestion: cosz = sin [z + - ]■
4. Find the «th derivatives of (a) cos x ; (6) sin ax ; (c) cos ax.
6. Find the nth derivatives of log x, log {x — 2)^.
69-71.] SUCCESSIVE DIFFERENTIATION. 109
112 a
6. Find the nth derivatives of
7. Find the nth derivatives of
a:' l + x' 3- x' (6 + ca;)'"
2 2x
1 - ^2 ' 1 _ xi
[Suggestion: Talie the partial fractions.]
70. Successive difierentials. In Art. 27 it has been shown that if
y =/W, (1)
then dy=fi{x)dx. (2)
The differential in (2) is, in general, also a function of x ; and its differ-
ential may be required. In obtaining successive differentials it is usual to
give a constant differential increment dx to x. Then (Art. 27), on taking
the differentials of the members in (2),
d{dy)= d'y {x)dx'] = \_f" {x)dx-]dx. (3)
On taking the differentials of the members of (3),
d{didy)} = d{lf"ix)dx1 dx} =f "{x)dx ■ dx ■ dx. (4)
It is customary to denote results (3) and (4) thus :
dhi=f"ix)dx'^ and d^y =f"i{x)d^.
In this notation the nth differential is written
in which /"(x) denotes the nth derivative of /(x), and da;" denotes (dx)".
71. The successive derivatives of y with respect to x when both
are functions of a third variable, i say.
An example will show the method of finding these derivatives.
EXAMPLES.
1. Find'^ and ^, when x = 2 + 5«-<2 (1)
dx dx^
and y = 8«-«i; (2)
also find x, w, -^, — ?, when t = 2.
dx dV^
From(l), ^ = 6-2«. (3)
(Jit
From (2), f^ = 8-3f^. (4)
dt
110 DIFFERENTIAL CALCULUS. [Ch. VI.
dy
■ gg=gl(Art.35)=8-3''.
dx d^^ ' 5-2t
at
(6)
...d^ = A/^Ul/^\ .* (Art. 34)=-^/'^U^ (Art. 36)
da:2 dx\dx.] dt\dx) dx dt\dx) dt
(6)
^6t'-30t + 16
(5-2 0^
If « = 2, then by (1), (2), (5), (6),
2. See Ex. 9, Art. 68.
3. Find iJ^j/ and DJ^y when a: = a — 6 cos ff and j/ = oS + 6 sin e.
4. Find ^ and — in the following cases :
dx dx^
1 * Of
(i)x=- — -, y = — —; (ii)x = acose, y = asin8: (iii) a: = a cos »,
^^ 1 + f l + «
y — bsm$; (iv) x = cot «, y — sin' (.
72. Leibnitz's theorem. This theorem gives a formula for the nth
derivative of the product of two variables. Suppose that u and v are func-
tions of X, and put y = uv.
Then, on performing successive differentiations,
Dy = u ■ Dv + V ■ Du ;
Dh) = u ■ DH + 2Du ■ Dv + v- D'^u ;
Ifiy = u- Dh) + 3Ihi- Dhi + S D^u ■ Dv + v ■ D'^u ;
D*y = u ■ D*v + 4 Du ■ DH + 6 Z)% ■ D'hi + i D'^u • Dv + v • D*u.
Thus far the numerical coefficients in these derivatives are the same as the
numerical coefficients in the expansions (u+v), (%i + vy, (« + »)', and
(jt + v)* respectively, and the orders of the derivatives of ?i and v are the
same as the exponents of u and v in those binomial expansions. Now sup-
pose that these laws (for the numerical coefficients and the orders) hold in
the case of the nth derivative of uv ; that is, suppose that
D"(«i)) = u • D^ + nDu • D"-H + "^" ~ ^^ D^u ■ D"-'hi + ■■■
1 ■ ^
n{n-l).:(n-r + 2) ^,_i^ . j5„-r+i^
] .2... (r- 1)
+ "(" - ^^ - ("->•+ n Qru . pn-'V + ...+V D'U. (1)
1 . 2 ■■• r
72, 73.] SUCCESSUTE BIFFEBENTIATION. Ill
Then these laws for the coefficients and the orders hold in the case of the
(n + l)th derivative of uv. For diSerentiation of both members of (1) gives
D»+i(Mt))= u . i)"+i» + (n + l)Du ■ D'v + ^" "*" ^^" D^u ■ i>»-'o + -.
^ (» + l)»(n-l)...(n-r + 2) ^,^ . ^_,,^ ^ ._^ ^ _ ^^,^
1 • 2 ••• (r— l)r
Hence, if formula (1) is true for the nth derivative of uv, a similar formula
holds for the (n + l)th derivative. But, as shown above, formula (1) is true
for the first, second, third, and fourth derivatives of uv ; hence it is true for
the fifth, and for each succeeding derivative.
Ex. 1. Find Di"y when y = x^'.
D-y = x"- ■ D-'ie') + nD(x^) ■ i>»-i(e') + "^" ~ ^) D^(x^) ■ D-'-\e') + •••
= e^x"^ + 2nx + n{n - 1)].
Ex. 2. Calculate the fourth x-derivative of a^ sin x by Leibnitz's theorem.
Ex. 3. Find D^'y when : (i) y =xe^ ; (ii) y = xe^.
Note. Reference for collateral reading on successive differentiation.
Echols, Calculus, Chap. IV., especially Art. 56.
73. Application of differentiation to elimination. It is shown in
algebra that one quantity can be eliminated between two inde-
pendent equations, two quantities between three equations, and
that 11 quantities can be eliminated between n + 1 independent
equations. The process of differentiation can be applied for the
elimination of arbitrary constants from a relation involving vari-
ables and the constants. For by differentiation a sufficient num-
ber of equations can be obtained between which and the original
equation the constants can be eliminated.
EXAMPLES.
1. Given that y = Acosx + Bsinx, (1)
in which A and B are arbitrary constants, eliminate A and B.
In order to render possible the elimination of these two constants, two
more equations are required. These equations can be obtained by diSeren-
tiation. Thus,
^ = -^sin3:-|-.Bcosx, (2)
dx
^ = — Acosx — B&inx. (3)
dx^ ^ '
112 DIFFERENTIAL CALCULUS. [Ch. VI.
On eliminating A and B between (1), (2), (3), there is obtained the relation
g + .=0. (4)
Note 1. Equation (4) is called a differential equation, as it involves a
derivative. It is the differential equation corresponding to, or expressing
the same relation as, the " integral " equation (1). The process of deducing
the integral equations (or solutions, as they are then called) of differential
equations is discussed, but for a very few cases only, in Chapter XXVII.
2. Eliminate the arbitrary constants m and 6 from the equation
y = mx + b. Ans. —^ = 0.
In this case the given equation represents all lines, m and b being arbi-
- trary. Accordingly the resulting equation is the differential equation of all
lilies. For the geometrical point of view see Art. 68, Note 3.
3. Eliminate the arbitrary constants a and 6 from each of the following
equations: (1) y = ax'' + b. (2) y = ax^ + bx. (3) (y- 6)^=4 02.
(4) y2-2ay + x^ = aK (5) y'^ = b{a^ - x^^).
4. Find the differential equations which have the following equations for
solutions, Ci and c» being arbitrary constants :
(l)y = ci. (2) y = cia;. (3) y = CiX + C2. (4) 2/ = Cie* + c^e-^
(5) y=Cie'"+C2e-"". (6) y=Ci cosnia; + C2 sinma:. (7) 2/=CiCos(TOa;+C2).
5. Obtain the differential equations of all circles of radius r: (1) which
have their centres on tlie z-axis ; (2) which have their centres on the y-axis ;
(3) which have their centres anywhere in the ajy-plane.
6. Show that the elimination of n arbitrary constants ci, C2, ■■., c„, from
an equation /(x, y, c\, cj, •■•, c„) = 0 gives rise to a differential equation
involving the rath derivative of y with respect to x.
Note 2. For geometrical explanations relating to differential equations
the student is referred to Murray, Differential Equations, Chap. I., which
may easily be read now. The reading will widen his mathematical outlook
at this stage.
CHAPTER VII.
FURTHER ANALYTICAL AND GEOMETRICAL
APPLICATIONS.
VARIATION OF FUNCTIONS. SKETCHING OF GRAPHS.
MAXIMA AND MINIMA. POINTS OF INFLEXION.
N.B. This chapter may be studied before Chapter V. is entered
upon.
74. Increasing and decreasing functions. When x changes con-
tinuously from one value to another, any continuous function of x,
say <^(a;), in general also changes. The function may either be
increasing or decreasing, or alternately increasing and decreas-
ing. By means of the calculus it is easy to find out how the
function behaves -when x passes through any value on its way
from — 00 to -f- 00 .
Let Ax be a positive increment of x, and A(t>(x) be the corre-
sponding increment of <j>(x). If <l>{x) continually increases when x
is changing from a; to a; -|- Aar, then A</>(a;) is positive ; and accord-
ingly, "^^ ' is positive. Moreover, this is positive for all posi-
tive values of Ax, however small; hence lim^^^^ "^ •', i.e. <l>'(x), is
... Aa;
positive or zero.
Similarly, if if>(x) continually decreases when x is increasing
from a; to a; -|- Aa-, <^'(a;) is negative or zero. In other words :
If (p(x) is increasing in an interval, <l>'{x) is positive or zero for values
of X in the interval ;
if <t>(x) is decreasing in an interval, (t>'''x) is negative or zero for values
of X in tlie interval.
On the other hand :
If tp'ix) is always positive in an interval, <f)(x) is constantly increas-'
ing in the interval ;
if <t>'(x) is always negative in an interval, (p(x) is constantly decreas-
ing in tlie interval.
113
A.
B.
114
DIFFERENTIAL CALCULUS.
[Ch. vn.
The case when <f>'{x) is zero will be discussed later.
Properties A and B are illustrated by Figs. 25 a,h,c; 26 a, h,
c,d,e,f.
Let <^(x) be graphically represented by the curve ABODE,
whose equation is
y = *(«)•
At any point on this curve, -^=<l)'(x).
dx
By Art. 24, the slope of the curve represents the derivative of
the function. Now at A, D, and E, the slope is negative, and the
ordinate y (the function) is evidently decreasing as x is passing in
the positive direction through the values of x at A, D, and E.
On the other hand, at B, C, and F, the slope is positive, and the
ordinate y is evidently increasing as x is passing in the positive
T
\
L
\
r
L,
3f.
N,
u
^
n — •
a
X o
K
jr
Fig. 25 6.
Fig. 25 c.
Fig.
direction through the values of x at B, C, and F. In Fig. 25 6
when a; is increasing from OL^ to OM^, the ordinate y is decreas-
ing from i,L to MiMand the slope at points on LM is negative;
when X is increasing from OMi to ON^, the ordinate is increasing
from MiM to .Ar,JV" and the slope at points on MN is positive.
Fig. 26 a shows functions increasing or decreasing in an inter-
val which have a zero derivative within the interval.
75. Maximum and minimum values of a function. Critical points
on the graph, and critical values of the variable. The values of the
function at points such as P^, P^, P^, M, and 7f (Art. 74), where
the function stops increasing and begins to decrease, or vice versa,
75.]
MAXIMUM AND MINIMUM.
115
may be called turning values of the function. When a function
ceases to increase and begins to decrease, as at Pj, P^, and K, it is
said to have a maximum value ; when a function ceases to decrease
and begins to increase, as at Pi, P3, and M, it is said to have a
minimum value. Therefore, at a point (on the graph) where the
function has a maximum value the slope changes from positive to
negative ; at a point where the function has a minimum value the
slope changes from negative to positive. (Examine Fig. 25.)
Accordingly, at each of these points the slope (i.e. the derivative of
the function) is generally (see Note 3) either zero or infinitely
gi-eat.
It should be observed that, although the derivative of a function
may be either zero or infinitely great for values of the variable for
which the function has a maximum or a minimum value, yet the
converse is not always the case. The function may not have a
maximum or minimum value when its derivative is zero or infinity.
^
0
Fig. 26 a.
O
Fig. 2fl 6.
This is exemplified by the functions whose graphs are given in
Figs. 26 a, h. Thus at P the slope is zero and the function is
increasing on each side of P; at Q the slope is zero and the
function is decreasing on each side of Q ; at P the slope is infi-
nitely great, and the function is increasing on each side of R ;
at S the slope is infinitely great and the function is decreasing
on each side of S.
Accordingly, a point where the slope of a graph of a function
is zero or infinitely great is, for the purpose of this chapter, called
a a-itical point. Such a point must be further criticised, or ex-
amined, in order to determine whether the ordinate has either a
maximum or a minimum value there. In other words, that value
116 DIFFERENTIAL CALCULUS. [Ch. VII.
of the variable for which the derivative of a function is zero or
infinitely great is called a critical value; further examination is
necessary in order to determine whether the function is a maxi-
mum or a minimum for that value of the variable.
Note 1. The pointa Q, P, B, S (Figs. 26 a, 6), are examples of what are
cahed. points of inflexion (see Art. 78).
Note 2. By saying that a function <j>(x) has a minimum value, for a; = a
say, it is not meant that (p(a) is the least possible value the function can
have. It is meant that the value of the function for a; = a is less than the
values of the function for values of x which are on opposite sides of a,
and as close as one pleases to a ; i.e. h being taken as small as one pleases,
0(a)<0(a— ft) and i^(a)<0(fl! + A). (See Pi in Fig. 25 a.) Likewise, if
0(r£) is a maximum for x = b, this means merely that <^(6) >(p(b — h) and
0(6) > 4>ib + ft), in which h is as small as one pleases. (See P2 in Fig. 25 a.)
EXAMPLES.
1. Examine sin x for critical values of the variable.
Here 4>(x) = smx.
The graph of this function is on page 459. In order to find the critical
points solve the equation
0'(a;) = cos X = 0.
•S fl- Stt
X
Accordingly, the critical values of x are -, — , — , •••.
2. Examine (x — 'iy{x + 3) for critical values of the
variable.
Here ^(x) = (a; - l^'Kx + 3).
The solution of 1^' (x) = (2 - 1) (3 x + 5) = 0,
gives the critical values of x, viz. 1, — |.
S. Examine (x— l)'-f-2 for critical values of the
variable.
Here 0(x) = (x - 1)3+ 2.
On sol ving <p' (x) = 3(x - ly = 0,
Fig. 26 d. the critical value of x is obtained, viz. x = I.
76.]
MAXIMUM AND MINIMUM.
117
4. Examine (x -2)^+3 for critical values of z.
Here ^(k) = (x - 2)*+ 3.
o
On solving <P'(x) = = = oo ,
3(x - 2)*
the critical value x = 2 is obtained.
5. Examine (x — 2)»+ 3 for critical values of x.
Here ^(a) = (x- 2)^ + 3;
1
and
*'(x) = -
3(x - 2)'
gives the critical value x = 2-
Fig. 26 /.
Note 8. A function may have a maximum or minimum value when its
derivative changes abruptly ; see Art. 164, Note 3, and Fig. 21 (c), Itifin. Cat.
76. Inspection of the critical values of the variable (or critical
points of the graph) for maximum or minimum values of the function.
Let the function be <^(x). The equation of its graph is y = <t>{x),
and the slope is -f- or ^'(x). The solutions of the equations
dx
</)'(x) = 0 and </)'(a;) = oo,
give the critical values of the variable.
Suppose that ABODE (Fig. 25 a) is the graph, and that the
critical values are x = a and x = b. There are three ways of
testing whether the critical values of the variable will give maxi-
mum or minimum values of the function, viz. :
(o) By examining the function itself at, and on each side of,
the critical value ;
(6) By examining the first derivative on each side of the
critical value ;
(c) By examining the second derivative (see Art. 68) at the
critical value.
Note 1. It follows from the definition of maximum and minimum values,
and Note 2, Art. 75, that if </>(a) is a maximum (or minimum) value of 0(x),
then 0(a)+TO, c<t>(a), v'0(a)i ^■'(o), •••, are maximum (or minimum)
118 DIFFERENTIAL CALCULUS. [Ch. VII.
values of <t>(x)+m, c<t>(x.), v'0(x), ify'^x), —, respectively. Accordingly,
the finding of critical values of x for one of these functions will give the
critical values for the other functions. It sometimes happens that it is much
easier to find the critical values for, say <p'^{x), than for <t>{x). In such a
case it is better to examine <t>\x) than to examine ^(x).
(a) Examination of tlie fnnction. Let <^(a;) denote the function,
and a; = a be the critical value of x.
In this test the value of <^(ffl) is compared with two values of
<^(a;), viz. when a; is a little less than a, and when a; is a little
greater than a^ say, when x = a—h and when x = a + h,m which
h is a small number.
If <l>(a) is greater than both <^(a — h) and 4>(a + h), <^ (a) is a maxi-
mum (as at Pj in Fig. 25 a and K in Fig. 25 c).
If <j>{a) is less than both tf>{a — h) and <^(a + h), <f>(a) is a minimum
(as at Pi and P^ in Fig. 25 a and JIf in Fig. 25 b).
If <f>{a) is greater than the one and less than the other of <f>{a — h)
and <l)(a + h), <f)(a) is neither a maximum nor a minimum (as
at P, Q, R, S, Figs. 26 a, b, and at a; = 1 in Fig. 26 d).
Ex. 1. In Ex. 1, Art. 75, examine the function at the critical value - of x.
Here sin( ^ — A )<sin -, and sin (^ + ft J < sin -• Accordingly, x=-
gives a maximum value of sin x.
Ex. 2. (a) In Ex. 2, Art. 75, examine the function at the critical value
x = l. Here0(l)=O, ^(1 - ft) = A-'(4-A),,^(l + ft)= A2(4 + ft). Accord-
ingly, ^(1 — A)>i^(l), and </>(l +ft)>i/>(l). Thus ^(1) is a minimum
value of 0(a;).
(6) Inspect this function at the critical value x =— f.
Ex. 3. In Ex. 3, Art. 75, examine the function at the critical value x = 1.
Here 0(1)= 2, 0(1 - h) = - ft' + 2, and 0(1 + ft)= ft' + 2. Accordingly,
0(1 — ft)<0(l)<0(l +ft), and thus 0(1) is not a turning value of the
function.
Ex. 4. Examine the functions in Exs. 4, 5, Art. 75, at the critical
values of x.
76.] MAXIMUM AND MINIMUM. 119
(6) Examination of tfcie first deriTative of the function. When
the derivative of a function is positive, the slope of its graph is
positive and the function is increasing; when the derivative is
negative, the slope of the graph is negative and the function is
decreasing (Art. 74). Hence, h being taken as small as one
pleases, if <l>'(a — Ji) is positive and <f>'{a + /') is negative, then <^(a)
is a maximum value of <t>(x). On the other hand, if ^'{a — /*) is
negative and <^'(a + K) is positive, then <t>{x) is decreasing when x
is approaching a, and <ft(x) is increasing when x is leaving a, and
accordingly <^(a) is a minimum value of <^(a;). Examine Figs. 25
at, and near, Pj, P^, P^, M, K.
Note 2. Test (6) is generally easier to apply than test (a). For test (a)
the functions <t>{a — h) and <t>{a + h) must be computed ; for test (6) merely
the algebraic signs of <t>'{a — h) and 0'(a + ft) are required.
Ex. 5. (a) InEx. 1, Art. 75, 0'(-— ft ) ispositiveand 0'( -+ ft ] isnega-
tive. Accordingly, 0(7], I'-e- sin — or 1 , is a maximum value of sin s.
(6) Apply this test at the other critical values in Ex. 1, Art. 75.
Ex. 6. (a) In Ex. 2, Art. 75, <t>'{\ — ft) is negative and ^'(1 + A) is posi-
tive. Accordingly ^(1), i.e. 0, is a minimum value of {x — l)2(x + 3).
(6) Apply this test at the other critical value in Ex. 2, Art. 75.
Ex. 7. In Ex. 3, Art. 75, <t>'(\ — ft) is positive and 0'(1 + ft) is positive.
Accordingly, <t>{V), or 2, is neither a maximum nor a minimum.
Ex. 8. Apply test (6) at the critical values of the functions in Exs. 4, 6,
Art. 75.
(c) Examination of the second derivative of tlie function. It has
been seen that the sign of the derivative of a function ^[x) changes
from positive to negative when the function is passing through a
maximum value. If the derivative <^'(a;) passes from a positive
value to zero, and then becomes negative, the derivative is contin-
ually decreasing, and hence its derivative, namely <t>"{x), must be
= , or <, 0 for the critical value of x. On the other hand, when
the function passes through a minimum value, the derivative
120 DIFFERENTIAL CALCULUS. [Ch. VII.
changes sign from negative to positive. If then the derivative
ff>\x) passes through zero, it is continually increasing, and hence
its derivative, namely 4>"{x), must be =, or >, 0 for the critical
value of a;. Therefore,
if <t>'{a) is zero and <^"(a) is negative, 4>{a) is a maxirmim value
of4>{x);
if <l>'{a) is zero and <^"(o) »'« positive, <t>(a) is a minimum value
of<^{x).
Note 3. When ^"(a) Is zero, one of the other tests can he used.
Another procedure that can be adopted when 0"(a) = 0, is discussed in
Art. 155.
Note 4. When the second derivative can be obtained readily, test (c) is
the easiest of the three tests to apply.
Note 5. Historical. Kepler (1-571-1630), the great astronomer, "was
the first to observe that the increment of a variable — the ordinate of a curve,
for example — is evanescent for values infinitely near a maximum or minimum
value of the variable." Pierre de Fermat (1601-1665), a celebrated French
mathematician, in 1629 found the values of the variable that make an ex-
pression a maximum or a minimum by a method which was practically the
calculus method (Art. 75).
Note 6. Many problems in maxima and minima may be solved by ele-
mentary algebra and trigonometry. For the algebraic treatment see
(among other works) Chrystal, Algebra, Part II., Chap. XXIV. ; William-
son, Diff. Cal., Arts. 133-137 ; Gibson, Calculus, § 76 ; Lamb, Calculus,
Art. 52.
Note 7. Maxima and minima of fanctions of two or more inde- -
pendent variables. For discussions of this topic see McMahon and Snyder,
Diff. Cal, Chap. X., pages 18.3-197; Lamb, Calculus, pages 135, 596-598;
Gibson, Calculus, §§ 159, 160 ; Echols, Calculus, Chap. XXX. ; and the
treatises of Todhunter and Williamson.
EXAMPLES.
9. (a) In Ex. 1, Art. 75, 0"(a:) =- sinx. Accordingly, *"(|)
negative, and thus <t>(—], i-e. sin -, is a maximum value of 0(x).
(6) Apply test (c) at the other critical values of sin x.
77.] PROBLEMS IN MAXIMA AND MINIMA. 121
10. (a) In Ex. 2, Art. 75, 0"(a;) = 2(3a: + 1). Accordingly, <p"{\) is
positive, and thus 0(1) is a minimum value of <t>{x).
(b) Apply test (c) at the other critical value in Ex. 2, Art. 75.
11. In Ex. 3, Art. 75, ^"(x)=6(x - 1). Here 0"(1)=O, and thus
test (c) fails to indicate whether ^(1) is a turning value of <i>(x). (See Note 4.)
12. Apply test (c) at the critical values of the functions in Exs. 4, 5,
Art. 75.
Note 8. Sketching of graphs. The ideas discussed in Arts. 74-76 are a
great aid in making graphs of functions, and in showing what is termed the
march of a function.
13. For each of the following functions find the critical values of x,
determine the maximum and minimum values, and sketch the graphs:
(1) 2x8 + 5x2-4a; + 2; (2) 5 + 12 x - x'' -23? ; (3) x^(x + l)(^x - 2)'>;
(4) (2;-2)8(x + l)2; (5) 2 + 3(x - 4)l + (x-4)i ; (6) 3x6-125xS+2160x;
(7) "'"^10^^: (8) f^^lp (9)^logx;(10)xi;(ll)28in2x + 8cos2x;
(12) sinxsin2x; (13) xcosx.
14. Show that a + (x — c)" is a minimum when x = c, if n is even ;
and that it has neither a maximum nor a minimum value, if n is odd.
15. (a) Show that (4 ac—b^) -r-ia is a maximum or a minimum value
of ax^ + bx + c, according as a is positive or negative. (B) Show that
ax^ -j- bx + c cannot have both a maximum and minimum value for any
values of a, 6, c.
16. Find the point of maximum on the curve x' + y' — 3 axy = 0.
Sketch the graph, taking a = 1.
17. In the case of the ellipse ax^ + 2 hxy + by^ + c = 0, show how to
find the highest and lowest points, and the points at the extreme right and
left.
77. Practical problems in maxima and minima. Some practical
applications of the principles of Arts. 75 and 76 will now be
given. In making these applications the student is in a position
analogous to his position in algebra when he applied his knowledge
about the solution of equations to solving "word problems." Here,
as in algebra, the most difficult part of the work is the mathe-
matical statement of the problem and the preparation of the data
for the application of the processes of Art. 76.
122
DIFFERENTIAL CALCULUS.
[Ch. VII.
EXAMPLES.
1. Find the area of the largest rectangle that can be inserted in a
given triaugle, when a side of the rectangle lies on a side of the triangle.
Let ABC be the given triangle, and let
the given values of the base AB and the
height CD be 6 and h respectively.
Suppose that MQ is the largest rectangle,
and let MN and NQ be denoted by y and x
respectively, and denote the area of MQ by u.
Then u = xy, which is to be a maximum.
It is first necessary to express «, the
quantity to be "maximised," in terms of a
Fig. 27. single variable.
M
/
\
P
1.
1
D
i
B
Now
du
MF:AB = CH:CD; i.e. x:b = h-y:h.
, a maximum.
a; = - (ft - y) ; accordingly, u = -y(h
h h
(ft — 2 y) = 0 ; whence y = ^h. Thus x = ^ 6, and area
dy h
MQ — \bh = one half the area of the triangle.
Note 1. It M be supposed to move along AC from Ato C, the rectangle
MQ increases from zero at A and finally decreases to zero at C. It Is thus
evident that for some point between A and C the rectangle has a maximum
value.
Note 2. In these examples it is necessary that the quantity to be maxi-
mised or minimised be expressed in terms of one variable. Conditions
sufficient for this must be provided.
2. Solve Ex. 1, expressing ii in terms of x.
3. A parabola y'' = Sx is revolved about the x-axis ; find the volume
of the largest cylinder that can be inscribed in the
paraboloid thus generated, the height of the parab-
oloid being 4 units.
Let OPL be the arc that revolves, LN be at
right angles to OX, and ON = 4. Take P(x, y),
a point in OL, and construct the rectangle FN.
When OFL generates the paraboloid, FN gen-
erates a cylinder. (As F moves along the curve
from O to X, the cylinder increases from zero at
0 and finally decreases to zero at L. Thus there
is evidently some position of P between O and L
for which the cylinder is a maximum.) Suppose Fig. 28.
77.]
EXAMPLES.
123
that PK generates the maximum cylinder, and denote its volume by V.
'^^^'^ r = irPG^ . GN = iry2(4 - X) = 8 iri(4 - z).
Accordingly, — = 8 ir (4 - 2 x) = 0.
dx
From this, x = 2 ; hence F= 100.53 cubic units.
Note 3. In the process of maximising in Exs. 1, 2, the constant factors -
and 8 ir may as well be dropped. (See Art. 76, Note 1.)
Note 4. In each of these examples it is well to perceive at the outset that
a maximum or a minimum exists.
4. A man in a boat 6 miles from shore wishes
to reach a village that is H miles distant along
the shore from the point nearest to him. He can
walk 4 miles an hour and row 3 miles an hour.
Where shoiild he land in order to reach the village
in the shortest possible time ? Calculate this
time. Let L be the position of the boat, M the
village, and N the nearest land to L. Then LN
is at right angles to NM. Let P denote the place
to land, and T denote the time (in hours) to go
over LP + PM, and denote NP by x.
PM
4
dx
iFP-6.8
Then
--¥-
V36 + x^ , 14 - X
3 +~i~'
-1 = 0.
a mmimum.
Hence,
D
3V36 + a;2
X = 6.8 miles, and r = 4.8 ••■ hours.
6. What must be the ratio of the height of a Norman window of given
perimeter to the width in order that the greatest possible amount of light may
be admitted ? (A Norman window consists
of a rectangle surmounted by a semicircle.)
Let m denote the given perimeter, 2 x the
width, and y the height of the rectangle in the
window desired ; let A denote the area of
the window.
Then .4 = 2 xy + J irx'.
Now 2x + 2y + TX = m.
.-. A = mx-2x'^-\irx^,
which is to be a maximum.
On finding the value of x for which A -is A
maximum, and then getting the corresponding value of y, it will appear that
x = y. Accordingly, the height MD = the width AB.
E
n
^
c
1
jjf
A
■r ,
i
Fig. 30.
124 DIFFERENTIAL CALCULUS. [Ch. VII.
6. Find the area of the largest rectangle that can be inscribed in an
ellipse. (First show that there evidently is such a rectangle.)
Sdggestions : Let the semiaxes of the ellipse be a and 6, and choose
axes of coordinates coincident with the axes of the ellipse. Let P(x, y) be a
vertex of the rectangle. Then area rectangle = 4xy = i- xy/a' — xK Maxi-
mise the last expression, or, better still, because it is easier to do, maximise
the square of a; Va^ - x'^, viz. x^(a^ - a;^). (See Art. 76, Note 1.) It will be
found that the area of the rectangle is 2 ab, half the area of the rectangle
circumscribing the ellipse.
7. Divide a number into two factors such that the sum of their squares
shall be as small as possible.
8. Two sides of & triangle are given: find, by the calculus, the angle
between them such that the area shall be as great as possible.
9. Find the largest rectangle that can be inscribed in a given circle.
10. Through a given point P(a, h) a line is drawn meeting the axes
in A and B\ 0 Ss the origin : Find (i) the least length that AB can have ;
(li) the least value of OA-^ OB; (iii) the least possible area of the triangle
OAB.
11. A and B are points on the same side of a straight line_21fiV:
determine the position of a point C in MN: (1) so that AC + Gn = a
minimum ; (2) so that AC + CB = a minimum.
IT.B. The cones and cylinders in the following examples are right circular :
12. (i) Find the height of the cone of greatest volume that can be in-
scribed in a sphere of radius r. (ii) Find the cone of greatest convex surface
that can be inscribed in this sphere.
18. Find the semi-vertical angle of the cone of least volume that can be
described about a sphere.
14. (i) Find the cylinder of greatest volume that can be inscribed in a
sphere of radius r. (ii) Find the cylinder of greatest curved surface that
can be inscribed in this sphere.
16. (i) Determine the maximum cylinder that can be inscribed in a
right circular cone of height 6 and radius of base a. (ii) Determine the
cylinder of greatest convex surface that can be inscribed in this cone.
16. What is the ratio of the height to the radius of an open cylindrical
can of given volume, when its surface is a minimum ?
17. A circular sector of given perimeter has the greatest area possible:
find the angle of the sector.
18. It is required to construct from two circular iron plates of radius
a a buoy, composed of two equal cones having a common base, which shall
have the greatest possible volume : find the radius of the base.
78.]
POINTS OF INFLEXION.
125
19. An open tank of assigned volume has a square base and vertical
sides : if the inner surface is the least possible, what is the ratio of the depth
to the width ?
20. From a given circular sheet of metal it is required to cut out a
sector so that the remainder can be formed into a conical vessel of maximum
capacity : show that the angle of the sector removed must be about 60°.
21. In a submarine telegraph cable the speed of signalling varies as
x'' log -, where x is the ratio of the radius of the core to that of the covering :
X
show that the speed is greatest when the radius of the covering is Ve times
the radius of the core.
22. Assuming that the power required to propel a steamer through still
water varies as the cube of the speed, find the most economical rate of
steaming against a current which is running at a given rate.
23. Assuming that the strength of a rectangular beam varies as the
product of the breadth and the square of the depth of its cross-section, find
the breadth and depth of the strongest rectangular beam that can be cut from
a cylindrical log, the diameter of whose cross-section is d inches.
24. Find the length of the shortest beam that can be used to brace a
vertical wall, if the beam must pass over another wall that is a feet high and
distant 6 feet from the first wall.
25. At what distance above the centre of a circle of radius a must an
electric light be placed in order that the brightness at the circumference of
the circle may be the greatest possible ? (Assume that the brightness of a
small surface A varies inversely as the square of the distance r from a source
of light, and directly as the cosine of the angle between r and the normal to
the surface at A.) (Gibson's Calculus.)
78. Points of inflexion: rectangular coordinates. As a point
moves along the curve LAM from L to M, the tangent at the
moving point changes from the position shown at L to that at A
T
0
FiQ. 31 6.
and then to that at M. In going from the position at L to the
position at A, the tangent turns in the direction opposite to that
in which the hands of a watch revolve ; in going from the position
126 DIFFERENTIAL CALCULUS. [Ch. VII.
at A to the position of M, the tangent turns in the same direction
as that in which the hands of a watch revolve. Points such as
A, D, H, 0 (Fig. 31), and Q, P, R, S (Figs. 26 a, b), at which the
tangent for the point moving along the curve ceases to turn in
one direction and begins to turn in the opposite direction, are
called poinds of inflexion.
Examination of the cm-ve for poiiits of inflexion. As the moving
point goes along the curve from L to A, -^ increases and accord-
ingly its derivative —^ is positive; as the moving point goes
along the curve from AtoM,-^ decreases, and accordingly — ^
''^ cPv . ■ .
is negative. Thus in the case of the curve LAM, — ^ is positive on
one side of A and negative on the other. Now -^ changes continu-
»2 Ct3j
ously from L to M; accordingly, at ^ t^ = 0- Hence, in order
uar
to find the points of inflection for a curve y=f(x), proceed as
follows :
cPy
CalculoUe ^ >
then solve the equation — ^ = 0<
dar
This will give critical values (or points) which are to be further
examined or tested. A critical point is tested by finding whether
— ^ has opposite signs on each side of the point. If — ^ has oppo-
site signs, the critical point is a point of inflexion; if — ^ has the same
^^^ - sign on both sides of the critical
•'■' ~~~~^ point, as in Fig. 31 c, the point is
*■ what is called o point of undulation.
Note 1. At a point of inflexion the tangent crosses the curve. The tan-
gent at an ordinary point on a curve is the limiting position of a secant when
two of the points of intersection of the
secant and the curve become coincident
(Art. 24). The tangent at a point of in-
flexion is the limiting position of a secant
which cuts the curve in more than two
points, when the secant revolves until three
points of intersection become coincident.
78.] EXAMPLES. 127
Thus PT, the tangent at the point of inflexion P, is the limiting position
of the secant MPQ when MPQ revolves about P until M and Q simultane-
ously coincide with P. At a point of undulation the tangent does not cross
the curve. The tangent at a point of inflexion is called an inflectional tan,-
gent; the tangent where y'' = 0 is called a stationary tangent.
Note 2. If /(z) is a rational integral function of degree n, the greatest
number of points of inflexion that the curve y =/(x) can have is n — 2.
Moreover the points of inflexion occur between points of maxima and minima.
[See F. G. Taylor's Calculus (Longmans, Green & Co.), Art. 206.]
Note 3. References for collateral reading. On maxima and minima of
functions of one variable, etc. . McMahon and Snyder, Dlff. Cal., Chap. VI. ;
Echols, Calculus, Chap. VIII. (in particular, Art. 85). On points of inflexion :
Williamson, Diff. Cal. (7th ed.), Arts. 221-224 ; Edwards, Treatise on Diff.
Cal., Arts. 274-279 ; Echols, Calculus, Chap. XI.
Note 4. Points of inflexion : polar coordinates. For a discussion of
this topic see Todhunter, Diff. Cal., Art. 294; Williamson, Dijf. Cal.,
Art. 242; F. G. Taylor, Calculus, Art. 276.
EXAMPLES.
1. In the following curves find the points of inflexion, and write the
equations of the inflexional tangents ; also sketch the curves and draw the
inflexional tangents : (1) y = i?; (2) a; - 3 = (y + 3)3 ; (3) y = a;2(4 - x) ;
(4) 12t, = a:»-6x= + 48; (5) 2,^-J-^; (C) ,=^^; (7) y = ^.
2. Find the points of inflexion on the following curves : (1) y =
x{x - a)« ; (2) xy'^ = a\a - x) ; (3) ax^ -xh/-a^y = 0; (4) j/ = 6 +
(c-x)«; (5) y = m-6(z-c)^; (6) x^ - 3 ftx^ + a=y = 0.
3. Show that the curve y = x^ has no point of inflexion. Sketch the
curve.
4. Show that the points where the curve y = bsin- meets the z-axis
are all points of inflexion.
6. Show that the curve (1 + x^)y = 1 - x has three points of inflexion,
and that they lie in a straight line.
6. Show why a conic section cannot have a point of inflexion.
7. Show, both geometrically and analytically, why points of inflexion
may be called points of maximum or minimum slope.
CHAPTER VIII.
DIFFERENTIATION OF FUNCTIONS OF SEVERAL
VARIABLES.
N. B. This chapter may be studied immediately after Chapter VII., or its
study may be postponed and taken up after any one of Chapters IX.-XVII.»
79. Partial derivatives. Notation. Thus far functions of one
independent variable have been treated; functions of two and
of more than two independent variables will now be considered.
^®* u=f{x,y) (1)
in which /(«, y) is a continuous function (see Note 2) of two
independent rariables x and y. The value of the function for a
pair of values of x and y is obtained by substituting these values
in f{x, y)-
Thus, if /(a;, y) = 3x - 2y + 7, /(I, 2) = 3 ■ 1 - 2 . 2 + 7 = 6.
z, Note 1. Geometrical
representation of a func-
tion of two variables.
The student knows how a
continuous function of one
variable can be represented
by a curve. A continuous
function of two variables
can be represented by a axir-
face. Thus the function z,
^^-^^ z=Ax,y), (2)
is represented by the sur-
face LEGS if MP, the per-
pendicular to the a;y-plane
erected at any point M{x, y)
on that plane and drawn to
meet the surface at P, is
equal tof{x, y).
Fio. 3a.
• See the order of the topics in Echols' Calculus.
128
79.] PAETIAL DERIVATIVES. 129
References for collateral reading. See chapters on the geometry of
three dimensions in text^books on Analytic Geometry, for instance, those of
Tanner and Allen, Ashton, Wentworth ; also Echols' Calculus, Chap. XXIV.
Note 2. Continuous function of two variables defined. A function
/(x, y) is said to be a continuous function of x and y within a certain range
of values of x and y, when : (i) /(x, y) does not become infinitely great, and
(ii) if, (a, 6) and (a + /i, 6 + A;) being any values of (x, y) within this
range, f{a + ft, b + k) can be made to approach as nearly as one pleases to
/(«, 6) by diminishing ft and k, and if f{a + h,h + k) becomes equal to /(a, 6),
no matter in what way ft and k approach to, and become equal to, zero.
This definition may be illustrated geometrically, thus : On the zy-plane
(Fig. 33) let M be (a, 6) and iV be (« + ft, 6 + A;), and let MP be f{a, h)
and NQ be /(a + ft, 6 + k). Then, if MP and NQ are finite, and if NQ
remains finite while JV approaches M, and becomes equal to MP when N
reaches M, no matter by what path of approach on the ary-plane, f{x, y) is
said to be a continuous function of x and yiorx = a and y = b.
In (1) suppose that x receives a change Aa; and that y remains
unchanged. Then u receives a corresponding change Am, and
M + Am =f(x + ^.x, y) ;
and Am = f(x + Aa;, y) —f{x, y).
. Am ^ fix + Aa;, y) -f{x, y)^
" Aa; Ax
and Y.^^^=M^,^ni±M,itzf(mi.
Ax Aa;
This limiting value is called the partial derivative of u with
respect to x, because there is a like derivative of u with respect
to y, namely, lim.^ f = lim.,^ .^C^. V + Ag -f{x, v).
These partial derivatives are usually written
Sw, 3«, (3)
dx dy
respectively, in order to distinguish them from derivatives (like
dM^ du^ ds^ ^^^ g^ ^^^ Qf functions of a single variable and from
da; dy dt
what are called total derivatives (see Art. 81). If u =f(x, y, z),
130 DIFFERENTIAL CALCULUS. [Ch. VIII.
the partial derivatives of the first order are — , — , and —
dx dy az
According to the above definition, the partial derivative with
respect to each variable is obtained by differentiating the func-
tion as if the other variable were constant. Notation (3) is very
commonly used, but vaiious other symbols for partial derivatives
are also employed.
Note 3. Gteometrical representation of partial derivatiTes of a func-
tion of two variables. Let f(x, y) be represented by the surface LEGS
(Fig. 33) whose equation is _/■/>,
Take P any point (x, y, z) on this surface. Through P pass planes parallel
to the planes ZOX and ZOY, and let them intersect the surface in the curves
LPG and BPS respectively. Along EPS, x remains constant; and along
LPO, y remains constant. Accordingly, from the definition above and
Art. 24 the partial z-derivative 2£ js the slope of LPG at P, and the
p, dx
partial j^-derivative — is the slope of EPS at P.
EXAMPLES.
1- K v = x^ + 2x-y + xy^-\-y* + e' + xcosy,
then ~ = Zx'^ + ixy + y> + e' + cosy,
ox
and — = 2 x2 + 3 a:v2 + 4 „3 _ J. sin „
dy
2. Find 2^, 2^, and ^, when u=x^ + 2y'^+3z^+e'siay+coszcosy.
ox dy dz
3. On the ellipsoid ^ + ^ + ^ = i : (a) find ^ and ^ at the point
16 2o 9 ^ ' dx dy
where x = 1 and y = 4 ; (6) find ^ and ^ at the point where « = 2 and
. - dz dy
z = 2 ; (c) find ^ and ^ at the point where 2=1 and 2 = 3. Make
(72 dx
figures for (a), (6), and (c), and show what these partial derivatives repre-
sent on the ellipsoid.
4. Verify the following :
(i) If u = log(e'-|-e»), i!i + §^ = i-
dx dy
(ii)If« = -?5^, f!^+^=(x + j,_l)„;
e'+ e" dx dy
(iii) If u = x^iy, x^ + y^=:(x + y + log u)u.
80.]
SUCCESSIVE PARTIAL DERIVATIVES.
131
80. Successive partial derivatives. The partial derivatives of
the first order described in Art. 79 are, in general, also continuous
functions of the variables, and their partial derivatives may also
be required. In the successive differentiation of functions of two
or more variables, the following is one of the systems of notation :
U'^\ is written g;
dx\dxj dv
d fdu\ . .^^ dhi
- / is written^;
d fdu\ . ... d'u
— ( — IS written ;
dy\Oxj dydx
d fdu\ . .^. d^u
— ( — ] IS written-—;
dx\dyj dxdy
V ^^M is written ^'^ ;
dz\dydxj dzdydx
l(S)— °^^
'(''-\ is written ^ ""- ;
dz \dx dzj dz dx dz
5 /'d''u\ . .^. d^u
5" Tl IS written -—- ;
dz\dy^J dzd'f
and so on.
Note 1. In this notation the symbol above the horizontal bar indicates
the order of the derivative, and the symbols below the bar, taken from right
to left, indicate the order in which the successive differentiations are to be
performed. Thus
^u
means that u is to be differentiated three times
dx'^dydz^
in succession with respect to z, and the result is then to be differentiated
with respect to y ; and the function thus obtained is then to he differentiated
twice in succession with respect to x.
Note 2. The adoption, by mathematicians, of the symbol d in the nota-
tion of partial differentiation was mainly due to the great mathematician,
Carl Gustav Jacob Jacobi (1804-1851), who decided, in 1841, to use 3 in
the manner which afterwards became the fashion. As to some points of
insufficiency and difficulty connected with this notation, see correspondence
between Thomas Muir and John Perry, Nature, Vol. 66, pages 53, 27i, 620.
Note 3. The order in which the snccessive differentiations are per-
formed does not affect the result (certain conditions being satisfied) ; e.g.
dxdy dydx dxdydz dzdxdy dydxdz
azazaz dz^dx 52 az"
This theorem is true in almost all cases which occur in practice ; e.g. see Exs. 1-8.
For a discussion and references see Infin. Cat., Art. 85. Also see Pierpont,
Functions of Ileal Variables, Vol. I., Art. 418, and Gibson, Calculus, x>a,ge 221.
132 DIFFERENTIAL CALCULUS. [Ch. VIII.
EXAMPLES.
1. Show that -^{Ax.'>y)=-^ (Ax^yO, in which A, m, and re
dxdy dydx m„ a2„
are constants. Then show that if a = ZAz'^y", " ^ = " ^ , and hence
ax dy dy dx
that the theorem in Note 3 is true for all algebraic functions.
2. In the following instances verify the fact that - — — = ;
u = sin (xy) :M = cos2-M = a;»;M = =^^ — — ; « = sec {ax -i-by) ; u = xlogy;
x' by — ax
u = a; sin 2/ + 2/ sin K ; u = y log (1 + xy) ; u = sin (x*) ; u = sin (x)'.
rl^U d^U d^u
3. In the following instances verify that -r^^-r- = . , ., ,= 3, -,,.0 =
(i) ?t = a tan-i f a j ; (ii) u = sin (xi/) + -^^^■
4. Show that -^ll— = ^ " , when u = cos (ax" + by").
dx^ dy'^ dy'^ ax''
6. If u = tan {y + ax) + {y- ax)^, show that sJi = a^^-
dx' dy^
6. If u = ^^, show that xf| + yJ!^ = 2|«, and that t,f«^ +
x + y 3x2 ^xay ax ar
j^_a%_ = 2^*.
ax aj/ a?/
7. If u = y/x^ + i/2, show that iS 5^' + 2 xy -^ + 3,2 3?^ = _ ? „
ax-^ dxdy dy^ 9
8. If u = (x2 + 2/2 + 02)-*, show that ^ + ^ + ^ = 0.
3x2 Qyi g^i
9. Show that a function of two independent variables has re + 1 partial
derivatives of order re.
81. Total rate of variation of a function of two or more variables.
N.B. Before reading this article and the next it is advisable to review
Arts. 25, 26.
Given that u —f{x, y), (1)
and that x and y vary independently of each other, it is required
to find the rate of variation of u in terms of the rates of variation
of X and y ■ i.e. to find — in terms of — and -i-
dt dt dt
In (1) let X and y receive increments -Aa; and Aj/ respectively, in
a time Ai say ; then u receives a corresponding increment Am, and
M + Am = f(x + Ax, y + A?/).
.-. Am =f{x + Ax, y + Ay) -f{x, y). (2)
81] TOTAL RATE OF VARIATION. 133
Hence, on introduction of -f{x,y + ^.y)+f{x,y + ^y) and
division by A<,
Am _ f{x + Ax, y + Ay) - /(a;, y + Ay) /(x, y + Ay) -f(x, y)
^t At "^ At
_/(x+Ax,y+Ay)-/(x,.v+Ay) Ax ^ /(g,y+Ay)-/(x,y) Ay
Ax At At/ 'At*
Now let At approach zero ; then Ax and Ay approach zero, and,
moreover (if a certain condition is satisfied),
^™^^ ^/(a; + Aa^, .V + Ay) -f(x, y + Ay) df(x, y)* 6«.
^^^ Ax ~ dx ' '-^-di'
and lim^^^^^LJLtMtzfi^ill^^.
Ay dy
Hence, du^dudx dud]i^ .„,
dt dx dt dy dt ^ '
In words : 7%e totoZ ?-ote o/ variation of a function of x and y is
equal to the partial x-derivative multiplied by the rate of variation of
X plus the partial y-derivative multiplied by the rate of variation ofy.
Similarly, if u =/(x, y, z),
du du dx , dudy , du dz ,,,
dt dx dt dy dt dz dt ^ ^
Results (3) and (4) can be extended to functions of any num-
ber of variables. (All derivatives herein are assumed to be con-
tinuous.)
Note 1. A function may remain constant while its variables change.
The total rate of variation of such a fanctiou is evidently zero. (See Art. 84.)
Note 2. Suppose that in {1) y is a function of x and that the derivative
of u with respect to 2 is required. This may be obtained either directly, as
(3) has been obtained, or by substituting x for t in (3) ; then
du_du,dudj^^ ,K\
dx dx dy dx ^ '
Result (5) may also be obtained by dividing both members of (3) by —
[Art. 34 (3)]. ''^
* For a discussion of the condition necessary and suflBcient for the passage
of the first member of this equation into the second, see W. B. Smith, Infini-
tesimal Analysis, Vol. I, Art. 205 (and also Arts. 206, 207).
134 DIFFERENTIAL CALCULUS. [Ch. VIII.
'6x
Note 3. In (5) -^ is tbe z-derivative of u when y is treated as a con-
stant, and — is tiie a>^erivative of u wlien y is treated as a function of x.
dz *
Here — is called the total a;-derivatiTe of m.
dx
Similarly the total a-derivatiye §^^du_^dudx_
dy dy dx dy
EXAMPLES.
1. Express result (5) in words.
2. Given z = 3x^ + iy% (1)
find — whena;=3, w=— 4, — = 2unitspersecond, and ^ = 3umtsper second.
dt " dt dt
On differentiation in (1), ^ = 6 a; ^ + 8 y^ = - 60.
dt dt dt
Geometrically this means that on the surface (1), which is an elliptic
paraboloid, if a point moves through the point (3, — 4, 91) in such a way
that the x and y coordinates of the moving point are there increasing at the
rates of 2 and 3 units per second respectively, then the 0-co6rdinate of the
moving point is, at the same place and moment, decreasing at the rate of
60 units per second.
N.B. Figures should be drawn for Ex. 2 and the following examples.
8. In Ex. 3 (n), Art. 79, find how the s-coordinate is changing when
the z-coordinate is increasing at the rate of 1 unit per second, and the
y-cobrdinate is deci'easing at the rate of 2 units per second.
4. In Ex. 3 (6), Art. 79, find how x is behaving when y is decreasing
at the rate of 2 units per second, and z is increasing at the rate of 3 units
per second.
82. Total differential. Let dx and dy be differentials of the x
and y in (1) Art. 81. They may be regarded as quantities such that
dx:dy = ^-M.
dt dt
Now let du be taken so that
As used in (1) ydx is called the partial x^ifferential ofu, —dy
is called the partial y-differential of n, and du is called the total
differential of u, and the complete differential of u.
82.] TOTAL DIFFERENTIAL. 135
Note 1. When y is a function of x, relation (1) follows directly from
Eq. (5), Art. 81, and definition (5), Art. 27.
Note 2. The partial differentials in (1) are also denoted by d^u and d^u,
and thus (1) may be written ^^ ^ ^^^ ^ ^^^
Note 3. In general the du in (1) is not exactly equal to the actual change
in u due to the changes dx and dy in x and y ; but the smaller dx and dy are
taken, the more nearly is du equal to the real change in u (see exercises below).
The differential du may be regarded as, and is very useful as, an approxima-
tion to the actual change in u. In some cases this change can be calculated
directly ; in others it can be found to as close an approximation as one pleases
by a series developed by means of the calculus. [See Chap. XVI., in par-
ticular. Art. 150, Eq. (10), and Art. 152, Note 5.]
EXAMPLES.
1. Express relation (1) in words.
2. Given u = Sx^ + 2y^, find du when x = 2, y = 5, dx= .01, and
dy = .02.
Here du = 6xdx + iydy =z .12 + .24 = .36.
The actual change in m is 3(2 • 01)= -|- 2(3 • 02)2 — (3 . 2^ -f 2 • 3^) = .3611.
3. As in Ex. 2 when dx = .001 and dy = .002. Also find the change in u.
4. Find the complete differential of each of the following functions :
(i) ta,n-^y.; (ii) y'; (iii) xn ; (iv) loga^; (v) M = a;iog».
6. Find dy when y = 8 cos A sin B, A = 40°, dA = 30', B = 65°,
dB = 20'.
Note 4. It may be said here that if LUGS (Fig. 38) be the surface
z =f(x, y), and if M be (a, y) and JV be (a; -|- dx, y + dy), and NQ be pro-
duced to meet in Qi the plane tangent to the surface at P, then the total
differential dz is equal to NQi — MP.
Ex. Prove this statement. (Suggestion : make a good figure.)
Similarly to (1), if u =f(x, y, z), and dx, dy, dz, be differentials
of X, y, z, respectively, and if du be taken so that
du = ^dx + ^dy + ^dz, (2)
dx dy dz ^ ^
du is called the total differential of u. Relation (2) is also written
du = d^u + djiU -f d,u.
Definitions (1) and (2) may be extended to functions of any
number of variables.
136 DIFFERENTIAL CALCULUS. [Ch. VIII.
6. Given u = x^ + y^ + 2z, find du when x = 2,y = 3, 2 = 4, di=.l,
dy = A, dz =— .3. Also find the actual change in u.
7. The numbers u, x, y, and z being as in Ex. 6, da; = .01, dy = M, and
dz = — .03, calculate the difference between du and the actual change in u.
8. Find du when u = a»'.
83. Approximate value of small errors. A practical application
of relations (1) and (2), Art. 82, may be made to the calculation
of approximate values of small errors. The ideas set forth in the
first part of Art. 65 may be applied to any number of variables.
If u = f{x,y,z,---),
and dx, dy, dz, •••, be regarded as errors in the assigned or measured
values of x, y, z, •••, then
du = —dx + — dy + —dz+--
ox ay dz
is, approximately, the value of the consequent error in the com-
puted value of u. Illustrations can be obtained by adapting
Exs. 2, 3, 5, 6, 7, Art. 82. In applying the calculus to the com-
putation of approximate values of errors it is usual to denote the
errors (or differences) in u, x, y, ••■, by Aw, Aa;, ^y, •••. rather than
by du, dx, dy, •••. Other notations are also used ; e.g. 8m, 8a;, hy, •••.
EXAMPLES.
1. In the cylinder in Ex. 3, Art. 65, give an approximate value of the
error in the computed volume due to errors AA in the height and Ar in
the radius.
Let F denote the volume. Then F= vr^h.
:. Ar = 2irrft ■ Ar + irr=- Aft.
The relative error is ^ = ?-^ + ^ .
V r h
2. Do as in Ex. 1 for a few concrete cases, and compare the above
approximate value of the error with the actual error. What is the difference
between the actual error in the volume in Ex. 1 and its approximate value
obtained by the method above ?
3. In the triangle in Ex. 7, Art. 66, let Aa, A6, AC, be small errors
made in the measurement of a, h, G : show that the approximate relative
error for the computed area .4 is — + — + cot C ■ AC.
a b
83, 84.] IMPLICIT FUNCTIONS. 137
Find, by the calculus, an approximate value of A^, given that a = 20 inches,
6 = 36 inches, C = 48° 30', Aa = .2 inch, A6 = . 1 inch, AC = 20'. How can
the actual error in the computed area be obtained ?
4. Show that for the area A of an ellipse when small errors are made
in the semiaxes a and 6, approximately — = — + _ .
A a b
In this general case, and in several concrete cases, compare the approxi-
mate error in the computed area with the actual error.
5. In the case described in Ex. 3 show that if Ac denote the consequent
error in the computed value of c, then, approximately,
Ac = cos 5 • Aa + cos .4 • A6 + a sin B ■ AC.
N.B. For remarks and examples on this topic see Lamb, Calculus,
pp. 138-142, Gibson, Calculus, pp. 258-260.
84. Differentiation of implicit functions, two variables. This
topic has been taken up in one way in Art. 56. Let the relation
connecting two variables x and y be in the implicit form
A^, y) = c, (1)
in which c denotes any constant, including zero. Let u denote the
function ^a;, y) ; then (1) may be written
u = c. (2)
Since u remains constant when x and y change, — = 0 ; i.e.
(Art. 81, Eq. 3, and Note 1)
dudx.dud/y_Q /Q\
5a; dt dy dt
dy du su
From (3), | = -|i whence [Art. 34, Eq. (3)], g = -g- (4)
dt dy ^y
Ex. 1. Express relation (4) in words.
Note. It should not be forgotten that the relation between the function
and the variable should be expressed in form (1) before (4) is applied.
Ex. 2. Do Exs. 13, 14, Art. 37, and exercises. Art. 56, by the method of
this article. Compare the methods of Arts. 37, 66, and 84.
138 DIFFERENTIAL CALCULUS. [Ch. VIII.
85< Condition that an expression of the form Pdx + Qdy be a total
difierential. This article may be regarded as supplementary to
Art. 82.
Suppose that /i (a, y) and /2(a;, y) are two arbitrarily chosen
functions : does a function exist which has /i (x, y) for its partial
x-derivative and f2(x, y) for its partial 2/-derivative ? A little
thought leads to the conclusion that in general such a function does
not exist. The condition that must be satisfied in order that there
may be such a function will now be found. Suppose that there is
such a function, and let it be denoted by u. Then, according to
the hypothesis,
-~=fi(^,y) and Y=f2{x,y). (1)
By Art. 80, Note 3, -^ = -^ • (2)
ay ax ax ay
Hence, from (1) and (2),
lf.{x,y)=lflx,y). (3)
Result (3) is directly applicable to the differential expression
Pdx+ Qdy on substituting P ioi fi(x, y) and Q lov f.i{x, y).
Otherwise : If Pdx 4- Qdy is a total differential, du say, then
^=Pand^=Q. (4)
dx ay '
Hence, from C2) and (4), ^ = ^. (5)
dy ox
When condition (5) is satisfied, Pdx + Qdy is also called an
exact differential.
Note 1. That this condition is not only necessary (as shown above), but
also sufficient, is shown in works on Difierential Equations. {E.g. see
Professor McMahon's proof in Murray, Diff. Eqs., Note E.)
Note 2. For the condition that an expression of the form Pdx + Qdy
+ Bdz (see Art. 82, Eq. 2) be a total differential, see works on Differential
Equations; e.g. Mun-ay, Diff. Eqs., Art. 102 and Art. 103, Note.
85,86.] PARTIAL DIFFERENTIALS. 139
Ex. 1. Apply test (5) in the following cases : (a) u = Sx'' + 2y^;
(6) « = tan 1^ ; (c) X dy + y dx ; (d) xdy — y dx.
Ex. 2. Illustrate by examples the phrase, " in general such a function
does not exist," which occurs in this article.
Note 3. On Evlcr's theorem on homogeneous equations and successive
total derivatives see Infin. Calculus, Arts. 87, 88.
86. Illustrations: partial differentials, total differentials, partial
derivatives. Illustrations of partial derivatives have already been
given in Art. 79, Note 3. Partial differentiation is often required
in engineering, physics, and other sciences. Accordingly, a stu-
dent should try to get a good understanding of the subject. The
interesting and peculiar relation h 0__^
shown in Illustration G impresses •b_' ' '
the necessity of having clearly in |
mind the conditions under which si
a partial derivative is obtained. |
Illustration A. Suppose that .j-
OABC is a rectangular plate ex- ' I*. x- *|«fci;>j
panding under the application of Fig. 34.
heat. Let x, y, denote its sides and u its area.
Then u = xy. (1)
From (1), on taking the partial derivatives (Art. 79),
'^=y, 1^ = ^. (2)
dx ay
.•.du = — dx + ~dy [Eq. (1), Art. 82]
dx dy
= ydx + xdy. (3)
In Pig. 34, AD, CH, denote dx, dy, the differentials of the
sides X, y ;
the partial a^differential of the area is ydx, i.e. BD ;
the partial y-differential of the area is xdy, i.e. HB;
the total differential of the area = ydx + xdy = BD + HB.
The difference in area = BD + HB + GE.
See Art. 82, Exs. 2-5.
140
DIFFERENTIAL CALCULUS.
[Ch. VIII.
87. lUustration B.
Note. In the case of a function y =/(x),
dy =f'(x)dx.
Draw the curve y =/(x), and at any point
P(x, ii) draw the tangent PT.
Draw PS parallel to OX.
Then
t3.nSPP=-^'-
dx
Let NM = dx, and draw the ordinate MQ meeting the tangent at R.
Then SR = PS tan SPB = /' (x) • dx.
Hence SR = dy,
and thus, as pointed out in Art. 27, Note 1, dy is the increment in the ordi-
nate drawn to the tangent corresponding to an increment dx in the abscissa.
At any point P(x, y, 2) on a
surface
z=f{x,y) (1)
let the tangent plane PSQR
be drawn. Draw PN parallel
to OZ meeting the a:t/-plane
in N{x, y). Now suppose
that X, y receive increments
dx and dy, as indicated in the
figure NLMG.
Draw LG, NM, meeting
in V. Through L, M, G, V,
draw lines parallel to 0.^and
meeting thg tangent plane in
B, Q, S, C, respectively.
(X -^ dx,y + dy'.
Fig. 36.
Through P pass the plane Pi^/iTfl" parallel to XOT.
By Art. 79, Note 3, tan FPE = — , tan HPS = — •
dy dx
Here
NP-.
GS--
LR--
■■z; MQ = MK+KQ
: GH+ HS = NP+ PH tan HPS
NP + KQ = z + KQ;
dz
dx
dz
-dx;
■- LF+ FR = NP+PFtss. FPU = z+ — dy.
dy
88.] PARTIAL DIFFERENTIALS. 141
Now CV=^^ + ^^; also CV = ^^±M.
2 2
.•.NP+MQ = GS + LR;
i.e. z + z + KQ = z + — dx + z+ — dy.
dx dy
.■.KQ = ^dx + ^dy.
dx dy
But, from (1) by definition, Art. 82,
dz = —^dx-\ dy.
ax By
.■.dz = KQ.
That is, if the surface z=/{x, y) be described, and a tan-
gent plane be drawn at a point {x, y, z), dz is the increment
in the length of the ordinate drawn to the tangent plane from the
a:y-plane when increments dx and dy are given to x and y.
88. Illustration C. In Fig. 37 let P . p,l
be the position of a moving point at any J?_Jl?_^rrj±?.
instant, and let its rectangular and polar y'\^^>^^^' ^"^
coordinates, chosen in the ordinary way, ^/^^^^ \y
be (x, y), (r, 6), respectively. The ^^P^ I
following relations hold : ^ ^^^ ^
x = rcos6, (1)
»-2 = .t2 + 2/2. (2)
When the point P moves, x, y, r, 6 (either severally or all),
change.
Note. Occasionally it is necessary to indicate the variable which is re-
garded as constant when a partial derivative is obtained. For this the fol-
lowing notation is sometimes employed :
The partial derivative of x with respect to r, d being kept constant, is
written (^] ;
the partial derivative of x with respect to r, y being kept constant, is
written (^] ■
From (1 ), by Art. 79, (—') = cos 61 = - • (3)
\drjg r
From (2), by Art. 79, f|^) =^- (4)
142 DIFFERENTIAL CALCULUS. [Ch. VIII.
Hence, from (3) and (4), in the case of a point moving in a
That is : the partial derivative of the abscissa with respect to the
distance when the argument * is kept constant,
is the reciprocal of
the partial derivative of the abscissa with respect to the distance
when the ordinate is kept constant.
This is a curious instance in which the partial derivative of one
variable with respect to a second under one condition, is the recip-
rocal of the partial derivative of the same variable with respect
to that second under another condition.
Geometric treatment of Illustration C. Relations (3) and (4),
from which (5) follows, can be shown geometrically.
lu Fig. 37 suppose that P moves to P,, say, 6 being kept constant.
Then r and x change by the amounts PPi and PN respectively.
Then in PP,N, cos 6 = ^^^ppJ^^ = (^)' (6)
Now suppose that P moves to Pj, say, y being kept constant.
Through Pj describe a circular arc about O, cutting OP in M.
Then r and x change by the amounts PM, PP-^, respectively.
Then, in a manner similar to that taken in Art. 63, it can be
shown that cos 6 = lim »„ ^( -— - j = ( -^ ) • (7)
' \PPJ \Bx),
Hence, from (6) and (7),
\prjg \dxjy l'dx\
{dr
EXAMPLES.
1. Given that (x, y), (r, 6) are the corresponding rectangular and polar
coordinates of a point P, show :
(a) {dx)^ + {dyy = (dr)2 + r\dBY,
(6) xdy — ydx = r^ de.
[ScGGESTiON. x = rcose, y =rsme ; see Art. 82, Eq. (1)].
2. Construct figures representing relations (a), (6), in Ex. 1.
of P.'
• 'The angle S' in the case of a point P{r, ff) is called 'the argument
P'
CHAPTER IX.
CHANGE OF VARIABLE.
If .B. If it is thought desirable, the study of this chapter may be post-
poned until some of the following chapters are read.
89. Change of variable. It is sometimes advisable to change
either, or both, of the variables in a derivative. If the relation
between the old and the new variables is known, the given
derivative can be expressed in terms of derivatives involving the
new variable, or variables. Arts. 91-93 are concerned with
showing how this may be done. In Art. 90 an expression for the
given derivative is found when the dependent and independent
variables are interchanged ; in Art. 91, when the dependent
variable is changed; in Art. 92, when the independent variable
is changed ; and in Art. 93, when both the dependent and the
independent variables are expressed in terras of a single new
variable. In Note 1, Art. 93, an example is worked in which the
dependent and the independent variables are both expressed in
terms of two new variables.
If .B. Principle (2) nf Art. 34 is repeatedly employed in Arts. 90-93.
90. Interchange of the dependent and independent variables. Let
y be the dependent and a; the independent variable. Also let y
be a continuous, and either an increasing or a decreasing, function
of X.
Then ^y=^0 when Aa; ^ 0, and — = - — (1)
Aa; Ax
Ay
Since y is continuous, Ay = 0 when Ax = 0 ; accordingly from (1),
^ = i ("if ^:^0\ (2)
dx dx \ dy J ' ^
dy
143
144
Again,
DIFFERENTXAL CALCULUS.
p.^^±m=±ff\.^ (Art.34)
ax- ax\dxj dy\dxj dx
[Ch, IX.
-A
dy
1_
dx
dy
dx
dy'
d^
df .
dxV
dy)
Ex. Express the third x-derivative of y in terms of ^-derivatives of x.
91. Change of the dependent variable. Let the dependent and
independent variables be denoted by y and x respectively. It is
required to express the successive derivatives of y with respect to
X, in terms of the derivatives of z with respect to x when
y = F{z).
dy_dy(h_p,,^^dz^
dx dz dx dx
d^y _ d fdy\ _ d
dx' dx\dxj dx
i^'(^)|]
dx
pi^ydh , dz d E„/.^ _ CT/»\ 'i'Z , dz dr-n,i/_s-i dz
=n^)i5+
da? dx dx
nz)=nzy^+
da? dx dz
[F'(z)] . ^
dx
^'dx" \dx)
Ex. 1. Given that y = F{z), show that
Ex. 2. Change the dependent variable from y to z in
given that 2/ = z^ + 2 z.
From (2),
Now
f^=2(z + l).
az
dy^dydz [Art. 34(1)] =2(z + l)^.
dx dz dx dx
A]-o
dx^ dx\dx) dxL dxJ ^ ^ dx^ \dx)
= ^(^\ = Ar2(z + l)^ + 2(!^Vl
dx^ dx dx^
(1)
(2)
(3)
(4)
(5)
(6)
81, 92.] CHANGE OF VARIABLE. 145
Substitution in (1) of the values of y and its derivatives, from C2), (4),
(5), (6), and reduction give
92. Change of the independent variable. Let the dependent and
indepeadent variables be denoted by y and x respectively. It is
required to express the successive derivatives of y with respect to
X, in terms of the derivatives of y with respect to z when
x=fiz).
Here — =f(z), and hence,
dz -^ ^ " ' dx f(z)
. dy _ dy dz _ 1 dy
' ' dx dz dx f\z) dz
^ _d_fdy\_d^fcly\ dz _ d /' 1 dyX dz
dar* dx\dx) dz\dxj dx dz\f'(z)dzj dx
(Vy f"(z) dy-\
f(z)
/\z) dz' U'(?)Y d^.
Ex. 1. Find 2J? when x =f(z).
t
Ex. 2. Change the independent variable from a; to « in
cPy 2x dy y _^
dx"^ ' l + X^dx ' (1 + k2)2 '
given that
X = tan «.
From (2),
^ = sec2«; whence ^ = ^-
dt dx sec^t
§l_dy.dt [Art. 34, (1)]= \ '{y.
dx dt dx eec^tdC
d^y d fdy\ d ldy\ dt _ d 1 1 dy\ dt
da;2 dx\dxj dt[dxj dx dl\sec^tdt) dx
1 1 d'y 2 tan tdy\ 1
{sec^ tdt^ sec^t dtjsec'^t
(1)
(2)
(3)
(4)
(5)
Substitution in (1) of the values of x, ^, ^ from (2), (4), (5), and
reduction give d^ "*" ^ ~ *^'
146 DIFFERENTIAL CALCULUS. [Ch. IX.
93. Dependent and independent variables both expressed in terms
of a single variable.
Let y = ^(f) and a; =/(«).
Then dy^d^ dx .^ g^ (3)] = #> •
dx dt dt'- ' ^ ^-' f(t)
d^^±fdy\^d^/dy\ dt ^d r<l>'(t)l _ 1
di?" dx\dx) dt\dx) ' dx dt[_f'{t)j ' f'(t)
_f'(t)r(t)-<t>'(t)f"it)
Similarly for higher derivatives.
See Art. 71, which is practically the same as this, and its
Exs. 1, 2.
EXAMPLES.
1. In the above case find J-
2. Given that x= a{9 — sine) and y = a(l — cos 5), calculate
['+(I)"]'-S- (*">=- ''.'^"■«»-)
3. Gi7en that x = a cos $ and y = a sin 9, calculate the same function as
in Ex. 2. What curve is denoted by these equations ?
4. Given that x = a cos ff and y = b sin e, calculate the same function as
in Ex. 2. What curve is denoted by these equations ?
Note 1. Both dependent and independent variables expressed in terms of
two new variables. Following is an example of this.
Ex. Given the transformation from rectangular to polar coordinates, viz.
X = rcos$, y= rsiae, (1)
express ^ and ^ in terms of r, 0, and the derivatives of r with respect to 8.
dx d3fl
From (1), — = cose — -r sin «, ^ = sin 9 — + rcosfl-
^ ^ de de de de
•••i=(Mi'--3^.-'^-(^))=
sine— + rcosfl
de
cose— -rsine
de
r2 + 2(^Y-r^
dx^-dx \dx) - de [dx) ■ dx - f^ ^dr _ ^ ^i^ ,y
S3.] CHANGE OF VARIABLE. 147
Note 2. For more complex cases of change of the variables in a deriva-
tive, see other text^books.
Note 3. References for collateral reading. Williamson, Diff. Cal,
Chap. XXII. ; McMahon and Snyder, Diff. Cal., Chap. XI. ; Edwards,
Treatise on Diff. Cal, Chap. XIX.; Gibson, Calculus, §§ 98, 99.
EXAMPLES.
N. B. In working these examples it is much better not to use the results
or formulas derived in Arts. 90-93, but to employ the method by which these
results have been obtained.
1. Change the independent variable from X to y in : (i) — "^2x1 — ] =0;
dx^ \dxJ
^ ^ \dx^l dxdx^ dx^\dx.j
2. In ^ = 1 4- -^ — i-^ ( — ) , change the dependent variable from y to
z, given that y = tan z.
3. Change the independent variable under the following conditions :
(i) a;2^ + a;^' + M =0, y = logx; (ii) (1 - a:2)f| - a;f^ + 5- = 0, X = cosj;
dx^ dx dx^ dx
(m)(l-x2)g-x| = 0,x = cos.;(iv)x^g + 2x| + g, = 0,x. = l;
(V)x3f^3+3x»f| + xf^ + y = 0,. = logx;(vi)x4^ + 6x»^ + 9x2f|
dx^ dx^ dx dui^ dx' dx^
+ 3x-^ + y = \ogx, X = e'-
ax
4. Find ^ and ^ when : (i) x = a(cos t + tBiat),y = a(8in « - ( cos «) ;
dx dx'
(ii) x = cot t,y = sin' (.
6. If X ^ - '^(^Y+ ^ = 0, and x = ye; show that j, f? + ^ = 0.
dx' y\dx/ dx dy^ dy
CHAPTER X.
CONCAVITY AND CONVEXITY. CONTACT AND CURVA-
TURE. EVOLUTES AND INVOLUTES.
94. Concavity and convexity of curves : rectangular coordinates.
Definition. At a point on a curve the curve is said to be con-
cave to a line {or to a point off the curve) when an infinitesimal arc
containing the point lies between the tangent at the point and the
given line (or point 'ofE the curve). If the tangent lies between
the line (or point) and the infinitesimal arc, the arc there is said
to be convex to the line (or point).
Thus, in Fig. 50 a, at Pthe curve ilfiVis concave to the line OX, and con-
cave to the point A ; in Fig. 50 6, at Pi the curve MN is convex to the line
OX, and convex to the point A. The arc on one side of a point of inflexion
is concave to a given line (or point), and the arc on the other side of the
point of inflexion is convex to this line (or point) (see Figs. 31 a, 6).
The curves passing through P and R have the concavity towards
the a>axis, and the curves passing through Q and S are convex
to the a^axis. At P y is positive;
and — ^ is negative, for — decreases
dar dx
as a point moves along the curve
towards the right through P. At M
y is negative; and — ^ is positive,
J dar
Fig. as ' for — increases as a point moves
dx
along the curve towards the right through R. Hence, at points
tt'here a curve is concave to the x-axis y -=-^ is negative. A similar
examination of the curves passing through Q and S shows that at
points ivhere a curve is convex to the x-axis y ^-^ is positive.
148
94, 95.]
CONTACT.
149
Ex. 1. Prove the theorem last stated.
Ex. 2. Test or verify the above theorems and Note 1 in the case of a num-
ber of the curves in the preceding chapters.
Note 1. The curves passing through P and S are concave downwards,
d^y
and here — j is negative. The curves passing through .B and Q are concave
upwards, and here -t-| is positive.
Note 2. A point where a curve stops bending in one direction and begins
to bend in the opposite direction as at L, A, D, H, G, P, Figs. 31 a, b, 32,
is called a point of inflexion.
Note 3. A curve /(r, 6) = 0 is concave or convex to the pole at the point
(r, $) according as u-\ — - is positive or negative, u denoting -. (See
d8'^ r
McMahon and Snyder, Diff. Cal, Art. 144.)
95. Order of contact. If two curves, y = 4,{x) and y=f(x),
intersect at a point at which a; = a, as in Fig. 39 a, then <f>(a) =/(a)
and <^'(a) ^^fla.)- If <^(a) =/(«) and <^'(a) =/'(«), then the curves
touch as in Fig. 39 b, and they are said to have contact of the first
order, provided that <^"(a) =/=/"(a). If </.(a) =f{a), i,'{a) =/'(o),
and <l>"(a) =f"{a), but <f,"'(a) ^f"(a), then the curves are said to
»-/(x)
»-«(*)
Fig. 39 a.
Fig. 39 6.
Fio. 39 c.
have contact of the second order, as in Fig. 39 c. And, in general,
if <f)(a) = f(a) and the respective successive derivatives of ^(x)
and f(x) up to and including the nth, but not including the
(n + l)th, are equal for x = a, then the curves are said to have con-
tact of the nth order. Hence, in order to find the order of contact
of two curves compare the respective successive derivatives of y
for the two curves at the points through which both curves pass.
150 DIFFERENTIAL CALCULUS. [Cii. X.
Note 1. Another way of regarding contact is the following. In analytic
geometry the tangent at P (Fig. 40 a) is defined as the limiting position
which the secant PQ takes when PQ revolves about P until the point of
intersection Q coincides with P. The line then has contact of the first order
with the curve. This notion of points of intersection of a line and a curve
becoming coincident will now be extended to curves in general. Two curves,
Fig. 40 a. Fio. 40 b.
Ci and Cj (Fig. 40 6) , are said to intersect when they have a point, as P, in
common. They are said to have contact of the first order at P when the
curves (see Fig. 40 c) have been modified in such a way that a second point
of intersection Q moves into coincidence with P. (The value of -==- at P is
then the same for both cufves, according to the definition of a tangent a.s
given above. ) The curves are said to have contact of the second order at P
when the curves have been further modified in such a way that a third point
of intersection S moves into coincidence with P and Q (see Fig. 40 d). (The
value of ■gzi^j' '■■^- j^i i* then the same for both curves at P.) And, in
general, the curves are said to have contact of the nth order at a point P when
n + 1 of their points of intersection have moved into coincidence with P.
(At P the respective derivatives of y up to the nth are then the same for both
curves.) See Echols, Calculus, Art. 98.
Note 2. In general a straight line cannot have contact of an order higher
than the first with a curve. For in order that a line have contact of the first
order with a curve at a given point, the ordinates of the line and the curve
must be equal there, and likewise their slopes ; thus two equations must be
satisfied. These equations suffice to determine the two arbitrary constants
appearing in the equation of a straight line. For example, if the line
y = mx + b has contact of the first order with the curve y = f(x) at the point
for which x = a, the following two equations are satisfied, viz. ;
/(a) = ma + b, f'(a) = m ;
from these equations m and 6 can be found.
This line and curve have contact of the second order in the particular (and
exceptional) case in which /"(a) =0; consequently (Art. 78), if there is a
95.] CONTACT. 161
point of inflexion on the curve y =/(z) where x = a, the tangent there has
contact of the second order.
The theorem at the beginning of this note is also evident from geometrical
considerations. Since, in general, a line can be passed through only two
arbitrarily chosen points of a curve, it is to be expected from Note 1 that in
general a line and a curve can have contact of the first order only.
Note 3. In general, a circle cannot have contact of an order higher than
the second with a curve. For in order that a circle have contact of the second
order with a curve at a given point, three equations must be satisfied, and
these equations just suffice to determine the three arbitrary constants that
appear in the general equation of a circle [see Eq. (2), Art. 96]. This
theorem is also evident from Note 1 and the fact that, in general, a circle can
be passed through only three arbitrarily chosen points of a curve. (In a few
very special instances a circle has contact of the third order with a curve.
See Ex. 4, Art. 101 )
Note 4. It js shown in Art. 166 that when two curves have contact of an
odd order, they do not cross "ach other at the point of contact ; but when they
have contact of an even order, they do cross there. Illustrations : the tangent
at an ordinary point on a curve, as shown in Figs. 15, 17 ; the tangent at a
point of inflexion, as in Figs. 26 a, 6, .31, .32 ; an ellipse and circles having
contact of second order therewith (see Ex. 4, Art. 101). This theorem may
also be deduced from geometry and the definitions given in Note 1.
N.B. As far as possible make good figures showing the curves, lines, and
points mentioned in the exercises in this chapter.
EXAMPLES.
1. Find the place and order of contact of (1) the curves y = 7? and
y = 6 a;2 — 9 X + 4 ; (2) the curves y = x' and i/ = 6 x^ _ 12 x + 8.
2. Determine the parabola which has its axis parallel to the y-axis, passes
through the point (0, 3), and has contact of the first order with the parab-
ola y = 2 x2 at the point (1, 2).
3. What must be the value of a in order that the parabola y = z + 1
+ o(x— 1)2 may have contact of the second order with the hyperbola
xy = 3 X - 1 ?
4. Find the parabola whose axis is parallel to the y-axis, and which has
contact of the second order with the cubical parabola y = x^ at the point
a !)•
6. Determine the parabola which has its axis parallel to the y-axis and has
contact of the second order with the hyperbola xj/ = 1 at the point (1, 1).
152
DIFFERENTIAL CALCULUS.
[Ch. X.
96. Osculating circle. It was pointed out in Art. 95, Note 3,
that contact of the second order is, in general, the closest contact
that a circle can have with a
cvLTve. A circle having contact
of the second order with a curve
at a point is called the osculating
circle at that point.
In Fig. 41 PT is tangent to the
curve C at P. Every circle which
passes through P and has its cen-
tre in the normal NM touches C
at P. One of these circles has
contact of the second order with
C at P; let this circle be denoted
by K. All the other circles, infinite in number, in general have
contact of the first order only.
Osculating circle: rectangular coordinates. The radius and the
centre of the osculating circle at any point P{x, y) on the curve
Fio. 41.
y=m
(1)
will now be obtained. Denote the centre and radius by (a, h)
and r. Then the equation of the osculating circle at the point
(x, y) is
{X-ay+{Y-hy = 7^.
(2)
For the moment, for the sake of distinction, x and y are used
to denote the coordinates of a point on the curve, and X and Y
are used to denote the coordinates of a point on the circle. Then
at the point 'where the circle and the curve have contact of the
second order „ „„ „
dY^^dy d^Y ^cPy
dX dx dX'~dx''
X=x, Y=y,
(3)
From (2), on differentiating twice in succession,
(4)
(5)
CONTACT.
86, 97.]
and X-a=^l + ^^^^--— •
L \dXJ ]dX dX*
Accordingly, from (3), (2), (6), (7),
and from (3), (6), (7),
'-"—W-'t-'"-^*-
dx'i
dx2
153
(6)
(7)
(8)
(9)
Note. For the osculating circle, polar coordinates being uaed, see Art.
102, Note 2.
Ex. 1. Determine the radius and the centre of the osculating circle for
each of the curves in Ex. 1 (1), Art. 9o, at their point of contact.
Ex. 2. Do as in Ex. 1 for the curves Ex. 1 (2), Art. 95.
97. The notion of curvature. Let the curves A, B, C, D have
a common tangent PT at P. At the point P the curve A, to use
the popular phrase, bends or curves more than the curves B, C,
and D ; and D bends or curves less than the curves A, B, and C
These four curves evidently differ in the rate at
which they bend, or turn away from the straight
line PT, at P. These ideas are sometimes ex-
pressed by saying that these curves differ in
curvature at P, and that there A has the greatest
and D the least curvature. In the case of two
circles, say one with a radius of an inch and the
other with a radius of a million miles, it is cus-
tomary to say that the second circle has a small
curvature, and that the first has a large curvature in comparison
with the second. An inspection of a figure consisting of a circle
and some of its tangents gives the impression that what is popu-
larly called the curvature is the same at all points of that circle.
Fio. 42.
154
DIFFERENTIAL CALCULUS.
[Cii. X-
On the other hand, an inspection of an elongated ellipse gives
the impression that the curvature is not the same at all points
of that ellipse, although at two particular points, or at four
particular points, it may be the same. Curvature will now be
given a precise mathematical definition and its measurement
will be explained.
Ex. 1. Draw an ellipse, and find by inspection the points where the curva-
ture is greatest and where it is least. Show how to obtain sets of four points
on the ellipse which have the same curvature.
Ex. 2. Discuss a parabola and an hyperbola in the manner of Ex. 1.
98. Total curvature. Average curvature. Curvature at a point.
At -4j the curve C has the direction A^T^, which makes the angle
<^i with the a^axis ; at A2 the
curve has the direction A2T2,
which makes an angle <^2 with the
aj-axis. The difference between
these directions represents the
angle by which the curve has
changed its direction from the
direction of the line A^T^ in
the interval of arc from A^ to
A^. Tliis difference, namely,
T^RT^ or <^2 — <^i, is called the
total mli~vature of the arc A^A^.
The average cuj-vature for this arc is
(<^2 — <^i) -^ length of arc AiA^.
(Here the angle is measured in radians.')
Accordingly, if (Fig. 44) A<^ is the angle between the tangents
at A and B, then A<f> is the total curva- y
ture of the arc AB; if As is the length
of the arc AB, then — = is the average
curvature of that arc. Now let B
approach A. The arc As and the angle
Ai^ then become infinitesimal ; and,
finally, when B reaches A, ^ has the
^« Fig. 44.
Fig. 43.
98, 100.] CURVATURE. 155
limiting value -^. The limit, ,^-^ at any point on a curve, i.e.
di> As
,* there, is called the curvature of the curve at that point. (The
phrase " curvature of a curve " means the curvature of the curve
at a particular- point.) In all curves, with the exception of
straight lines and circles, the curvature, in general, varies from
point to point.
99. The curvature of a circle. Let A and B be two points on
a circle having its centre at 0. In
Fig. 45 the angle between the direc-
tions of the tangents AT^ and BT^ is
A<^, say. Let As denote the length of
the arc AB. Then AOB= T^RT2=C^^.
Hence, by trigonometry, As = >-A</>.
From this,
As~r' '^^^^'^^ ds - r ^^ F'°- ^■
That is, the curvature of a circle is constant and is the reciprocal
of (the measure of) the radius.
Note. When the radius increases beyond all bounds, the curvature
approaches zero, and the circle approaches a straight line as its limiting
position. When the radius decreases, the curvature increases ; as the radius
approaches zero and the circle thus shrinks towards a point, the curvature
approaches an infinitely great value.
It is shown in Ex. 5, Art. 227, that all curves of constant curvature are
circles.
Ex. Compare the curvatures of circles of radii 2 inches, 2 feet, 5 yards,
2 miles, 10 miles, 100 miles, and 1,000,000 miles.
100. To find the curvature at any point of a curve : rectangular
coordinates. Let the curve in Fig. 44 be y=f{x), and let its
curvature at any point A(x, y) be required. Let k denote the
curvature at A, and <^ denote tlie angle which the tangent at A
makes with the a^axis. Take an arc AB and denote its length
by As, and denote the angle between the tangents at A and B by
A^. Then, by the definition in Art. 98,
K ^ — — at -o-
as
156 DIFFERENTIAL CALCULUS. [Ch. X.
Now (Art. 59), tan <^ = ^. .-.4, = tan"'^-
A; =
da; ' ' dx
d^
d<l> d f _idy\ d ( -\dy\ dx dx' _ ds
Ytaa-^VI-rtan-'^
i\ dxj dx\ c
ds ds \ dxJ dx \ dxj ds ^ (djt^ ' ^^
\dx
.-. [Art, 67 c(2)], fc = ^ 5. (1)
This, by (1) Art. 99 and (8) Art. 96, is the same as the curva-
ture of the osculating circle.
In order to find the curvature at a definite point (a;,, yi) it is
only necessary to substitute the coordinates a;,, y,, in the general
result (1).
Ex. 1. Compute and compare the curvatures of the two curves in Ex. 1 (1),
Art. 95, at their point of contact.
Ex. 2. Find the curvature of the curve y = x^ — 2x'' + 1 x at the origm.
Determine the radius and centre of its osculating circle at that point.
101. The circle of curvature at any point on a curve : rectangular
coordinates. The circle of curvature at a point on a curve is the
circle which passes through the point and has the same tangent
and the same curvature as the curve has there. The radius of
this circle is called the radius of curvature at the point, and the
centre of the circle is called the centre of curvature for the point.
The radius of curvature. Let It denote the radius of curvature
and (o, ^) denote the centre of curvature for any point (x, y) on
the curve y =f(x). Then it follows from Art. 99, and Art. 100,
Eq. 1, that
b<m
(That is, R is the value of this expression at that point.)
Note 1. There is an infinite number of circles that can pass through a
given point on a curve and have the same tangent as the curve has there but
not the same curvature, and there is an infinite number of circles that can
101.]
CURVATURE.
157
pass through this point and have the same curvature but not the same tangent
as the curve has there ; but there is only one circle passing through the point
that has there both the same tangent and the same curvature as the curve.
Ex. 1. Illustrate Note 1 by figures.
The centre of curvature. Since at any point on a curve the circle
of curvature and the curve have the same tangent and curvature,
it follows that — and — ^ are respectively the same for the circle
and the curve at that point. Accordingly (Art. 95, Note 3) the
circle of curvature has, in general,* contact of the second order
with the curve, and thus (Art. 9G) coincides with the osculating
circle passing through the point. Accordingly (Art. 96, Eq. 9)
1 +
(djf_y
\dx)
dx'
fi = y +
d^y
dx^
(2)
Note 2. The coordinates of the centre of curvature may also be obtained
in the following manner.
Let C be the centre of the circle of cur-
vature of the curve PL at P, and let the
tangent PT make the angle <t> with the
i-axis. Draw the ordinates PM and CiV,
and draw PB parallel to OX. Let B
denote the radius of curvature. Then
NCP = <t>, and tan 0 = ^-
dx
In Fig. 88
a= 0N= OM-BP = x-Rsin(f>
y
C«r.p) L
/
B
■TV
r
=^
PU.U)
0
I
f/T A
I
X
FiQ. 46.
[■^(1)']
s
dx
-(:
[•-©']
i
dx
1 +
Also, p = NC=MP-\- BC = y + Rcosit> =
The results for Fig. 88 are true for all figures.
+ -
(!)■
dhi
dx'
(3)
(4)
• For an exception see the circles of curvature at the ends of the axes of
an ellipse. (See Ex. 4 following.)
158 DIFFERENTIAL CALCULUS. [Ch. X.
Ex. 2. Verify the last statement by drawing the radii of curvature at points
on each side of points of maximum and minimum in the curves in Fig. 80
and carefully noting the algebraic signs of ^ and ^ at these points.
dx d'x
Note 3. A glance at Fig. 38 shows that at P and H the normal (Art. 62)
and the radius of curvature have the same direction, and at Q and S they
have opposite directions. Hence (see Art. 94) the normal and the radius of
curvature at a point on a curve have the same or opposite directions accord-
ing as y — - there is respectively negative or positive.
.dx^
Note 4. At a point of inflexion, according to Art. 78, and Art. 100, Eq. (1),
the curvature is zero.
Note 5. A centre of curvature is the limiting position of the intersection
of tuio infinitely near normals to the curve. For a consideration of this im-
portant geometrical fact, see Williamson, Diff. Cal. (7th ed.). Art. 229;
Lamb, Calculus, Art. 150 ; Gibson, Calculus, Art. 141.
EXAMPLES.
3. Find the radius of curvature and the centre of curvature at any point
on the parabola y'^ — ipx. What are they for the vertex ?
Apply the general results just obtained to particular cases, by giving p par-
ticular values, e.g. 1, 2, etc., and taking particular points on the curves,
and make the cori'esponding figures.
N.B. As in Ex. H, apply the general results obtained in the following
examples to particular cases.
4. As in Ex. 3 for the ellipse b'hc^ + a^y^ = aV. Find the radii of cur-
vature at the ends of the axes. Show that this radius at an extremity of
the major axis is equal to half the latus rectum. Illustrate Note 4, Art. 95,
by drawing an ellipse and the circles of curvature at various points on it.
Show that the circles of curvature for an ellipse, at the ends of the axes, have
contact of the third order with the ellipse.
5. Find the radius and centre of curvature at any point of each of the fol-
lowing curves : (1) The hyperbola hV - ah/'^ = d^h'^. (2) The hyperbola
* -- 2 2 2
xy = aK (3) The catenary j/ = ^ («« -|- e »). (4) The astroid x^ + y^= a*.
(5) The astroid x = a<ios^e, y = asm^e. (6) The semi-cubical parabola
x» = ay'^. (7) The curve xh/ = a\x - y) where x = a. (8) The cycloid
X = a(8 — smB), y = a(l — cos B). In this cycloid show that the length of
the radius of curvature at any point is twice the length of the normal.
6. Find the radius of curvature at any point of each of the following
curves : (1) The parabola Vi -I- Vy = Vu. In this curve show that a + p —
3(x -I- y). (2) The cubical parabola a'^y = i-\ (3) The catenary of uniform
102.] CURVATURE. 159
strength y = clog sec (*). (4) The witch xy" = a''(a - x) at the vertex.
(5) The parabola x = a cot^ ij/, y = 2acot <l/. (6) The ellipse x = a cos <f>,
y = 6 sin 4>. (7) The hyperbola a; = a sec 0, 2^ = 6 tan 0. (8) The catenary
a; = a log (sec e + ia.n0), y = aaece.
102. The radius of curvature : polar coordinates. This can be deduced
(a) directly from the definition of curvature (Art. 98) and the definition of
radius of curvature (Art. 101) ; and (6) from form (1), Art. 101, by the
usual substitution for transformation of coordinates, namely, x = rcose,
y = rsinS.
(a) By Art. 63 (2), 0 = e + ^.
Now A = ^(Art.98)=^.^ = fl+^
ds^ ^ de ds \ d6
Also, taji<l^ = r~ (Art 63). .-. ^ = tan-»
r
dr
Hence H -'.- ^ JfJ\.-
\de) de^
^de
)["<i)'f-
[Art. 67 d, Eq. (3).]
d'f' _ \d0J dfi
(1)
(6) The deduction of (1) from (1), Art. 101, by the transformation of coor-
dinates is left as an exercise for the student.
Note 1. On the substitution of u for - in (1), i? =
["■^(l)t
•■(•+S)
Note 2. Since the osculating circle and the circle of curvature coincide,
the forms just found for B give the radius of the osculating circle.
Note 3. For other expressions for B see Todhunter, Diff. Cal., Art. 321,
and Ex. 4, page 352 ; Williamson, Diff. Cal. (7th ed.). Art. 236. Also see
F. G. Taylor, Calculus, Arts. 288-290.
EXAMPLES.
1. Find the radius of curvature at any point of each of the following
curves: (1) The circlesr = a and r= 2 6cosfl. (2) The parabola r(l + cos«)
= 2 a. (3) The cardioid r = a(l + cos ff). (4) The equilateral hyperbola
r2 cos 2 e = a'^. (5) The lemniscate r^ = a^ cos 2 e. (6) The logarithmic
spiral r = e"*. (7) The spiral of Archimedes r = o0. (8) The general
spiral r = aip".
2. Derive the expression for B in Note 1.
160
DIFFERENTIAL CALCULUS.
[Ch. X.
103. Evolute of a curve. Corresponding to each point on a
given curve there is a centre of curvature. The locus of the
centres of curvature for all the points on the curve, is called
ilie evolute of the curve.
Thus, if AA^ be the
given curve and Ci,
t'2i ('Si
be respec-
and let A{x, y) be any point on it.
ture for the point A, and denote
Eq. (2)],
1 +
tively the centres of
curvature for any
points Ai, A.2, A3, •■•,
on the given curve,
the curve C1C2C3 is
the evolute of AAy
To find the equation
of the evolute of the
curve. Let the equar
tion of the given
curve be
y=A^), (1)
Let C be the centre of eurva-
C by (a, P). Then [Art. 101,
dx
y-P = -
dPy
dy
dx'
(2)
(3)
On the elimination of x and y from equations (1), (2), (3), there
will appear an equation which is satisfied by a and j8, the coordi-
nates of the point C. But A is any point on the given curve, and,
accordingly, C is any of the centres of curvature for the points on
AAi. Accordingly, the equation found as indicated is the equa-
tion of the evolute.
Note. The algebraic process of eliminating x and y from (1), (2), and
(3) depends on the form of these equations.
103, 104.J CIRCLE OF CURVATURE. 161
EXAMPLES.
1. Find thf- fvolute of the parabola
2^ = 4p2. (Fig. 48 a.) (i)
Here by Ex. 3, Art. 101, a = -2p + 3x; (2)
The elimination of z and y between equations (1), (2), (3), gives the
equation of the evolute, viz. the semi-cubical parabola
4(a-2p)3 = 27p^;
i.e. on using the ordinary notation for the coordinates,
i(x-2py = 2Tpy\
2. Find the evolute of the ellipse 6V ^ 32^2 _ (,2j^_ (pig_ ^g j^ ^i\
Here, by Ex. 4, Art. 101, a = (^^:-^\ifi, (2)
The elimination of x and y between equations (1), (2), (3), gives the equa-
tion of the evolute, viz. :
(aa)i + (6/S)* = (o2 - 62)i
i.e. on using the ordinary notation for coordinates,
(az)f+ (62/)^ = (a2 - 62)i
8. Find the evolute of the following curves : (1) the hyperbola b'h:'^ — ah/^
= 0^6^. (2) The equilateral hyperbola xy = a'. (3) The four-cusped hypo-
cycloid xi + yi = o*.
4. Find both geometrically and analytically the evolute of a circle.
6. Show that the evolute of a complete arch of a cycloid consists of the
halves of an equal cycloid. [Suggestion : see Ex. 5 (8), Art. 101.]
104. Properties of the evolute. The two most important proper-
tie.s of the evolute of a curve are the following :
(a) The normal at any point of a given cwve is a tangent to the
evolute, and any tangent to the evolute is a normal to the given curve.
(6) TTte length of an arc of an evolute, provided that the curvar
ture varies continuously from point to point along this arc, is
equal to the difference between the lengths of the two radii of curvature
drawn from the given curve to the extremities of the arc.
162
DIFFERENTIAL CALCULUS.
[Ch. X.
Proof of (a). Let AA^ (Fig. 47) be the given curve, and let its
equation be y =f(x), and let CC^ be its evolute. Let C(a, /3) be
the centre of curvature for any point A{x, y).
dy
dx
The slope of the given curve at A is "^, and the slope of the
d§
da
tion and reduction,
evolute at C is ^- From Equations (2), Art. 101, on differentia-
da
dp _ dx\darj
dx
H%
n^y
d^
'djyy
d^j
da
dx
_dy(^dy^(Py
dx i dx\dx'
'■) \ ^\dx) \do^]
d^J
From (1) and (2), and Art. 34 (3),
do, \ dx dx
dx
dy
(1)
(2)
(3)
dx .
But — — is the slope of the normal at A{x, y). Hence, the
normal at A and the tangent to the evolute at C coincide.
/A X
Fig. 48 a.
Fig. 48 6.
Note 1. Thus, in Fig. 47, AC is the radius of curvature for A on AAj,
AC is normal to AAi at A, and AC touches the evolute COi at C. In Figs.
48 a, 48 6, PiCi, P2C2, are normal to the parabola and tangent to its evolute ;
PC is normal to the ellipse and tangent to its evolute.
104.] THE EVOLUTE. 163
Note 2. On account of property (a) the evolute is sometimes defined as
the envelope (see Art. 120) of the normals of the curve. See Art. 123 (Ex. 2
and Notes 4, 5) and Art. 124, Ex. 1. Also see Echols, Calculus, Arts.
106-108.
Proof of (6). In Fig. 47 AA-^ is the given curve, CC^ is its evo-
lute, and C'(«, P) is the centre of curvature corresponding to the
point A{x, y).
Let ds denote the differential of the arc of the evolute CCi-
Then, by Eq. (5), Art. 67 (c),
'^aRI
dp
dP; (4)
ds ^ U.fdaY d§
" dx V \dpj dx '
from (1) and (3)
(5)
, sdlfd^yf-t.fdy
da-''
\dxj ■ fdY\
(6)
\dx-J
Differentiation of B in Art. 101, Eq. (1), gives
dR
-— - = the second member in (6).
da; '
Hence ^s^dR^
dx dx '
This means that at any point on the evolute CCi the rate of
change of the length of the arc with respect to the abscissa x, is
the same as the rate of change of the length of the radius of cur-
vature at the corresponding point on AA^ (Art. 26). It follows
that on starting from two corresponding points (viz. a point on
the curve and its centre of curvature) these lengths change by
the same amount. Accordingly,
the length of an arc of the evolute is equal to the difference between
the lengths of the radii of curvature which touch this arc at its
extremities; or, in other words, the difference between the radii
of curvature at two points on a curve is equal to the arc of the
evolute intercepted between the centres of curvature of these points.
164 DIFFERENTIAL CALCULUS. [Ch. X.
Thus in Fig. 47, arc CCi = A^d - AC; arc dC^ = A^Cs - AJd.
Note 1. Property (6) is also shown in Art. 214.
Note 2. Property (6) should not be applied thoughtlessly ; for in certain
circumstances, for either the curve or its evolute, the property does not hold.
Thus in the case of the curve ay''' = x', the theorem is true only for points on
the curve which are either both above the avaxis or both below. Again, in
Fig. 48 a the theorem is true only for arcs of the evolute which are altogether
above or altogether below the i-axis. For instance, if (Fig. 48 a) F\C\ =
P2C2, a reckless application of the theorem obtains the result
arc C1/SC2 = P2C2 - PiCx = 0,
which is obviously absurd.
Note 3. See Echols, Calculus, Art. 170 and Chap. XIV.
Ex. 1. Show that the total length of the evolute of the ellipse whose
semi-axes are a and 6, is — ^ ■
ab
Ex. 2. Show that the length of the evolute of the parabola j/^ = 4px that
is intercepted by the parabola (i.e. 2 SB, Fig. 48 a) is 4p (SVS - 1).
105. Involutes of a curve. In Fig. 47 the curve CCj is the
evolute of the curve AA^. Suppose that a string is stretched
tightly along the curve CCi and held taut in the position
Z1C1C2C3C, the portion LCi thus being tangent to the evolute
at C,. Now, a point A^ being taken in the string, let it be
unwound from C\C. It follows from properties (a) and (6),
Art. 104, that, as the string is unwound from the evolute CiC, Ai
will describe the curve AjA. It is on account of this property
that CCi is called the evolute of AAi. On the other hand, AAi
is called an involute of CC,. " An involute," because CCi has an
infinitely great number of involutes. For, when the string is
unwound from the evolute CiC an involute will be traced out
by each point like A^ taken in the string LA1C1C2C3. These
involutes are parallel curves * ; for (1) they have the same normals,
namely, the tangents of their common evolute, and (2) the dis-
tance between any two of them along these normals is constant,
* Two curves are said to be parallel when they have common normals
always differing in length by the same amount.
105.] SXAMPLES. 165
namely, the distance between the two points originally taken on
the string that is being unwound. Figure 47 shows three involutes
of CC^.
EXAMPLES.
1. Construct several involutes of the evolute of the parabola whose latus
rectum is 8 (besides the parabola itself).
2. Construct several involutes of the evolute of the ellipse whose axes
are 9 and 25.
3. Given a cycloid, construct the involute that is traced out by the point
at the vertex in the course of "the unwinding."
4. Given a circle, construct the involute that is traced out by any point
on the circle in the course of "the unwinding." (In the case of a circle
all such involutes are identically equal. Accordingly, such an involute is
usually termed "tfte involute of the circle.")
6. Construct several involutes of an ellipse, and several involutes of a
parabola.
CHAPTER XI.
ROLLE'S THEOREM. THEOREMS OF MEAN VALUE.
■APPROXIMATE SOLUTION OF EQUATIONS.
106. Tn this chapter two theorems of great value in the cal-
culus are discussed, viz. Rolle's Theorem and the Theorem of
Mean Value. The truth of the latter theorem is made manifest
in a geometrical or intuitional manner in Art. 108 ; in Art. 110 it
is deduced from Rolle's Theorem. Since there are several mean-
value theorems in the calculus, the Theorem in Arts. 108, 110, 111
may be called the First Mean-valne Theorem. Another mean-
value theorem is given in The Integral Calculus, Art. 213.
Kolle's theorem and the first mean-value theorem are funda-
mental, and play a highly important part in the modern rigorous
exposition of the calculus. Two other mean-value theorems are
deduced in Arts. 112, 113. The theorem in Art. 113 is required
in Chapter XVI. An application of the mean-value theorem is
made to the approximate solution of equations in Art. 109.
107. Rolle's Theorem.
Note 1. Progressive and regressive derivative. In Art. 22 the derivative
of /(x) was defined as
r lim^^/(^ + A^)-/W. (1)
Ax
The process of evaluating (1) is equivalent
to the geometrical process of revolving the
chord PQ of tl e curve y =f{x) about P until
Q coincides with P, and thus PQ becomes the
tangent PT. If in this curve a chord PS be
drawn, and RP be revolved about P until B
coincides with P, then JtP will finally take
the position PT. The slope of the tangent
obtained by thus revolving RP is evidently
Fig. 49.
Ax —Ax
166
106, 107. ]
bolle's theorem.
167
It is customary to call (1) the progressive derivative, and (2) the regressive
derivative.* In general these derivatives are equal ; tliat is, in general the
tangent on the representative curve is the same, whether the secant which is
revolved until it assumes a tangential position be drawn forward or backward
from the point under consideration. In some cases, however, these deriva-
tives are not equal ; such a case is represented at P on Fig. 51 c, where the
two revolving secants give two diflerent tanjrents. In such a case the deriva-
tive is discontinuous at P, for its value suddenly changes from the slope of
TP to the slope of LP.
Theorem. If a function f{x) and its derivative f(x) are continu-
OMS for all values of x between a and b, and if f(a)=f(b), then
f'(x) = Ofor at least one value of x between a and b.
Following is a geometrical proof f and representation of this
theorem. Let the curve MX (Figs. 50 a, b, c) represent the
function f{x).
At M and X let x = a and .r = 6 respectively. Since the ordi-
nates AM and JB^are equal, it is evident that there must be at
least one point between M and JV" where the function ceases to
increase and begins to decrease, or ceases to decrease and begins
to increase. There may be several such points, as in Fig. 50 c.t
But at such a point, for instance F, or Pi, or P^, or Pj, the value
of the first derivative, which is continuous by hypothesis, must
be zero.
Fig. 50 a.
Fig. 50 b.
Fig. 50 c.
•They are also called right- and left-hand derivatives.
t An analytical discussion will be found in the collateral reading suggested
in Note 3, Art. 108.
J Here functions having only a finite number of oscillations between M
and X are dealt with. On the relation between RoUe's theorem and func-
tions having an infinite number of oscillations between M and If, see Pier-
pont. Functions of Seal Variables, Vol. I., Arts. 394-396.
168
DIFFERENTIAL CALCULUS.
[Ch. XI.
A special case of this theoiem is that in which /(a) = 0 and
/(b) =0. The student may construct the figure for himself by
merely moving OX to the position M2^. The statement of the
theorem for this case is usually taken as the general statement
of the theorem. It is as follows :
Bolle's Theorem (second statement) :
If f(x) is zero when x — a and when x = b, and f{x) and its de-
rivative fix) are continuous for all values of x between a and b,
then f'{x) will be zero for at least one value of x between a and h.
Note 2. The necessity of the condition relating to continuity is evident
from rigs. 51 a, 6, c, d.
Fig. 51 a.
Fio. 51 6.
Fig. 51 c.
Fig. 51 d.
For a value of x between x = a and a; = 6 : in Fig. 51 a, f{x) is infinite ;
in Fig. 51 6, f{x) is discontinuous ; in Fig. 51 c, f'(x) is discontinuous ; in
Fig. 51 d,f'(_x) is infinite.
Note 3. The theorem does not necessarily fail if f(x') is infinitely great
for some value of x between u and b. For instance, if there is a vertical
tangent at a point of inflexion between P2 and Pg or at a point between P3
and Pi, Fig. 50 c (tangents as in Fig. 26 6), the theorem still holds true.
Not« 4. Algebraic application of RoUe's Theorem.
An important application of Rolle's Theorem may be made to the
theory of equations. According to the theorem, geometrically,
'f{x)-o nx)-o\x
Fig. 52 a.
Fig. 52 b.
107, 108.]
THEOREM OF MEAN VALUE.
169
the slope of a curve y = f(x) is zero once at least, between the
points where the curve crosses the a>axis. Hence, at least one
real root of the equation /'(«)= 0 lies between any two real roots
of the equation /(x)=0. In the theory of equations this is
called Rolle's Theorem, after Michel Roire (1652-1719).
Note 5. According to this principle r real roots of an equation f(x) = 0
have at least (r — 1) roots of /(z)= 0 between them. Now, if the >• roots
coalesce and thus make an r-tuple root, the (»• — 1) roots must also coalesce
and thus make an (r — l)-tuple root of /(x) = 0. (See An. 66a.)
Ex. Verify RoUe's Theorem in each of the following equations /(a;) = 0 ;
also sketch the curve y =f{x):
(1) a;2 + a; _ 6 = 0 ; (2) a:'' + 2 1'^ - 5 a; - 6 = 0.
108. Theorem of mean value. If a function f(x) and its derivative
f(x) are continuous for all values of x from x = a to x = b, then
there is at least one value of x, say Xi, betimen a and b such that
f(b)-f(a).
b — a
-f'i^O;
i.e. such that/(6)=/(a) + (6 - a)f'(x,).
Following is a geometrical proof * and explanation of this theorem.
Let the curve ilfJV (Fig. 53 a or Fig. 53 b) represent the func-
tion f(x). Draw the ordinates AP and BQ at A and B, where
Fig. 5.3 6.
x = a and x = b respectively. Draw PQ and draw PR parallel to
OX. Then AP = f(n), BQ=f(b).
* For an analytical deduction of the theorem of mean value from RoUe's
Theorem, see Art. 110.
170 DIFFERENTIAL CALCULUS. [Ch. XI.
Hence BQ = f{b)-f(a),
and t.nEPQ = ^ = f<^>^^I^.
PR b — a
Now the chord PQ and the tangent ST drawn at some point V
(or Vi and Fj) between P and Q evidently must be parallel. At
F let a; = x^, x^ thus being between a and b ; then tan RPQ=f'{xi).
Hence ^^^ = //(x,). (1)
Since a;, is between a and b, a;, = a + 6(b — a), in which $
denotes some number between 0 and 1 {i.e. 0<6<T). Accord-
ingly, theorem (1) may be expressed
f(b)=f{a) + (b - a)fla + 6{b - a)]. (2)
lib — a = h, then 6 = a*+ h, and (2) is written
Aa + h)=fia)+ hf'ia + 9ft). (3)
Kesult (3) has important applications. It is very useful for
finding an approximate value of f(a + h) when f{x), a, and h, are
given. A closer approximation to the value of f{a + h) can be
found by Taylor's formula, Art. 150.
Note 1. The necessity for the condition relating to continuity can be
made evident by figures similar to Figs. 61 a, b, c, d.
Note 2. The remark in Note 3, Art. 107, applies also to the mean-value
theorem. In cases, however, in which /'(x) may be infinite for values of x
between a and 6, Xi in (1) must be such that/'(xi) is finite.
Note 3. References for collateral reading on Molle's theorem and the
theorem of mean value: McMahon and Snyder, Diff. Cat, Arts. 59, 66;
Lamb, Calculus, Arts. 48, 49, 56; Gibson, Calculus, §§ 72, 73; Harnack,
Calculus, Art. 22 ; Echols, Calculus, Chap. V. The last mentioned text has
a particularly full and valuable account of these theorems. Also see Pierpont,
Functions of Meal Variables, Vol. I., Arts. 39-3-404 ; Goursat-Hedrick, Math-
ematical Analysis, Vol. I., Arts. 7, 8 ; Osgood, Calculus, Chap. XI.
EXAMPLES.
1. Find by relation (3) an approximate value of sin 32° 20' taking a = 32° :
(1) putting e = 0, (2) putting e = I ; and compare the calculated results
with that given in the tables.
108,109.] APPROXIMATE SOLUTION OF EQUATIONS. 171
2. If /(a;)= 2a:2 _ X + 5, find what e must be in order tiiat relation (3) be
satisfied : (1) when a = 3 and A = 1 ; (2) when n = 10 and h=2.
3. Show that for any quadratic function, say /(x) = Vj? + mx + n,
f(a + h) will be obtained by putting 9 = J in relation (3). What geometrical
property of the parabola corresponds to this ? (Deduce the value ofO.)
4. If/(a:)= a;'', find what e must be in order that relation (3) be satisfied
when a = 3 and h = \. Wiiat problem in connection with the cubical
parabola y = x^ is the correlative of this?
109. Approximate solution of equations. The real roots of an
equation can generally be found to as close an approximation as
one pleases by the help of the calculus.
I^et f(x) = 0 (1)
be the equation. Suppose that an approximate value of a root of
(1) has been found, by substitution or otherwise, and suppose this
value, say the nearest integral number in the root, is a.
Suppose the corresponding root of (1) is a + ^.
Then f(a + h) = 0. (2)
But, by Art. 108, result (3),
f(a + h) = f(a) +hf{a + eh),l~l<e<ll (3)
From (2) and (3),
f{a)+hf{a + eh)=0.
An approximate value of h, say hj, may be found by taking
6 = 1, and putting „, , , , ,
t,..0 f{a) + hj'ia) = 0. (4)
. This gives h.=-f(^.
Accordingly, a second (and, in general, a closer) approximation
to a root of (1) is «_£«!. ...
f'(a) ^^)
On starting with this value as an approximate value of the root,
and again proceeding in a similar way, a still closer approxima-
tion to the root may be found. This process may be repeated as
often as may be deemed necessary.*
* This method of finding an approximate solution of an equation is called
Newton's method.
ERRATUM
172 DIFFERENTIAL CALCULUS. [Ch. XI.
EXAMPLES.
1. Find approximately a root of the equation
3;3 + 2a;-19=0. (6)
Here /(2) = — 7, and /(3) = + 14. -Accordingly, at least one root of the
equation lies between 2 and 3.* Since 2 is evidently nearer the value of
the root than 3 is,t let the number 2 be chosen as the first approximation to
the root.
In this example, f{x) = a;' + 2 a; — 19.
Hence /'(a;)= 3a;2 + 2,
and /(2) = -7, /'(2) = 14.
_ 7
.-. by (5) , a closer approximation to the root than 2 is 2 i.e. 2.5.
Now taking 2.5 as an approximate value of a root of (6),
a closer approximation = 2.5 - i^MI = 2.5 - i^ = 2.5 - .07 = 2.43.
/'(2.5) 20.75
Using 2.43 as an approximate value,
a closer approximation = 2.43 - /(M§1 =2.43 - :?2§22Z
/'(2.43) 19.7147
= 2-43 - .0106 = 2.4194.
2. Find a root of a? - «2 _ 2 = 0.
Substitution gives/(l) = - 2, /(2) = + 2. Accordingly, a root lies be-
tween 1 and 2.
Here /(a;)= a:'' - a:^ - 2.
. ./'(a;)=3a;2-2a;.
It will be found better to take 2 for a first approximation to the root.
A second approximation = 2 — -^^ — '- = 2 — 1 = 1.75.
A third approximation = 1.75 - iXLiiil = 1.75 - :?25§I^
/'(1-75) 5.6875
= 1.75- .05219
= 1.698.
If 1 be taken as an approximation instead of 2, the process for finding the
next approximation gives 3, which is farther from the root than 1 or 2. Thus :
second approximation = 1 — •' ' ' = 1 — ^^-=. = 3.
/'(I) 1
An explanation of this result is given in Note 1.
* In this case, when z changes from 2 to 3, f{x) changes from — 7 to + 14.
Now /(a;) is a continuous function of a;. Accordingly, /(a;) must pass through
zero once at least when it is changing from the negative value ( - 7) to the
positive value (+14). t For — 7 is nearer zero than + 14 is.
109.]
EXAMPLES.
173
Ex. Taking 3 as an approximation to a root of the above equation,
derive successive approximations therefrom.
Note 1. Suppose a; = ii is taken as an approximation to a root of the
equation ^, ,
/W = o.
Consider the equation of the tangent to the curve
at the point whose abscissa is xi, say the point (xi, yi). Here yi =zf(xi).
The equation of tliis tangent is
y -yi =f'ixi)(x-xi.
On proceeding as shown in analytic geometry, it is found that this line
crosses the a-axis where
■ Xi-
Accordingly [see (5)], the above nietliod of finding a second approxima-
tion to a root of f(x) = 0, on starting with an approximation xi, is practically
the same as finding where the tangent at
(xi, yi) on the curve y = /(a;) intersects
the a-axis.
When the abscissa of this intersection
is outside the limits between wliith the
root is known to lie, the nietliod fails.
This is shown in Fig. 54, which illustrates
Ex. 2.
SL is the curve y = x^— x'' — 2.
At A, x= 1; 3.1 B, X = 2. The curve
crosses the a;-axis at D, between A and JJ,
The abscissa OD represents the real root of
the equation
a^ - a;2 - 2 = 0.
On proceeding as shown in Art. 61, it
will be found that :
Y
A
L
r
ha
/
0
/
1
y
/C X
y
^
/
Fig. 54.
the tangent PT, at P where i = 1, crosses tiie a:-axis at C where x = Z;
the tangent §fl, at Q where z = 2, crosses the z-axis at F where x = 1.75.
Note 2. Another method of finding an approximate solution, when the
equation is algebraic, is Humerus * method. This is described in text-books
on algebra.
* Also see pages 247, 256.
17-4 DIFFERENTIAL CALCULUS. [Ch. XI.
Yet another method of finding an approximate solution of an equation is
the graphical method. This is described in various textr-hooks. Thus, to
solve the equation
2^ - a:2 _ -2 = 0,
carefully plot the curves y = x^,
y = x--i--2,
and obtain the abscissa of their point of intersection. At this point
i3 = i2 + 2, i.e. x^-3?-2= 0.
Another example : to solve the equation
X = 3 sin X,
carefully plot the curves ^ ~ q '
y = sin X,
and obtain the abscissa of their point of intersection. At this point - = sin x,
i.e. X = /! sin I.
Ex. Solve these examples by the graphical method.
Note 3. In connection with this article, see Osgood, Calculus, Chap. XX.,
Arts. 1-5.
EXAMPLES.
Find approximate solutions of the following equations :
1. a^- 12a; + 6 = 0. 6. i^ + 43.2 + j. .,. 1 _ 0.
2. x? + x^-10x + 9 = 0. 7. 1^ = 5.
3. X* — 12a;2 + i2x_ 3 = 0. 8. x^ — ix — 2 = 0.
4. z'' + 3 1 - 20 = 0. 9. 2x'^ + x^ - 15 a; - 59 = 0.
6. e'(^l +x^)=40. 10. x3 — 0a:- + 3x + 5 =0.
11. a;3-33--4 =0.
110. Theorem of mean value derived from Rolle's Theorem. Let
/(j) and its first derivative /'(x) be continuous in the interval
from a; = a to a; = 6.
Fig. 55.*
Consider the quantity Q which represents the difference-quotient
in the equation, f(f>)-.f(o) ^ q (l^
b — a
rrom(l), /(6)_/(a)-(6_a)Q = 0. (2)
* In connection with Figs. 55-60, see Art. 15 a and Fig. 5, footnote.
110, 111.] THEOREM OF MEAN VALUE. 175
Let F{x) denote the function formed by replacing 6 by a; in the
first member of (2) ; that is, let
F{x)=f{x)-f{a)-{x-a)Q. (3)
Then, F{h) = f{h)-f{a)-{h-a)Q = Q,hy{2); (4)
also, i^(a) = /(a)-/(a)-(a-a)Q = 0, identically. (6)
Novf /(x) and/'(a;) by hypothesis are continuous in the interval
(a, b); also (x — a)Q is a continuous function, and its derivative
Q is a constant. Accordingly, from these facts and equation (3)
it follows that
F(x) and its derivative F'(x) are continuous in the interval (a, b).
Also, F(x) is zero -when x = a and when x = b. [Eqs. (4), (5).]
Thus the conditions of Eolle's Theorem (second statement) are
satisfied by F(x), and therefore
F'(x) will be zero for at least one value of x, Xi say, between a
and b;
that is F'{xi) = 0, in which a<Xi<b (see Fig. 55). (6)
From (3), on difEereutiation, F'(x) = f'(x)-Q. (7)
.-. on substitution of a;, in (7), F' (x^) = f (x^ — Q; (8)
whence by (6) and (8), Q =/'(a;,), a<x^< b. (9)
Substitution from. (9) in (1) gives
f(^l^^=f(aei-),a<x,<b. (10)
m Another form for the theorem of mean value.
From Art. 110 (10), f(b) = f(a) + (b- a)f\x,), a<x,<b. (1)
Suppose b — a = h. (See Fig. 55.)
Then b = a + h;
and, since x^ is between a and b,
Xi = a + Oh,
in which 6 denotes a proper fraction, i.e. 0<6<1.
Then (1) can be written
f{a + h) = f(^a)+ hf{a + ih), 0<^<1.
(See Art. 108, Eq. (3) and on.)
176 DIFFERENTIAL CALCULUS. [i'». XI.
112. Second theorem of mean value. Jf a function of f(x) and
its first and second derivatives, f'(x), f"{x), are conlinuouH for all
values of x from x = a to x = b, then there is at least one value of x,
say X2, between a and b such that
f{b) = f{a) + (b - a)f'(a) + iib - aff"(x,).
The proof proceeds on lines similar to those in Art. 110.
? ?2 Ji +
Fia. m.
Consider the quantity R in the equation
f(b)-f(a)-(b-a)f'(a)^^(h-afB = 0. (1)
Let F(x) denote the function formed by replacing b by x in the
first member of (1) ; that is, let
Fix) =f(x) - fia) -ix- a) fia) - ^ ix - a fit. (2;
Then Fia) = 0, identically ; and F(b) = 0, by (1 j.
Also, it follows from equation (2) and the hypothesis of the
continuity oi f(x) and f'(x) that F(x) and F'(x) are continuous
in the interval (a, b). Thus the conditions of KoHc's theorem
are satisfied by F(x), and therefore
F'ix) will be zero for at least one value of x, x, say, between
a and b ;
that is F'ixi) = 0, in which a < «i < 6. Ci)
From (2), on differentiation,
F'(x) = f'(x) - f'(o.) -(x- a)R. (4)
Hence, from (■'>) and the substitution of x, in (A),
F'(x,) = f'(x,)-f'(aj-(x,-a)Ii = i). (0;
Also, from (A), F'(a) = f'(a) - fia) - (a -a)R = 0. (1)
Further, it follows from equation ("4), and the continuity of
f(x), fix) and F'ix), that F"{x) is continuous in the interval
(a, b). Thus the conditions of Rolle's theorem are satisfied by
F'(x) in the interval io., a;,), and therefore
112, 113.] EXTENDED THEOREMS OF MEAN VALUE. 177
F"{x) will be zero for at least one value of x, x.^ say, between a
and a;,, and thus between a and b ; that is
F"(x,) = 0, a<Xi<h. (8)
From (4), on differentiation, F"(x) = f"{x)- R; (9)
whence, on substitution of x^, F"{x^ = f"{x^—R. (10)
From (10), by (8), R = f'(x,), a<x,<b. (11)
Substitution of this value of R in (1), and transposition, give
/(6) = /(«) + (6 -«)/'(«)+ ^(6 -a)V"(a;2), a<x^<b. ,(12)
Another form of theorem (12).
On denoting the interval b — a by h, and proceeding as in
Art. Ill, relation (12) will take the form
f{a ■¥ h} = f{a)+ hf'(ia)+ lh^f>'(a + Bih), 0<ei<l. (13)
113. Extended theorem of mean value. -4. First method. Sup-
pose that /(x) and its first three derivatives /'(a;), /"(x), /'"(x),
are continuous in the interval from x = a to x=b. Ky the same
method as that used in Art. 112 a number S can be considered
which satisfies the equation
f{b) -f{a) - (6 - a)f'{a) - ^ (6 - a)T'(«) " g^ (^ " «)''^ = «• (1)
It will be found that S —f"{x^, in which x^ is a value of x
between a and 6.
Substitution of this value of S in (1) and transposition give
f(p) =/(a) + (& - a)f{a) + ^^/"(a) + ff^/"'(=«3). (2)
in which a < Xj < 6.
Suppose that f(x) and its first n derivatives are continuous in the
interval from x = a to x=b. By following this method succes-
sively there will at last be obtained the extended theorem of mean
value :
/(6) = /(«) + (6 - «)/'(«) + ^^f^f'W + ^^^f^f'W + -
+ i&zu^/(»)(a,„), (3)
n!
in which a<x„<h.
178 DIFFERENTIAL CALCULUS. [Ch. XI.
g X, b Since x„ is between a and h, a;„ = a +
j,jg gy ^ (& — a), in which 0 < S < 1.
On denoting h — a by ^, and proceed-
ing as in Art. Ill, result (3) will take the form
/(a + ft) = /(a)+ A/'(a)+ I?/ '(«) + |^/"'(a)+ ...
+ ^/.»X« + e„A), (4)
in which 6^ is a fraction between 0 and 1, i.e. 0<fl„< 1.
B, Second method. Theorem (3) can also be obtained by a
single application of Rolle's theorem.
Let /(a;) and its first n derivatives be continuous in the inter-
val from x = ato x = h. Consider i?„ in the equation
f{b) -f{a) -(b- a)f'(a) - K^ - «)y"(«) - • • •
~ fa"l'i)7-^'"'"(") - (* - ")"^" = ^- ^^>
Let J'(x) denote the function formed by replacing a by a; in the
first member of (5) ; that is, let
F(x} =/(6) -f{x) -(b- x)f\x) -\{h- xff '(X) -f . . .
- J^_^^^f" ""(^) -0>- ^)"^« = 0. (6)
Since /(a;) and its first n derivatives are continuous in the in-
terval from x = ato x = b, it follows from equation (6) that
F(x) and F'{x) are continuous in this interval.
Also, F(a) = 0, by (5) ; and F{b) = 0, identically.
Thus the conditions of Rolle's theorem are satisfied by F(x),
and therefore F'{x) will be zero for at least one value of x, x„ say,
between a and b ; that is
F'ix,)=0, a<x„<b. (7)
From (6), on differentiation and reduction,
F'ix)=- (A^rV")(x) + n(6 - xy-'R„ ;
113.] EXTENDED THEOREMS OF MEAN VALUE. 179
whence, on substitution of a;„ for x,
F'i^n) = - ^—^P'^x^) +n{b- x^r-'R.. (8)
(n - 1) !
From (8) it follows, by virtue of (7), that
R„=^f\x„). (9)
Substitution of this value of R„ in (5) and transposition give
formula (3) above.
N. B. Another theorem of mean value commonly called the
Generalized Theorem of Mean Yalne is given in Art. 116, Chap. XIII.,
where it is needed for immediate application.
CHAPTER XII.
INDETERMINATE FORMS.
114. Indeterminate Forms. Functions sometimes take peculiar
x' — i
forms. For instance, — ,
X — /
when x = 2,
has the meaningless form -•
Special instances in which this form presents itself have been
considered in preceding articles ; e.g. — and — in Chap. I., and
Ax Af
in Arts. 22, 24, 25; 5^, ^, in Exs. 7, 8, Art. 14.
9 0
When x = 0 the function x cot x has the form 0 • oo ;
when x = - the function (tana;)°°" has the form oo".
2 ^ ^
Cases like these, and others to be mentioned, require further
special examination. These peculiar forms are called indetermi-
nate forms. They are also called illusory forms. The object of
this chapter is to show the calculus method of giving a definite,
a determinate, value to a so-called indeterminate form.
There are various other methods, which are sometimes simpler
than the method of tlie calculus, for " evaluating " functions
when they take illusory forms.* All the methods, however, start
* " In the present chapter we propose to deal specially with these critical
cases of algebraical operation, to which the generic name of "Indeterminate
Forms " has been given. The subject is one of the highest importance, inas-
much as it forms the basis of two of the most extensive branches of modern
mathematics — namely, the Differential Calculus and the Theory of Infinite
Series (including from one point of view the Integral Calculus). It is too
180
114,115.] INDETERMINATE FORMS. 181
with the same fundamental principle, or rather with the same
definition, concerning what is to be taken as the value (sometimes
called ' the true value ') of an indeterminate form. The princi-
ple on which a value is assigned is illustrated in Arts. 117, 118.
Briefly stated, the principle is this :
Suppose a function f(x) taken an indeterminate form when
x = a.
Tlie value off (a) is defined as
the limit* of the value otf(x) when x approaches a.
A.
Note 1. Definition A really takes that value for/(x) which makes the
function f{x) continuous when x = a. This may be indicated arithmetically
in the case of the function ^ ~ • For, when
X--2
X takes the values 1, 1 • 5, 1 ■ 7, 1 . 9, 2, 2 . 1, 2 . 2, 2 • 3, ••. successively, the
function takes the values 3, 3 . 5, .3 . 7, .3 ■ 9, 4, 4 . 1, 4 • 2, 4 ■ 3, •■■ successively.
The calculus method for obtaining the value 4 for the function when i = 2,
is shown in Art. 117, Ex. 1.
115. Classification of indeterminate forms. The following seven
cases of indeterminate forms occur in elementai-y mathematics.
/-.s 0 sin X , „
(1) n ' ^S- ' ^^hen x = 0.
0 X
/ox OC loSX ,
(2) — ; e.g. — i^^— , when a;= oo .
^ ^ 00 ' " X
(3) 00 — oc ; e.g. sec x — tan x, when a; = - •
LI
(4) 0 • 00 ; e.g. j ; — x\ tan x, when a; = -
much the habit in English courses to postpone the thorough discussion of
indeterminate forms until the student has mastered the notation of the dif-
ferential calculus. This, for several reasons, is a mistake. In the first place,
the definition of a difEerentiaJ coefBcient involves the evaluation of an inde-
terminate form ; and no one can make intelligent applications of the differ-
ential calculus who is not familiar beforehand with the notion of a limit.
Again, the methods of the differential calculus for evaluating indeterminate
forms are often less effective than the more elementary methods which we
shall discuss below, and are always more powerful in combination with them."
Chrystal, Algebra^ Part II., Chap. XXV., § 1. * If there is such a limit.
182 DIFFERENTIAL CALCULUS. [Ch. XII.
(5) V^ ; e.g. ( 1 + - ] , when a; = oo .
(6) O" ; e.g. af , when x = 0.
(7) ceO; e.g. (cotx)""'^, when a; = 0.
The ' evaluation ' of forms (3)-(7) can be reduced to the evalua-
tion of either (1) or (2).
In this book the method of the calculus for evaluating forms
(1) and (2) is made to depend upon an important mean-value
theorem — the generalised theorem of mean value. This theorem is
given in the next article.
116. Generalized theorem of mean value. Iff(x), F(p), o,nd their
derivatives f (x), F'(x), are continuous in the interval from x=^ato
x = b, and if F'(x) is not zero tohen x is between a and h, then
(1)
f(b)-f(a) ^/'(a^i)
in which a<xi<b. a fi >
Fio. 58.
Consider the function <^(x) in the equation
Since f(x), F(x), f'{x), F'(x) are continuous in the interval (a, b),
it is apparent on an inspection of (2) that the function <t>{x) and
its derivative <f>'(x) are continuous in this interval.
Also, from (2), </>(«)= 0, identically; and <^(t)=0, identically.
Thus <f>(x) satisfies the conditions of Rolle's theorem.
.•. <t>'{x) will be zero for at least one value of x, x^ say, between a
and 6 ; that is 't>'(^i) = ^> ill which a < Xi < 6. (3)
From (2), on differentiation,
^'M = /(^)-/(«) F'(x) - f'(x) ; (4)
^^ ' F{b)-F{a) W ./ w. \ J
whence, on substitution of x^ for x,
^'(a:i)= /(^)-./"(a) F'(x{)-f'(x{). (5)
116, 117.] INDETERMINATE FORMS. 183
From (5) it follows, by virtue of (3), that
/[&)! jS) = J-%> ''^ ^^^^°^ a<^^<b. (6)
117. Evaluation of functions when thev take the form - • Refer-
0
ring to definition A, Art. 114, the determination of the limit
mentioned there is called the evaluation of the function.
Suppose f{x) and F{x) both vanish when x = a; that is,
suppose /(a) = 0 and F{a) = 0. (1)
According to definition A, Art. 114,
value of ^^ is defined as "'"«=« ^'■^' (2)
Suppose that a is finite.
In the generalised theorem of mean value, Art. 116, Eq. 6,
substitute x for b.
Here x and a^ must be such that
a<x^& and a<x^<x. ? f' ft
Fig. 59.
Then the theorem takes the form
F{x)~F{a) F'{x,y ' ^ ^
Since /(a) = 0 and F(a) = 0, this becomes
fM.^f(^ a<x,<x. (4)
F{x) F\x,)' ' ^'
Now let X approach the limit a. Then, since a;, lies between a
and X, Xi must also approach the same limit a, and x and x^ must
reach the limit a together.
■ lin, iM-lim /M-/M (4)
• ■^''^-'F{x)'''^F\x,)- F\ay ^*^
184 DIFFERENTIAL CALCULUS. [Ch. XIL
It sometimes happens that /'(a) and F'{a) are both zero. When
this is the case, the application of the same reasoning and process
f(x\
to the function ■' ^ ' when x approaches o, leads to the result
F'{x) ^^
value of /M = />I. (6)
F\a) F"{a) ^ '
If the second member of (6) also has the same indeterminate
form, the fraction formed by the third derivatives is required;
and so on. It thus becomes evident that :
If, for x = a, f{x) and F(x) and all their derivatives up to and
including their nth derivatives, are zero, while f^"'^\a) and J<'<"+'\a)
are not both zero, then
theTalueof^=^'""V(«). (7)
Result (7) may also be expressed thus :
If a is infinite, substitiite - for x and evaluate for « = 0.
z
It can be shown that this is practically the same as to put
a = oo in relations (5) or (7).
Note. In virtue of definition A, Art. 114, the following expressions may
be regarded as synonymous in the case of a function /(i), which takes an
indeterminate form when x = a; viz.
" find the value of /(i) when x = a ; " "evaluate f{x) when a: = a ; "
" find the limit off(x) when x approaches a " (i.e. "find limi=ia/(x)").
EXAMPLES.
1. Evaluate ^ ~^ when x = 2. (See Art. 114, Note 1.)
Valuers 5i^ = value.^^ D{x^~i) ^ ^^^ 2x^^
X — 2 D{x - 2) 1
2. Evaluate (x — sinx; ~ x' when x = 0. In this case,
,!_ X — sinx_i;„ 1 — cosx* ,.„ sinx* ,.„ cosx 1
x' 3x^ 6x a 6
* Which is in the form 0 -=- 0.
117, 118.] INDETERMINATE FORMS. 185
Note. The labour of evaluating f{a)-^ 0(a) may be lightened in the fol-
lowing cases :
(o) If, in the course of the reduction a factor, say ^(x)i appears in both
the numerator and the denominator, this common factor may be cancelled.
(6) If at any stage during the process of evaluation a factor, say ^(x),
appears only in the numerator or only in the denominator, and ^(a) is not
zero, the value of ^(a) may be substituted immediately for ^{x). This will
lessen the labor in the succeeding differentiations.
3. Evaluate the following : (1) "' ~ '''. when a; = 0; (2) Sillily, when
X X
x = 0; (3) ?^l^=^,when3-=a; (4) f' ' <'~\ when a: = 0; (5) lrL££i^
X - a sin X z'
sm {x — 2)
when z :
= 0.
4. Find the
following :
(1)
lim^io
(a- — 5)2 sin I.
X
(2) lim
(3)
lim,^
e' + «-' + 2 cos X
-4
(4) lim
X*
'
(5) 1
im,.iO
1 — cos a;
cos X sin- X
tan y — sin x
) ; ;
X — sm X
lAnswers : Exs. 3. log 2, l, na" ^ 2, J ; Exs. 4. 26, - 9, J, 3, J.]
b
118. Evaluation of functions when they take the form ^. Sup-
pose f(x) and F{x) are both infinite when x = a\ that is, suppose
f(a)=(x> and F{a) = x.
Let the limiting value of
F{a)
be required.
Suppose that a is finite. Suppose that the conditions for the
generalised theorem of mean value, Art. 116, are satisfied in an
interval {x, b), in which x is some number such that
a < -r < b.
For the interval {x, b) then, Theorem (6), Art. 116, has the form,
f(b)-f(x-) _/'(Ja) n)
Fib)-FCx) F'(:x,y ^
1 — I I 1
in which a < .r < a;, < b. fig. 60.
186 DIFFERENTIAL CALCULUH. [Cu. XII.
On changing signs and multiplying up, (1) becomes
/W-/W = ^in^)-P'(^)l- (2)
It is also supposed that F'ix^) is not zero, in the interval (a, b).
On division of the members of (2) by F(x),
fix) fib) ^ fix,) I
F(x) F(x) F'{x,)
1 Fjb)]
F{x)l
(3)
Now let X approach a as a limit. Then, since
fiPl = 0 and ^^ = 0 (because F(a)= oo and/(6) and F{b)^<K),
F{a) -F(ffl)
equation (3) takes the form
fia) ^fi^) u)
Fia) F'(xi)
The first member in (4) has the form — , and the Xi in the sec-
ond member is any number in the interval (a, 6). The value ob-
tained for the second member by letting x^ approach a as a limit,
is taken as the value of the first member ; that is
value of ^ = lim,^„^^;
F{a) F'{x) F'{a) ^^
If ■' '"^ is also indeterminate in form, similar reasoning to
F'{a) ^
that in Art. 117 leads to the same general result (6) of that arti-
cle. If a is infinite, the remarks made in Art. 117 for the same
condition apply.
It thus appears that the illusory forms in Arts. 117, 118, both
are evaluated by the same process in the calculus.
* For more rigorous derivations of the fact that the second member of (5)
is the limiting value of • '^^ when x = a, see Gibson, Calculus, pages 420,
421 ; Pierpont, Functions of Beat Variables, Vol. I., Art. 452.
118, 119.] INDETERMINATE F0SM8. 187
EXAMPLES.
1. Evaluate -^ = when x = x> . (See Art. 8, Note 2.)
liniiioc, = liiDiia, - = linixia, x =co .
logx 1
X
2. Evaluate—, -, — , wlienx = oo.
e-' e" e'
3. Find: (l)lim^'i^; (2) lim^ir i5^ ; (3) Um^,*^!!!^.
cotx 2sec3x 2 tanx
[Answers; Exs. 2. 0, 0, 0 ; Exs. 3. 0, —3, ^.]
119. Evaluation of other indeterminate forms. The evaluation
of these forms can be made to depend on Arts. 117, 118.
(a) The form 0 • oc . Let f(x) and F{x) be two functions such
that /(«) = 0 and F(a) = x ,
and let the limiting value of /(a;) • F(x) for x= a be required.
• Now f(x) ■ F(x) = ^^ ■
F(wj
tion has the form - when a
0
Art. 117.
This fraction has the form - when x = a, which was discussed in
Also, f(x).F(x)=^,
which has the form — when x = a, that was discussed in Art. 118.
EXAMPLES.
1. Liin,eM)(x • cota;)= liniiio-^ I J-e. - ) = lim,^ — -— =z 1.
tanx\ 0/ sec^x
2. Determine: (1) lim^" (^ — x j tanx; (2) liu)x=
(3) liniiii (x — 1) tan — • \ Answers: 1, to, |-
(6) The form oc-ac. By combining terms and simplifying, an
expression having the form oo — oo may be reduced to a definite
value, or to one of the preceding illusory forms.
188 DIFFERENTIAL CALCULUS. [Ch. XII.
/ 2 1 \ ,- 2x-a:2 2 — 2z 1
\x^ — i X — 2J x^ — 4 2z 2
4. Find: Um^if— ^ T^] ^ liin^ I i - -, log (1 + x) 1 ,
\x - 1 \ogxJ [x X- J
lim,^(i - \/r<;2 - a^) . [Answers : J, |, 0.]
(c) The forms 1*, ao", 0®. Suppose the function
takes one of these forms when x = a.
Put u=[f(x)Y^''\ (1)
Then \ogu = F{x).]ogif{x)l (2)
The function in the second member of (2) has one of the forms
± 0 • 00 , 00 • 0, when a; = a.
Hence the limiting value of log u can be evaluated as in case (a)
above. From this value, the limiting value of u can be derived.
I
6. Evaluate (1 — x)' when x = 0. (The form then is 1".)
Put M =(1 - a:)' ; then log u = ^°- ^^ ~ '^) •
x
Accordingly, liinjio log u = lim,^o( ~ I = — 1. .: u = - when x = 0.
\l-x; e
6. Find lim^ioCa:'). (This form is 0".)
Put V = X'; then log u = x log x.
Accordingly,
1
limjiologu = lim,io— rj- = linii=o 3j = linii=()(— x)= 0;
consequently, u = ef = 1 when a; = 0.
when a; = 0. (The form then is ao".)
Put M=(a;)'"'".
Then logit = tan a; -log [ - j = — tana;- logo;
limi=o log M = lim^io ( - tan x • log x) = limj^io I — ■^^]
\ cot a;/
= limi:io
cosec^ x
2 sin a; cos a;
sin^j;
= liniiio
X
T. ^ hill X lUhX rt
= limiio = 0.
.•. lim 1=0 K = 1-
119.] INDETERMINATE FORMS. 189
8. Evaluate the following: (1) [l+ij when x = oo ; (2) sin z"""
when x = 0; (3) x^ when a = oo ; (5) (1 — x)^ when x = ao ;
<.,(..!)
1\^
/ IV — —
whenx = ao; (6) (1+ — ) when i = co ; (7) x'-' when x = 1 ; (8) x*-i
whenx = oo; (9) x"'"' when i = 0. [Answers: (1) e, (2)1, (3)1, (4)1,
(5) 00, (6) 1, (7) e, (8) 1, (9) 1.]
9. Evaluate the following: (1) x tan x — — sec x when x= — •
^ ^ 2 2 '
(2) *5Il£zi£whenx = 0; (3) «ec° <> - 2 tan 0 ^j^gn^^i (4) 8in:lx-x
^x-sinx ^^ l + cos4« 4^^ 3z^
when X = 0 ; (5) -^5E*. when e = - ; (6) - - cot^x when x = 0;
^ ^ tanSe 3' ^ x2
7r<6
A "I"
1 when 0 = 1; (8) (sec^)''"* when ^ = 0. lAnswers:
0) (tan!j
(1) -1; (2) 2; (3) J; (4) i; (5) 3; (6) f; (7) 1; (8) 1.]
e
Note. References for collateral reading on illusory forms. For a
fuller discussion on the evaluation of expressions in these forms, and for
many examples,. see McMahon and Snyder, Diff. Cal., Chap. V., pages 11&-
131 ; F. G. Taylor, Calculus, Chap. XII., pages 136-148; Echols, Calculus,
Chap. VII. ; also Gibson, Calculus, Arts. 161, 162. For a general treatment
of the subject see Chrystal, Algebra, Vol. II., Chap. XXV. For a rigorous
and critical treatment by the method of the calculus see Pierpont, Functions
of Real Variables, Vol. I., Chap. X. Also Osgood, Calculus, Chap. XI.
CHAPTER Xm.
SPECIAL, TOPICS RELATING TO CURVES.
ENVELOPES, ASYMPTOTES, SINGULAR POINTS, CURVE TRACING.
Envelopes.
120. Family of curves. Envelope of a family of curves. The
idea of a family of curves may be introduced by an example.
The equation
(x-cy+f=4:
is the equation of a circle of radius 2 whose centre is at (c, 0).
If c be given particular values, say 2, 3, —5, the equations of
particular circles are obtained. Thus Equation (1) really repre-
sents a family of circles, viz. the circles (see Fig. 61) whose radii
Fig. 61.
are 2 and whose centres are on the x-axis. The individnal
members of the family are obtained by letting c change its values
from — 00 to + 00. A number such as c, whose different values
serve to distinguish the individual members of a family of curves,
is called the parameter of the family. Thus, to take another
example, the equation y = 2x + b represents the family of straight
lines having the slope 2 ; and y = 2x + 5, y = 2x — 7, are particu-
lar lines of the family. (Let a figure be constructed.) In this
case the parameter 6 can take all values from — oo to -|- oo.
190
liiO, 121.] ENVELOPES. 191
To generalize : f{x, y, a) = 0 (2)
is the equation of a family of curves whose parameter is a. The
individual members or curves of the family are obtained by giving
particular values to a. These curves are all of the same kind,
but differ in various ways ; for instance, in position, shape, or
enclosed area. A family of curves may have two or more param-
eters. Thus, y = mx + b, in which m and b may take any values,
has two parameters m and b, and represents all lines. The equa-
tion (x — hy + {y — ky = 25, in which h and k may take any
values, represents all circles of radius 5. The equation (x — hy
+ {y — ky = r^, in which h, k, and r may each take any value,
represents all circles.
Envelope. The envelope of a family of curves is the curve, or
consists of the set of curves, which touches every member of the
family and which, at each point, is touched by some member of
the family. For example, the envelope of the family of circles
in Fig. 61 evidently consists of the two lines «/— 2=0 and y+2=r0.
On the other hand, the family of parallel straight lines y=2x+b
does not have an envelope ; and, obviously, a family of concentric
circles cannot have an envelope.
EXAMPLES.
1. Say what family of curves is represented by each of the following
equations, and in each instance make a sketch showing several members of
the family :
(a) x'^ + y^ = r^, parameter r. (6) y = mx + 4, parameter m.
(c) y^ = ipx, parameter p. (d) y^ = i a{x + a), parameter a.
(e) — + 2- = 1, parameter a. (f) 1 — = 1, parameter k.
(g) y = mx H — , parameter m. (A) y = mx + V25 m'^ + 16, parameter m.
m
2. Express opinions as to which of the families in Ex. 1 have envelopes,
and as to what these envelopes may be.
121. Locus of the ultimate intersections of the curves of a family.
In Eq. (2), Art. 120, the equation of a family of curves, let a be
given the particular value a, ; then there is obtained the equation
of a particular member of that family, viz.
fix,y,a,)=0. (1)
192 DIFFERENTIAL CALCULUS. [Ch. XIII.
Also, /(x, y, a, + /t) = 0
is the equation of another member of the family. Let I. and II.
be these curves. The smaller h becomes, the more nearly does
curve II. come into coincidence with curve I. Moreover, as h be-
comes smaller and approaches zero, A, the point of intersection of
these curves, approaches a
definite limiting position. For ^w^v^^il^
example, if (Fig. 61) the centre
L approaches nearer to C, then
K, the point of intersection of
the circles whose centres are
at C and L, moves nearer to ^^ yio. 62.
P; and finally, when L reaches
C, K arrives at the definite position P. The locus of the limiting
position of the point (or points) of intersection of two curves of
a family which are approaching coincidence is called the locus of
ultimate intersections of the curves of the family. For instance, in
the case of the family of circles in Fig. 61, this locus evidently
consists of the lines y — 2 = 0 and y + 2 = 0.
Note. The last-mentioned locus may also be derived analytically.
Let (a; - Ci)2 + 2/^ = 4 (1)
and (x - Ci - hy -\-y^ = i (2)
be two of the circles. On solving these equations simultaneously in order to
find the point of intersection, there is obtained
(a;- ci)2-(x-ci- A)2 = 0; whence h(2x-2ci- h) = 0,
and, accordingly, x-= Ci-\ —
An ultimate point of intersection is obtained by letting h approach zero.
If A = 0, then x = Ci, and by (1) !/ = ± 2. Thus y =±2 at the ultimate
points of intersection, and therefore the locus of these points is the pair of
lines y =±2.
N.B. In the following articles "the locus of ultimate intersections" is
denoted by I. u. i.
121, 122.]
ENVELOPES.
193
122. Theorem. In general, the locus of the ultimate intersections
touches each member of the family. Let I., II., III. be any three
members of the family, and let I. and II. intersect at F, and II.
and III. at Q. When the curve I. approaches coincidence with
II., the point P approaches a definite position on I. u. i. of the
curves of the fauiily. When the curve III. approaches coincidence
with II., Q approaches a definite position on I. «. i. When I. and
III. both approach coincidence with II., P and Q approach each
other along II., and at the same time approach I. u. i. When P
and Q finally reach each other on II., they are also on I. u. i. More-
over, when P and Q come together, the tangent to II. at P and the
tangent to II. at Q come into coincidence as a line which is at the
same time a tangent to curve II. and a tangent to I. u. i. at the point
■where P and Q, meet. Thus the curve II. and I. u. i. have a com-
mon tangent at their common point. Similarly it can be shown
that I. u. i. touches every other curve of the family. Since, in gen-
eral, each point of I. u. i. may be approaclied in the manner indicated
in this article, the above theorem may be thus supplemented: In
general, I. u. i. is touched at each of its points by some member of
the family.
Note 1. The family of circles, Fig. 61, will serve to illustrate this theorem.
Note 2. An analytical proof oi the theorem is given in Art. 12.3, Note 3.
Note 3. It is necessary to use the qualifying phrase in general in the
enunciation of the theorem, for there are some families of curves (viz. curves
having double points and cusps, see Arts. 129, 130), in which a part of I. u. i.
may not touch any member of the family. It is beyond the scope of this
book to go into these cases in detail. (See Edwards, Treatise on the Diff. Cal.,
Art. 365 ; Murray, Differential Equations, Chap. IV.) Illustrations may be
obtained by sketching some curves of the families (y -H c)^ = z» and
<]/ -h c)2 = x(x - 3)2.
194 DIFFERENTIAL CALCULUS. [Ch. XIII.
123. To find the envelope of a family of curves having one pa-
rameter. It is in accordance with the definitions and theorem
in Arts. 120-122 to say that the envelope of a family of curves
f(x, y, a) = 0, if there be an envelope, is, in general, the locus of the
limiting position of the intersection of any one of the curves of the
family, sa.y the cnive ., . n n\
f{x, y,a)=0 (1)
with another curve of the family, viz.
f{x,y,a + ^a) = 0 (2)
when the second curve approaches coincidence with the first; that
is, when Aa approaches zero.
From (1) and (2), /(», y,a + Aa) -f{x, y,d) = 0;
hence fix,y,a + Aa)-fix,y,a) ^^_
Aa
Now Equations (1) and (3) may be used, instead of (1) and (2),
to find the points of intersection of curves (1) and (2). If Aa = 0,
the point of intersection approaches an ultimate point of inter-
section. When (Arts. 22, 79) Aa = 0, Equation (3) becomes
£f{x,y,a)=0. (4)
Thus the coordinates x and y of the point of ultimate inter-
section of curves (1) and (2) satisfy Equations (1) and (4) ; and,
accordingly, satisfy the relation which is deduced from (1) and
(4) by the elimination of a. Hence, in order to find the equation
of I. u. i. of the family of curves f{x, y,a) = 0 eliminate a betioeen
the equations
f{x, y,a)=0 and ^ f{x, y, a) = 0. (5)
The result obtained is, in general, also the equation of the
envelope.
Note 1. A slightly difierent way of making the above deduction is as
follows. Let the equations of two curves of the family be
/(x, y, a) = 0 (6), and /(x, y,a + h) = 0. (7)
'^3.] ENVELOPES. 195
By Art. 108, Eq. (3), Equation (7) may be written
n
f(x, y,a) + h-^ f(x, y,a + eh) = 0, in which | # |< 1. (8)
By virtue of (6) this becomes ^f(x, y, a + eh) = 0. (9)
Accordingly, the coordinates of the intersection of curves (6) and (7)
satisfy (6) and (9). When h becomes zero, the point of intersection becomes
an ultimate point of intersection. Hence the ultimate points of intersection
satisfy equations /(x, y, a) = 0 and -- f(x, y, a) = 0, and, accordingly, the
a-eliminant of these equations.*
Note 2. For an interesting and useful derivation of result (5) for cases
in which /(x, y, a) is a rational integral function of a, see Lamb's Calculus,
Art. 157.
Note 3. To show that, in general, the a-eliminant of Equations (5) touches
any curve of the family.
Let the second of Equations (5) on being solved for a give a = <i>{x, y).
Then the equation of the I. u. i. of the family of curves /(«, j/, o) = 0 is
f{x, y, a) = 0 in which a = <p(x, y). (10)
The slope ^ of any one of the family of curves /(a;, y,a) = Oi& given (see
Art. 56), by the equation ^f r^f j„
The slope — of the I. u. i. is obtained from Equations (10). On taking
the total z-derivative in the first of these equations,
dl,Bfdjy^dlda_
dx^ dy dx^ da dx~ ^ ■'
But by the second of (6), -^ = 0, and accordingly, (12) reduces to
Thus the slope of the I. u. i. and the slope of any member of the family
are both given by the same equation. Hence, at a point common to any
curve and the I. u. i., the slopes of both are the same, and accordingly, the
curve and the I. u. i. touch at that point. *
Sometimes the value of -^ obtained from (11) is indeterminate in form,
dx
and the slopes of the curve and I. u. i. may not be the same. See Arts. 131,
122 (Note 3), and Lamb, Calculus, Art. 158.
• This method of finding envelopes appears to be due to Leibnitz.
196 DIFFEIiENTIAL CALCULUS. [Cii. XIII.
EXAMPLES.
1. Find the envelope of the family of circles (see Art. 120)
(I - cy + y = 4. (1)
Here, on differentiation with respect to the parameter c,
2 (a; - c) = 0. (2)
The elimination of c between these equations gives
2,2 = 4,
which represents the two straight lines i/ = 2, y = — 2.
2. Find the envelope of the family of lines
y = mx — 2 pm — pm^, (1)
in which m is the parameter. (This is the equation of the general normal of
the parabola y' = ipx ; see works on analytic geometry.) On differentiation
with respect to the parameter m,
0 = x-2p-3pm2. (2)
The OT-eliminant of (1) and (2) is the equation of the envelope.
On taking the value of m in (2) and substituting it in (1), and simplifying
and removing the radicals, there is obtained
27p!/2 = 4(x-2p)». (3-)
Note 4. In Art. 104 it is shown that the normals to a curve touch its
evolute. It also appears from Art. 104 that each tangent to an evolute is
normal to the original cur\-e. Accordingly, it may be said that the evolute
of a curve is the envelope of its normals, and likewise that the evolute of a
curve is the I. u. i. of Us {family of) normals. (See Art. 104, Note 2, and
Art. 101, Note 5.)
Note 6. Compare Ex. 1, Art. 103, Ex. 2 above, and Ex. 1, Art. 124.
8. If A, B, C are functions of the coordinates of a point and m a
variable parameter, show that the envelope of Am^ + Bm + C = 0 is
Note 6. The result in Ex. S is the same inform as the condition that the
roots of the quadratic equation in m be equal. This result is immediately
applicable in many instances. It is very easily deduced on taking the point
of view explained in the article mentioned in Note 2.
4. Deduce the result in Ex. 3 without reference to the calculus.
Apply this result to Ex. 1.
123, 124.] ENVELOPES. 197
N.B. Make figures for the following examples.
6. Find the curves whose tangents have the following general equations,
in which m is the variable parameter :
(1) y = mx + a vT+n?. (2) y = mx + x'aHi^ + bK
(8) y = m,x± Vara'^ + bm + c. (4) y = mx + aVm.
(5) mH = my + a. (6) !/ — 6 = m(a; - a) + rVl + m^.
6. Find the envelopes of the following lines :
(1) X sin 5 — J/ cos 9 + a = 0, parameter 6. (2) a; + y sin e = a cos 9,
parameter B. (3) ax sec a — by cosec a = a'' — 6^, parameter a.
7. Find the envelopes of (1) the parabolas j/' = 4 a(x — a), parameter a ;
(2) the parabolas cy'^ — a^(x — a), parameter a.
8. Show that if A, B, C are functions of the coordinates of a point, and
a a variable parameter, the envelope of A cos a + Bsin u = CisA' +B'^ = C.
9. Find the evolute of the ellipse x = a cos ip, y = b sin (j>, considering
the evolute of a curve as the envelope of its normals.
10. One of the lines about a right angle passes through a fixed point, and
the vertex of the angle moves along a fixed straight line : find the envelope
of the other line.
11. From a fixed point on the circumference of a circle, cliords are
drawn, and on these as diameters circles are described. Show that they
envelop a cardioid.
124. To find the envelope of a family of curves having two parame-
ters. Let -, ,> rt
f{x, y, a, 6) = 0
be a family of curves which has two parameters. If there is a
given relation between these parameters, say
F(a,, 6) = 0,
then the two parameters practically come to one, and accordingly,
the case reduces to that considered in Art. 123.
EXAMPLES.
1. Find the envelope of the normals to the parabola }f^-=^px. The
equation of the normal at any point (Xi, yi) on this parabola is
y-yx+^{x-x{) = Q.
dyi
198 DIFFERENTIAL CALCULUS. [Ch. XIH.
This reduces to 2py — 2py<. + xyi — xiyi = Q. (1)
Here there are two parameters, xi and yy. They are connected by the
'«>^"°° y^^ = ipx^.
Hence (1) becomes 2^)^ — 2^yi + a;yi — ^ = 0, (2)
ip
which involves only a single parameter y\. On differentiating in (2) with
respect to the parameter yi and then eliminating yi, there will appear the
equation of the envelope, viz.
21py'^ = i(x-2pY.
Compare Ex. 1 with Ex. 1, Art. 103, and Ex. 2, Art. 123.
Note. This problem may be expressed : Find the envelope of the line
(1), given that the point (zi, yy) moves along the parabola y'^ = ipx.
2. Find the envelope of the line
^ + 1 = 1 (1)
when the sum of its intercepts on the axes is always equal to a constant c.
Since a + b = c, (2)
Equation (1) may be written - -\ V— = 1,
a c — a
i.e. (c — a)x + ay = ac — a^. (3)
Thus (1) is transformed into an equation involving a single parameter a.
On differentiating in (3) with respect to the parameter a,
— x + y = c — 2a. (4)
The elimination of a between (3) and (4) gives
x2 + y2 + c2 = 2 d + 2 xy + 2 cj/.
This reduces to Vx + Vy = Vc.
See Ex. 7, Art. 62.
The elimination of a and b can also be performed thus ;
Differentiation in (1) and (2) with respect to a gives
_i_l^ = Oand 1+^=0.
a^ b'^ da da
On equating the values of — ,
da
^ = 1; whence * = ^. (5)
a« 6= a Vx
124, 125.] ASYMPTOTES. 199
From (2) and (6), a = f ^ _, 6= f^^ .
Vz + Vy Vx + v^
On substitution in (1) and reduction, Vx + Vy = Vc.
Tills second method is generally more useful tiian that used in Ex. 1 and
in the first way of working Ex. 2, in cases when the two parameters are
involved symmetrically in the equation and in the expression of the relation
between the parameters.
3. Find the envelope of the straight lines the product of whose intercepts
on the axes of coordinates is equal to a'.
4. Find the envelope of a straight line of fixed length a which moves with
its extremities in two lines at right angles to each other.
6. A set of ellipses which have a common centre and axes, and in
which the sum of the semi-axes is equal to a constant a, is drawn : find the
envelope of the ellipses.
6. Show that the envelope of a family of co-axial ellipses having the
same area consists of two conjugate rectangular hyperbolas.
7. Circles are described on the double ordinates of the parabola
y'^ = iax as diameters : show that the envelope is the equal parabola
2/2 = ia(x + a).
8. Circles are described having for diameters the double ordinates of
the ellipse whose semi-axes are a and b : show that their envelope is the
co-axial ellipse whose semi-axes are Va' -t- 6^ and 6.
9. About the points on a fixed ellipse as centre, ellipses are described
having axes equal and parallel to the axes of the fixed ellipse : show that
their envelope is an ellipse whose axes are double those of the fixed ellipse.
10. A straight line moves so that the sum of the squares of the perpen-
diculars on it from two fixed points (± c, 0) is constant (= 2 A;^) ; show that
x^ tp-
its envelope is the conic — -|- f- = 1.
k'' — c' k'
11. If the difference of the squares in Ex. 10 is constant, show that the
envelope is a parabola.
12. Show that if the comer of a rectangular piece of paper be folded
down so that the sum of the edges left unfolded is constant, the crease will
envelop a parabola.
Asymptotes.
125. Rectilinear asymptotes. In preceding studies acquaint-
ance has been made with two lines related to the hyperbola,
called asymptotes and possessing the following properties:
(a) These lines are the limiting positions which the tangents to
the hyperbola approach when the points of contact recede for an
200 DIFFERENTIAL CALCULUS. [Ch. XIII.
infinite distance along the curve (or, as it may be expressed,
recede towards infinity) ; (h) the lines themselves do not lie
altogether at infinity. (This is the mathematical way of saying
that the lines run across the field of view ; in fact, in the case of
the hyperbola they pass through the centre of the curve.)
Besides hyperbolas there are many other curves which have
branches extending to an infinite distance and which have associ-
ated with them certain lines having properties like (a) and (6) ;
namely, lines : (1) that are the limiting positions which the tan-
gents to the infinite branches approach when the points of contact
recede towards infinity ; (2) that do not lie altogether at infinity ;
for instance, using rectangular coordinates, lines that pass within
a finite distance of the origin.
Lines having properties (1) and (2) are called asymptotes of the
curves. Thus an ellipse cannot have an asymptote, since it has
no branch extending to infinity (see Ex. 3, Art. 127). Again
the parabola y^ = 4pa; has no asymptote, for (see Ex. 4, Art. 127)
the tangent at an infinitely distant point of this parabola crosses
each of the axes of coordinates at an infinite distance from the
origin, and, accordingly, no part of this tangent can be in sight ;
i.e. it lies wholly at infinity. (The asymptotes are apparent in
the figures on pages 460-461.)
It will now be shown how an examination may be made for the
asymptotes of curves whose equations have the form
F(x,y)=0, (1)
where F{x, y) is a rational integral function of x and y. For this
it is necessary to call to mind the algebraic property stated in the
following note.
Algebraic Note. On substituting - for x in the rational integral equation
Col" + cii"-' + csa:"-^ + ■•• + c„_ij; -f- c„ = 0, (a)
and clearing of fractions, it becomes
Co + cit + C2«2 + ... + c„_i("-i -I- c„«» = 0. (6)
It is shown in algebra that if a root of Equation (6) approaches zero, Co
approaches zero ; and that if a second root also approaches zero, c\ also
approaches zero. But, since x = -, when a root of (6) approaches zero, a
125, 120.] ASYMPTOTES. 201
root of (a) increases beyond all bounds, i.e., to use a common phrase, it
approEiches infinity. Hence, the condition that a root of (a) approach
infinity is that co approach zero, and the condition that a second root of (a)
at the same time approach infinity is that ci also approach zero ; and so on
for other roots approaching infinity. This is briefly expressed by saying that
equation (o) has a root equal to infinity when co = 0, and has two roots
equal to infinity when co = 0 and ci = 0.
126. To find asymptotes which are parallel to the axes of coordi-
nates. Suppose that the equation of the curve F(x, y) = 0 [Art.
125 (1)] is of the nth degree, and that the terms in the first mem-
ber of this equation are arranged according to decreasing powers
of y. Then the equation has the form
p<,y" + pa"' + pa"''' + — + Pn-^y +Pn = o. (i)
Here, po is a constant ; pi may be an expression in x of the first
degree at most, say ax + b; p.^ may be of tlie second degree at
most, say ca^ + dx + e ; p.^ may be of the third degree in x at
most ; • ■ ■ ; and p„ may be of the nth degree in x at most. For
if any one of the respective p's were of a higher degree than that
specified above, F{x, y) would be of a higher degree than the nth.
Ex. 1. Arrange the first members of the following equations (a) in
de-scending powers of x ; (6) in descending powers of y :
(1) xy-ay-bx = 0. (2) a;^ + xy^ + 2 1^ - 2 j/2 - 7 a; + 4 j^ - 11 = 0.
(3) 2 zi/2 - x^y + .3 !/2 _ 3 a;2 + 4 a;j, _ 2 X + 7 y + 1 = 0.
(4) y^+x^y + x^ + 2xy + 7 x-\- 2=0.
Now suppose that in (1) Po = (^; then (1) may be written
0-y'' + (ax+ 6).v"-' -t- {cx^ + dx + e)y''-^ +p^'-^ + —
+ Pn-iy+P. = 0. (2)
If this be regarded as an equation of the nth degree in y, then
to any finite value of x there correspond n values of y, one of
which is infinitely great. If also ax + b = 0, i.e. if x = , a
second of the n values of y is infinitely great. In a similar way
points whose abscissas are infinitely great and whose ordinatea are
finite may be found.
202 DIFFERENTIAL CALCULUS. [Ch. XIII.
Ex. 2. Thus in Ex. 1 (1) the equation, which is of the second degree, may
be written y(x — a) — 6a; = 0. Accordingly one value of y is infinite ; a second
value of y is infinite when a; = a.
Ex. 3. Show that a second value of x is infinite when y = b.
It will now be shown that an infinite ordinate whose distance
from the origin is finite is tangent to the curve at the infinitely dis-
tant point.
On difierentiating in (2) with respect to x and solving for -^,
dz
dy_ ay~' + (2 ca: + (?)y"~'' + ••• + p'n
dx~ (u - \){ax+b)y'-^+(n-2){cx' + dx + e)y-^-{ 1- i)„-i
When a; = — -, the numerator in the second member is an infinity of an
a
order at least two higher than the denominator, and hence the value of the
fraction is then infinite. Hence the line a; = — is a tangent at any point
b "
for which x = — and y = ao.
a
In a similar way it can be shown that if one of the values of x in Equa-
tion (1), Art. 125, is infinite when y = c, in which c is finite, then y = c is
a tangent at any point for which x = cd and y - c.
Note 1. If [see Eq. (2)] x = — - also satisfies cx^ + dx + e = 0, then
a
three values of y in F{x, y) = 0 are infinitely great for this value of x. The
line X = is then an inflexional tangent (see Art. 78, Note 1) at infinity.
Note 2. This method of finding asymptotes parallel to the axes can be
applied to curves whgse equations are not of the kind considered above.
Instances are given in Exs. 7, 8 (6), (9) that follow.
EXAMPLES.
4. Find the asymptotes of the curves in Ex. 1.
6. Determine the finite points (if they exist) in which each asymptote
in Ex. 4 meets the curve to which it belongs.
6. Show that the line z = a is an asymptote of the curve y = £i^
when <l>{a) and 0'(a) are finite. x — a
Here,lim..„s^ = «. Also ^ = (^ - a)»'(^)- 0(a:) . whence Um,.„^ = ».
dx (x — ay dx
Hence a; = o is a tangent at an infinitely distant point (x = a, j/ = oo).
7. Examine y = tan x for asymptotes.
Here y = + a> when x = I, ii, ^JL, ....
" 2 2 2
Also, ^ = 8ec2 X. Hence ^ = « when x = -, — , ^, ....
dx dx 2 2 2
.*. X = -, X = — , X = — , •■•, are asymptotes.
126, 127.] ASYMPTOTES. 203
8. Determine the asymptotes of the following curves : (1) The hyper-
bola xy = a^. (2) The cissoid «2 = —^ — (S) The witch u = ^°° .
2a-x ^ ■^ " x^ + ia^
(4) (a;2 - a2) (y^ - b^) = a^bK (5) aH = y(x-ay. (6) y = log x. (7) jy = e-.
(8) The probability curve y = e-". (9) y = sec x.
127. Oblique asymptotes. There are asymptotes which are not
parallel to either axis. The method of finding them can best be
shown by an example.
EXAMPLES.
1. Find the asymptotes of the folium of Descartes (see page 463)
x' + y" = S a xy. (1)
First find the intersections of this curve and the line
y = mx + b. (2)
On solving these equations simultaneously,
(1 + m')x^ + 3 (to26 - am)!" + 3 (nifts - ab)x + 6' = 0.
Line (2) is a tangent to the curve (1) at an infinitely distant point, if two
roots of this equation are infinitely great. That is, if
1 + to8 = 0, and m% - am = 0. (3)
That is, on solving Equations (3) for m and 6, if
m = —l, and b = — a.
Hence, the asymptote is y + x + a = 0.
Note 1. A curve whose equation is of the nth degree has n asymptotes,
real or imaginary. This may be apparent from the preceding discussion.
For proof of this theorem see references for collateral reading, Art. 128.
In Ex. 1 two values of m in Equations (3) are imaginary ; thus curve (1)
has one real and two imaginary asymptotes.
2. Find the asymptotes of the hyperbola b'hfl — aV = ^''fi^-
3. Show by the method used in Ex. 1 that the ellipse ft^x' + aV = a'S"
has no real asymptotes.
4. Show by the method used in Ex. 1 that the parabola y^ = ^px
does not have an asymptote.
204 DIFFERENTIAL CALCULUS. [Ch. XIII.
6. Find the asymptotes of the following curves : (1) y> = x^ + x.
(2) X* - y* - 3 a;8 - X3/2 _ 2 X + 1 = 0. (3) xyiy - x) = 3 x^ + 2 j/2.
(4) (x2 _ 2,2)2 _ 4 y2 + y + 2 X + 3 = 0. (5) x^ - 8 i/' + 3 x^ - xj( - 2 2^2 = 0.
Note 2. Other methods of flading asymptotes.
a. Find the values of the intercepts on the axes of coordinates of the
tangent at a point (x', y') on a curve [see Art. 61, Equation (3)], when
x' = CO, or y' = 00, or both x' and y' are Infinitely great. If one or both of
these intercepts is finite, the tangent is an asymptote. Its equation can be
written on finding its intercepts.
6. Apply this method to Exs. 2, 4, above. [See Note, p. 212.]
6.' Find the length of the perpendicular from the origin to the tangent
at (i', y') when x' = 00, 01 y' = a>, or both x' and y' are infinitely great.
If this length is finite, the tangent is an asymptote.
7. Do Exs. 2, 4, by this method. [See Note, p. 212.]
c. By means of the equation of the curve express y in terms of a series
in decreasing powers of x, or express x in terms of a series in decreasing
powers of y. From one of these expressions there may sometimes be de-
duced the equation of a straight line which, for infinitely distant points,
closely approximates to the equation of the curve.
8. Thus, in the hyperbola in Ex. 2,
' a' a \ x^J
a \ 2x' / ax4
x»
It is apparent from this that the farther away the points on the lines
fix
y = ± — are taken, the more nearly will they satisfy the equation of the
a
hyperbola, and that when x increases beyond all bounds, the points on these
lines satisfy the equation of the hyperbola. Accordingly, these lines are
asymptotes.
Note 3. Curvilinear asymptotes. Expansion may sometimes reveal
the equation of a curve of higher degree than the first whose infinitely distant
points also satisfy the equation of the given curve. Accordingly the two
curves coincide at infinitely distant points. The two curves are said to be
asymptotic, and the new curve is called a curvilinear asymptote of the
original curve. For a discussion on curvilinear asymptotes see Frost's Curve
Tracing, Chaps. VII. and VIIL
127, 128.]
ASrMPTOTES.
205
128. Rectilinear asymptotes : polar coordinates. In order to find
the asymptotes of the curve
f{r,e)=0 (1)
a method similar to that outlined in Art. 127, Note 2 (6), can be
used. First find the value of 0
in Equation (1) for when the
radius vector r is infinitely great.
Suppose that this value of 6 is
^1. Thus the point (r=oo, 6=61)
is an infinitely distant point of
the curve. If the tangent TN at
this infinitely distant point is
an asymptote, it passes within
a finite distance from O. Accord-
ingly, TN is parallel to the radius
d6
vector, and the subtangent OM, viz. r^ — (Art. 64) is finite for
(r = oo,d=ex). '*'■
EXAMPLES.
1. Find and draw the asymptote to the reciprocal spiral rS = a.
Fig. 64.
Here
Also
r = — .-. r i CO when 9 = 0.
e
dr'
dr
',. (See Fig., page
464.)
Hence the asymptote is parallel to the initial line and at a distance a to
the left of one who is looking along the initial line in the positive direction.
Note 1. The convention used in Ex. 1 is as follows : A positive subtan-
gent is measured to the right of a person who may be looking along the
infinite radius vector in its positive direction, and a negative subtangent is
measured toward the left.
2. Find and draw the asymptotes to the following curves : (1) r sin 0
= aB. (2) r cos 9 = a cos 2 9. C3) r sin - = a.
Note 2. Circnlar asymptotes. If the radius vector r approaches a fixed
limit, a say, when 6 increases beyond all. bounds, then as 6 increases, the curve
approaches nearer to coincidence with the circle whose centre is at the pole
and whose radius is o. This circle, whose equation is r = a, is said to be a
circular asymptote, or the asymptotic circle, of the curve.
206 DIFFERENTIAL CALCULUS. [Ch. XIII.
3. In the reciprocal spiral, Ex. 1, if 9 = oo, then r = 0. Hence the
asymptotic circle is a circle of zero radius, viz. the pole.
Q
4. Find the rectilinear and the circular asymptote of r =
e~\
References for collateral reading on asymptotes. McMahon and
Snyder's Diff. Cal., Chap. XIV., pages 221-242 ; F. G. Taylor's Calculus,
Chap. XVI., pages 228-249, and Edwards's Treatise on the Differential Cal-
culus, Chap. VIII., pages 182-210, contain interesting discussions on asymp-
totes, with many illustrative examples. For a more extended account of
asymptotes see Frost's Curve Tracing, Chaps. VI.-VIII. , pages 76-129.
Singular Points.
129. Singular points. On some curves there are particular
points at which the curves have certain peculiar properties which
they do not possess at their points in general. For instance, there
are points of maximum or minimum ordinates (Art. 75), points of
inflexion (Art. 78), and points of undulation (Art. 78). There are
also points through which a curve passes twice or more than twice
(see Figs. 65 a, b, c), and at which it has two or more different
tangents ; there are points through which pass two branches of a
curve that have a common tangent (Figs. 66 a, b, c, d) ; and there
are other peculiar points hereafter described. Points of maximum
and minimum ordinates depend on the relative position of a curve
and the axes of coordinates ; the peculiarities at the other points
referred to above are independent of the axes and belong to the
curve whatever be its situation. Points at which a curve has
peculiarities of this kind are called singular points. Some of these
singular points are considered in Arts. 130, 131 .
130. Multiple points. Double points. Cusps. Isolated points.
Mnltiple points are those through which a point moving along the
curve, while changing the direction of its motion continuously,
can pass two or more times, and at which the curve may have two
or more different tangents.
For example, in moving from L to M along the curves in Figs.
65 a, b, c, a point passes through A and C three times and through
B and D twice. At A there are three different tangents, at C
there are three, and at B and D there are two each. Points, such
128, 100.]
SINGULAR POINTS.
207
as B and D, through which the point moving along the curve,
while continuously changing the direction of its motion, can pass
M
Fig. 6."io.
Fig. (ioi.
Fio. 65 c.
twice, are called double points; points such as A and C are called
triple points. The curve r=a sin 2 6 (see p. 464) has a quadruple
point.
Note 1. Multiple points are also called nodes. (Latin nodus, a knot.)
Cusps are points where two branches of the curve have the same
tangent. See Figs. 66 a, b, c, d.
In Fig. 66 a both branches of the curve stop at A and lie on
opposite sides of their common tangent at A. In Fig. 66 b both
branches stop at B and lie on the same side of the tangent at B.
Both branches of the curve pass through C Accordingly C is
sometimes called a double cusp. If a point is moving along a
curve LKM which has a single cusp at iL'(Fig. 66 d), there is an
Fig 66 a.
Fig. 66 6.
Fio. m c.
Fio. 66 d.
abrupt (or discontinuous) change made in the direction of its
motion on its passing through K. On arriving at K from L the
moving point is going in the direction a; on leaving if for ilf the
moving point is going in the direction b. Thus at if it has sud-
denly changed the direction of its motion by the angle it.
Note 2. A cusp such as K (Fig. 66 d) may be supposed to be the final
(or limiting) condition of a double point like D (Fig. 65 c) when the loop
DB dwindles to zero and the two tangents at D become coincident.
208 hlFFEBENTIAL CALCULUS. [Ch. XIII.
Isolated or coi^n^te points are individual points which satisfy
the equation of the curve but which are isolated from (i.e. at a
finite distance from) all other points satisfying the equation.
EXAMPLES.
1. Sketch the curve y^ = (x — a){x — b){x — c), in which a, 6, and c,
are positive and a<ib<,c.
2. Sketch the curve y^ = (x — a)(x — b)^, in which a<b and both
are positive.
3. Sketch the curve y^ = (x — ay^x — 6), in which a and 6 are as in Ex. 2.
4. Sketch the curve y'^ = (x — a)^, in which a is positive.
The sketch in Ex. 1 will show an oval from x = a to i = 6, a blank space
from x=b to x=c, and a curve extending from x = c to the right. The sketch
in Ex. 2 will show a curve having a double point at (6, 0). The sketch in
Ex. 3 will show a conjugate point at (a, 0), a blank space from x=a to x = b,
and a curve extending from x = bto the right. The sketch in Ex. 4 will show
a curve having a cusp at (a, 0).
Note 3. Other singular points. There also are points called salient
points, like D (Fig. 98), for instance, where two branches of the curve stop
but do not have a common tangent. In these
cases the slope of the tangent changes abruptly.
Accordingly, ii y = (p(x) be the equation of the
curve, 0'(a;) is discontinuous at the salient
points. (See Exs. 5, 6, below.) A salient point
such as D may be considered to ba the limiting condition of a double
point like D (Fig. 96 c), when the loop DR dwindles to zero but the two
tangents at D do not become coincident. (Compare
"A. Note 2.)
There are also stop points, as A, Fig. 68, where the
Fio. 68. curve stops and has but one branch. See Ex. 7.
1
6. In the curve y(l + e*) = x show that when x approaches the origin
from the positive side, the slope is zero ; if from the negative side, the slope
is 1. The origin is thus a salient point. Suggestion : The slope at the
origin may be taken as lim^o -• | Find the angle between the branches at the
origin. ^ -^
6. In the curve y = x^ " show that when x approaches the* origin
e'+ 1
from the positive side the slope is + 1, and if from the negative side, the
slope is — 1. The origin thus is a salient point : find the angle between the
branches there.
7. Show that the origin is a stop point in the curve y = x log x.
130, 131.] SINGULAR POINTS. 209
131. To find multiple points, cusps, and isolated points. From
Art. 130 it is evident that in order to determine tiie character of
a point on a curve, it is first of all necessary to examine the tan-
gent (or tangents) there. Let the equation of the curve be
f{^,y) = o, (1)
and let f{x, y) be a rational integral function of x and y. Then
|=-|- [Art. 84, (4).J (2)
dy
Now at a multiple point or a cusp -^ has not a single definite
dx
value, and, accordingly, at such points — in (2) must have an
indefinite form, viz. the form -•* Hence, at a multiple point of
curve (1)
Si = 0 and SL = 0. (3)
dx dy ^ ■'
The solutions of Equations (3) will indicate the points which it
is necessary to examine, t At these points
dx 0' ^ ^
the indefinite form in the second member can be evaluated by the
method explained in Chapter XII., Art. 117, and applied in Note
below.l Suppose that the second member of (4) has been evaluated
and the resulting equation solved for — • Then : If — has two
dx dx
real and different values at the point under consideration, the
point is a double point or a salient point; if — has three real
dx
and different values there, it is a triple point ; and so on. If -^
dx
* This is frequently called an '^indeterminate" form. The evaluation of
(so-called) ^indeterminate forms " is discussed in Chapter XII.
t The values of x and y that satisfy Equations (3) may give points that
are not on the curve. Of course these points need not be examined further.
t Or by other methods referred to in Art. 114.
210
DIFFERENTIAL CALCULUS.
[Ch. XIII.
has two real and equal values at the point which is being examined,
the point is a cusp,
is an isolated point.
If -^ has imaginary values at the point, it
If the point is a cusp, the kind of cusp can be found bj' further examina-
tion of the curve in the neighborhood of
the point. For example, if {xi, yi) is
known to be a cusp and it is found that
ioT X = Xi — h {h being infinitesimal), y is
imaginary, then the curve does not extend
through (xi, y\) to the left, and thus the
cusp is not a double cusp. If for x=xi-\-h,
the value of the ordinate of the tangent at
(s^i) 2/1) is less than the ordiuates of both
branches of the curve, the cusp is as in Fig.
69. In a similar way tests may be devised
and applied in special cases as they arise.
Fig. 6<J.
Note. The evaluation of the second member of Equation (2) gives,
by Art. 117, and Art. 81, (5)
dy
dx'
SV JV_dy
dx:^ dy dx dx
dV , dVdy'
(5)
- +
dx dy dy'' dx
If the second member of (5) is not indefinite in form, this equation, on
clearing of fractions and combining, becomes
dVfdyy dV dy ey_
dy'' \dxj "*" dydx dx "^ dx'' ~ '
(6)
a quadratic equation in
dy
dx
By the theory of^uadratic equations, the two
values of ^ are real and different, real and equal, or imaginary, according as
/ ffif \2 "^ 32/ yif
( ^-^ 1 IS respectively greater than, equal to, or less than ^ • ^. Hence,
the point is a double point, a cusp, or a conjugate point, according as
\dydxj ^'-' ^dy^ dx^
If the second member of (5) also is indefinite in form, proceed as required
by Art. 117, remembering that =^ here is constant. The resulting equation
dx
will be of the third degree in ^ .
dx
131, 132.] SINGCLAR POIXTS. 211
EXAMPLES.
1. Examine the curve i* - y' — 7 1^ + 4 y + 15 2 — 13 = 0 for singular
points.
Here ^^_3x^ - 14x + 15,
dx -•2y + i ^ '
On giving each member tlie indefinite form -, and solving the equations
3 x» - 14 z + 15 = 0,
-2y + i = 0,
it results that x = S or f , and y = 2.
Substitution in the equation of the curve shows that x = |, y = 2, do not
satisfy the equation, and that x = 3, y = 2 do. Accordingly, the point (3, 2)
is the point to be further examined.
On evaluating, by the method shown in Chap. XII., the second member
of (1) for the values x = 3, y = 2, it is found that
dy _ 6 X - 14
dx~ dy
; whence (!)'= 2. and § = ±v^.
Thus the curve has a double point at (3, 2), and the slopes of the tangent
there are + V2 and — v'2.
[The curve consists of an oval between the points (1, 2), and (3, 2), and
two branches extending to infinity to the right of (3, 2).]
3. Sketch the curve in Ex. 1.
3. Examine the following curves for singular points :
(1) a5y» = x2(a> - x^). (2) x^ + 9 x^ - y' + 27 x + 2 y + 26 = 0.
(3) ys - xs - 3 y2 + 3 y + 4 X ^ 5 = 0. (4) The curve in Ex. 5 (5), Art. 127.
(5) I* + y* + 3 x-y + 3 xy= - 10 y» - 16 xy - 10 x^ + 25 1 + 29 y - 28 = 0.
(6) xS-y^- 10x3*+ 33 X- 36 = 0.
132. Curve tracing. Some of the matters involved in curve
tracing have been discussed in Arts. 75-78. 125-131. To do more
than this is beyond the scope of a primary text^book on the
calculus. The topic is mentioned here merely for the purpose of
giving a few exercises whose solutions require the simultaneous
application of methods for finding points of maximum and mini-
mum, asymptotes, and singular points.
212 DIFFERENTIAL CALCULUS. [Ch. XIII.
Note 1. For a fuller elementary treatment of singular points and curve
tracing, see McMahon and Snyder, Dif. Cal, Chaps. XVII., XVIII.,
pp. 275-306 ; F. G. Taylor, Calculus, Chaps. XVII., XVIII., pp. 250-278 ;
Edwards, Treatise on Diff. Cal., Chaps. IX., XII., XIII.; Echols, Calculus,
Chaps. XV., XXXI., pp. 147-164, 329-346. The classic English work on the
subject is Frost's Curve Tracing (MacmlUan & Co.), a treatise which is
highly praised both from the theoretical and the practical point of view.*
Note 2. For the application of the calculus to the study of surfaces (their
tangent lines and planes, curvature, envelopes, etc.) and curves in space, see
Echols, Calculus, Chaps. XXXII. -XXXV., pp. 847-390, and the treatises of
W. S. Aldis and C. Smith on Solid Oeometry.
EXAMPLES.
1. Trace the curves in Ex. 8, Art. 160; in Ex. 5, Art. 161; in Ex. 2,
Art. 162 ; in Ex. 3, Art. 165.
2. Trace the following curves :
(1) y2 = 3^(i_a;2). (2) !/2 = j;2(i_x). (3) x«-4j;2j,_2a;!/2 + 4j/2 = 0.
(4) 'IxC = ixy — x^. (5) r = ocos4ff.
133. NOTE SUPPLEMENTARY TO ART. 127.
(Ill this Note parts of Exs. 6, 7, Art. 127, are worked. Figures should be
drawn by the student.)
Ex. 6. Find the asymptotes of the hyperbola
bV - aV = a^V^ (1)
by method (a) Art. 127.
The equation of the tangent at a point P(x\, y{) on (1) is (Art. 61)
y-yy=^i.x-x,).
Hence the K-intercept of the tangent
b^'xi b'^xi xi' ^ ^
and the y-intercept of the tangent
a^b' a^yi yi ^ ^
* A recent important work on curves is Loria's Special Plane Curves, a
German translation of which (xxi. + 744 pp.) is published by B. G. Teubner,
Leipzig.
132, 133] SINGULAR POINTS. 213
When the point P(xi, yi) recedes to an infinite distance along the hyper-
bola, xi and yi each increases beyond all bounds. Accordingly the intercepts
in (3) and (4) both approach zero as a limit. Hence a tangent which touches
the hyperbola (1) at an infinitely distant point passes through the origin.
The equation of the line through the origin (0, 0) and P(zi, j/i) is
!' = !". (5;
X Xi
If line (2) is an asymptote, it passes through the origin ; substitution of
(0, 0) and solution for Vl gives
Xi
Ji a
.-. from (5) and (6) the equations of the asymptotes of the hyperbola are
y = ±^x.
a
Ex. 7 . Examine for asymptotes the parabola
y'' = ipx, (7)
by method (6), Art. 127.
The equation of the tangent at a point P(xi, Vi) on (7) is (Art. 61)
y-yi=^(x-xi). (8)
By analytic geometry, the length of the perpendicular from a point (A, k) to
a line ax + by + c = 0\a
ah + bk + c
length of perpendicular from the origin (0, 0) on the tangent (8)
, 2»x
- J'l + „
y\ _2pxi — yi'
V
l + iPi Vy,2 + ipl'
Since yi" = 4pxi, this reduces to
2/)Xi _ Vp-Xl _ "^PXl rQ\
2 % p Vjti +p Vzi +p Ji^.P-
> Xi
When the point P(xi, yi) recedes to an infinite distance along the pa-
rabola, Xi increases beyond all bounds. Hence, length (9) increases beyond
all bounds. Accordingly, the tangent which touches parabola (7) at an in-
finitely distant point is itself at an infinite distance from the origin, and thus
is not an asymptote.
CHAPTER XIV.
APPLICATIONS TO MOTION. PRELIMINARY NOTE.
134. Speed, displacement, velocity. Suppose a point moves from
0 to P, througli a distance As, in a time AA, either along a
straight line or along any curve (Figs. 70, 71).
O A«
As_
Fio. 70. Fio. 71.
The mean speed of the moving point during the time Af =
As
The speed of the moving point at any instant* = lim^,io —
_ds
~ dt'
(This has been shown in Art. 26.)
The rate of change of speed = — (speed) = ;r ( -r )
dis
dt^'
Displacement. If a point moves from one point to another, no
matter by what path. Us change
of position (only its original and
final positions and no intermedi-
ate position being considered) is
called its displacement.
^ According to this definition,
if a point moves from P to Pi
along any path PAP^ say, its
Y displacement is known com-
Fio. 72. pletely when the length and
• One may also say the speed of the moving point at any point in its path.
214
134.] APPLICATIONS TO MOTION. 215
direction of the straight line PPi are known. A displacement
thus involves both distance and direction. The length of the
line PPi is called the magnitude of the displacement ; the direc-
tion of the line PPj is called the direction of the displacement.
Thus the straight line PPi represents the displacement which a
point has when its position shifts from P to Py
Mean velocity. Velocity. The mean velocity of a moving point
which has a certain displacement in a time Ai
_ its displacement in time A<
Thus the mean velocity, since it depends on a displacement,
takes account of direction. E.g. in Fig. 72, if a point moves along
the curve from P to P-i in a time A<,
its mean speed =?:^5^5^;
At
its mean velocity ^ chord PP,
That is, on denoting the arc and the chord in Fig. 72 by As and
Ac, respectively,
mean speed = — ; (1)
At
Ac
mean velocity = (2)
The velocity of a moving point at any instant *
= lin,^,.„ displacement ^3^
At
This velocity can be represented by the displacement that
would be made in a unit of time were the velocity to remain
unchanged during that time (or remain uniform, as it is termed).
From the above definitions it follows that :
speed involves merely distance and time ;
velocity involves direction as well as distance and time.
• One may also say the velocity at any point.
216
DIFFERENTIAL CALCULUS.
[Ch. XIV.
135. To find for any instant (or at any point) the velocity of
a point which is moving along a curve. It has been shown in
Art. 134, result (2) (see Fig. 72), that when a point moves along
the curve from P to P,,
Ac
its mean velocity
a;
Ac
velocity a,t P — lim^j^o —
= limA,=nf —
Ac As
Now,
As A«_
T Ac ,. As*
As At
= 1 • - [See Arts. 25, 67 (c), (d).J
dt'
Thus the magnitude of the velocity at P is the same as the
magnitude of the speed at P. The direction of the velocity at
P is the same as the direction of the tangent at P; since the
chord PPi approaches the tangent as its limiting position when
A« = 0.
Note. Velocity may change owing to a change in the direction of motion,
or to a change in speed, or to changes in both direction and speed. Thus
the velocity of a point moving in a straight line with ever increasing speed is
changing ; the velocity of a body moving in a circle with uniform speed is
changing ; the velocity of a body moving with changing speed along any
curve is changing.
136. Composition of displacements. Suppose a particle has
successively the displacements a and b.
Fio. 73.
Fig. 74.
* As is not zero when At is not zero.
135, 136.] APPLICATIONS TO MOTION. 217
The resultant of these two displacements can be shown thus :
Through any point 0 draw OA parallel and equal to a ; through
A draw AB parallel and equal to b. A particle which, starting at
0, undergoes successively the displacements a and h, must arrive
at B. The particle would also have arrived at B, if, instead of
having these displacements, it had the displacement represented
by OB. The displacement OB (or a displacement equal and
parallel to OB) is called accordingly the resultant of the displace-
ments a and b.
Fig. 74 shows that " if two sides of a triangle taken the same
way round represent the two successive displacements of a moving
point, th§ third side taken the opposite way round will represent
the resultant displacement."
When there are more than two successive displacements, the
resultant is obtained in a manner similar to the above. Thus,
for example, let a, b, c, represent three successive displacements
of a moving point.
Through any point 0 draw OA parallel and equal to a, through
A draw AB parallel and equal to 6, through B draw BC parallel
and equal to c. A particle which, starting at 0, undergoes suc-
cessively the displacements a, b, c, must arrive at C. The particle
would also have arrived at C, if instead of having these displace-
ments it had the displacement represented by OC. The single
displacement 00 (or a displacement equal and parallel to OC) is
accordingly called the resultant of the displacements a, b, c. The
resultant of any finite number of displacements can be found by
an extended use of the methods used in the preceding cases.
EXAMPLES
1. A point undergoes two displacements, 40 ft. E. and 30 ft. N. Find
the resultant displacement.
218
DIFFERENTIAL CALCULUS.
[Ch. XIV
2. A point undergoes two displacements, 60 ft. W. 30° S. and 30 ft. N.
Find the resultant displacement.
3. A point undergoes three displacements, 12 ft. W., 20 ft. N. W., and
60 ft. N. E. Find the resultant displacement.
4. To an observer in a balloon his starting point bears N. 20° E., and is
depressed 30° below the horizontal plane ; while a place known to be on the
same level as the starting point and 10 miles from it is seen to be vertically
below him. Find the component displacements of the balloon in southerly,
westerly, and upward directions.
137. Resolution of a displacement into components. A displace-
ment can be resolved into component displacements (or, briefly,
components) which have that displacement as their resultant.
This may be done in an unlimited number of ways. For instance,
in Figs. 76, 77, 78, various pairs of components (in light Hues)
are shown for the displacement a.
Fio. 76.
Fig. 77.
Fio. 78.
The components are often represented by drawing them from
0; thus corresponding to Figs. 76, 77, 78, are Figs. 79, 80, 81,
respectively.
Fig. 79.
*P
Fig. 80.
Fig. 81.
Components which are at right angles to one another, like
those shown in Figs. 78, 81, are called rectangnlar components.
If a displacement a is inclined at an angle 0 to its horizontal
projection, the horizontal and vertical components of the displace-
ment (as is evident from Figs. 78, 81) are respectively
a cos e, a sin 6.
136, 138.] APPLICATIONS TO MOTION. 219
EXAMPLES.
1. A particle has a displacement of 12 feet in a direction making an
angle of 86° with the horizon. What are the horizontal and vertical com-
ponents of the displacement ?
2. The vertical component of a displacement of 35 ft. is 24 ft. Find the
horizontal component and the direction of the displacement.
3. The horizontal component of a displacement is 300 ft., and the direc-
tion of the displacement is inclined 37" 20' to the horizon. Find the ver-
tical component of the displacement and the displacement itself.
4. One component of a displacement of 162 ft. is a displacement of 236 ft.
inclined at the angle 78° 40' to the given displacement. Find the other
component.
138. Composition and resolution of velocities. It has been re-
marked in Art. 134 that the velocity of a moving particle at any
instant may be represented by the displacement which the parti-
cle would have in a unit of time were the velocity to become and
remain uniform. Accordingly, velocities may be combined, and
may be resolved into components, in precisely the same manner
as displacements (Arts. 136, 137).
EXAMPLES.
1. A book is moved along a table in an easterly direction at the rate of 2
ft. a second ; at the same time the table is moved across the floor at the rate
of 1 ft. a second in a southerly direction. Find the resultant velocity of the
book with respect to the floor.
2. A steamer is going in a direction N. 37° Fi. at the rate of 18 miles per
hour, and a man is walking on the deck in a direction N. 74° E. at a rate of
3 miles per hour. Find the resultant velocity of the man over the sea.
3. A river one mile broad is running at the rate of 4 miles per hour, and
a steamer which can make 8 miles per hour in still water is to go straight
across. In what direction must she be steered ?
4. A man is driving at a rate of 12 miles per hour in a direction N.
18° 40' E. Find the rate at which he is proceeding towards the north and
towards the east respectively.
8. A train is running in the direction S. 48° 17' AV. at a rate of 32.4 miles
per hour. Find the rates at which it is changing its latitude and longitude
respectively.
220
DIFFERENTIAL CALCULUS.
[Ch. XIV.
139. Component velocities of a point moving along a curve. Let
the rectangular and polar
T coordinates of the point be
// as in Pig. 82.
/ (a) Components parallel to
the axes. It has been seen
in Art. 135 that the velocity
V of the moving point when
it is passing through P has
the direction of the tangent
at P and that in magnitude
ds
v = —
Fig. 82. ^^
When the point moves, its abscissa and coordinate generally
change.
dx
The rate of change of the abscissa x =
dt'
the rate of change of the ordinate y = — -
dt
These are the components of v along the axes ; and thus
,dt
dt
(1)
If the direction of motion PT makes an angle a with the x-axis,
dx
dt
= -y cos a,
dy
~dt'
■■ V sin a.
(b) Components along, and at right
angles to, the radius vector.
In Fig. 82, x = r cos 6, y — r sin 6.
."., on differentiation,
dx adr ■ adO
— = cos 9 r sin 6 —
dt dt dt
dy ■ a'lr , ^d6
-^ = sin 5 — + r cos ^ _ .
dt
dt
dt
(2)
139.]
APPLICATIONS TO MOTION.
221
Now, as is apparent from Fig. 84,
vel. along radius vector 0/*= component
.dx
of — alongit + component of — ^ along it
dy .
dt
dt
:^xcose + ^sin«
dt dt
dr
dt
[from (2) and (3)].
(3) 0
Similarly, it may be seen
[Fig. 85] that
vel. at right angles to radius vector
dx
= — cos 6 ■
dt
di
dt
•sin^
(5)
^ =r^ [from (2) and (5)]. (6)
dt
^'°- ^- From (1) on the substitution
flOT (77/
of the values of — , — , from (2), or, directly from (4) and (6),
dt dt
\dtj \dtl \ dt)
(7)
(9)
Note. The equality of the second members of (3), (4), and the equality
of the second members of (5), (6), can also be deduced from the relations
(see Fig. 82)
r2 = x2 + 2/2 (8) ;
For, from (8), on difierentiation,
whence
e=tan-i^-
X
whence
dt
dt
*'i'
dr _
dt
xdx
r dt
-li-
dr _
dt
cosfl
dt dt
ferent
latioi
x^
1
" dt
x^-
y^
de
dt
dt
"dt
dt~
X-'
+ f'
r^
dt
xdy
rdt
_ydx
rdt
=
cos 6
dt
dt
222
DIFFERENTIAL CALCULUS.
[Ch. XIV.
EXAMPLES.
Note. See Examples, Art. 65.
1. A point is moving away from the cusp along the first quadrant branch
of the curve y'' = a;' at a uniform speed of 6 in. per second. Find the respec-
tive rates at which its ordinate and abscissa are increasing when the moving
point is passing through the point (4, 8). Also find the rate at which its dis-
tance from the cusp is increasing.
Since y^ = a^,
dt
3x2^
dt
at every point on the curve.
Also
On solving (1) and (2),
16^ = 48-,
dt dt
dy_ndx
dt ~ dt
2
Hence at (4, 8)
dx
:36.
(■i=«)-
(1)
(2)
— = 1.897 in. per second ; ® = 5.69 in. per second.
dt dt
Also
in which
dx ^
> + ^sine,
dt
■ tan-i 2.
; 1.897 X -L-t-5.69 :
V5
[Eqs. (3), (4).]
(See Fig. 86.)
: 6.94 in. per second.
dr
dt dt
e = tan-
dr_
dt V5 V5
2. In each of Exs. 1, 2, Art. 65, find the rate at which the moving particle
is increasing its distance from the vertex of the parabola.
3. In each case in Exs. 'i, 5, Art. 65, find the rate at which the moving
particle is increasing its distance from the origin of coordinates.
4. The radius vector in the cardioid r = a(l — cos 6) revolves at a uniform
rate about the pole : investigate the motion of the point at the extremity of
the radius vector. Apply the results to determining the motion of this point
at the following points on the cardioid in which a = 10 inches, when the
radius vector makes a complete revolution in 12 sec, viz. at the points
(1) 10,
i)^
(2)
^''3
(.3) 15
27r\.
3 /'
(4) (20,^).
[Suggestion. Find (a) the velocity of the moving point toward or away
from the pole ; (6) the velocity of the moving point at right angles to the
radius vector ; (c) the velocity of the moving point along the cardioid.]
139, 140.] APPLICATIONS TO MOTION. 223
140. Acceleration. The rate at which a body is moving may-
change, either becoming greater or becoming less ; the direction
of its motion may also change ; again, both the rate and the direc-
tion of its motion may change.
E.g. a train may be moving at one instant at a rate of 10 miles per hour ;
ten minutes later it may be moving at a rate of 40 miles an hour. The rate
at which the train moves has thus increased by 30 miles an hour in ten
minutes.
The change made during an interval of time in the velocity of
a body is called the total acceleration, and also the integral acceler-
ation for that interval. Thus, suppose (Fig. 87 u) a body at one
Fig. 87 a. Fio. 87 b.
moment has a velocity Vi, and at another moment some time later
has a velocity v^. Fig. 87 b shows that the velocity v^ can be
obtained by compounding the velocity AB with the velocity Vi.
Thus AB represents the change that must be made in the velocity
v-i in order that the velocity of the body may become v^. In this
instance AB is called the integral, or total, acceleration of the body.
The mean (or average) acceleration of a body is the result obtained
by dividing the integral acceleration by the number of units of
time that has elapsed while the integral acceleration was in the
making. Thus if (Figs. 87a, b) Vi changed to v^ during an interval
of t seconds,
the mean acceleration = — •
t
This may be called the change in the velocity per unit of time.
The direction of the mean acceleration is the direction of the
integral acceleration.
The instantaneous acceleration of a moving point at any moment,
usually called 'the acceleration,' is the limit, in magnitude and
direction, of the mean acceleration when the interval of time, t, is
taken as approaching zero. The acceleration is usually denoted
by the letter a.
224 DIFFERENTIAL CALCULUS. [Ch. XIV.
In symbols : if the velocity v has a change Au in. a time At,
the acceleration = lim^,^ — ;
At
«=— . (1)
Accelerations have direction and magnitude ; accordingly, they
can be represented by straight lines. Accelerations may be com-
bined and may be resolved into components, in precisely the same
way as displacements and velocities.
Note. Another form for the acceleration o is
„ dv dv ds ^, dv ^9-,
dt ds dt ds
EXAMPLES.
1. The initial and final velocities of a moving point during an interval of
3 hours are 20 miles per hour W. and 16 miles per hour N. 43° W. Find
(o) the integral and (6) the mean acceleration. Also find the easterly and
northerly components of these accelerations.
2. A particle is moving downvpards in a direction making 36° with the
vertical, and the vertical component of its acceleration is 80 ft. per second
per second. Find (a) acceleration in the path of motion and (6) the hori-
zontal component of its acceleration.
141. Acceleration : particular cases.
(a) Acceleration of a point moving in a straight line.
By Art. 140, (1) a = ^.
at
Now v = — ;
dt
d ,. d fds\ d^8 ,^ .
Note. In the case of a point that is moving on a curve, the direction of
the velocity at any point of the curve is along the tangent at that point and
the velocity (Art. 136) is — Accordingly in this case — represents merely
dt dt^
the acceleration of the moving point in the direction of the tangent, the
tangential acceleration, as it Is termed. This is also shown in (6) following.
140, 141.]
APPLICATIONS TO MOTION.
225
EXAMPLES.
1. In the case of a body falling vertically from rest, the distance s fallen
through in t seconds is given by the formula s = J gfi. Show tbat the accel-
eration is g.
2. A point P is moving at a uniform rate round a vertical circle. An
ordinate PM is drawn to meet the horizontal diameter in M. Find the
acceleration of M with respect to the centre of the circle.
3. Suppose that the circle in Ex. 2 has a radius 3 ft. and that P goes
round the circle 25 times per second. Find the acceleration of M: (a) when
P is 20° above the horizon ; (6) when P is 66° above the horizon.
(6) Acceleration of a point moving in a plane curre.* In order to
determine this acceleration at any point two rectangular components
of it are first found ; namely, the
acceleration along the tangent
at the point and the acceleration
along the normal. These are
called the tangential and the nor-
mal accelerations.
Suppose a point moves along
the curve in Fig. 88 from P, to
Pi in a time A<, and let its veloci-
ties at Pi and P^ be v and v + Av,
respectively.
Let PiRi and PiR^ represent these velocities in magnitude
and direction.
Draw P,S equal and parallel to P2R2 and join R^S.
Then R^S represents in magnitude and direction the change in
velocity. A?;, made during the time At. From S draw SQ at right
angles to PiQ, the normal at P^ and draw ;ST at right angles to
P\R\T, the tangent at Pj. Denote the arc P^P^ by As, and the
angle between P^Ri and P2B2 ('-fi- angle TP^S) by A<^.
Denote the tangential acceleration by «,, and the normal accel-
eration by a„. The components of R^S, in the directions of the
tangent and normal at P,, respectively, are RiT and TS, the
latter of which is equal to P\Q.
* See Campbell's Calculus, Art. 25.3.
226
Then
DIFFEBENTIAL CALCULUS.
[Ch. XIV.
a.
FEiT
M
= lim.
= liiiiA
"PiScosA<^-P;
A<
(v + ^v) cos A<^ ■
At
1^1
.J
^]
"■;; (cos A<^ — 1) + A^) cos A<^
■^i^^sin^iA^ A^^ 1
At At ^ J
"sini_A^(— vsin^A^) A(j> ^ Aii
+ ^^^ cos A <i
At At
= 1.0.^ + ^
dt dt
.dv_dS8
dt iiH
(2)
P 0
Further a„ = limA,^— i-^ = li
At
:liin.
At
'/ , . sSinA</) A</> A.s
•^ A<^ As At
At
ds dt
-■v-1- — • V
ds
Arts. 98-101
)
(3)
in which r denotes the radius of curvature at the point.
.-. the actual, or resultant, acceleration
'4
dty j-2 ■
Special case. When a point is moving uniformly in a circle,
there is no tangential acceleration. The acceleration at any point
is then wholly directed towards the centre and its magnitude is — .
r
Ex. Show that when a point moving with uniform speed goes
round a circle of radius r in time t, its acceleration at any instant
has the magnitude ^ ■
141.] APPLICATIONS TO MOTION. 227
EXAMPLES.
4. A circvis rider is moving with the uniform .speed of a mile in 2 min.
40 sec. round a ring of 100 ft. radius: find liis acceleration towards the centre.
5. A point moving in a circular path, of radius 8 in., has at a given posi-
tion a speed of 4 in. per second which is changing at the rate of 6 in. per
second per second. Find (a) the tangential acceleration ; (6) the normal
acceleration ; (c) the resultant acceleration.
6. A particle is moving along a parabola j/^ =4x, tlie latus rectum of
which is 4 inches in length, and when it is passing through the point P (4, 4)
its speed, which is there 6 in. per second, is increasing at the rate of 2 in. per
second per second. Find at P, (a) its tangential acceleration; (6) its normal
acceleration ; (c) its integral acceleration.
7. If the particle in Kx. 6 were moving at a uniform rate of 6 in. per
second, what would be its acceleration at P'/
Note 1. When a point is moving along a curve, the coordinates x, y
of its position are continually changing. The components of its acceleration
at P (a;, y) which are parallel to the x and y axes are respectively [compare
Art. 139(a)] ^ ^
dt^' dt^'
If the tangent to the curve at P makes an angle a with the a;-axis, then,
as is apparent from a figure, the tangential acceleration
d^3 (Pa; „„„ , d^y ■ ,^,
— = — cos o H sm a (4)
Relation (4) Mows also from result (1) Art. 139 (a), viz.,
\dt) \dt) \dtj '
For, on difierentiation.
ds ^ _ ^ ^ 4. ^ . ^ . (■§•)
dt' dt^~ dt' dfi dt' df^'
whence ^^dx ,^ d^,^
dfl ds dfi ds dfi'
*•«• ^=cosa.^ + sina.^. (6)
dfl d«2 dt^
Note 2. Afl^lar Telocity. Angular acceleration. The mean rate at
which a straight line revolves about a given point (i.e. mean rate at which
it describes an angle from a certain initial position) is called the mean angu-
lar velocity of revolution.
E.g. if a straight line revolving about a point describes the angle - in
o
4 sees. , its mean angular velocity per second is ^ -h 4, i-e. — radians per second.
228
DIFFERENTIAL CALCULUS.
[Ch. XIV.
The instantaneous angular velocity, commonly called the angular veloo-
ity, at a particular moment, A9 denoting the angle described in a time At,
,. A9 d9
The angular acceleration at any moment is the rate of change of the
angular velocity. Accordingly, ,
angular acceleration = — ( — 1= — z'
(7)
EXAMPLES.
8. A wheel is rolled at a uniform rate along a straight line; investigate
the motion of a fixed particular point P on the vrheel.
Tlie particular point P on the wheel describes a cycloid. If the axes be
chosen in the usual way, the equations of the cycloid are
x = a{e -sine) 1
y = a(l - cosff) / ^ -'
in which a denotes the radius of the wheel and 6 denotes the angle through
which the radius through P turns after P has been on the straight line.
Fig. 89.
It is required to investigate the motion of the point P of the wheel at any
point on Its cycloidal path.
tiff
Since the wheel is rolling at a uniform rate, — has a constant value and
dt
accordingly — =0.
dt^
In Fig. 89 PT is the tangent to the cycloid at P, and PN is the normal.
From (8), on differentiation,
— = (1(1 — cos 9) —
dt dt
dy _ „!„ a d0
-s- = a sin 6 —
dt dt
(9)
— = a sm d\ — I
dt^ \dtj
d^
dt^'
: a COS $
dev
(I)'
Hence, on substitution in Art. 139, Eq. 1,
velocity u a« P = - = 2 a sin - . — .
dt 2 dt
(10)
(11)
141] APPLICATIONS TO MOTION. 229
From (11), on differentiation, and Art. 141, Eq. 2.
the tangential acceleration at P, a, = — = a cos - f — V ri2^
df' 2\dt] ^ '
Result (12) can also be derived from Eq. (5), Note 1, on substitution of
the values of the derivatives from (9), (10), (11), above.
Result (12) can also be derived from Eq. (4), Note 1, on observing that the
tangent Pr makes an angle 90 — - with the a;-axis.
2
The radius of curvature rat P [Art. 101, Ex. 5 (8)] = 4 a sin -■ (13)
Hence by Art. 141 and Eqs. (11) and (13) above,
) ide\
\\dtl
, t ■ .-,9 /rtS\2
4 a^'sm^- ' ^
the normal acceleration at P, a„ = — = .
4 a sin -
2
="'"I(S)' <»'
(15)
. ■. integral acceleration at P = Vai' + </„- = a 1 — 1
\dtl
On making a figure showing the accelerations ( 12) and (14), which are
directed along PT and PN respectively, it will be apparent that acceler-
ation (12) makes an angle - witli the resultant acceleration. Accordingly,
the resultant acceleration of the point on the wheel at any point on its
cycloidal path is constant, and is always directed towards the centre of the
wheel.
9. Suppose the wheel in Ex. 8 has radius 2 feet, and is pushed along at a
rate of 3 miles an hour. Calculate the velocity and the tangential, normal,
and integral accelerations of a point on the wheel the radius to which makes
an angle of 60° with the vertical radius downward from the centre.
10. K the wheel in Ex. 8 is not rolling at a uniform rate, show in each of
the three ways indicated for deriving result (12) in that example, that the
tangential acceleration at P is
2„sin^^+acosfW.
2 dt'^ 2 \dtj
CHAPTER XV.
INFINITE SERIES.
EXPANSION OF FUNCTIONS IN INFINITE SERIES. DIFFEREN-
TIATION OF INFINITE SERIES. SERIES OBTAINED BY
DIFFERENTIATION.
N.B. There are some students whose time is limited and who require to
obtain as speedily as may be a working knowledge of Taylor's and Mac-
laurin's expansions. These students had better proceed at once to Arts. 149,
154, work the examples in Arts. 160 and 152, and then take up Art. 148.
It is, perhaps, advisable in any case to do tliis before reading this chapter and
the other articles in Chapter XVI. Those who are studying the calculus as
a "culture" subject should become acquainted with the ideas and principles
described, or referred to, in Chapters XV., XVI. A thorough understand-
ing of these ideas and principles is absolutely essential for any one who
intends to enter upon the study of higher mathematics.
142. Infinite series : definitions, notation. An infinite series
consists of a set of quantities, infinite in number, which are con-
nected by the signs of addition and subtraction, and which suc-
ceed one another according to some law. A few infinite series of
a simple kind occur in elementary arithmetic and algebra.
For instance, the geometrical series
' + i + i + - + 2^ + l^ + 2^+-^ (^)
the geometrical series
'i.+x + x^+-: + x''-'^ + x'' + x"+i+..-, (2)
which may also be obtained by performing the division indicated in —^ ;
the geometrical series I — x
1.-x + x''+..-+(-l)'>x^-' + ..., (.3)
which may also be obtained by performing the division indicated in — — ;
the geometrical series ^ + x
a + ar + ar^ + ■•■ + ar"-^ + ar" + ar'+'^ -f •■• ; (4)
the series i -i- i- + J--i- ... j. J_ j. ... /■fi^
1" 2p^p up ' ^'
2ao
1*2, 143.] INFINITE SERIES. 231
The successive quantities in an infinite series, beginning with
the first quantity, are usually denoted by
Mo, Ml, Mj, •••, M„_i, M„, tJ„+i, •••;
or, in order to show a variable, x say, by
u„(x), Ui(x), ic,(x), ■■; M„_,(.'<;), M„(a;), «„+,(«), ....
Then the series is
?<0 + Ml + M2 H h M„_i + M„ + M„+, + .... (6)
The value of the series is often denoted by s ; and the symbol »„
is generally used to denote the sum or value of the series obtained
by taking the first n terms of the infinite series ; thus,
Sn = Wo + Ml + M2 -I h M„_i.
The value of the infinite series (6) is the limit of the sum of the
quantities in the series; i.e. the value of the series is the limit of
the sum of n terms of the series when n increases beyond all
bounds.* This is expressed in mathematical symbols
s = liin„it„ s„. (7)
(This limit s is frequently, but not quite correctly, called " the
sum of the series " or " the sum of the series to infinity.")
Thus, in (1), s„=l+l + l+... + J_ = 2/'l-— V
and hence
s = liin,^^aoS„ = 2 ;
(7)
in (2),
s„ = 1 + a; + a;2 + ... + x»-i = ?!!^,
and hence
s = lin],^„ s„ = 00 when x^l and x S -
-1,
(8)
= — - — when — 1 < 2;< 1.
(9)
143. Questions concerning infinite series. The subject of infinite
series is highly important in mathematics. Such questions as the
following arise and require to be answered :
(a) Under what conditions may infinite series be employed in
mathematical investigation and used in practical work ?
* Thus s is not the sum of an infinite number of terms of the series, but is
the limiting value of tliat sum.
232 DIFFERENTIAL CALCULUS. [Ch. XV.
(6) Under what conditions may an infinite series be used to
define a function or employed to represent a function ?
Thus, in Art. 167, result (8) shows that series (2) does not represent the
function when x is greater than 1 or less than — 1 or equal to 1 or — 1.
i — X
This is obvious on a glance at the series ; in fact, the greater the number of
terms of (2) that are taken, the greater is the error committed in taking the
series to represent the function. (For instance, put a; = 2 ; then the func-
tion is — 1 and the series is + oo.) On the other hand, the infinite series (2)
does represent the function when x lies between — 1 and + 1 ; the
1 — X
greater the number of terms that are taken, the more nearly will the sum of
these terms come to the value of the function. The limit of the sum of these
terms when the number of them is infinite is the function.
(c) May two infinite series be added like two finite series ? In
other words, if
W = Wj 4- «j + M2 + • • •
and V = Vo + Vx + Vi-\ ,
is u-irV-Ua + v^ + Ui + Vi-\ (1)
a true equation; and under what conditions is (1) a true equation?
(d) May two infinite series be multiplied together like two
finite series ? In other words, u and v being as in (c), is
UV = U^Vo + UffVi + '\(iVo + U^V^ + UfVi + U,Vi + • •• (2)
a true equation; and under what conditions is (2) a true equation ?
(e) May the principles of Art. 31 and Art. 174 A, namely, that
the derivative and the integral of the sum of a finite number of
terms are respectively equal to the sum of the derivatives and the
sum of the integrals of these terms (to a constant), be extended
to infinite series ? That is, Wqi '^d Wj, ■••, being functions of x, if
S = Mo + W1 + M2+ •",
are | sdx = | v^dx + j Uidx + j u^x + •••, (3)
-
dx\
143. 144.] INFINITE SERIES. 233
true equations; and what are the conditions which must be
satisfied in order that these equations be true ? Equations (3) and
(4) may be expressed :
J lim„=i<. s„{x) \c}x = lim,^ J s„(a;)da; 1,
'-\ lim^s„(a!) =lim^ di*"*^'^M*
The above questions then may be stated thus : Is the integral
of the limit of the sum of an infinite number of quantities equal to
the limit of the sum of the integrals of the quantities ; and is it
likewise in the case of the differentials ?
For instance, given that — — =1 + x + z^ + x^ + •■•,
1 — X
dx\l-xjL (l-x)'J
and is r-^r,-.e. log -JL-"|=x +?!+?-%...?
144. study of infinite series. Knowledge, elementary knowledge at
least, of the theory of infinite series, and practice in their use are necessary in
applied mathematics. Infinite series frequently present themselves in the
theory and applications of the calculus, and accordingly the subject should
be studied, to some extent at least, in an introductory course in calculus.
The better text-books on algebra, for instance, among others, Chrystal's
Algebra (Vol. II., Ed. 1889, Chap. XXVI., etc.), Hall and Knight's Higher
Algebra (Chap. XXI.), contain discussions on infinite series and examples for
practice.* Osgood's pamphlet, Introduction to Infinite Series (71 pages,
Harvard University Publications), gives a simple, elementary, and excellent
account of infinite series. "This pamphlet is designed to form a supplemen-
tary chapter on Infinite Series to accompany the text-book used in the course
in calculus." Recent text-bonks on the calculus, in particular those of
McMahon and Snyder, Lamb, and Gibson, contain definitions and theorems
on infinite series ; they will especially well repay consultation. More
elaborate expositions of the properties of infinite series, which form parts of
introductory courses in modern higher analysis, are given in Harkness and
Morley, Introduction to the Theory of Analytic Functions, in particular
• Also see Hobson, A Treatise on Plane Trigonometry, Chap. XIV., and
following chapters.
234 DIFFERENTIAL CALCULUS. [Ch. XV.
Chaps. VIII -XI. , and in Whittaker, Modern Analysis, in particular Chaps.
II. -VIII. These discussions can be read, in large part, by one who possesses
a knowledge of merely elementary mathematics.
A statement of a few of the principal definitions and theorems which are
necessary for an elementary use of infinite series is given in Arts. 145-147.
145. Definitions. Algebraic properties of infinite series. An
infinite sesies has been defined in Art. 142. If (see Art. 142)
lim„i„ s„ is a definite.finite quantity, U say, the series is called
a convergent series, and is said to converge to the value U. If s„
does not approach a definite finite value -when n approaches
infinity, the series is called a divergent series. In a divergent
series, when n approaches infinity, s„ may either approach infinity,
or remain finite but approach no definite value.
Thus, in Art. 142, series (1) is convergent; series (2) is convergent for
values of x between — 1 and + 1, for then s = ; series (4) is convergent
1 — x
when r lies between — 1 and + 1, for then s = " ■ Series (5) is con-
1 - r
vergent for;; > 1, and divergent forp = 1 and for;) < 1. (Hall and Knight,
Algebra, p. 235.)
[Note 1. The harmonic series. When j) = 1, series (5) is
1+UUUU... + 1 + - 1 ■
2 .3 4 6 n n + 1
This series is called the harmonic series.'}
The series 1 + 2 + 3 + ^-n+ — is divergent. The series 1 — 1 + 1 — 1 +
... + (-1)""' + .-•, obtained by putting a; = 1 in series (3), is divergent ; for
its limit is 0 or 1 according as n is even or odd. (A series that behaves like
this is said to oscillate. Some writers do not include oscillatory series among
the divergent series.)
In general only convergent series are regarded as of service in
applied mathematics. (For the necessity of the qualifying phrase
" in general," see Note 2.) A series may be employed to represent
a function, or, what comes to the same thing, a function may be
defined by a series, if the series is convergent. Thus series (2),
Art. 142, may be used to represent or to define , if x lies
1—x
between — 1 and + 1. [See questions (a) and (6), Art. 143.*]
• Carl Friedrich Gauss (1777-1855), the great mathematician and astrono-
mer of Gottingen, and Augustin-Louis Cauchy (1789-1857), professor at the
145.] INFINITE SERIES. 235
Note 2. On diTcrgent series. Those who apply mathematics, astrono-
mers in particular, have frequently obtained suflBciently good approximations
to true results by means of divergent series. Such series, however, " cannot,
except in special cases, and under special precautions, be employed in mathe-
matical reasoning" (Chrystal, Algebra, Vol. II., p. 102). At the present
time considerable attention is being paid by mathematicians to divergent
aeries and to investigations of the fundamental operations of algebra and the
calculus upon them. A vfork on the subject has recently appeared, viz.
Leqons sur les series divergentes, par ifemile Borel (Paris, Gauthier-Villars,
1901, pp. vi-i-182). "It is safe to say that no previous book upon diver-
gent series has ever been vrritten." Interesting and instructive information
concerning divergent series will be found in reviews on this book, by G. B.
Mathews {Nature, Nov. 7, 1901), and E. B. Van Vleck (Science, March 28,
1902).
Absolutely convergent series. A series the absolute values (see
Art. 8, Note 1) of whose terms make a convergent series is said
to be absolutely or unconditionally convergent; other convergent
series are said to be conditionally convergent.
Ex. 1. Series (1), Art. 142, is an absolutely convergent series.
Ex.2. The series l-J + i-J + i (a)
may be written (1 - i)-(-(i - i)-f (i - i)+ -, i.e. 1 + ^t + -,\ + —.
Series (o) may also be written
i-a-i)-a-i)-. »■•«• i-i-A---
Thus the value of the series (a), the terms being taken in the order indi-
cated, is less than 1 and greater than J. It can also be shown that this series
converges to a definite value. On the other hand (see Note 1, and the state-
ment just preceding Note 1), the series
l+i+i+i+-
is divergent. Thus series (a) is a conditionally convergent series.
Theorems. (1) If a series is absolutely convergent, it is obvious
that any series formed from it by changing the signs of any of
the terms is also convergent.
Polytechnic School at Paris, who did mucli to make mathematics more rigor-
ous than it had been during its rapid development in the eighteenth century,
may be regarded as the founders of the modern theory of convergent series.
James Gregory, professor of mathematics at Edinburgh, introduced the terms
convergent and divergent in connection with infinite series in 1668.
236 DIFFERENTIAL CALCULUS. [Ch. XV.
(2) la a conditionally convergent series it is possible to rearrange
the terms so that the new series will converge toward an arbitrary
preassigned value.
(3) In an absolutely convergent series the terms can be rearranged
at pleasure without altering the value of the series.
(4) If (see Art. 143) u and v are any two convergent series, they
can be added term by term; that is, Equation (1), Art. 143, is true.
(5) If M and v are any two absolutely convergent series, they
can be multiplied together like sums of a finite number of quanti-
ties ; that is, Equation (2), Art. 143, is true.
For proofs and examples of these theorems see Osgood, Intro-
duction to Infinite Series, Arts. 34, 35 ; Chrystal, Algebra, Vol. II.,
Chap. XXVI., §§ 12-14.
In a convergent series as n increases, s„ may either: (a) con-
tinually increase toward the limiting value of the series ; or
(b) decrease toward this limit ; or (c) be alternately greater than
and less than its limit.
Thus in series (1), Art. 142, s„ continually increases toward its limit (2);
in the series 1 1- — — ; -|- ■••, «» is alternately greater than and less than
its limit f 2 2 2
Remainder after n terms. The symbol r„ or JJ„ is often used to
denote the series (and also to denote the value of the series)
formed by taking the terms after the nth, thus
'V = «n + «»+l + Un+2 H .
This is usually called the remainder after n terms. Let a func-
tion be represented by a convergent series ; i.e. let the value of
the function be equivalent to the value of this convergent series.
Then since ^.x t ^- ^■
the function = lim„i„ s^
it follows that lini„i„ r„ = 0.
Interval of convergence. In general a convergent series, in a
variable, x say, is convergent only for values of a; in a certain
interval, say from x = a to x=b. The series is then said to con-
verge within the interval (a, &), and this interval is called the
interval of convergence.
145, 146.] INFINITE SERIES. 237
Thus in series (2), Art. 142, the interval of convergence extends from
a; = — 1 to a; = + 1. In this case, as in many others, the series is not conv^f-
gent for the values of x (in this case — 1 and + 1) at the extremes of the
interval. In some cases series are convergent for the values of the variable at
the extremes of the interval of convergence as well as for the values between ;
in other cases a series may be convergent for the value of the variable at one
extreme of the interval but not for the value at the other.
Power series. Series of the type
a© + ajx + 02x2 + ••• + a„x» •••,
in which the terms are arranged in ascending integral powers of x
and the coefficients are independent of x, are called power series
in X. A power series may converge for all values of x, but in
general it will converge for some values of x and diverge for others.
Theorem. In the latter case the interval of convergence ex-
tends from some value a; = — ?• to the value x = + r; i.e. the value
a; = 0 is midway between the values of x at the extremes of the
Dlveroent Convergent Divergent
-r o +*■
Fig. 90.
interval of convergence. Thus in the power series (2), Art. 142,
the interval of convergence extends from — 1 to +1. This theo-
rem may be graphically represented, or illustrated, by Fig. 90.
(For proof of the theorem see Osgood, Infinite Series, Art. 18.)
146. Tests for convergence. Two simple tests for convergence
will now be shown. For nearly all the infinite series occurring
in elementary mathematics these tests will suffice to determine
whether a series is convergent or divergent. These two tests are :
(A) the comparison test and (B) the test-ratio test.
A. The comparison test. Let there be two infinite series,
and v^ + v^ + v.A l-'y„-i + v„ + -••. (2)
If series (1) is convergent, and if each term of series (2) is not
greater than the corresponding term of series (1) (i.e. if v„ ^ u„
for each value of n), then series (2) is convergent. If series (1)
238 DIFFERENTIAL CALCULUS. [Ch. XV.
is divergent, and if each term of series (2) is greater than the
corresponding term of series (1), then series (2) is divergent.
Two series which are very useful for purposes of comparison are :
(a) The geometric series
which is convergent when | r | < 1, divergent when | »• | ^ 1.
(6) The series l + l+l+i + ...,
which is convergent when p > 1, divergent when ^ ^ 1 (see Art.
145).
Ex. 1. The series 1 + i + fj + s^ + -
is convergent, for it is term by term not greater than the geometric con-
vergent series j + j + ^^ ^ ^ + ....
B. The test-ratio test. In series (6), Art. 142, the ratio
M«+l /3^
is commonly called the test-ratio. If when n increases beyond all
bounds this ratio approaches a definite limit which is less than 1,
then the series is convergent. For, suppose that ratio (3) is finite
for all values of n, and suppose that after a certain finite number
of terms, say m terms, it is less than a fixed number R which is
less than 1. Now
S = Ml + M2 + ... + M„ + U„+-^ + M„+j + ....
The sum of the first m terms is finite. Since
it follows that the series beginning with u„ is less than the
geometric series ^__^(i + ^ + ^.+ ...)^
and, accordingly, is less than
1
"l-.B
140.] INFINITE SERIES. 239
Hence s < s„ + m„ -»
1 — it
and thus the series is convergent.
If when n increases beyond all bounds the test-ratio approaches
a definite limit which is greater than 1, the series is divergent.
Ex. 2. Prove the last statement.
If the limiting value of the test-ratio is + 1 or — 1, further special investi-
gation is necessary in order to determine whether the series is convergent or
divergent. •
Thug the quality of the series, as regards its convergency or
divergency, depends upon
lim . ^«+l
Hin,^x> •
Un
EXAMPLES.
3. Find whether the following series are convergent or divergent :
(1) ^— + ^— + ^- -I- ^— -I- -, (2) 1 -I- — + — + — -I- ....
*■ -^ 1 -2 2 .3 3 -4 4 -5 '^^ 21 3! 41 '
4. Examine the following series for convergency :
(1) l-)-3x-)-5a:2-|-7x» + 9a^-|--, (2) V + 2-^x^2,^ x-^ + ^^x^-y b'^oi^ + -.,
(3) ^+T + ^^3!'^4I+ ' ^' 1.2 + 3.4 + 5.6+7.8^ '
^'^ i+i+l+io+-+;;^+-- ^'^ ^-3i+5-i-r.+--
* A series in which the absolute value of the test-ratio tends to the limit
unity as n increases, will be absolutely convergent if, for all values of n after
some fixed value, 1 + c
this absolute value < 1 ^,
— n
where c is a positive quantity independent of n. (For a proof of this general
theorem, see Whittaker, Modern Analysis, Art. 13.)
240 DIFFERENTIAL CALCULUS. [Cb. XV.
147. Differentiation of infinite series term by term. It is be-
yond the limits of a short course in Calculus to investigate the
conditions under which an infinite series can properly be differ-
entiated term by term ; in other words, to determine what condi-
tions must be satisfied in order that Equation (4), Art. 143, (e),
may be true.*
It must suffice here merely to state the theorem that applies to
most of the series that are ordinarily met in elementary mathe-
matics, viz. :
A power series t can he differentiated term by term for any value
of X within, but not necessarily for a value at, the extremities of its
interval of convergence. (For proof see Osgood, Infinite Series,
Art. 41.) See Art. 197.
148. Examples in the differentiation of series.
In this article the results are obtained by application of the
theorem in Art. 147.
EXAMPLES.
1. It Is known that (see Art. 152, Ex. 7)
the second member of (1) is a, power-series ; accordingly, the theorem of
Art. 147 applies.
On differentiation of each member of (1),
£^-)
= 1+.+!^+.
,..
= e*, as already known.
2.
It is known that
(see Art.
sin a; =
162, Ex. 2)
3! 5!
..(1).
On differentiation,
cos X -
1 a;2 X*
..(2). (See
Art.
152,
Ex.
6.)
3. Derive expansion (1) from (2) of Ex. 2 by differentiation.
4. When - 1< a; < 1,
-l-=l+x + x'i + x?+..: (1)
1 — X
* On this, see Infinitesimal Calculus, Art. 173, especially Note 2 of that
article for references. t See page 237.
147, 148.] INFINITE SERIES. 241
On diSerentiation,
1
■ = l+2x + 3a;2 + 4x8 + .
(1 - ^Y
On difierentiation and division by 2,
^ , = 1(1. 2 + 2. 32 + 3.4*2 + ...).
{\-xy 2'
5. Show by successive diSerentiation of the members of Ex. 4 (1) that
(l-x)" ^ 1.2 1.2.3
6. It is known that (see Art. 150, Ex. 2)
log(l + a;)=x- Jx2 + |aJ'..., (1)
a series which is convergent if — 1 < a; ^ 1.
On diSerentiation in (1),
-L.=l-x + x-^...; (2)
1 + X
which is true if — 1 < i< 1, but not if z = 1.
CHAPTER XVI.
TAYLOR'S THEOREM.
(See N.B. at beginning of Chapter XV.)
,149. Taylor's theorem is one of the most important theorems
in the calculus. It has a wide application, and several important
series, for example, the binomial series (see Ex. 6, Art. 150) can
be derived by means of it. Let fix) be a function of x which is
continuous throughout the interval from x = atox=h, and which
also has all its derivatives continuous in this interval. Now let
X receive an increment h. Taylor's theorem is a theorem which
gives the development of the function f{x + li) in a power series
in h. The power series itself is called Taylor^s series. (See Note
2, Art. 152.)
N.B. In reading this chapter it is better to take up Art. 154
first.
150. Derivation of Taylor's theorem. Let /(x) and its first 7i
derivatives be continuous in the interval from x = a to x=b. It
has been proved in the extended theorem of mean value (Art. 113,
Eq. 4) that, on denoting
b — a hj h,
f(a + h) =f(d) + hf{a) + ^f"(a) + 11/"' (a) + ■
+ ^r"\a + eji).
(8)
If X and x + h denote any values in the interval for which f{x)
and its first n derivatives are continuous,
i.e. ii a-^x^b, and a < x + h -^b,
242
149,150.] TAYLOR'S THEOREM. 243
then theorem (8) holds true for f{x-\-h). On replacing a in
(8) by X there is obtained
/(as + h) =/(x) + hf'(x) + ^/"(x) + ... + , ^""' /"-^(a;)
+ ^|/'"K» + eft),o<e<i. (9)
This is Taylor's theorem with the remainder, the last term of the
second member being denoted as the remainder. In formula (9)
X and x + h must both be in the interval of continuity ; in any
particular application of this formula, x has a fixed value and h
varies. Theorem [or formula] (9) is true for all functions which,
with their first n-derivatives, are continuous in the assigned inter-
val of continuity. If all the derivatiues of f(x) are continuous in
the interval, and if
n !
then f{x + K)= fix) + hfKx) + |^/"(aj) + |^/"'(a;) + .... (10)
For (by Art. 145) the infinite series in the second member converges
to the value of /(x -f- K) and, accordingly, represents the function
/(a; + A). Formula (10) is called Taylor's theorem, and the
series is called Taylor's series. In (9) and (10) h, may be positive
or negative, so long as a; and x + A are in the interval of con-
tinuity. " The remainder" the last term in (9), represents the
limit of the sum of all the terms after the nth term of the infinite
series in (10) ; it is the amount of the error that is made when
the sum of the first m-terms of the series is taken as the value of
the function.
Note. The method in Art. 110 of proving the theorem of mean value was
first given by Joseph Alfred Serret (1819-1885), professor of the Sorbonne in
Paris, in his Cours de calcul differentiel et integral, 2« 6d., t. I., page 17 seg.
The above proof of Taylor's theorem appears in Hamack's Calculus (Cath-
cart's translation, Williams and Norgate), pages 65, 66, and in Gibson's
Calculus, pages R90S93. The proof in Echols's Calculus (p. 82) is likewise
based on the theorem of mean value.
Taylor's theorem and series are important in the theory of functions of
a complex variable, and are more fully investigated in that subject.
244 DIFFERENTIAL CALCULUS. [Ch. XVI.
EXAMPLES.
1.
Express log (x
+ A) by an infinite series in ascending
powers
of ft.
Here
/(x + A)
= log (X + A).
.-.fix)
= logx,
fl{x) :
_1^
X
/"(x)
1
X2'
/'"W
= |,etc.
.Mog(x + ft)=logx + ^--!^^ + J^-^+....
X 2 x^ 3 x^ 4 X*
Here x must not be 0, for then /(x) =— oo, and thus is discontinuous for
X = 0. The series is evidently more rapidly convergent the smaller is h and
the larger is x.
On putting x = 1 and A = 1, this result gives
log2 = l-J+l-J + ...,
as found in Ex. 3, Art. 198.
It the finite series in (9) is used, then
log(x + ft)=logx + ^ + ^^+... + (-l).-i_^-^,0<.<l.
Here, if x = ft = 1,
log2 = 1 - i + i - i + ... + (- l)-i . ^ .
On interchanging h and x in formula (10), if that can be done
in the interval of continuity, there is obtained the following
form of Taylor's theorem :
/(« + ft) = /(ft) + xf{h) + |H /"(ft) + 1? /'"(ft) + ..., (11)
a form which is often useful. Similarly in the case of formula (9).
2. Express log (x + ft) by an infinite series in ascending powers of x.
Here /(x + ft) = log (x + ft). .-. /(ft) = log ft, /'(ft)= \, f>\h) = - f , etc
ft h^
.Mog(x + ft)=logft + |-^ + ^-....
Ifft = l, log(l + x) = x-|' + |-^+-,
as otherwise obtained in Ex. 3, Art. 198.
sin fi: + -LV = sin ^ + i cos ^ !^ sin ^ i cos ^ + ....
V3 100/ 3 100 " ^- " ""•
150.] TAYLORS THEOREM. 245
3. Represent sin (x + K) by an infinite series in ascending powers in h.
Here f{x + A) = sin (z + A). .-. f{x) = sin x, f'(x) = cos z, f"{x) = - sinx,
etc.
Hence, on using formula (10),
hi hi fit
sin (X + fe) = sin X + ft cos z — — sin I — — cos a; + — sin x + ....
2! 3! 41
Let X = ^, and ft = jjj of a radian (i.e. 34' 22".66).
Then
J_
3 (100)'i2!"^"'3 (100)83! '"3
This is a rapidly convergent series.
Now sin ^ = .86603, cos ^ = .50000. On making the computations, it will
be found that, to Jive places of decimals, sin 60^' 34' 22". 65 = .87099.
Note. The last exercise is an example of one of the most useful practical
applications of Taylor's tlieorem. Namely, if a value of a function is
known fur a particular value of the variable, then the value of the function
for a slightly different value of the variable can he computed from the known
value by Taylor'' s formula. (See Art. 27, Notes 1,3; Art. 82, Note 3.)
4. Expand sin (z + ft) in a series in ascending powers of z.
In this case form (11) is to be used. Here /(z + ft)= sin (z + ft).
.-. /(ft) = sin ft, /'(ft) = cos ft, /"(ft) = - sin ft, /"'(A) = - cos ft, etc.
.•. sin (x + A) = sin ft + a; cos ft — — sin ft cos ft H — .
On letting ft = 0, the following important series is obtained :
sinz = x- — + — .
31 51
5. Expand cos (x + ft) in series, (a) in ascending powers of ft, (6) in
ascending powers of x. From the latter form deduce the series
1 x2 z«
cosx = l-- + --....
6. Expand (x + ft)" by Taylor's formula in a power series in ft, and
thus obtain the Binomial Expansion
(X + ft)" = x" + TOX"-ift + '"•"'~^ z"-2ft2 + ....
(This series is convergent for ft < 1, divergent for ft > 1. The case in which
h = ±l requires special investigation.)
246 DIFFERENTIAL CALCULUS. [Ch. XVI.
7. Given that f(x) = 4x^ -3x^+7 x + 5, develop f(x + 2) and /(a; - 3)
by Taylor's expansion. Then find /(x + 2) and /(x - 3) by the usual
algebraic method, and thus verify the results.
8. (1) Assuming sin 42°, compute sin 44° and sin 47° by Taylor's
expansion. (2) Assuming cos 32°, compute cos 34° and cos 37° by Taylor's
expansion. (3) Do further exercises like (1) and (2).
9. Derivelog(a; + A) = logft + |-^, + ^3-^, + -, when |a:l<l;
log(^ + A)=loga: + ^-^^ + ^,— •, when |xl>l.
10. Show that
fj'2 ftS COS X
log sin (x + a) = log sin x + a cot x — — esc- x + — -^- — I- ■•■■
o - ' ■> " 2 3 sin^ X
151. Another form of Taylor's theorem. This form expresses
f(x) as a series in ascending powers of (x — a). On writing x for
b in Art. 113, Eq. (3), and in the value of x„, two lines after that
equation, there is obtained
/(a;)=/(a)+(x-a)/'(a)+ kx-a)2/"(a) + -+ ^^=^J^/(»-iHa)
+ ^^=^V»)[a+e(x-a)],0<e<l. (1)
n!
If all the derivatives of f(x) are continuous in the assigned
interval, and
lim„,„ (^^Ii^>"'[« + Oix - a)] = 0,
n !
then (Art. 145) the infinite series /(a) + (a;-a)/'(a)+Ka' -«)'/"(«)
+ ••• represents the f unction /(x) * ; i.e.
fix) = f(a) + {x- a)f'(a) +(«'-«)V"(a) + («'-«)V'"(a) + ...
^(x,zSL)lfin)^a)+.... (2)
n 1
Forms (1) and (2) for Taylor's theorem and series, are fre-
quently useful. The last term in the finite series (1) is Lagrange's
form of the remainder in Taylor's series. (See Note 4, Art. 152.)
• Except in some rare cases.
150,152.] TAYLOR'S TIIEOUEM. 247
EXAMPLES.
1. Express 5 a;^ + 7 x + 3 in powers of i — 2.
Here f{x) = 5 x^ + 7 a: + 3, .-. /(2) = 37,
f<{x) = \Ox + ^, /'(2)=27,
/"(x) = 10, /"(2) = 10,
/"'(a!)=0, /"'(2)=0.
Now by (2), f{x) =/(2) + (x - 2)/'(2) + (2-=^/"(2) + ....
.-. 5 x2 + 7 X +3 = 37 + 27(x - 2) + 5(x - 2)2.
2. Express 4x' — 17x- + llx + 2 in powers of x + 3, in powers of
X — 5, and in powers of x — 4, and verify the results.
3. Express 5 y* + 6 r/' — 17 y- + 18 y — 20 in powers of y — i and in
powers of y + 4, and verify tlie results.
Note. Exs. 1-3 can be solved, perhaps more rapidly, by Horner'' s process.
(See text-books on algebra, e.g. Hall and Knight's Algebra, § 549, 4th edition,
1889.)
4. Develop e' in powers of x — 1.
6. Show that i= i - — (x - a) + - (x - a)^ - — (x- a)' + •••, when x
X a a^ a* a*
varies from x = 0 to x = 2 a.
6. Show that loga;= (x- 1) - K«- 1)H i(x- 1)'- - is true for
values of i between 0 and 2.
152. Maclaurin's theorem and series. This is a theorem for
expanding a function in a power series in x. As will be seen
presently, it is really a special case, of Taylor's theorem.
Let f{x) and its first n derivatives be finite f or x = 0 and be
continuous for values of x in the neighborhood of a; = 0.
In form (9), Art. 150, put a; = 0 ; then
y(A)=/(0)+A/'(0) + |:/"(0)+ ... +^-^/»-)(0) +^.f»'(«A).
On writing x for h, this becomes
248 DIFFERENTIAL CALCULUS. [Ch. XVI.
lij{x) and all its derivatives are finite for a; = 0, and if
\hn^^f^0(x) = O, then
71 !
Ax) =/(0) +«/'(0) +|^/"(0) +... + ^/v«)(0) + .... (2)
This is known as Ma«lanrin's theorem, and the series is called
Maclaurin's series. The last term in (1) is called the remainder in
Madaurin's series. It is the limit of the sum of the terms of the
series after the »jth term.
EXAMPLES.
1. Show that formula (2) comes from form (11), Art. 160, on putting
ft = 0 ; show that this has practically been done in the derivation above.
Show that formula (2) comes from form (2), Art. 151, on putting a = 0.
2. Develop sin x in a power series in x.
Here f{x) = sin x. .: /(O) = 0,
.•./'(a:) = cosa;, /'(0) = 1,
f"{x)=-8mx, /"(0) = 0,
/"'(a:)=-cosa;, /"'(0)=-l,
/i»(a;) = sin x, /'(O) = 0,
etc. etc.
(Compare Ex. 2 above and Ex. 4, Art. 150.)
On applying the method of Art. 146 it will be found that the interval of
convergence is from — to to + oo.
3. Calculate sin (^ radian), i.e. sin 5° 43' 46".5.
By A, sin (.1 radian) = .1 - i^ + i^ = .09983.
4. Calculate sin (..S') and sin (.2') to 5 places of decimals. (For results,
see Trigonometric Tables.)
6. Show that cosx = 1 -^ + ^,-~+ •", (B)
2 ! 4 ! 61
and show that the interval of convergence is from — ao to + oo.
6. To 4 places of decimals calculate the following; sin (.3'), cos (.2)',
sin (.4'), cos (.4'). (See values in Trigonometric Tables.)
152.] TAYLOR'S THEOREM. 249
7. Showthate»' = l+a; + f? + ^+-, (C)
^ I o I
and show that this series is convergent for every finite value of x.
8. Substitute 1 for x in C, and thus deduce 2.71828 as an approximate
value of e.
9. Assuming A and B deduce that the sine of the angle of magnitude zero,
is zero, and that the cosine of this angle is unity.
Note 1. Expansions A and B were first given by Newton in 1669. He
also first established series C. These expansions can also be obtained by tlie
ordinary methods of algebra, without the aid of the calculus. For this
derivation see Chrystal, Algebra, Part II., Chap. XXIX., § 14, Chap.
XXVIII., § 5, and the texts of Colenso, Hobson, Locke, Loney, and others,
on what is frequently termed Analytical Trigonometry, or Higher Trigo-
nometry. ['This subject is rather to be regarded as a part of algebra
(Chrystal, Algebra, Part II., p. vii).] Also see article "Trigonometry"
(Ency. Brit., 9th ed.).
10. Develop the following functions in ascending powers in x : (1) secx;
(2) log sec z; (3) log (1 + x), tan-iz, sin-iz (see Art. 198, Eis. 1, 2, 3.)
11. Show that tan X = X 4 i x» + t". x^ + ,,1/- x' + ....
By this series compute tan (.5') , tan 15°, tan 25°.
12. Find: (_!) (e'coaxdx; (2) T^tto; (3) ('e-''dx.
Note 1 a. The integral in Ex. 12 (3) is important in the theory of probabili-
ties. If the end-value x is «, the value of the integral is iVr. (Williamson,
Integral Calculus, Ex. 4, Art. 116.)
13. Assuming the series for sinx, prove Huyhen's rule for calculating
approximately the length of a circular arc, viz. : From eight times the chord
of half the arc subtract the chord of the whole arc, and divide the result by
three.
14. State Maclaurin's theorem, and from the expansion for tanx find
the value of tan x to three places of decimals when x = 10°.
15. Show that cos- x = 1 - .^ x^ + ^'^^ " ~ ^) x* .
Note 2. Historical. Taylor's theorem, or formula, was discovered by
Dr. Brook Taylor (1685-1781), an English jurist, and published in his Metho-
dus Incrementnrum in 1715. It was given as a corollary from a theorem in
Finite Differences, and appeared without qualifications, there being no refer-
ence to a remainder. The formula remained almost unnoticed until Lagrange
(1736-1813) discovered its great value, investigated it, and found for the
250 DIFFERENTIAL CALCULUS. [Ch. XVI,
remainder the expression called by his name. His investigation was pub-
lished in the Memoires de VAcademie de Sciences a Berlin in 1772. "Since
then it has been regarded as the most important formula in the calculus."
Maclauriri' s formula was named after Colin Maolauriu (1698-1746), pro-
fessor of mathematics at Aberdeen 1718 ?-1725, and at Edinburgh, 1725-1745,
who published it in his I'reatise on Fluxions in 1742. It should rather be
called Stirling''s theorem, after James Stirling (1690-1772), who first an-
nounced it in 1717 and published it in his Methodus Differentialis in 1730.
Maclaurin recognized it as a special case of Taylor's theorem, and stated
that it was known to Stirling ; Stirling also credits it to Taylor.
Note 3. Taylor's and Maclaurin's theorems are virtually identical. It
has been shown in Art. 152 that Maclaurin's formula can be deduced from
Taylor's. On the other hand, Taylor's formula can be deduced from Mac-
laurin's ; e.g. see Lamb's Calculus, page 667, and Edwards's TYeatise on
Differencial Calculus, page 81.
Note 4. FuT-ms of the remainder for Taylor's series (2), Art. (151).
Lagrange's form of the remainder has already been noticed in Art. 151.
Another form, viz.
(n - 1) !
was found by Cauchy (1789-1857), and first published in his Le(;ons sur le
Calcnl infinitesimal in 1826. A more general form of the remainder is the
Schlomilch-Boche form, devised subsequently, viz.
i^^^^lUllzilTf/wta + eix - a)], O<0<1.
(n- 1) ! p
This includes the forms of Lagrange and Cauchy ; for these forms are ob-
tained on substituting n and 1 respectively for p. (The d's in these forms
are not the same, but are alike in being numbers between 0 and 1.) In par-
ticular expansions some one of these forms may be better than the others for
investigating the series after the first n terms.
Note 5. Extension of Taylor's theorem to fnnctions of two or more
rariables. For discussions on this topic see McMahon and Snyder's Calcu-
lus, Art. 103 ; Lamb's Calculus, Art. 211 ; Gibson's Calculus, § 157.
Note 6. References for collateral reading on Taylor's theorem.
Lamb, Calculus, Chap. XIV. ; McMahon and Snyder, Diff. Cah, Chap. IV. ;
Gibson, Calculus, Chaps. XVIII., XIX. ; Echols, Calculus, Chap. VI.
153. Relations between trigonometric (or circular) functions and expo-
nential functions. The following important relations, which are extremely
useful and frequently applied, can be deduced from the expansions for sin x,
cos X, and e' in Art. 162.
152, 153.] TAYLOR^ S THEOREM. 251
The substitution of ix for a; in C gives
e** = 1 - |i + ^ - ... + lYa: - 1^^ + 1^ - ...\ = cosx + i sinx. (1)
The substitution of — ix for x in C gives
e-'' = l-f^ + f-^---i[x~f, + f,--) = cmx-isinx. (2)
From (1) and (2), on addition and subtraction,
COS a; = ^ + ^ (3), 8iiix=? — -^ — . (4)
2 2 1
On putting ir for x in (1), there is obtained the striking relation
e*"' = - 1. (See Art. 38, Note on e.)
Note 1. The remarkable relations (l)-(4), by vrhich the sine and cosine
of an angle can be expressed in terms of certain exponential functions of the
angle (measured in radians), and conversely, were first given by Euler
(1707-1783). (In connection with the expansions in Arts. 162, 153, see the
historical sketch in Murray's Plane Trigonometry, Appendix, Note A ; in
particular pp. 168, 169.)
NoTB 2. Results (l)-(4) can also be deduced by the methods of ordinary
algebra; see Note 1, Art. 152, the references therein, and Chrystal's Algebra,
Part II., Chap. XXIX., § 23.
EXAMPLES.
1. From (3) and (4) deduce that cos" x + sin" a; = 1.
2. Show that tan a; = ^" ~ ^''' •
3. Express cot x, sec x, cosec x, in terms of exponential functions of x.
Note 3. Since, by (1), e"* = cos <f> + t sin ip, and e'"' = cos nip + isin n^,
and since («'*)" = e'"*, it is evident that
(cos <H- i sin +) " = c«8 n<|> + f sin »n|>,
for all values of n, positive or negative, integral or fractional.
This very important theorem is called De Moivre^s theorem, after its dis-
coverer Abraham de Moivre (1667-1754), a French mathematician who
settled in England. It first appeared in his Miscellanea Analytica (London,
1730), a work in which "he created ' imaginary trigonometry.'" [On De
Moivre''s theorem, and results (l)-(4), see Murray, Plane Trigonometry,
Art. 98, and Appendix, Note D ; and other text-books on Trigonometry.]
N.B. The article on Hyperbolic Functions, Appendix, Note A, may be
conveniently read at this time.
252 DIFFERENTIAL CALCULUS. [Ch. XVI.
154. Another method of deriving Taylor's and Maclaurin's series.
Following is a method which is more generally employed than
that in Arts. 150 and 162 for finding the forms of the series of
Taylor and Maclaurin.
A. Maclaurin's series. Let fix) and its derivatives be con-
tinuous in the neighbourhood of a; = 0, say from x = — a\ax = a.
Suppose that f{x) can be expressed in a power series in x conver-
gent in the interval —a to -f-a. That is, assume that (for
— a<x<:ia) there can be an identically true equation of the
■fo™ J{x)^A + Ax + A.p:' + A.^+-+A^x"-\--. (1)
The coefficients A^, A^ A^, ••-, A^, •■■, will now be found. It
has been seen in Art. 147 that if Equation (1) is identically true,
then the equation obtained by differentiating both members of (1),
'^i^- f<{x) = Ai-\- 2 A^ + 3 A^ + ■■■ +nA„af-^ + ■■;
also is identically true for values of x in some interval that
includes zero. For the same reason the following equations,
obtained by successive differentiation, are also identical in inter-
vals that include zero, viz. :
f"(x) = 2A, + 2-3A,x+...+ n(« - l)A„3f-' + -,
f"(x) = 2.3-A,+ -+n(n-l)(n-2)A„xi'--+-,
/(»>(x) = n-n-l-n-2 2-lA„ + •••,
On putting « = 0 in each of these identities it is found that
A=/(0), A=/'(0), A = ^, A = ^^, ..., A = =^, -.
Hence, on substitution in (1),
f(x)=f(0)+xf'(0)+^f"(0) + ^f"\0)+ ... -K^/(-)(0)-F ..., (2)
which is Maclaurin's series (Art. 152).
B. Taylor's series. Let f(x) and its derivatives be continuous
in the neighbourhood of x = a, say from x = a — h to x = a + h.
Suppose that f(x) can be expressed in a power series in a; — a
154.] taylob's theorem. 253
which is convergent in the neighbourhood of x = a. In other
words, suppose that there is an identically true equation of the
form
/(«) = Ao + Ai(x-a) + A2(x - ay + As(x- ay + •••
+ A„(x-ay+-. (3)
Then, as in case A, the following equations, which are obtained
by successive differentiation, also are identically true for values
of X near x = a, viz. :
f'(x)=A, + 2A,ix-a)+3A,{x-ay+-+nA,(x-ay-' + --,
f'(x) =2 A,+2 ■ 3 A^(x-a) + - +n • n -1 ■ A„{x - ay-'+-;
f"\x)=2- 3 . A^+.-+n- n-l-n-2- A^(x-ay-'+ •-,
.f"{x)=n-n-l .n-2-:.2-l-A„+-,
On putting a; = o in each of these identities it is found that
A=/(a), A=f'(a), A = rM, ^3 = =^, -,
Hence, on substitution in (3),
fix) =f{a) +{x- a)f'{a) + ^^^/"(a) + -
+ ^^^/'"'(a) + -, (4)
n !
which is series (2), Art. 151.
If in (4) X is changed into x-\-a, then
fix + a) =fia) + xf'ia) + ff'ia) +..■ + ^./^"'(a) + -, (5)
which is series (11), Art. 150, with a written for h. On inter-
changing a and x in (5), form (10), Art. 150, is obtained.
Note. On the proof of Taylor's theorem. The above merely shows the
derivation of the form of Taylor's series. It is still necessary to examine into
the convergency or divergency of the series and to determine the remainder
254 DIFFERENTIAL CALCULUS. [Ch. XVI.
after any number of terms. The investigation of the validity of the series is
a very important matter in the calculus. For this investigation see, among
other works, Todhunter, Diff. Cal., Chap. VI. ; Williamson, IH_f. Cal.,
Arts. 73-77 ; Edwards, Treatise on Diff. Cal., Arts. 130-142 ; McMahon
and Snyder, Diff. Cal, Chap. IV. ; Lamb, Calculus, Arts. 203, 204; article,
"Infinitesimal Calculus" (^Ency. Brit., 9th ed., §§ 46-52).
155. Application of Taylor's theorem to the determination of con-
ditions for maxima and minima. This article is supplementary to
Art. 76. Let/(a;) be a function of x such that f(a + h) and f(a — 7i)
can be developed in Taylor's series; and let it be required to
determine "whether /(a) is a maximum or minimum value of /(x).
On developing f(a — h) and /(a + h) by formula (9), Art. 150,
f(a-h) =f(a)-hf'(a) + f^f"{a)-^f"'ia) + .••
+ ^^>'(a-W. (1)
/(« + h) =/(a) + hf{a) + 1!/" (a) + 1!/" (a) + ■ • •
+ -/"Ka + eji), (2)
n I
in which 0, and O^ lie between 0 and 1.
Suppose that the first n — 1 derivatives of f(x) are zero when
x^=a, and that the nth derivative does not vanish for x = a. Then
/(a-70-/(a) = t^7(»)(a-e,;i), (3)
/(a + h) -f{a) = -f'"\a + 6^). (4)
n !
It follows from the hypothesis concerning f(x) that the signs of
/'"'(a — ^1^) and/<">(a + 6ji), for infinitesimal values of h, are the
same as the sign of /'"'(a). From (3), (4), and the definitions of
maxima and minima, it is obvious that :
(a) Ifn is odd, the first members of (3) and (4) have opposite
signs, and consequently, f(a) is neither a maximum nor a minimum
value of fix);
(6) Ifnis even and /<"' (a) is positive, the first members of (3)
and (4) are both positive, and consequently, f(a) is a minimum
value of fix) ;
156, 156.] TATLOB'8 THEOSEM. 255
(c) If n is even andf^"^{a) is negative, the first members of (3)
and (4) are both negative, and consequently, /(a) is a maximum
value off(x).
The condition for maxima and minima that was deduced in
Art. 76, (c), is a special case of this, viz. the case in which n = 2.
156. Application of Taylor's theorem to the deduction of a theorem
on contact of curves. This article is supplementary to Art. 95.
(See Art. 95, Note 4.)
Theorem. If two curves have contact of an even order, they cross
each other at the jjoint of content; if two curves have contact of an
odd order, they do not cross each other at the point of contact.
Let the two curves y = <^(x) and y = i/f(x) (1)
have contact of the nth order at a; = a. Then
,Ka) = ^(a), <#.'(a) = ^'(«), <^"(«) = 'A"(«), • • •, «/><"'(«) = >/'<"'(«)• (2)
Now compare the ordinates of these curves a.t x = a — h, i.e. com-
pare tft(a — h) and ^|/(a — h); also compare the ordinates a.tx = a + h,
i.e. compare </>(a + h) and ^(a + h). Let it be further premised
that <^a ± h) and i/'(a ± h) can be expanded in Taylor's series. On
using Taylor's theorem (form 9, Art. 150), and remembering
hypothesis (2), it will be found that
^a-h)- ^(a -h) = 1^1^ [<^"'+»(a - 6,h) - ^<-+>'(« - dji)], (3)
<^(a + h)- ^a + h) = -^j [,^<"+»(a - OJi) - ^<-+"(a - e,h)l (4)
in which the four O's all lie between 0 and 1.
Let h approach zero; then, by the premise above, the signs
of the expressions in brackets are the same as the signs of
[<^"'+i>(a) - !/'•'"*''*(«)]. Hence, ifn is odd, the first members of (3)
and (4) have the same sign, and, accordingly, the curves do not
cross; if n is even, these first members have opposite signs, and,
accordingly, the curves do cross.
Ex. Accompany the proof of this theorem with illustrative figures.
256 DIFFERENTIAL CALCULUS. [Cii. XVI.
157. Applications of Taylor's theorem in elementary algebra. Let
f(x) be a rational integral function of x, of the nth degree say.
Then /•"''■"(a;) and the following derivatives are all zero. Hence,
Taylor's series for f(x + h) in ascending powers of either /* or x
[see forms (10) and (11), Art. 150] is finite. That is,
f(x + h)=f(x) + hfix) + '^f'(x)+... +JfJ/'"'(4 (1)
/(a: + K)=f(h) + xf'(li) + ^f"(h)+- +^/""(/0- (2)
A rational integral function f{x) of the nth degree can also be
expressed in a finite series in ascending powers of x — a [see
form {2), Art. 161]. That is,
/(^) = /(a) + (x-a)/'(a) + (^^>'(«)+ - +^~^f'\a). (3)
Exercise. See Ex. 7, Art. 150, and Exs. 1, 2, 3, Art. 151.
Note 1. Let f(x) be as specified above. In general the calculation of
f{x + h) and the expression of /(x) in terms of a; — a, can be more speedily
efiected by Horner's process.* This process is shown in various texts on
algebra ; e.g. Hall and Knight's Algebra (4th edition), Arts. 549, 572.
Note 2. For an application of Taylor's theorem to interpolation,
see McMahon and Snyder, Calculus, Note, pp. 326, 326.
Note 3. In expansion (10), Art. 160, if ft is a differential dx of x, then
h, A^ A', ••-, are respectively differentials of x of the first, second, thii-d, •••,
orders; and A/(x), Ay"(x), fty"(x), •••, are respectively differentials of
/(x) of the first, second, third, ••-, orders. If h (or dx) is an infinitesimal,
these differentials are also infinitesimals of the respective orders mentioned.
* William George Horner (1786-18.37), an English mathematician, who
discovered a very important method of finding approximate solutions of
numerical equations of any degree.
CHAPTER XVII.
APPLICATIONS TO SURFACES AND TWISTED CURVES.
158. Introductory.
(a)' Plane curves of one parameter. In the case of a circle
x2 + ^/2 = a^ (1)
the varying positions of a point (x, y) on the
circle may be described by giving values to
6 in the equations
x = a cos 6,
y =a sin 0.
Here 6 denotes the angle made with the
X-axis by the radius drawn from the centre
to the point.
In the case of the ellipse
$4 = i. (3)
the varying positions of a point (x, y) may be described by giving
values to <^ in the equations
(2)
Fig. 91.
r
(4)
(5)
y = h sin <^
The equations of the cycloid,
x — ai6 — sm 6),
y = a(l — cos ff)
have been used in several preceding articles;
Variable numbers such as 6, <^, 6, used in equations (2), (4),
(5), are called parameters. Curves, such as the above, in whose
equations only one parameter appears, are called curves of one
parameter.
* See text-books on analytic geometry.
257
258
DIFFERENTIAL CALCULUS.
[Ch. XVII.
(6) Twisted cnrres or skew cnrres. A twisted curve, also called
a skew curve, is a curve which does not lie in a plane. Thus the
curve which is drawn on the surface of a right circular cylinder
crossing the elements of the cylinder at any constant angle not a
right angle, is a skew curve.
Skew curves sometimes may be expressed in terms of one param-
eter. Thus the equations of the curve just described, a helix, are
x = a cos 6, y— a sin 6, z = bd.
Here a is the radius of the cylinder, 6 at any point is the angle
which the projection of the radius vector of the point makes with
the a^axis on the xy-plane, and 6 is a constant depending on a and
the constant angle at which the curve crosses the elements of the
cylinder. (See Tig. 150, Note C. Here 6 = a tan a.)
Another example of equations of a skew curve of one parame-
ter ia
a; = 2 a cos t, y = 2asmt, 2 = ct'.
Tangent to a skeiv curve. A method of finding the direction of
the tangent to a plane curve y=f(x) at any point has been shown
in Arts. 24, 59. The method was founded on the definition that a
tangent at any point of the curve is the limiting position of a se-
cant drawn through that point when a neighboring point of inter-
section of the secant with the curve approaches the first point.
A like definition will be used in finding the direction of the tan-
gent to a skew curve.
(c) Direction cosines of a line. Let the line OP (or any parallel
line BS) make angles «, /3, y, with the
axes OX, OY, OZ, respectively. Then
cos «, cos /8, cos y
are called the direction cosines of the
line.
The direction of a line is known
when two of them are given ; since, as
shown in analytic geometry,
cos^ a -f cos^ p 4- cos^ y = 1-
Fig. 92.
158, 159.] SURFACES AND TWISTED CURVES.
259
(«?) It is shown in analytic geometry that if a, b, c are propor-
tional to the direction cosines of a line ; that is, if
a:b:c = cos a : cos ^ : cos y,
then the values of the direction cosines are respectively,
a b c
Va' + b' + c'' Va'T¥+?' ■Va' + b'+e''
159. Tangent line to a twisted curve
curve be
x = <l>(t),]
z = F(t).
Take any
(1)
Let the equations of the
+ Ay,*i+ A*)
point
P on the curve ; let
its coordinates be
(ai, Vv Zi)- Through
P draw any secant
meeting the curve
in Q. Denote the
coordinates of Q as F'g- 93-
(Xj 4- Ax, !/i + Ay, «! + Ax). Denote the value of < at P as t^, and
the value of < at Q as fi + A(. Thus Ax, Ay, Aa, At are the corre-
sponding differences between the coordinates and the parameter t
respectively, at P and Q.
The direction cosines of the secant PQ are proportional to
Ax, Ay, Az ; *
and hence proportional to
Ax Ay
At' 'At'
A«_
At'
(2)
Now suppose the secant PQ turns about P, Q moving along the
curve until it comes to P. 77ie limiting position of PQ when Q
thus arrives at P is the tangent line to the curve at P. When Q ap-
proaches P, At approaches zero, and the quantities (2) approach
* It is shown in analytic geometry that the direction cosines of the line
passing through the points {xu r/i, Zi), (X2, 1/2, zz), are proportional to xj — Xi,
3^2 — Vi, Z2 — zi, respectively.
260 DIFFERENTIAL CALCULUS. [Ch. XVII.
the values —, ^, — • Accordingly, the direction cosines of the
dt dt dt
tangent to the curve at a point P{xi, y^, z^ are proportional to the
values of — , -^, -^ at (a;,, y^, «i).
dt dt dt
These values may be denoted by ^, -fj, -^•
dt dt dt
It is shown in analytic geometry that the equations of a line
passing through the point (xj, y^, z^ and having the direction
cosines proportional to I, m, n, are
I m n
The equations of the tangent line drawn to the curve at (y\, ?/i, Zi)
are accordingly ^_^ y_y^ ^_^^
dxx dyi dzj
dt dt dt
(4)
160. Equations of a plane normal to a skew curve of one param-
eter. A plane is said to be normal to a skew curve at a point
when it is normal to the tangent line to the curve at that point.
It is shown in analytic geometry that if the direction cosines
of a line are proportional to I, m, n, the equation of the plane
which passes, through a point (xi, y,, Zi) and is at right angles to
that line, is , , v , / x . / x a /i \
l(x-x^) + m(y-yi) + n(z-Zi)=0. (1)
Hence, from this property, the preceding definition, and equa-
tions (4), Art. 159, the equation of the plane which is normal to
the skew curve (1), Art. 159, at the point (a^, y,, z,) is
EXAMPLES.
1. Find the equations of the tangent line and the equation of the normal
plane which are drawn to the curve
x = 2acost, y = 2 a sin {, z = cfi:
(1) at any point (a;i, yi, Zi) ; (2) at the point for which t =-; (3) at the
point for which t = r.
159, 160.] SURFACES AND TWISTED CURVES. 261
(1) Here, — =-2 asint = - yi,
dt
-^ = 2 a cos ( = ail,
dt
— ^2ct = i\/cz[.
dt
Hence the equations of the tangent line at (ki, j/i, z-i) are
x — xi _ y—vi _ z- zi _
- 2'i *i 2 Veil
The equation of the normal plane at (xi, yuZi) is
-yi{x- xi) + «! (y - yi) + 2 \/c«7(« - Zi) = 0.
This reduces to
Xiy -yiX + 2\/czi(x-Zi)=0. (fc)
(2) When« = I, the point (xu j/i, 2i) is [o, 2a, — V
Equations (a) then have the form
K y — 2 a 4
(a)
- 2 a 0 TTC ■
whence irca; + 2 az - ^^!^^ = 0 and y =2 a.
z
Equation (6) then is
-2ax-\-i^c{z- — ^ = 0.
(3) When t = ir, the point (a;i, ^i, 2i) is (— 2 a, 0, t'^'c).
The equations of the tangent line are i + 2 a = 0, ircy + az = w'^ac.
The equation of the normal plane is 2 ay = irc (2 — t^c).
2. Find the equations of the tangent and the equation of the normal
plane to the helix x = a cos 6, y = asine, z = b9:
(a) at any point (xi, j/i, zi) ; (6) when $ = 2ir.
Ans. (a) ^~^^ = y^lll = ?^lll , equations of tangent line ;
-yi xi b
— yi{x — X\) + x\{y — y\)+ b(z — zi)= 0, equation of normal plane.
(6) X = a, by = az — 2 abir, equations of tangent line ;
ay + bz — 2 6V = 0, equation of normal plane.
(See Granville, Calculus, p. 272, Ex. 1. )
262 DIFFERENTIAL CALCULUS. [Ch. XVII.
161. Tangent lines to a surface at any point. Tangent plane to
a surface at any point. Suppose a straight line is drawn through
a point on a surface and any neighboring point, and that the
latter point moves towards the first point along the surface.
The limiting position of the line as the moving point approaches
the fixed point is said to be a tangent line to the surface at this
point.* A neighboring point may be chosen in an unlimited
number of ways, and moreover it can approach the fixed point
by any one of an unlimited number of paths on the surface. It
is evident, accordingly, that through any ordinary point on a sur-
face an unlimited number of tangent lines can be drawn.
Theorem. All the tangent lines that may be drawn through an
ordinary (i.e. a non-singular) point on a surface lie in a plane.
Let the equation of the surface be
F(x,y,z)=0. (1)
Suppose that
x=f{t),y = ,i>{t),z = ^{f), (2)
~ are the equations of a curve C drawn on
the surface through a point P(xi, y^, «,).
Then at P, tlie total ^-derivative of
Fio, 94_ F(x, y, z), by (1), must be zero ; that
is, from (1) and (2),
dF dx dF dy OF dz_
dx dt '^ dy' dt'^ dz'dt~ *- ■'
For P(xi, 2/i, Zi) equation (3) may be written
dF dx, dF dy, dF d^_
dx, dt "^ fli/i ■ dt "^ a«i ' dt~ ' ^^
in which — - denotes the value of — when x„ y„ z, are substi-
oxi dx
tuted for x, y, z, and — ' denotes the value of — at P.
dt dt
* This definition of a tangent line to a surface applies only to ordinary
points on the surface. "Singular points" on a surface are not discussed
here.
161.J SURFACES AND TWISTED CURVES. 263
According to the definitions in Arts. 159, 161, the tangent line,
T say, drawn to the curve C at P must be a tangent line to the
surface. By Art. 159 the direction cosines of the tangent line
to the curve C at P are proportional to
dx^ dy^ dz, ,_
dt' dt' dt ^ ^
Equation (4) shows * therefore that the tangent line T is per-
pendicular to a line through P, N say, whose direction cosines are
proportional to q^ q^ q^
dxi ' dy^ ' dz^
But T is any tangent line through P; accordingly the line N
is perpendicular to all the tangent lines through P. There-
fore, all these lines lie in a plane, viz. the plane passing through
P at right angles to N. This plane is called the tangent plane at P.
The line N, from fact (6), is perpendicular to the plane
through P(sc„ y^, z,) whose equation is f
(.-.,)|^+(,-.0g+(-^0|f=0; (7)
this, accordingly, is the equation of the tangent plane at P.
Should the equation of the surface be in the form
^=f{^.y), (8)
this can be put in form (1), viz. :
f(x,y)-z = Q. (9)
„,, dF dF dF
^^^^ T-' ^' T-'
oxi ayi oZi
are respectively — . -^ . — 1,
dxi dy^
» It is shown in analytic geometry that if two lines are perpendicular
to one another and their direction cosines are proportional to ?, m, n, and
^1, mi, J!i, respectively, then
III + niTOi -I- «»i = 0.
t By analytic geometry, the equation of a plane through a point (xi, t/i, zi)
at right angles to a line whose direction cosines are proportional to I, m, n, is
l(x - xi) + m{y - t/i) -I- n(2 - z{) = 0.
261 DIFFERENTIAL CALCULUS. [Cb. XVII.
ote, Sz, ^
and (7), the equation of the tangent plane at (a^, y^, z^) becomes
(x-x{)^+{y-yO^-(z'-zO=0. (10)
dXi ayi
Note. For another derivation of (10) see Osgood, Calculus, pp. 288, 280.
162. Normal line to a surface at any point. A line which is
drawn through a point on a surface at right angles to the tangent
plane passing through the point is said to be a normal to the surface.
It has been seen in Art. 161 that the line N, which is drawn
through the point P{Xi, y\, Zj) and whose direction cosines are
proportional to — , — , — .is at right angles to the tangent
dxi flj/i dzi
plane at P. Accordingly, N is & normal to the surface at P.
Its equations, since it passes through that point with those direc-
tion cosines, are ^_^ ^ ^_^ ^ ^_^ ^
dF ~ BF ~ dF ' (1)
dxi dyi dzi
Otherwise : Since the normal at P is perpendicular to the tan-
gent plane at P, whose equation is (7), Art. 161, the equations of
the normal are (1).*
When the equation of the surface has the form
^=f(x, y),
the equations of the normal at (aj,, y^, z,) [see Art. 161, (8)-(10)] are
x — x^_y-y^ z — z,
dl df -l' (2)
These are the same as '- = = /„\
dzy dz^ —1 \p)
dXi 9y,
* By analytic geometry the equations of the line drawn through a point
(ii, 1^1, 2i) at right angles to a plane Ix + my + nz +p = 0, are
X- xi _ y-yi _ z-zi
I TO n
dyi
-1
y —
Vi
0-Z,
162.] SURFACES AND TWISTED CURVES. 265
EXAMPLES.
1. Find the equation of tlie tangent plane and the equations of the
normal line to the ellipsoid
at the point (2, 3, 1).
Here ^=2x, ^ = iy, ^=8z.
dx dy dz
At (2, 3, 1) these values are
dxi dyi dzi
The equation of the tangent plane, by substitution in (7), Art. 161, is
(a; - 2)4 +(y- 3)12 + (2-1)8 = 0,
i.e. ix + 12y + 8z = 62.
The equations of the normal line, by substitution in (1), Art. 162, are
x-2 _ y -S _ g- 1
4 12 8 '
which simplify to 3 a; = y + 3, 2y = 3z+Z.
2. Find the equation of the tangent plane and the equations of the normal
line to each of the following surfaces :
(a) the sphere x^ -\- y^ + z^ + Sx - eiy + 4 z = n at the point (2, 4,1);
(b) the hyperboloid of one sheet 2 a;2 + 3 ?/2 - 7 a^ = 38
at the point ( — 3, 4, 2) ;
(c) the hyperboloid of two sheets x'' - iy^ -3z^ + 12 = 0
at the point (8, — 4, 2)
(d) the elliptic paraboloid z = x- + Hy^ at the point (2, - 3, 31)
(e) the sphere x' + y^ + z^-12x — 4y -6z = 0 at the origin
(/) the surface x^ -f j/2 _ 4 ^-2 - le at the point (8, 4, 4).
3. Show that the sum of the squares of the intercepts on the axes made
by any tangent plane to the surface
x^ + y^ + z^ = a\
is constant.
4. Show that the volume of the tetrahedron formed by the coordinate
planes and any tangent plane to the surface
xyz = a',
is constant.
266
DIFFERENTIAL CALCULUS.
[Ch. XVII.
163. Equations of the tangent line and the normal plane to a
skew curve.*
A curve may be the common intersection of two surfaces, e.g.
of a cone and a cyliuder.
In such a case the curve
is given by the equations
of the two surfaces ;t say
F(x, y,z) =0,U
.f,ix,y,z)=0.l ^^
The tangent line to this
curve, at any point on it,
is the intersection of the
two tangent planes, one
for each surface, passing
through the point. Ac-
cordingly [by Art. 161, Equation (7)], the equations
of the tangent line drawn through a point (a^ y^, Zj)
on the curve given by equations (1), are
Fig. 95.
By I
•«l)
(2)
Equations (2), as may be seen on solving them for the values
of the ratios
z — z,' z — z,'
may be transformed into
y-Vi
(3)
dFd^_dFd^ dFd^_dFd^ dF dj, dF d<f>
dy^ dzi dz^ 5?/, 5z, dxi fix, dzi dx^ dy^ dy^ dx^
In Fig. 95, APB is the curve, LP the tangent line, NP the normal plane
* This Article is .supplementary to Arts. 159, 160.
t Since the coordinates of any point on it satisfy the equation of each
surface.
t For example, see in Fig. 125 the curve BVR, which is the intersection
of the sphere x'^ + %/'■ + z^ = a^ and the cylinder x^ + y^ = ax.
163.]
SUBFACES AND TWISTED CURVES.
267
From equations (3) and the principle quoted in the second
footnote on page 265 the equation of the normal plane to the curve
(1) at the point (a^, j/,, Zj) is
Note. The expressions in the denominators in (3) may be
expressed in the determinant forms:
dF
dF
52i
dF
dz:
dF
dF
dxi'
dF
9^
84,
!
d<f,
dz,'
d<t>
dxi
?
d<f,
dx^'
d^,
Syi
EXAMPLES.
1. Find the equations of the tangent line and the normal plane at the
point (1, 6, — 5) to the curve of intersection of the sphere x^ + y'^ + z^ —
iix + 4z — S6 = 0 and the plane x + Sy — 2z = 20.
Here F(_x,y, z) = x^ + y^ + z^ -6x + iz -36,
<p (x, y, z) = X + 3y - 2 z - 29.
Accordingly,
M:=2x-6, f =2,,
50
= 1,
8<P
^=2z + i,
d±__
= 3, 5JS = _2
dx dy dz
At the point (1, 6, — 5), xi = 1, 2/1 = 6, Z\=— 5.
The values of the above derivatives at (1,6, — 5) are thus :
d<t> _■,
dl
dy\
= 12,
dF.
azi
6,
50 _ 3 d^ _.
5^1 dyi 52i
The equations of the tangent line at (1, 6, — 5), on substitution in result
(2), are thus :
{X - 1)(- 4) + (y - 6) 12 + (2 + 5)(- 6) = 0, ■
(x-l) X l+(y-6)3 + (z+6)(-2)=0.
These simplify to
4a;-12j/ + 62 + 98=0, ■
x + 3y — 2s-29=0.
268 DIFFERENTIAL CALCULUS. [C'h. XVII.
The equation of the normal plane to the curve at (1, 6, - 5), on substitu-
tion in result (4) and simplification, is thus :
3 a- + 7?/ + 12 3 + 15 = 0.
2. Find the equations of the tangent plane and the equations of the nor-
mal at the point (6, 4, 12) to the surface
9 z2- 4 1-^ = 288?/.
Also find the equations of the tangent line and the equation of the normal
plane to the curve of intersection made with that surface at that point by "
(a) the plane Zx-2y + z = 22;
(ft) the plane ix + y -3z + S = 0.
3. As in Ex. 2 at the point (5, 4, 2) on the surface
2/2 + 02 = 4 a:,
taking for the planes of intersection :
(a) 7x-2y-z = 25,
(6) 2x + 3y + z = 2i.
4. As in Ex. 2 at the point (4, - 6, 3) on the surface
4 a;2 + 9 8/2 _ 16 z2 = 244,
taking for the planes of Intersection :
(a) 3x-2y-3z = 16,
(6) x + 2y + iz = i.
6. Find the equations of the tangent line and the equation of the
normal plane at the point (6, 4, 12) to the curve of intersection of the
surfaces
9z2-4a;2 = 288j/*1
x^ + y^ + z^ = 196. J
6. Find the equations of the tangent line and the equation of the normal
plane at the point (5, 4, 2) to the curve of intersection of the surfaces.
i/+z'^ = 4 x,1
2x2 + 4y2 + 322 _ 126.
N.B. For other examples, see Granville, Calculus, pp. 276, 278, 279.
•See Ex. 2. t See Ex. 3.
INTEGRAL CALCULUS.
CHAPTER XVIII.
INTEGRATION.
If.B. If thought desirable, Art. 167 may be studied before Arts. 165, 166.
(Remarlcs relating to the order of study are in the preface.)
164. Integration and integral defined. Notation. In Chapter III.
a fundamental process of the calculus, namely, differentiation,
was explained. In this chapter two other fundamental processes
of the calculus, each called integration, are discussed. The
process of differentiation is used for finding derivatives and
differentials of functions ; that is, for obtaining from a function,
say F{x), its derivative F'{x), and its differential F'{x)dx. On
the other hand the process of integration is used :
(a) For finding the limit of the sum of an infinite number of
infinitesimals which are in the differential form, f{x) dx (see Art. 166) ;
(6) For finding functions whose derivatives or differentials are
gioeii ; that is, for finding anti-derivatives and anti-differentials
(see Arts. 27 a, 167).
Briefly, integration may be either (a) a process of summation,
or Q)) a process which is the inverse of differentiation, and'which,
accordingly, may be called anti-differentiation. Integration, as a
process of summation, was invented before differentiation. It
arose out of the endeavor to calculate plane areas bounded by
curves. An area was (supposed to be) divided into infinitesimal
strips, and the limit of the sum of these was found. The result
was the whole (area) ; accordingly it received the name integral,
and the process of finding it was called integration. In many
practical applications integration is used for purposes of sum-
mation. In many other practical applications it is not a sum
but an anti-differential that is required. It will be seen in Art. 166
that a knowledge of anti-differentiation is exceedingly useful in
the process of summation. Exercises on anti-differentiation have
appeared in preceding articles.
269
270 INTEGRAL CALCULUS. [Ch. XVIII.
Note. The part of the calculus which deals with differentiation and its im-
mediate applications is usually called The Differential Calculus, and the part
of the calculus which deals with integration is called The Integral Calculus.
With Leibnitz (1646-1716), the differential calculus originated in the problem
of constructing the tangent at any point of a curve whose equation is given.
This problem and its inverse, namely, the problem of determining a curve
when the slope of its tangent at any point is known, and also the problem of
determining the areas of curves, are discussed by Leibnitz in manuscripts
written in 1673 and subsequent yeai's. He first published the principles of
the calculus, using the notation still employed, in the periodical. Acta
Eruditorum, at Leipzig in 1684, in a paper entitled Nova methodus pro
maximis et minimis, itemque tangentibus, quae nee fractas nee irrationales
quantitates moratur, et singulare pro ilHs calculi genus. Isaac Newton
(1642-1727) was led to the invention of the same calculus by the study of
problems in mechanics and in the areas of curves. He gives some description
of his method in his correspondence from 1669 to 1672. His treatise,
Methodus fluxionum et serierum infinitarum, cum ejusdem applicatione ad
curvarum geometriam, was written in 1671, but was not published until 1736.
The principles of his calculus were first published in 1687 in his Principia
(Fhilosophiae Naturalis Principia Mathematica). It is now generally
agreed that Newton and Leibnitz invented the calculus independently of each
other. For an account of the invention of the calculus by Newton and
Leibnitz, see Cajori, History of Mathematics, pp. 199-236, and Cantor,
Qeschichte der Mathematik, Vol. 3, pp. 150-172.
" There are certain focal points in history toward which the lines of past
progress converge, and from which radiate the advances of the future. Such
was the age of Newton and Leibnitz in the history of mathematics. During
fifty years preceding this era several of the brightest and acutest mathe-
maticians bent the force of their genius in a direction which finally led to the
discovery of the infinitesimal calculus by Newton and Leibnitz. Cavalieri,
Eoberval, Fermat, Descartes, Wallis, and others, had each contributed to
the new geometry. So great was the advance made, and so near was their
approach toward the invention of the infinitesimal analysis, that both
Lagrange and Laplace pronounced their countryman, Fermat, to be the true
inventor of it. The differential calculus, therefore, was not so much an
individual discovery as the grand result of a succession of discoveries by
different minds." (Cajori, History of Mathematics, p. 200.)
Also see the " Historical Introduction " in the article. Infinitesimal Cal-
culus (Ency. Brit., 9th edition), and, at the end of that article, the list of
works bearing on the infinitesimal method before the invention of the
calculus.
Notation. In differentiation d and D are used as symbols ; thus,
df{x) is read " the differential of f(x)," and D/(x) is read " the
164, 165.]
INTEGRATION.
271
derivative of /(a;)." In integration, whether the object be sum-
mation or anti-differentiation, the sign j is most generally used
as the symbol; thus, \f{x)dx is Y&!t.di" the integral of f{x)dx."*
Other symbols, viz. d-^f(x)dx and 2)-'/(a;), are used occasionally
(see Art. 167, Note 2). The quantity f{x) which appears " under
the integration sign," as the mathematical phrase goes, is called
the integrand.
165. Examples of the summation of infinitesimals. These examples
are given in order to help the student to understand clearly what
the phrase " to find the limit of the sum of a set of infinitesimals
of the foTm/(x)dx {i.e. a set of infinitesimal differentials)" means.
(a) Find the area between t?ie line y = mx, the x-axis, and the ordinates
drawn to the line at
x = a and x = b.
Let PQ be the line
whose equation is
y = mx, OA = a, and
OB = b. Draw the
ordinates .4 P and BQ ;
it is required to find
the area APQB.
Suppose that AB
is divided into n equal
parts each equal to Az,
X so that
n . Ax = b — a.
Draw the ordinates at each point of division, Mi, Mi, •••, ilf„_i ; complete
the inner rectangles PMi, Pi, Mt, •••, P„-iB ; and complete the outer rectan-
gles PiA, PzMi, ..., QM„-i. The area APQB is evidently greater than the
sum of the inner rectangles and less than the sum of the outer rectangles ; i.e.
sum of inner rectangles < APQB < sum of outer rectangles.
• The word integral appeared first in a solution of James Bernoulli (1654-
1705), which was first published in the Acta Eruditorum in 1690. Leibnitz
had called the integral calculus calculus summalorius, but in 1696 the term
calculus integralis was agreed upon by Leibnitz and John Bernoulli (1667-
1748). The sign \ was first used in 1675, and is due to Leibnitz. It is
merely the long S which is the initial letter of summa, and was used by
earlier writers to denote " the sum of."
272 INTEGRAL CALCULITS. [Ch. XVIII.
The difference between the sum of the inner and the sum of the outer rectangles
is the sum of the rectangles PPi, Pi Pa, •••, P"-^Q. The latter sum is evidently
equal to the rectangle QS, i.e. to CQ ■ Ax. This approaches zero when Ax
approaches zero. Therefore APQB is the limit of the sum of either set of
rectangles when Ax approaches zero. The limit of the sum of the inner
rectangles will now be found.
At^,
X = a,
and hence,
AP = ma ;
atJlfj,
x = a + Ax,
and hence,
MiPi = TO(a + Ax) ;
atJifs,
x = a + 2Ax,
and hence,
M2P2 = m(a + 2 Ai) ;
at Mn-i, x = a + n—1 Ax, and hence, M„-iP„-i = m(a + n — i. ■ Ax),
.'. sum of inner rectangles
— ma ■ Ax + m(a + Ax) • Ax + m{a + 2 Ax) • Ax + •••
+ m(a + n — I • Ax) ■ Ax.
.: area APQB = lim^,^lmaAx+m(a + Ax)Ax+...+m(_a + n — l-Ax)Ax]
= lim^;e=oi»[a+(o + Aa;)+(o+2 Aa;)H H(a + n — 1 • Aa;)]Aa;.
Hence, on summation of the arithmetic series in brackets,
mn Ax,,
area APQB = lim^z^ !5!L^{2 a + n - 1 . Ai}.
On giving n Aa; its value b — a, this becomes
area APQB = limi^ m(b - a) (j + „ _ ^)
-(f-f)-
Note 1. In this example the element of area, as it is called, is a rectangle
of height y and width Ax when Ax is made infinitesimal, i.e. the element
of area is y dx or mx dx in which dx == 0. (See Art. 27, Notes 3, 4, and
Art. 67(1.)
Note 2. It may be observed in passing that on taking the anti-differential
oimxdx, namely — — , substituting 6 and a in turn for x therein, and taking
the difference between the results, the required area is obtained.
Ei. Eind the limit of the sum of the outer rectangles when Ax approaches
zero.
(4) Find the area between the parabola y = x^, the x-axis, and the ordinate!
atx = a and x = b.
166.]
INTEGRATION.
273
Let LOQ be the parabola, OA = a, OB = b; draw the ordinates AP
and BQ ; the area APQB is
f
required. As in the preceding
problem, divide AB into n
parts each equal to Ax, so that
■^2 *na X
nAx = 6 — a ;
draw ordinates at the points
of division, and construct the
set of inner rectangles and
the set of outer rectangles.
As in (a), it can be seen that
sum of inner rectangles <
area APQB < sum of outer rectangles ; and also that
(sum of outer rectangles) — (sum of inner rectangles) = CQ ■ Ax,
which approaches zero when Ax approaches zero. Hence the area APQB is
the limit of the sum of either set of rectangles when Ax approaches zero.
The limit of the sum of the inner rectangles will now be found.
At A, X = a, and hence, AP = a* ;
at Ml, x = a + Ax, and hence. Mi Pi = (a + Ax)^ ;
at Ml, X = a + 2 Ax, and hence, M^P^ = (a + 2 Ax)^ ;
at Jtf„.i, X = « + » - 1 . Ax, and hence, itf„-iP„.i = (a + n - 1 • Ax)«.
.•. sum of inner rectangles = a^Ax + (a + Ax)2Ax + (a + 2 Ax)2Ax + •••
Now
and
+ (a + n - 1 • Ax)2Ax.
, area APQB = limA«io{a2 + (o + Ax)2 + (a + 2 Ax)2 + •
-\-(_a-^n -1- Ax)2}Ax
= liraAxifl{na2 + 2aAx(l + 2 + 3 + ••■ + n - 1)
+ (Ax)2(12 + 22 + 32 + ••• + n - r)}Ax.
1 + 2 + 3 + ••• + n - 1 = J n(ji - 1) ;
la + 22 + 32 + - + S^^" = \{n- l)n(2 n - 1).»
, area APQB = limAcio n Ax {a" + an Ax — a Ax + |(n Ax)'
- J n (Ax)2 + J (Ax)2}.
* It is shown in algebra that the sum of the squares of the first n natural
numbers, viz. l^, 22, 3', -, n\ is J n(n + 1)(2 n + 1).
274 INTEGRAL CALCULUS. [Ch. XVIII.
But n Aa; = 6 — a, no matter what n and Aa; may be.
.-. area APQB = liniAzio (b - a){a'^ + a{b - a) - aAx + ^(b - a)^
+ i (6 - a)Ax + i (Axy]
_ 6' a' >
Note 1. In this example the element of area is a rectangle of height y
and width Ax, when Ax becomes infinitesimal, i.e. the element of area is
y dx, i.e. x'' dx, in which dx = 0.
Note 2. It may be observed in passing that the result (1) can be ob-
tained by taking the anti-diflerential of z^ fj^, namely — , substituting 6 and
o
a in turn for x therein, and calculating the difference -- — — •
o o
Ex. Find the limit of the sum of outer rectangles.
(c) Find the distance through which a body falls from rest in ti seconds,
it being known that the speed acquired in falling for t seconds is gt feet per
second. [Here g represents a number whose approximate value is 32.2.]
Note 1. If the speed of a body is v feet per second and the speed remains
uniform, the distance passed over in t seconds is vc feet.
Let the time ti seconds be divided into n intervals each equal to At, so that
nAt = ti.
The speed of the falling body at the beginning of each of these successive
intervals of time is
0, g ■ At, 2 g ■ At, ■■■, (n — l)g • At, respectively ;
the speed of the falling body at the end of each successive interval of time is
g ■ At, 2 g ■ At, 3 g ■ At, •••, ng ■ At, respectively.
For any interval of time the speed of the falling body at the beginning is
less, and the speed at the end is greater, than the speed at any other moment
of the interval. Now let the distance be computed which would be passed
over by the body if it successively had the speeds at the beginnings of the
intervals ; and then let the distance be computed which would be passed over
by the body if it successively had the speeds at the ends of the intervals.
The first distance = 0 + giAty + 2g{Aty + ■■■ +(n - 1)?(A0*
= [0 + 1 + 2 + ... +(n - 1)]^(A0''
= in(n-l)g{Aty.
INTEGRA TION.
275
165, 166.]
The second distance =[l + 2 + 3 + -.- + n]g{My
The actual distance fallen through, which may he denoted by s, e>ridently
lies between these two distances ; i.e.
J n(n - \)g{^ty < s < J n{n + \)g(^^ty.
On putting ti for its equal, n At, this becomes
hgti'-igh ■ At<s<\gh^ + \gti,- M.
On letting At approach zero these three distiinces approach equality, and
hence s = J gti^.
Note 2. For two other examples see Art. 166, Note 4.
166. Integration as summation. The definite integral. It will
now be shown, geometrically, how integration is a process of sum-
mation. Let /(x) denote any function of x which is continuous
from x=^ a to a; = & and geometri-
cally representable. Let its graph
be the curve K whose equation is
accordingly ^ ^^^^^_
Suppose that OA — a and OB = b,
and draw the ordinates AP and BQ.
Divide AlB into n parts, each equal
* to Aa; ; accordingly,
nAx=b — a. (1)
At the points of division erect ordinates, and construct inner
and outer rectangles as in Art. 165 (o), (&). It can be shown, as
in the examples in Art. 165, that the difference between the set of
the inner rectangles and the set of the outer rectangles is CQ • Ax
{CQ being equal to BQ — AP), a difference which approaches
zero when Aa; approaches zero. The area APQB lies between
these sets and evidently is the limit of the sum of either set of
rectangles when Aa; approaches zero. The Umit of the sum of
inner rectangles will now be found.
27G INTEGRAL CALCULUS. [Ch. XVIII.
At ^, x = a, and hence, AP=f(a);
at Ml, X = a + Aa;, and hence, -Mi -Pi = /(« + Ax) ;
at Mi, x = a + 2 Ax, and hence, M^P^ —f{a + 2 Ax) ;
at iW„_i, x = b — Ax, and hence, ilf,.!/^,.! =/(6 — Ax).
.-. area APQB = lim^,^
i /(a) Ax + /(a + Ax) Ax + /(« + 2 Ax) Ax H h /(6 - Ax) Ax | . (2)
The second member, which is the sum of the values, infinite in
number, that /(x)Ax takes when x increases from a to 6 by equal
infinitesimal increments Ax, may be written (i.e. denoted by)
Iim^^^/(x)Ax.*
It is the custom, however, to denote the second member of (2)
by putting the old-fashioned long S before f(x)dx and writing at
the bottom and top of the S respectively the values of x at which
the summation begins and ends ; thus
f{x)dx; or, more briefly, I f{x)dic. (3)
This symbol is read " the integral of /(x) dx between the limits
a and h," or " the integral of f(x)dx from x = a to x = 6."
Note 1. The numbers a and 6 are usually called the lower and upper
limits of X. It would be better, perhaps, not to use the word limit in this
connection, but to say "the initial and final values of x," or simply, "the
end-values of a;." t
Note 2. The infinitesimal difierential f(x)dx is called an element of
the integral. It is the area of an infinitesimal rectangle of altitude /(x) and
infinitesimal base dx.
* The latter part of this symbol denotes, and is to be read, "the sum of
all quantities of the type " [or " form "] "/(x)Ax, from x = a to x = b"
[or " between x = a and x = b "].
t Joseph Fourier (1768-1830) first devised the way shown in (3) of indi-
cating the end-values of x.
166.] INTEGRATION. 277
Note 3. It is not necessary that the infinitesimal bases, i.e. the increments
Ax of X, be all equal ; but for purposes of elementary explanation it is some-
what simpler to take them as all equal. (See Lamb, Calculus, Arts. 86, 87,
and the references in Art. 167, Note 6 ; also Snyder and Hutchinson, CalctUvs,
Art. 150.)
Note 4. For the calculation of ( e'dx and \ sin zdx by the process
shown in Art. 165, see Echols, Calculus, Art. 125.
The sum in brackets in (2) will now be calculated, and then its
limit, which is indicated by the symbol (3), will bs found.
Let the anti-diiTerential (Art. 27 a) oi f(x)dx* be denoted by
^(x); that is, let f(^,^ax=di>ix).
Then, by the elementary principle of differentiation (see Art. 22,
Note 3) for all values of x from a to 6,
.l.(x + Ax)-<l>(x) ^^^^^ _^ ^^ (^^
Ax
in which e denotes a function whose value varies with the value
of X, and which approaches zero when Ax approaches zero. On
clearing of fractions and transposing, (4) becomes
/(x)Aa; = </)(x + Ax) — <^(a;) — e- Ax. (5)
On substituting a, a + Ax, a + 2 Ax, •••, 6 — Ax in turn for x in
(5), and denoting the corresponding values of e by e,, e^, e^, •••, e,,
respectively, there is obtained :
/(a)Ax = </>(a-|- Ax) — <^(a) — ei-Ax,
/(a + Ax) Ax = <^ (a + 2 Ax) — <^ (a + Ax) — e^ ■ Ax,
/(a + 2 Ar) Ar = ^ (a + 3 Ax) - <^ (a + 2 Ax) - e, • Aa;,
f(b-Ax)Ax=il>(b) — <^(6-Ax) — e.-Ax.
• If /(x) is a continuous function of x, /(x) dx has an anti-differential. For
proof see Picard, Traite d' Analyse, t. I. No. 4 ; also see Echols, Calculus,
Appendix, Note 9.
278 INTEGRAL CALCULUS. [Ch. XVIII.
Addition gives
/(a) Aa; +f{a + Aa;) Aa; +f(a + 2Ax)Ax-\ 1-/(6 - Ax)
= <^ (6) - </, (a) - (ei + ej + 63 + • • • + e„) Aa;. (6)
On taking the limit of each member of (6) when Aa; approaches
zero,
J^/(x)dx = <^(6)-<^(a)-lim^^(ei + e2+- +e,)Aa;. (7)
Let Cj be one of the e's which has an absolute value E not less
than any of the others ; then evidently
(ei -h % H h e„) Aa; < nEAx;
i.e. by (1), (ei + fij + • • • + e„) Aa; < (6 - o) E.
Hence, lim^,^(ei + 62 H h e„) Aa; = 0, since E approaches zero
when Ax approaches zero ; and therefore,
J^VCa;)*** = ♦(&)-♦(«). (8)
That is, expressing (8) in words : The integral ( /(x) dx, which
• /a-
is the limit of the sum of aM the values, infinite in number, that
f(x) dx takes as x varies by infinitesimal increments from a to b, is
obtained by finding the anti^lifferential, <(> (x), of f(x) dx, and then
calculating <f>(b) — <t> (a).
Note 5. Many practical problems, such as finding areas, lengths of curves,
volumes and surfaces of solids, and so on, can be reduced to finding the limit
of the sum of an infinite number of infinitesimals of the form /(x) dx. (See
Arts. 181, 182, 207-212.) Aa has been seen above, the anti-diSerential
of /(a:) dx is of great service in determining this limit ; accordingly, con-
siderable attention must be given to mastering methods for finding anti-
differentials.
Note 6. The process of finding the anti-diSerential of f(x) dx is nearly
always more difficult than the direct process of differentiation, and frequently
the deduction of an anti-differential is impossible. When the anti-differential
of f(x) dx cannot be found in a finite form in terms of ordinary functions,
approximate values of the definite integral can be found by methods dis-
cussed in Chapter XXII. The impossibility of evaluating the first member of
(8) in terms of the ordinary functions has sometimes furnished an occasion
for defining a new function, whose properties are investigated in higher
mathematics. (On this point see Snyder and Hutchinson, Calculus, Art. 123,
166.]
INTEGRATION.
279
fooUnote.) For instance, the subject of Elliptic Functions arose out of the
study of what are called the elliptic integrals (see Art. 209, Ex. 4, Art. 199,
Note 4, Art. 192, Note 4).
(The ordinary elementary functions can be defined by means of the
calculus, and their properties thence developed.)
Note 7. At the beginning of this article the principle was enunciated
that the area bounded by a smooth curve PQ (Fig. 98), the i-axis, and a pair
of ordinates, is the limit of the sum of certain inner, or outer, rectangles
constructed between the ordinates. The student can easily show that this
principle holds for the smooth curves in Figs. 99 a, 6, c.
O B X
FlO. 99 6.
B X
FiG. 99 c.
Note 8. This article shows that a definite integral may be represented
geometrically as an area. For a general analytical exposition of integration
as a summation, see Snyder and Hutchinson, Calculus, Art. 148. Their
exposition depends on Taylor's theorem (Art. 150). Also see the references
mentioned in Art. 167, Note 5.
Ex. Show that the calculus method of computing the area in Fig. 99 c
bounded by PMNSQ, AB, AP, and BQ really gives area APM+ area. It QB
- area MNB.
[As a point moves along the curve from P to Q, dx is always positive. In
APM y is positive, in MNB negative, in BQB positive. Accordingly, the
elements of area, /(x) dxorydx, are positive in A PM and R QB, and negative
in MNR.^
EXAMPLES.
JT.B. The knowledge already obtained in Chapter IV. about anti-differen-
tials is sufficient for the solution of the following examples. It is advisable
to make drawings of the curves and the figures whose areas are required.
1. Find the area between the cubical parabola y = x* (Fig., p. 462), the
X-axis, and the ordinates for which x = 1, x = 3.
280 INTEGRAL CALCULUS. [Ch. XVIII.
According to (3) and (8), the area required = C j^dx
= *^ + c-(i + c)
= 20 sq. units of area.
2. Find the area between the curve in Ex. 1, The a-axis, and the ordi-
nates for which x = ~ 2, x = 3. Ans. 16J sq. units.
3. Explain the apparent contradiction between the results in Exs. 1, 2.
4. Find the actual number of square units in the figure whose boundaries
are given in Ex. 2. Ans. 24J sq. units.
5. Find the area between the parabola 2 j/ = 7 a;^, the z-axis, and the
ordinates for which : (1) x = 2, x = i ; (2) x = — 3, x = 5.
Ans. (1) 65Jsq. units; (2) 177J sq. units.
N.B. A table of square roots will save time and trouble.
6. Find the area between the parabola y^ = 8x, the a;-axis, and the
ordinates for which : (1) x = 0, a; = 3 ; (2) x = 2, z = 7.
Ans. (1) 9.798 sq. units ; (2) 29.59 sq. units.
7. Find the area of the figure bounded by the parabola y'^ — 6x and
the chord perpendicular to the x-axis at x = 4. Ans. 26.128 sq. units.
8. Find, by the calculus, the area bounded by the line y = 3x, the
X-axis, and the ordinate for which x = 4. Ans. 24 sq. units.
9. (1) Find, by the calculus, the area of the figure bounded by the line
y = 3 X, the x-axis, and the ordinates for which x = 4, x = — 4. (2) How
many sq. units of gold leaf are required to cover this figure ?
Ans. (1) 0 ; (2) 48 sq. units.
10. (1) Find the area between a semi-undulation of the curve y = sin x
and the x-axis. (2) Find the area of the figure bounded by a complete
undulation of this curve and the x-axis. (3) How many sq. units of gold-
leaf are required to cover this figure. Ans. (1) 2 ; (2) 0 ; (3) 4.
11. Compute the area enclosed by the parabola y^ =:4x and the lines
z =z 2, X = 5. Ans. 22.27 sq. units.
12. Compute the area enclosed by the parabola y = x^ and the lines
y = 1, y = i. Ans. 9 J sq. units.
18. Find the area between the parabolas x^ = y and y^ = Sx.
Ans. 2§ sq. units.
14. Find the area between the curves : (1) y^ = x and y^ = 3^; (2) x^ = y
and j/2 = x*. (Make figures.) Ans. (1) y'ysq. units; (2) y'j sq. units.
15. Find the area bounded by the curves in Ex. 14 (2) and the lines
X = 2, X = 4. Ans. 8.129 sq. units.
N.B. Art. 181 may be taken up now.
166, 167.] INTEGRATION. 281
167. Integration as the inverse of differentiation. The indefinite
integral. Constant of integration. Particular integrals. In many
cases there is required, not the limit of the sum of an infinite
number of infinitesimals of the form f(x)dx, but the function
whose derivative or differential is given. The following is an
instance from geometry. When a curve's equation, y =f(x), is
known, differentiation gives the slope at any point on the curve
in terms of the abscissa x,- namely, -^=/'(x) (Art. 24). On the
other hand, if this slope is given, integration affords a means of
finding the equation of the curve (or curves) satisfying the given
condition as to slope. Again, an instance from mechanics : if a
quantity changes with time in an assigned way, differentiation
determines the rate of change for any instant (Art. 25). On the
other hand, if this rate of change is known, integration provides
a means for determining the quantity in terms of the time. (See
Art. 22, Notes 1, 2, and Art. 27 a.)
EXAMPLES.
Ex. 1. The slope at any point (x, y) of the cubical parabola y = x» is 3x' ;'
that is, at all points on this curve, -^ = 3 a;^ and dy = Sx' Ox.
dx
Now suppose it is known that a curve satisfies the following condition,
namely, that its slope at any point (x, y) is Sz^ | j.e. that for this curve,
^ = 3 a;2, (whence, dy = 3 x^dx).
dx
Then, evidently, y = sfi + e,
in which els a constant which can take any arbitrarily assigned value. This
number c is called o constant of integration ; its geometrical meaning is
explained in Art. 99. Since c denotes any constant, there is evidently an
infinite number of curves (cubical parabolas, y = x* + 2, y = z' — 10, y = i*
+ 7, etc., etc.) vsrhich satisfy the given condition. If a second condition is
imposed, the constant c will have a definite and particular value. For
instance, let the curve be required to pass through the point (2, 1). Then,
1 = 2' + fl ; vfhence c = — 7, and the equation of the curve satisfying both
the conditions above is y = x' — 7. (Also see Ex. 17, Art. 37.)
2. Suppose that a body is moving in a straight line in such a way that
(the number of units in) its distance from a fixed point on the line is always
282 INTEGRAL CALCULUS. [Cii. XVII r.
(the number of units in) the logarithm of the number of seconds, t say, since
the motion began : i.e. so that
s = log t.
Then, the speed, ^ = 1 , and ds = --
dt t t
Now suppose it is known that at any time after the beginning of its
motion, after t seconds say, the speed of a moving body is -; i.e. that
^ = 1, (whence, (to-=*V
dt t \ t I
Then, evidently, s = log J + c,
in which c is an arbitrary constant. If a second condition is imposed, the
constant c will take a definite value. For instance, let the body be 4 units
from the starting-point at the end of 2 seconds, i.e. let s = 4 when ( = 2.
'^ '^*" 4 = log 2 + c ; whence c = 4 - log 2,
and s = log < + 4 — log 2.
3. In Ex. 1 determine c so that the cubical parabola shall go through
(a) the point (0, 0); (6) the point (7, -4); (c) the point (-8, 2); (d) the
point (h, k). Draw the curves for (a), (6), (c).
4. Find the curves for which the slope at any point is 4. Determine
the particular curves which pass through the points (0, 0), (2, 3), (—7, 1),
respectively. Draw these curves.
6. Find the curves for which (the number of units in) the slope at
any point is 8 times (the number of units in) the abscissa of the point.
Determine the particular curves whicli pass through the points (0, 0), (1, 2),
(2, 3), (—4, 2), respectively. Draw these curves.
6. How are the curves in Exs. 4, 1, 3, 5, respectively, affected when
the constants of integration are changed ?
7. If at any moment the velocity in feet per second at which a body
is falling is 32 times the number of seconds elapsed since it began to fall from
rest, what is the general formula for its distance, at any instant, from a point
on the line of fall ?
ds
In this instance, — = 32 f, (whence, ds = S2tdt).
dt
Hence s=iet^ + c.
8. In Ex. 7, at the end of t seconds what is the distance measured
from the starting-point ? What is the distance at the end of 2 seconds ? of
4 seconds ? of 5 seconds ? What are the distances, in these respective dis-
tances, measured from a point 10 feet above the starting-point ? If at the
time of the beginning of fall, the body is 20 feet below the point from which
167.] INTEGRATION. 283
distance is measured, wtiat is its distance below this point at the end of t
seconds ? Explain the meaning of the constant of integration in the general
formula derived in Ex. 7 ? Derive the results in Ex. 8 from this general
formula.
Suppose that d(t>(x)=f(x)dx; (1)
then also (Art. 29), d{4>{x) +cl=J{x)dx, _ (2)
in which c is any constant. Hence, if <^(ic) is an anti-differential
of f(x)dx, <f>(x) + c is also an anti-differential of f(x)dx. That is,
if d<l>{x) = f(x)dx,
then (f(x)dx = +(«:) + c, (3)
in which c is an ai'bitrary constant. Thus the anti-differential of
f(x)dx is indefinite, so far as an added arbitrary constant is con-
cerned. (This has already been pointed out in Art. 29, Note 6.)
On this account the anti-diiferential is called the indefinite inte^al.
The arbitrary constant is called the constant of integration. The
indefinite integral is often called the general integral. If the
constant of integration be given a particular value, as ^, — 2,
100, etc., the integral is called a particniar integral. Tor instance,
the indefinite, or general, integral of x^dx, i.e. I a^dx is \a^+c;
and particular integrals of a^dx are | a^ -f- 5, |- a;* — 11, etc
9. Name the indefinite (or general) integrals and the particular integrals
appearing in Exs. 1-8.
10. How many particular integrals (anti-differentials) can a function
have ? What must the difference between any pair of them be ?
Note 1. It should be noted that the indeflniteness in the integral does
not extend to the terms involving the variable. For instance,
|(
(i+ l)dx = ii^-f a; + c,
and j'(z4 l)dx= ( (x+ l}d(x + 1)* = i(x + '[y + k = ix^ + x + l + k;
thus the terms involving x are the same.
Note 2. The origin of the words integral and integration has been
indicated in Art. 164. It is, in a measure, to be regretted that the terra
integral and the symbol I , which both imply summation, should also be
used to denote an anti-differential. In accordance with the fashion in vogue
►Since d(x + \) = dx.
284 INTEGRAL CALCULUS. [Ch. XVIII.
in trigonometry for denoting inverse functions {e.g. sin x and sin-' x for sine
of X and anti-sine, or inverse sine, of x, respectively*) the anti-derivative
of/(x) and the anti-differential olf(x) dx are sometimes denoted by D~^f{x)
and d-^f(x)dx respectively. Tliu.s \f{x)dx, d-^f{x)dx, and D-'^f(x), are
equivalent.
NoTK 3. If dip (») =f(,x) dx, then (Art. 166) f f(x) dx = 4> {x) - 0(a).
If the upper 'end-value x is variable, and the lower end-value a is arbitrary,
then this integral Is indefinite and of the form 0 (i) + c. Accordingly, the
indefinite integral may be regarded as in the form of a definite integral whose
upper end-value is the variable, and whose lower end-value Is arbitrary.
Note 4. Result (8), Art. 166 for the area of APQB (Fig. 98) can also be
derived by a method which is founded on the notion of the indefinite integral.
For instance, see Todhunter, Integral Calculus, Art. 128, or Murray, Integral
Calculus, Art. 13.
Note 5. References for collateral reading on the notions of integra-
tion, definite integral, and indefinite integral. Gibson, Calculus, §§ 82, 110,
124-128 ; Williamson, Integral Calculus, Arts. 1, 90, 91, 126 ; Harnack,
Calculus (Cathcart's translation), §§ 100-106 ; Echols, Calculus, Chap. XVI. ;
Lamb, Calculus, Arts. 71, 72, 86-93.
168. Geometric or graphical representation of definite integrals.
Properties of definite integrals. It has been seen (Art. 166) that
if PQ (Fig. 98) is the curve whose equation is
tlien the integral j f(x)
dx
gives the area bounded by the curve, the ovaxis, and the ordinates
for vyhich x = a and x = b respgctively. Accordingly, the figure
thus bounded may be said, and may be used, to represent the
integral graphically. Hence, in order to represent an integral,
(f>(x)dx say (no matter whether this integral be an area, or a
length, or a volume, or a mass, etc.), draw the curve whose
equation is y = <t)(x), and draw the ordinates for which x = l and
x = m respectively. The figure bounded by the curve, the a^axis,
and these ordinates, is the graphical representative of the integral,
and (Art. 166) the number of units in the area of this figure is the
same as the number of units in the integral.
* See Art. 12, Note.
167, 1B8.]
INTEGRATION.
285
Tlie folloioing properties of definite integrals are important. Prop
crties (6) and (c) are easily deduced by using the graphical
representatives of the integrals.
(a) If d<l>(x)=f{x)dx, then (Art. 166)
('■f(x)dx = <f>{b) — <t>(a) and ("f{x) dx = <l,(a) - 4>(b) ;
and hence, t f(x)dx = — | f{x)dx.
Therefore, if the end-values of the variable in an integral be
interchanged, the algebraic sign of the integral will be changed.
Ex. Give several concrete illustrations of this property.
(6) \ f{x)dx= \°f{x)dx+ I f{x)dx, whatever c may be.
Draw the curve yz=f(x), and draw ordinates AP, BQ, CR, for
which X = a, X = b, X = c, respectively. Then :
C X
FiQ. 100 a.
In Fig. 100 a,
C f{x)dx = area APQB
= area APBC + area CRQB
=jy(,x)dx+jy(x)dx.
Fig. 100 b.
In Fig. 100 b,
C f(x)dx = area APQB
= area APRC - area BQRG
= Cf{x)dx- fy{x)dx
Ja Ji
=£f{x)dx+£f{x)dx.
286 INTEGRAL CALCULUS. [Oh. XVIII.
Similarly, it can be shown that
{ f{x)dx=: rf{x)dx+ r/(x)da; + ...+ Cf{x)dx+ Cf{x)dx.
That is, a definite integral can be broken up into any number of
similar definite integrals that differ only in their end-values.
(Similar definite integrals are those in which the same integrand
appears.)
Ex. 1. Prove the principle just enunciated.
Ex. 2. Give concrete illustrations of the principles in (6).
(c) The mean value of f(x) for all values of x from a to b.
(That is, the mean value of /(») when x varies continuously
from a to b.) Draw the curve y=f(x), and at A and B erect
the ordinates for which x = a and x= b respectively. Then
r f{x)dx = area APQB.
Now, evidently, on the base AB there can be a rectangle whose
area is the same as the area of APQB. Let ALMB, which has
an altitude CR, be this rectangle ; then
j f{x)dx = area ALMB = area AB • CR
= (b-a)' length CR. (1)
168, 169.] INTEGRATION. 287
The length CR is said to be the mean value of the ordinates
f{x) from a; = a to a; = 6. Hence, from (1),
Mean value of /(«) from i _ J^/Wt^a; ^
x = a to x = b / b-a ' '^
In words, «Ae mean value off{x) when x varies continuously from
a to h, is equal to the integral of f{x)dx from the end-value a to
the end-value b, divided by the difference between these end-values.
EXAMPLES.
1. Make a graphical representation of each of the integrals appearing
in Exs. 2-5 below.
2. Find the mean length of the ordinates of the parabola y = x' from
a; = 1 to X = 3. -a
i x^dx
Mean length = 'l^ = 4i.
3-1 ^
3. In the parabola y = x'', find the mean length of the ordinates of the
arc between a; = 0 and x = 2 ; and find the mean length of the ordinates
from a; = — 2 to a; = 2. Explain, with the help of a figure, why these mean
lengths are the same.
4. In the cubical parabola y = i?.
6. In the line y = ix.
169. Geometric (or graphical) representation of indefinite integrals.
Geometric meaning of tlie constant of integration. If
d <^{x) = f{x) dx,
then (Art. 167) (f{x) dx = ^(x) + c, (1)
in which c is an arbitrary constant. Draw the curve
y = <l>{x) ; (2)
let AB be the curve. Give c the particular values 2 and 10, and
draw the curves, y = <t>{x) + 2 (3)
and y = <t>{x) + 10. (4)
•For clear proof that this is the mean value, see Art. 213, where the
topic of mean values is more fully discussed, and Echols, Calculus, Art. 150
(and Arts. 151, 152).
288
INTEGRAL CALCULUS.
[Ch. XVIII.
Let CD and EF be these curves
curves obtained by giving
particular values to c,
In the case of each one of the
dy_
dx
m;
and hence, at points having
the same abscissa the tan-
gents to these curves have
the same slope, and, accord-
ingly, are parallel. For in-
stance, on each curve, at
the point whose abscissa is
m the slope of the tangent is /(m).
Moreover, the distance between any two curves obtained by
giving c particular values, measured along any ordinate, is always
the same. For, draw the ordinates KR and ST at a; = m and
x = n, respectively, as in the figure. Then, by Equations (3)
FlQ. 102.
ME=<i>(m) + 2; NS = <l>(7i) + 2 ;
and
and (4),
MR = <t>{m) + 10 ; NT = 4,{n) + 10.
Hence KR = S, and ST=».
Accordingly, the graphical representation of the indefinite integral,
I f{x) dx, consists of the family of curves, infinite in number,
whose equations are of the form y = <^(a;) + c, and which are
severally obtained by giving c particular values ; and the effect of
changing c is to move the curve in a direction parallel to the
y-axis. (Also see Art. 29, Note 2.)
Ex. 1. How many difierent values can be assigned to c ? How many
particular integrals are included in the general integral ? How many different
curves can represent the indefinite integral ?
Ex. 2. Write the equations of several curves representing each of the
following integrals, viz. . \x dx, Xx'' dx, \3xdx, isdx, ( (2 1 -|- 5) dx.
Draw the curves.
169, 170.]
INTEGRATION.
289
170. Integral curves. It d<l){x)=f{x)dx,
then (Art. 166) ("/(x) da; = ^ (a) - <^ (0).
The curve whose equation is
y = ^(x)-^ (0), i.e. y = )f{x) dx,
Jo
(1)
which is one of the particular curves representing y = <^(x)-\-c
(see Art. 169), is called the first integral curve for the curve y =f{x).
Since the area of the figure bounded by the curve y =f(x), the
a>axis, and the ordinates at a; = 0 and a; = a;, is <i>{x) — <^(0) (Art.
166), the number of units of length in the ordinate at the point of
abscissa x on the curve (1), is the same as the number of units
of area in this figure. Accordingly, if the first integral curve of
a given curve be drawn, the area bounded by the given curve, the
axes, and the ordinate at any point on the a;-axis, can be obtained
merely by measuring the length of the ordinate drawn from the
same point to the integral curve. Consequently, it may be said
that this ordinate graphically represents the area, and thus, the
integral.
Note 1. The original curve y =f(x) is the derived or differential curve
of curve (1).
Ex. For instance, for the line y = J a: + 3 ; (2)
since j ( J x + 3) ds = J a;^ + 3 x,
the first integral curve of curve (2) is tlie parabola y = J a:' + 3 x. (3)
Tliese two curves are shovrti
here. If M be any point on tlie
X-axis, and OM=m units of length,
and the ordinate MLO be drawn,
(the number of units of length
in Jlfff) = (the number of units of
area in OKLM).
For, length MG, by (3), is J m'
+3m; and
area OKLM
Fio. 103. Jo ^^ ^ ■' *
290 INTEGRAL CALCULUS. [Ch. XVIII.
Just as a given curve — it may be called the original or the
fundamental curve — has a first integral curve, this first integral
curve also has an integral curve. The latter curve is called the
second integral curve of the fundamental curve. Again, the second
integral curve has an integral curve; this is said to be the third
integral curve of the fundamental curve. On proceeding in this
way a system of any number of successive integral curves may
be constructed belonging to a given fundamental curve.
Note 2. The integral curve can be drawn mechanically from its funda-
mental by means of an instrument called the integraph, invented by a
Russian engineer, Abdank-Abakanowicz.
Note 3. Integral curves are of great assistance in obtaining graphical
solutions of practical problems in mechanics and physics. For further in-
formation about integral curves and their uses and the theory of the integraph,
and for other references, see Gibson, Calculus, §§ 83, 84 ; Murray, Integral
Calculus, Art. 15, Chap. XII., pp. 190-200 (integral curves). Appendix,
Note G (on integral curves), pp. 240-245 ; M. Abdank-Abakanowicz, Les
Integraphes : la courbe integrale et ses applications (Paris, Gauthier-Villars),
or BitterlVs German translation of the same, with additional notes (Leipzig,
Teubner). Also see catalogues of dealers in mathematical and drawing
instruments.
EXAMPLES.
1. Show that, for the same abscissa, the number of units of length in
the ordinate of the fundamental curve is the same as the number of units in
llie slope of its first integral curve.
2. Does the first integral curve belong to the family of curves referred to
in Art. 99 ?
• 3. Show how the members of the family of curves in Art. 169 may be
easily drawn when an integraph is available.
4. Write the equations of the first, second, and third integral curves
of the following curves : (a) y = x ; (b) y = 2x + 6; (c) y = smx; (d) y — <>*.
Draw all these fundamental and integral curves. Can the curve x?y = 1 be
treated in a similar manner ?
6. Find and draw the curve of slopes for each of the curves («), (6),
(c), (d), Ex. 4. Then find and draw the first, second, and third integral
curves of each of these curves of slope.
171. Summary. The two processes of the infinitesimal calculus,
namely, differentiation and integration, have now been briefly
described.
170, 171.] INTEGRATION. 291
The process of differentiation is used in solving this problem,
among others : the function of a variable being given, find the
limiting value of the ratio of the increment of the function to the
increment of the variable when the increment of the variable
approaches zero (Art. 22). This problem is equivalent to finding
the ratio of the rate of increase of the function to the rate of
increase of the variable (Art. 26). If the function be represented
by a curve, the problem is equivalent to finding the slope of the
curve at any point (Art. 24).
The process of integration may be regarded as either :
(a) a process of summation ; or
(6) a process which is the inverse of differentiation.
Integration is used in solving both of the following problems,
viz. :
(1) To find the limit of the sum of infinitesimals of the form
f(x) dx, X being given definite values at which the summation
begins and ends (Arts. 164-166) ;
(2) To find the anti-differential of a given differential f(x) dx
(Art. 167).
Problem (1) is equivalent to finding a certain area; problem
(2) is equivalent to finding a curve when its slope at every point
is known.
In solving problem (1) the anti-differential of f(x) dx is required
(Art. 166). Hence, in both problems (1) and (2) it is necessary to
find the anti-differentials of various functions of the form f(x) dx.
Chapters XIX. and XXI. are devoted to showing how anti-differ-
entials may be found in the case of several of the comparatively
small number of functions for which this is possible. It may be
stated here that, in general, integration is more difficult than the
direct process of differentiation.
CHAPTER XIX.
ELEMENTARY INTEGRALS.
172. In this chapter the elementary or fundamental integrals
(anti-differentials) are obtained, and some general theorems and
particular methods which are useful in the process of anti-differ-
entiation are described. There is one general fundamental process
(Art. 22) by which the differential of a function can be obtained.
On the other hand, there is no general process by which the anti-
differential of a function can be found.* The simplest integrals,
which are given in Art. 173, are discovered by means of results
made known in differentiation.
In Art. 174 certain general theorems in integration are deduced.
Two particular processes, or methods, of integration which are
very serviceable and frequently used, are described in Arts. 175,
176. A further set of fundamental integrals is derived in Art.
177. When f{x) is a rational fraction iu x, the anti-differential
of f(x)dx may be found by means of the results in Arts. 173, 177 ;
for this reason examples involving rational fractions are given in
Art. 178. The integration of a total differential is considered in
Art. 179.
So far as finding anti-differentials is concerned, this is the most
important chapter in the book. The student is strongly recom-
mended to make himself thoroughly familiar with the chapter
and to work a large number of examples, so that he can apply its
results readily and accurately. T7ie list of formulas, I. to XXVI.
(Arts. 173, 177), should be memorized. Every function, f(x)dx,
whose integral cau be expressed in finite form in terms of the
functions in elementary mathematics, is reducible to one or more
of the forms in this list. It is often necessary to make reductions
of this kind. A ready knowledge of these forms is not only useful
* There is a general process by which the value of a definite integral can
be found approximately, as described in Art. 193.
292
172, 173.] ELEMENTAUT INTEGRALS. 293
for integrating them immediately when presented, but is also a
great aid in indicating the form at which to aim, when it is neces-
sary to reduce a complicated expression.
173. Elementary integrals. The following formulas in integra-
tion come directly from the results in Arts. 37-55, and can be
verified by differentiation. Here w denotes a function of any
variable, and c, Cq, Cj, denote arbitrary constants.
I. r«"dM = — — - + c, in which w is a constant.
Note 1. This result is applicable in tlie case of all constant values of n,
excepting n = — 1. The latter case is given in II.
II. f— = logM-|-C0 = l0g« + lOgC = IogCM.
•' u
Note 2. The various ways in which the constant of integration can
appear in this integral, should be noted.
NoTB 3. Formula IL can also be derived by means of I. (See Murray,
Integral Calculus, p. 37, foot-note.)
III. (a^du = :^^ + c.
J log a
IV. j'e»dM = c« + c.
T. I sin u du = — cos m + c.
VI. jcos u du = sin M -I- c.
VII. \sei^udu = ta,au + c.
VIII. fcsc* MrfM = - cot u-)-c.
IX. Jsec u tan u du = secu + c,
X. fc9CMC0t«dM = - C8CU + C.
XI. C — gM — = sin-i u + c=- cos-^ u + c^.
[Remark. By trigonometry sin-i m = — cos"' a -)- 2 mr -f- - • See Art. 167,
Ex. 10 and Note 1.] ^
294 INTEGRAL CALCULUS. [Ch. XIX.
XII. f-^ = tan-iM + c.
XIII. ( — ^ — = sec-^ u + c.
■' U^u^ - 1
XIT. f ^^ =yer8-^u + c.
Note 4. Integrals XII., XIII., XIV., may also be written — cot-iu + c,
— csc-i « + c, — covers"' u + c, respectively.
174. General theorems in integration.
A. Let fix), F(x), <t>(x), —, denote functions of x, finite in
number. By Arts. 29, 31, 167, the differentials of
("[/(«) + -F'(x;+4>(a5) +•••]«'« + Co and
^f{x)dx +(F(x)dx +(^(x)dx + - + cj
are each jXx) dx + F{x) dx + >f,{x)dx -\ .
Hence, the integral of the sum of a finite number of functions and
the sum of the integrals of the several functions are the same in the
terms depending on the variable, and can differ at most only by an
arbitrary constant.
(For integration of the sum of an infinite number oi fvmctions, see
Art. 197.)
EXAMPLES.
1. I (x' + cosx + e')dx = ijfidx + \cosxdx + i e'dx ■+ co
= Jx* + sin2 + e* + c. (1)
Note 1. Each integral in the second member in Ex. 1 has an arbitrary
constant of integration ; but all these constants can be combined into one.
2. \ (jfi — sinx + Bec'^x)dx = Jsc" + cosx + tanx + c.
B. The differentials of
Ctnu dx + cq and m\udx + Cj
are each mudx. Hence,
a constant factor can be moved from either side of the integration
sign to the other without affecting the terms of the integral which
depend on the variable.
1"4-] ELEMENTARY INTEGRALS. 295
C. The differentials of
J M da; + Co, tn\ — dx-\-Ci, — \tnu dx + ca,
are each udx. Hence,
the terms of the integral xchich depend on the variable are not affected,
if a. constant is introduced at the same time as a multiplier on one
side of the integration sign and as a divisor on the other.
Note 2. Theorems B and C are useful in simplifying integrations.
3. (1) (3xdx = s(zdx = ix^ + c.
(2) f^^ fz-Ma^ ^"° +c= i- + c.
^ ' Jx* J -4 + 1 3x8
4. j 2 sin xdx — 2 I sin xdx = — 2 cos x+ c.
6. Tsin 2xdx= ^ \2sm2xdx = i fsin 2 x d(2 x) = — J cos 2 x + e.
Note 3. A factor involving the variable cannot be moved, or introduced,
in the manyier described in theorems B and C. Tlius, I x^'dx = Jx' + c;
hut x\xdx = l^ + c. Also, \x^dx = \iifi + c; hut- \x'^dx = lx' + c.
_ r , rsin u . rd (cos «) , , , ,
6. I tan u du = 1 du = - \ — = - log (cos u) + c
J J cos M J cos U
= log (sec «) + c.
r , rcosu , /•(J(sin u) , 1 , • V ,
7. I cot ad« = ( -: — du = ( -^^ '-+ c = log (am u) + c.
J J sin u J sin u
1 12
9. Write the anti-derivatives of i", 6x", 2z*% 4z-", 5x-", -j. -gi
3xt, x^2, 6^, 2^, A, -^,
Vx Vifi 7Vi5»
10. Write the anti-differentials of u^dB, iVfidt, —du,— — ds.
11. Find fox"<ii;, Kcy/P'dt, Kly/^dv, \r\/vo'dw.
296 INTEGRAL CALCULUS. [Ch. XIX.
^^ ) v' JJ+2' JT-a;-' J4«2-3« + ll*-
13. (efdt, (te^'dx, (ie'^xdx, (i'dx, (lO^dx.
14. jsin3zda;, iicosT xdx, 0 \ sec'^ d x dx, l8m(x + a)dx,
(cos{2x+ a)dx, (aec^(^ + ^\dx.
15. C sec 2 a; Un 2 Ida;, (sec^xts^n^xdx, f '^ . C xdx ^
J •' •' Vl - «- •' Vl - r*
r 7f?j: r_5i^dx_ r dp r td< r 2dx r dt
^ v'l -2ox-^' -^ \/]r^' -^ vFTV^' Ji + «'' J 1 + 4x2' Jtvj^^^'
r dx r xdx C dx r dx
•^xV'Jx^^T' -' x2 v'l* - 1 ' -' \/0x-9x-' •' Vb X - 16 x^
16. r(t2-4)2d<, r(ai + xJ)'dx, (e^^'dx, f (cos ax + sin nx) dx.
17. Express formula II. in words.
175. Integration aided by substitution. Integration can often be
facilitated by the substitution of a new variable for some function
of the given independent variable; in other words, by changing
the independent variable. Experience is the best guide as to
what substitution is likely to transform the given expression into
another that is more readily integrable. The advantage of such
change or substitution has been made manifest in working some
of the examples in Art. 174, e.g. Exs. 6, 6, 7, 8, etc.
EXAMPLES.
1. I (x + a)''dx, in which n is any constant, excepting — 1.
Put x + a = z ; then dx = dz, and
C(x + a)»dx= ('z"dc = -?^+c = ^^i2i^ + c.
J J n + 1 n + 1
This may be integrated without explicitly changing the variable. For, since
dx = d(x-i-a), ('(x + a)"dx= ("(x + a)»d(x+ a)= i^-±-^i^+ c.
J J n + 1
3. C(x + a)-idx=C-^^=f^l2^t«l = iog(x + a) + c.
J Jx+aJx+a
174, 175.] ELEMENTARY INTEGRALS. 297
8. f_^
3x
Put 4 + 3 X = z2 ; then x = ^(2^ — 4) , and dx = izdz. Heuoe, on denoting
the integral by/, ,^ , r^^^ 1 r ^_L.__JL_U
Jz^-4: 2J\z-2 2 + 2/
= ilog^ + c = ilog^^^l§^ + .
3 + 2 V4 + 3 X + 2
4. f-^—
Put X = a sin 9. Then dx = a cos S d9, and
•^ Va2 - x2 -' Vo'' - o2 sin-^ e -^ a
This integral may be found by another substitution. For, put x = az; then
<to=„d,,and f ^ ^CaJ^^C^^
= sin-i z + c = sin-i - + c.
6. 1 Va2 - x2 dx.
Ihit X = a sin 9. Then dx = acos9de ; and
J Vo^-x^ (ix= f Va^-a^ sin^ e- acosede=a' (cos'ede=^ f (1 + C082 9)£»
= ^f sin-i ^ + gJ"^-^'\ + c = i(a2 sin-i ^ + x Va^ - x^ + c.
2\, aa'a^/ a
This important integral may also be obtained in other ways ; see Ex. 4,
Art. 188, and Ex. 5, Art. 176.
6. C—^ — (Put u = az.) Ans. i-Un-i - + c.
J a' + u'^ a a
7. C "^^ (Put u = az.) A71S. - sec-i - + c
-' « Vu^ — o2 a a
8. C '^" — (Put u = 02.) ^JiS. vers-i - + c
„ C xdx
^ Vx+T
Put Vx+1=2. Thenx+1=22, dx=22d2,and C ''■^ ^r{z^-\)2zdz
•^ Vx + 1 -^ »
= 2 f (Z2 - l)d2 = f 2(22 - 3) + C = |(a; _ 2) Vx+1 + c.
298 INTEGRAL CALCULUS. [Ch. XIX.
10. (ft^dx.
■^ Vsin X
Put sinx = «. Then cosatix = dt, co^xdx = cos" a; ■coBxdx = (l — e')dt.
•' Vsinx ji
= f <^(4 - (2) + c = I sin^ a:(4 - sin^ x).
1. fsin^ X cos K dx, | tan^xsec'ida;, ( sec'' (4 — ' x) (Jx, fe-2«dx.
,3. ('\/(x + a)2dx, f v'(nH-Jix)8dx, C-=^=, f '^^
•' -' -^ VS - 7 X -' -^(4 + 5 y)3
J J J (H-x2)taii-ix' J X
6. f«(«-l)^d«, j'(o+62/)^dj/, r(m + z)'d0, Tcosfzdx.
,6. jcos'xdx, lsec*idx, Isin^xdx, j sec'' ( - |dfl.
- C sinxdx r cosxdx C sec^ x dx f sec'^xdx
J3 + 7cosx' J9-2sinx' J V4 - 3 tan x -' VlO - 3 sec^'
C_xdx_^ C(a2_x'')^xdx, CV(5M^).x<2x, f '"^ ■
-' Va-* + x--' J ^ J (a2-x2)*
176. Integration by parts. Let u and i; denote functions of a
variable, say x; then [Art. 32 (7)]
d (uv) = udv + v du,
whence udv = d (uv) — v du.
Hence, on integration of both members,
\u dv = uv — \v du, (1)
If an expression f{x) dx is not readily integrable, it may be
divided into two factors, u and dv say. The application of
formula (1) will lead to the integral | v du, and it may happen
that this integral can easily be found.
Note 1. The method of integrating by the application of formula (1) is
called integration by parts. This is one of the most important of the par-
ticular methods of integration.
175, 176.] ELEMENTARY INTEGRALS. 299
EXAMPLES.
1. Find Ixe^dx.
Put u = x; then dv = e' dx,
du = dx, and v = e'.
.: i ate* dx = xe» — j e' dx = xe' — e* + c.
2. Find
in~i X dx.
Put u = sin~i X ;
then (2v = dx,
d«- ^ ,
and r = X.
Vl-X2
.'. I sin-1 xdx = x sin"' x — l — i^^n
= X sin-i X + Vl -x2 + c. (See Ex. 18, Art. 175.)
3. Find I x cos x dx.
Put M = cos X ; then d« = x dx,
du = — sin X dx, and » = J x^.
.-. 4 X cos X dx = i x'' cos X + J 4 x^ sin x dx.
Here the integral in the second member is not as simple a form, from the
point of view of integration, as the given form in the first member. Accord-
ingly, it is necessary to try another choice of the factors u and dv.
Put u = X ; then dv = cos x dx,
du = dx, and v = sin x.
.•. jxcosxdx = xsinx — | sinxdx = xsinx + cosx + c.
4. Find 1 x* cos x dx.
Put u = i' ; then dv = cos x dx,
du = 3 x^ dx, and v = sin x.
.•. jx' cos X dx = x' sin X — 3 j x2 sin x dx. (1)
It is now necessary to find ( x^ sin x dx.
Put j« = x'' ; then do = sin x dx,
du = 2 X dx, and » = — cos x.
/. f x" sin X dx = - x^ cos X + 2 Jx cos x dx. (2)
300 INTEGRAL CALCULUS. [Ch. XIX.
It is no w necessary to find | x cos x dx.
By Ex. 3, \x cosxdx = a; sinx + cosx + c
Substitution of this result in (2), and then substitution of result (2) in
(1), gives
I x^ cosxdx = x'sinx + Sx^'cosx — Gxsinx — 6cosx + ci.
When the operation of integrating by parts has to be performed several
times in succession, neatness in arranging work is a great aid in preventing
mistakes. The virork above may be arranged much more neatly; thus:
( ifi coBxdx = x^ sin X — 3 4 x^ sin X dx
= x' sin X — 3 — x^ cos x + 2 l x cos x dx
= x' sin X — 3 [ — x^ cos X + 2(x sin x + cos x + c)]
= x^ sin X + 3 x^ cos x — 6 x sin x — 6 cos x + C
= x(x2 - 6) sin X + 3(x2 - 2) cosx + C.
The subsidiary vrork may be kept in another place.
6. Find j"Va--x^dx. (See Ex. 5, Art. 176.)
Put u = Va-' — x'^ ; then dv = dx,
du = "^ . and v = x.
Va?- - x2
. . f Va' - x' dx ;= X Vo^ - x^ + C_-g!JS — (1)
'' J \Ja:^ - x2
Now Va^ — x'^ :
a'-
Va' - x2 Va2 - X.' Va^ - x"
hence . '^ = ° - Va^ - x^.
Va^ - x2 Va^ - x'-i
Substitution in (1) gives
f Vg-^ - xMx = xVa' - x' + f-"!^ f Va' -x'dx. , (2)
•' -', Va2 - x2 -^ , ■*,,,
Hence, on transposition of the last integral in (2) to the first member,
division by 2, and Ex. 4, Art. 175,
f Va2 - x2 (ix = 1 [x v/a2 - x^ f a" sin-i ^V
176, 177.] ELEMENTARY INTEGRALS. 301
6. \ e' cos xdx — le* (sin x + cos x).
(Integrate, putting u = e'; then integrate, putting m = cosi. Take half
the sum of the two results.)
' sin X dx.
I «'«" dx.
7. (xeo'dx. 11. filogz^a;. 16. fa;=siii
8. (xe"dx. 12. (x^\ogxdx. 16. ("«
9. ix^e'dx. 18. Jtan-'xdj;. 17. r^sinicosidx.
10. (logxdx. 14. r3;tan-i2dx. 18. f? ''"'"' ^<te.
^ J •' V 1 - k2
19. Derive I e* sin X da; = J e' (sin a — cos a). (See Ex. 6.)
177. Further elementary integrals. A further list of elementary
integrals is given here. They can be verified by diiferentiation.
Some of the ways in which they may be derived are indicated in
the latter part of the article.
XV. { ta.n u du = log see u + c,
XYI. I cot udu = log sin u + c,
XTIl. i sec udu = log (sec u + tan u) + c,
= logtan(| + =) + c.
XVIII. Jcosec M dM = log tan ^ + c.
XIX. f *^^ =sin-^M + c.
XX. f--^=ltan-i«+c.
XXI
r du ^lgec-i« + c.
XXII. C '^^ =Terg-i^+c.
Jf.B. -See iVot« 1.
302 INTEGRAL CALCULUS. [Cii. XIX.
XXIV. C — ^ — = log (u + Vu^ + «2) + c,
= log«±2:«i±«-%c'.
XXT. f "" = log (M + v^M^ _ a2) + c^
= log^+^^'-"-+c'.
a
XXVI. f Va^ - M« dM = i ^M VaS _ ^2 + a2 gj^-i «\ + c.
Integral XXII. is also reducible to form XIX. For 2au — u^
= a^ — {u — ay, and du = d{u — a);
. C du r d(u-a) gin-i«zi«+c..
^ y/2au- 1*2 J Va^-(M-a)''"^ "
Ex. Show that this result and that in XXII. are equivalent.
Bemarks on integrals XV. to XXVI.
Formulas XV., XVI. For derivation, see Exs. 6, 7, Art. 174.
Formulas XVII., XVIII.
Since coseo u = cosec « cosec^-cota^
cosec u — cot u
J„™o« .. J.. C - cosec u cot u + cosec'' « ,..
cosec uau = \ ■ du
J cosec u — cot u
= C<i (cosec «- cot u) ^ i^g ^^^gg^ ^ _ ^^t „)
J cosec u — cot «
2 sin2 !^
= log 1-lC2i« = log 2_ ^ ,^g t^^ «.
""" 2sm^cos^ 2
2 2
Substitution ol u + - for « in the last two lines gives
Jcosec In + - j du = log tan ( - + - V ie- Csec u du = log tan f - + j^ J ;
= log j cosec ( u+- )— cotj U+-) |. = log (secu+tanu).
There are various methods of deriving XVII. and XVIII.
177.] ELEMENTABT INTEGRALS. 303
Formulas XIX., XX., XXI., XXII., XXIII. For derivation,
see Exs. 4, 6, 7, 8, Ait. 175, and the following suggestion :
Suggestion: = — ( ) ; — -= — ( 1 ].
u^ — d' 2a\u — a u + aj a' — u' 2a\a + u a — uj
rorraula XXIV.
Put u^ + a^ = e^; then udu = z dz, whence — = — .
z u
„ du du dz
Hence, -—^^r:^. = — = —
^ ... du du -'r dz d(u + z)
On composition, — ^^^^= = = —^ — ■ — '—
Vu'-! + a^ u + z u + z
... C '^" = CilE±ll = log (u + 2) + c = log (k + Vu^ + aO + c.
The last result may be written
log (« + v'uS'+a'O - log a + c' , i. e. log " + "^"'' "*" - + c',
a
a form which is convenient for some purposes. See Note 3.
Formula XXV. can be derived in the same way as XXIV.
Formula XXVI. For derivation, see Ex. 5, Art. 175, and Ex.
5, Art. 176.
Note 1. Integrals XIX., XX., XXI., XXII., may be respectively written
_ cos-i - + c', -- cot-i - + c', - - csc-i - + c', - covers-i - + &.
a a a a a a
Ex. Show this.
Note 2. Integrals XXIII., XXIV., XXV., may be written thus :
r_^ = 1 hy tan-i ^ + c'(ti^ < a^),
■J a^ — u' o, a
•^ u^ — a^ a «
(• '^^ =hysin '^+r,
C ^" =±hycos-i^ + C.
•' V«2 _ a -• a.
304 INTEGHAL CALCULUS. [Ch. XIX.
The functions whose symbols are here indicated are the inverse hyperbolic
tangent of -, the inverse hyperbolic sine of _, and the inverse hyperbolic
a a
cosine of -. For a note on hyperbolic functions see Appendix, Note A.
a
The close similarity between XX. and these forms of XXIII. may be remarked ;
so also, between the forms of XIX. and these forms of XXIV. and XXV.
Note 3. The same integral may be obtained by various substitutions, and
may be expressed in a variety of forms. Instances of this have already been
given ; another example is the following : Integral XXIV. can also be derived
by changing the variable from w to z by means of the substitution Vu^ + a'
= z — u; this leads to the form
J;
dn
■_ = log (a + V«2 + a^) + c.
The first member can also be integrated by changing the integral from a
to z by means of the substitution Vm^ + a^ = zu ; this leads to the form
C '^" =iog |V"-^ + «-^ + n^+c.
•' Vu2 + a^ '■v/uM^-u
It is left as an exercise for the student, to employ these substitutions in
the integration of XXIV., and, the arbitrary constants of integration being
excepted, to show the identity of the various forms obtained for the integral.
EXAMPLES.
Ji + x^ J\i + x^ i + x'l 22
2. C±tULdx= C( * +-l^\dx=4sin-ig-7(4-x')^c.
J ^/^:^^i J \^/4-x' Vi-x^J ^
3 f ^ ^C d(x + 2) ^ltan-i^±J + c.
Jx'^ + ix + 20 J{x + 2y^ + 16 4 4
4a, f _^^____= f — ±(.x + 2) ^io^(x+2+Vx^+4x+20) + c.
^ V'x2+4a:+20 •' V(a; + 2)-i+ jo
4 6. C dx ^C d(x + 2) ^,i„-igJi2^^
-' \/l2 -x--ix -' VlG -{x + 2)2 *
Notice should be taken of the aid afforded (e.^r. in Exs. 3, 4 o, 4 6) by
completing a square involving the terms in x.
6 r ^^ =lf ^(2x) ^±sec-i2x + e.
J7a:v'4a;2-9 7 J 2a:v'(2i)2 - 3^ 21 3
177, 178.] ELEMENTARY INTEGRALS. 305
dx
' h
Vie - x^
t
Put a; = i- Then dx = --dt, and
C '^ f ^JL—.--l f(l6«2-l)~id(16«2-l)
•' ^Witi - x^ •' Vl6 «^ - 1 ii^ •'
16^ ^ 16x
_ ,,v r (fa . i-.TN r dx , ,gN r dx
J 22 + 61+17' J VlT + tix-K^' •' Vx-' + Gi + lO
8 fn r t^ (•2') c ^ (s-) r <^^
^M7-6x-x2' ^ W VT-ox-x^' ' ' J Vx2-6x+7
9. (1) r — ^5 — ; (2) c — ^5 — ; (3) r '^^
10. (1) C 1^ ; (2) f <^ : (3) (■ ^
^ '^ j4x2-5x+6' ■' v/9-5x-4x' J7-5x-4x2
11. (1) C dx . ^,) f '^^ ; (.3) f ^
-'Vdx-X'' •'V9x-4x-^ •'5xV9x2-a5
(1) C ^__^; (2) CVQ^^dx; (3) fV25-x2dii.
•^ (x-l)Vx-2-2x-3 -' Jo
. (1) j* v'36-4x-(ix ; (2) ("sec 3 x dx ; (3) f cosec (4 x - a) dx.
14. (1) (ta.n(3x+a)dx; (2) ("cot (4x2+o2)x(2s ; (3) rsec2xdx. '
16. Derive integrals 62 a, h, 63 a, b, p. 406.
16, C^^^E^dx, (■— ^^-, f- <^*
12.
13
(4 + x^)^ X V12 X — x'
178. Integration of f{x)dx when /(.r) is a rational fraction.
la order to find {f{x)dx when f{x) is a rational fraction, the
procedure is as follows :
Resolve fix) into component fractions, and integrate the differ-
entials involving the component fractions.
NoTB. It is here taken for granted that in his course in algebra the
student has been made familiar with the decomposition of a rational fraction
into component fractions, or, as it is usually termed, the resolution of a
rational fraction into partial fractions. Reference may be made to works
on algebra, e.g. Chrystal, Algebra, Part I., Chap. VIII. ; also to texts on
calculus, e.g. Snyder and Hutchinson, Calculus, Arts. 132-137.
306 INTEGRAL CALCULUS. [Ch. XIX.
Examples 1, 2, 4 will serve to recall to mind the practical
points that are necessary for present purposes.
J
EXAMPLES.
3?-Zx''-\-ix + 14^
Z2 -I- z _ 6
Here x^ - 3a=^ + 4x + 14^^ _ 14^ - 10 ,
x^ + a-ti x^ + x-6
The fraction in the second member is a proper fraction, and is in its
lowest terms. Accordingly, the work of resolving it into fractions having
denominators of lovrer degree than the second, may be proceeded with.
Since its denominator, x^ + x — ti, i.e. (x — 2)(x + 3), is the common denom-
inator of the component fractions, one of the latter evidently must have a
denominator x — 2, and the other a denominator x + 3. Since these frac-
tions must be proper fractions, their numerators must be of lower degrees
than the denominators, and, accordingly, must be constants.
Accordingly, put
14x-10^/ 14X-10 \ A B ,j.
Z2 + X-6 \(x-2)(x + 3)/ 1-2 x-t-3"
Here A and B are to be determined so that the two members of (1) shall
be identically equal.
On clearing of fractions,
14x-10 = ^(x + 3)+.B(x-2). (2)
Since the members of (2) are to be identically equal, the coefficients of
like powers of x must be equal. That is,
A + B = \i,
3 ^ - 2 B = - 10.
On solving these equations, A = '^, B = V-
C^-3x^ + 4x + 14^^^r/ _ 18 52 \^
J x« -t- X - 6 J \ 6(x - 2) 5(x + 3) /
= i x2 _ 4 j; + ;^ log (x - 2) + ^ log (X + 3) -I- c.
Another way of finding A and B in (2) is the following :
The two members of (2) are to be identically equal, and accordingly equal
for all values of x.
Now, put X = — 3 ; then — 5 B = — 52 ; whence, B = -=/.
Put x = 2; then. 5^ = 18; whence, ^ = ^^
Note 1. Any other values, e.g. 3 and 7, may be assigned to x ; in this
case, however, the values 2 and — 3 give the most convenient equations for
determining A and B.
Note 2. For a more rapid way of finding A and B in such cases as (1),
Bee Murray, Integral Calculus, Appendix, Note A.
178.] ELEMENTARY INTEGRALS. 307
x2 + 21a;- 10
J
-dx.
a;' + a;''' - 5 X + 3
The fraction in the integrand is a proper fraction, and is in its lowest
terms. Accordingly, the work of decomposing it into fractious having de-
nominators of degrees lower than the third may be proceeded with. Since
the denominator x^ + ai' — bx + 3, i.e. (x — l)2(a; + 3) is the common
denominator of the component fractious, one of the latter evidently must
have a denominator x + 3, and another must have a denominator (x — J)^.
It is also possible that there may be a component fraction having the denom-
inator X — 1 ; for, if there is such a fraction, it does not affect the given
common denominator. Accordingly, put
x2-|-21x-10 A , B ,_C ,„,
(x - l)-^(x + 3) X -I- 3 (x - 1)2 X - 1
in which A, B, C are constants to be determined.
On clearing of fractions, equating like powers ot x (for reasons indicated
in Ex. 1), and solving for A, B, C, it is found that
A--i, B = S, C=5.
. r x^ -f- 21 X - 10 ^ ^ r/ - 4 3 6 \ .
" J x^ + x'-6x+3 J\x + 3 (^x-iy x-ll
= 5 log (X - 1) - 4 log (X -I- 3) §- -I- c = log C^^lil! - -J- + c.
X — 1 {x + S)* X - 1
Note 3. It may be asked why the numerator assigned to the quadratic
denominator (x — l)'^ in the second member of (3) is not an expression of
the first degree in z,»say Bx + D, instead of a constant. The reason is, that
if such a numerator were assigned, the fraction would immediately reduce to
the forms in (3). For
Bx + D _ .BCx-l)-f- D + B _ B , D+ B ^
(x-iy (.X-iy X-l (X-l)2'
forms which appear in (3).
Note 4. If a factor of the form (x — ay appears among the factors of the
denominator of the fraction to be resolved, there evidently must be a com-
ponent fraction having (x — ay for its denominator. There may also possi-
bly be fractions having as denominators (x — a) of various powers less than
r, e.g. (x — a)'-', (x — ay-', ■■•, x — a. Accordingly, in such a case it is
necessary to allow also for the possibility of the existence o^ fractions of the
forms ,, „
M F L
(x-a)'-! (x-a)'-2'
in which M, F, •••, L, are constants.
308 INTEGRAL CALCULUS. [Ch. XIX.
xs _ 8 a; - 10
dx. (Compare denominators in £zs. 2, 3.)
8. f^^
Ja;3-x2 + 4a;-4
Tlie fraction in the integrand is a proper fraction and is in its lowest terms.
If it were not so, division as in Ex. 1 and reduction would be necessary.
Since the denominator x^ — x- + ix — i, i.e. (t^ + 4)(x — 1), is the com-
mon denominator of the component fractions, one of the latter must have a
denominator x- + 4, and the other, a denominator x — 1. Accordingly, put
6x^ + 3x + n _Ax + B,C_
(x^-|-4)(x- 1) x^ + i
in which A, B, C, are constants to be determined.
On clearing of fractions, equating coefficients of like powers of x, and
solving for A, B, C, it is fnund that
^ = 0, B = 3, 0 = 6.
Jx^-x'^ + ix-i J\x^ + 4 x-l/
= ?tan-i5+51og(x-l) + c.
Note 5. The expression x'^ + 4 has factors x + 2 i, x — 2 i (i = V— 1) ;
if these be taken, component fractions imaginary in fonn, are obtained. It
is usual, however, not to carry the decomposition of a fraction as far as the
stage in which component fractions imaginary in form may appear.
Note 6. The numerator .<4x + B is assigned above ; for the numerator
over a quadratic denominator whose factors are imaginary, may have the
form of the most general expression of the first degree in x.
Note 7. When a quadratic expression x'' + px + q has imaginary factors
and is repeated r times in the denominator of a fraction, in the process of
decomposition of this fraction allowance must be made for fractions of the
forms, ^^+^ _Cx_+D ..^ _Mx + N__
(x^+j-x + g)'' (x'^+iJX + g)'-'' ' x^+px + q
6. (1) I— ; :!— — dx; (2) V— ; dx. (Compare the
Jx^-x- + 4x-i ^Jx3-x2 + 4x-4 ^ ^
denominators in Exs. 4, 5.)
178, 179.] ELEMENTARY INTEGRALS. 309
Find the anti-derivatiTes of the following fractions :
8x+ 1 J a;3 - 3 a: + 3
2a;2-9a; — 35' ' 2(3:2 + 3)
x^-2K2-i jg 12 -X -3;
x^-1 ' (3x-2)(2-^ + 5)
x' - X x' + X
10. ^'-'0^-5 21 '''-^
11.
x(2x2 4.3j;- 5) ■ xi' + 3x
x^+pq 22 2x' + 3x + 6^
x(i-p)(x + g)' ■ x3 + 3x
12 llx3-llxg-74x + 84 7x' + 9
x«- 13x2 + 36 • ^- x8 + 3x'
13 3=+ 1 24 ^ J:" - a:' + 8 X + 12
(x-l)2' ■ x-^(x2 + 4)
14. -8i+5_. 26. ^ + ^^-x'
(4x+5)2 ■ (X- l)(x^-2x + 5)
5 x' + X - 10 1 + 7 X 4- X- + x3
". ",",7 -.":"• 26
16.
x2(2 X + 6) (a;2 + 1)-^
30x2 + 43x-8
(x + 4)(3x + 2)2
Ex. 27. Show that any expression of the form r_C^E±i5l^ in which
J ax^ -4- fix 4- c
m, n, a, 6, and c are constants, is integrable. -r ".x -r i-
179. Integration of a total differential. In Art. 85 it has been
shown that the necessartf condition for the existence of a function
^^'^"^ Pdx + Qdy (1)
for its differential, is that |^= |^* (2)
It has also been stated (Art. 85, Note 1) that condition (2) is
sufficient for the existence of such a function. In other words,
if the expression (1) has an anti-differential (or integral), relation
(2) must be satisfied; conversely, if relation (2) is satisfied, the
expression (1) has an integral. Accordingly, relation (2) is called
the criterion of integrability for the expression (1). If this criterion
310 INTEGRAL CALCULUS. [Ch. XIX.
is satisfied, the expression (1) is said to be a complete differential,
a total differential, aud also an e-ract differential.
If test (2) is satisfied, the integral of (1) can easily be found.
This integral's partial x-diilerential, Pdx, can only come from
terms containing x (Art. 79). Hence, the integral of Pdx with
respect to x, namely, />
jPdx + c, (3)
must yield all the terms of the required integral that contain x.
Also, Qdi/ can only come from terms containing y. Hence the
integral of Q dy with respect to y, namely,
/
Qdy + c (4)
must yield all the terms of the required integral that contain y.
Some of these terms may contain x: if so, they have already been
obtained in (3), and need not be taken this second time. Hence,
if the integral of a differential of the form
Pdx+ Qdy
is required, apply the test for integrability, namely,
dP^dQ_
dy dx '
if this test is satisfied, integrate Pdx tcith respect to x ; then integrate
Qdy with respect to y, neglecting terms already obtained in j Pdx ;
add the residts and the arbitrary constant of integration.
EXAMPLES.
1. Integrate (2 ry + 2 + 3 y' + 12 2) di + (i» + 6 xy + 4 {/») dy.
Here P=2xy + 2 + 3y2 + 12z, and Q = x2 + 6iy + 4y».
.-. ^=2z + 6y, andi2 = 2z + 6y.
dy dx
Thus the criterion of integrability is satisfied.
Also (pdx = x'h/ + 2x + Sxy^ + 6x^;
and I Q dy = xh/ + 3 xy^ + y*, in which y* has not been already obtained
in j Pdx. Hence the integral is
2=y + 2x + 3y2 + 6z> + yi + c.
l''-'] ELEMENTABY INTEGRALS. 311
2. Verify the result in Ex. 1 by differentiation.
8. Find i 'x dy — y dxj.
Here ^ = 1, and 2— = — 1 ; hence the test for integrability is not satie-
rjx dy
fied, and there U not an anti-differential.
4. {l)(e'{cosydx-wiydy). C^j (['?jZ^^fixy+ijdx+(i-/^-6)dy'\.
6. Integrate : (1) cob x sec^ y dy — (sin z tan y + co8 x) dx.
(2) (xe> - ^ z; dj^ + C«» - 2 y -r 2 z; (Ir.. {Zj {3 - i x - yjdx - (x + y) dy.
K.B. An accurate and ready memory of the fundamental inte-
grals (Arts. 173, 177;, resourcefulness in making substitutions
(Art. 17.5^, and quickness in integrating by parts CArt. 176;, are
three very important things to cultivate in order to insure com-
fortable progress in the study of the calculus.
EXAMPLES.
1. (in^hx^^^'dz, ((a + bjx?"'-'''''dx, i (r + a)Z'~'*''ds. (rh^y^-'dy,
Jo t + 2 'J ^ + 2 J xJ -2 ' J9t^ + 20
C^^, C<J + yh'dy, (-^^, f^i^, f ^-^ dz.
2. f tan (mz + n; dz, C (Bec3x + 2ydx, j"*tan2»de, f "sinf j-(- ^jtW.
6
8. ("cos-la; das, (eecr^zdx, (cof^xdx, ((logxydx, \3?e''dx,
Cx^e-'dz, fsinalogcosz, iz'logx.
* f-b^-' /?^-' ^".-ar ^^s-
Jo 2 Jo e'' Jo Ji y/i _ x^
r asinede ^ r(l + coBe)de r dx reecxUnxdx
' J m + ncoee J sin * J sin z + cos 2 J (tan'^'z — 3f
r d0 riog' (mz 4- n) ^,^ f dz f_J?_,
312 INTEGBAL CALCULUS. [Ch. XIX.
. sin-dx
c (fa c de c
J e^ - 6-2^' J COS'' 26- sin2 2 e' J . x
sin -"V cos?
4 \ 2
( [(1 — sinzoos!^) dx — (coszsiny + %y) dy'],
\ [(1 — sin a; sill J/) (fa +(cosa;cos!/ — 1) dy'\.
7. Derive the following integrals :
(3) (x{a^ - x^y dx = - (°'^-^')"^'. (4) f -^^^ = - V^fl^^. '
^ J ^ ' 2(n + l) -J Va2"^^
8. Derive the following integrals :
(1) f-^ = ilog(a + 6a;). (2) ("(a + 6a;)»<fa =-^2Jl6£^, when n
J a -\- hx b J o{n + 1)
is different from - 1. (3) C_5_^ = 1 [a + ftx _ a log (a + 6a;)].
J a + bx b'^
W r^-^ = I [J(a + fca;)^ - 2 a(a + 6a:) + o21og (a + 6a;)]. (5) f— -^^
J a + 6a; 6' ^ a:(a + 6a;)
=_iiogi±^. (6)r_^£_=-i+Aiog«+^. (7) r ^'^^
= l[log(a + 6a;)+-^J.
9. Derive the following integrals :
Wi'V^ = .J^;l°g^^- (2) f_^^ = ^tan-ixJ^when
J a' — b-x^ 2 ab a — bx J a + bx^ ^/^ > a
a>0an(i6>0. (3) f-i^ = J- log faj^ + ?V (4)f^i^=?-
^^Ja + 6a;2 26'\^6y ^^Ja + 6a;^6
2f^^. (5) r^ = Alog_?!_. (6) f ^ = --L
hJa + bx^ ^'Jx{a + bx^) 2 a a + bx^ ^ Jx^(^a + bx^) ax
_6 r dx ,., C xdx 1
aJ a + bx'^' J (a + 6a:0» " 2 b(n - l)(a + 6a;2)''->"
10. Derive the following integrals :
(1) r:.V5T6Sdx=-2C2a::LlM:^5Z±E2. (2) rx=v/IT6Sdx =
-^ 16 b'^ J
2(8 a'- 12 a 6a;+ 15 ft'a;') VCa+6z)3 ,„. r xda; ^ 2(2 a - 6a;) /—-r-
10563 ■ ^^Jv^Tbi 362 + ■
(4) C x^dx ^2(8a2-4a6a; + 3 62a;')^^q:^ (•__*
•^ VaT^6£_ 15 6' Jx^/^.
± log ^^'^ + ^-^. for a > 0 ; -2- tan'i J^L+&?, for a < 0.
va Va + 6a; + Va V— a ' — a
<fa
CHAPTER XX.
SIMPLE GEOMETRICAL APPLICATIONS OF
INTEGRATION.
180. This chapter treats of some simple geometrical applica-
tions of integration. Examples of some of these applications
have already appeared in Arts. 166, 167. In Art. 181 integra-
tion is used in measuring plane areas, in Art. 182 in measuring
the volumes of solids of revolution. In Art. 183 the equations
of curves are deduced from given properties whose expression
involves derivatives or differentials.
Tf.B. The student is strongly recommended to draw the figure for each
example. In the case of examples which are solved in the text he will find
it extremely beneficial to solve, or try to solve, the examples independently
of the book.
181. Areas of curves : Cartesian coordinates.
A, Bectangnlar axes. In Art. 166 it has been shown that for
a figure bounded by the curve
the a;-axis, and the two ordinates for which x = a and x = b respec-
tively, the axes being rectangular, area of figure = limit of sum of
quantities y A j; (or /(a;) Ax) when Ax approaches zero and x varies
continuously from a to b. This limit is denoted by | ydx or
f{x) dx; it is obtained by finding the anti-differential oif(x) dx,
substituting b and a in turn for x in this anti-differential, and
taking the difference between the results of the substitutions.
In fewer words : tJie number of units in the area is the same as the
number of units in a certain definite integral ; namely,
area of flgnre = f ydx= \' f{x) dx. (1)
The infinitesimal differential ydx is called an element of area.
:!13
314
INTEGRAL CALCULUS.
[Ch. XX.
N.B. It will be found that in many problems it ia necessary :
(1) To find a differential expression for an infinitesimal element of area,
or volume, or length, etc. , as the case may be.
(2) To reduce this expression to another involving only a single variable.
(3) To integrate the second expression between limits (end-values of the
variable), which are either assigned or determinable.
S, Obliqae axes. Suppose that the axes are inclined at an
angle <u, and that the area of the
figure bounded by the curve vi^hose
equation is y=f(x), the a>axis, and
the ordinates AP and BQ (for which
x = a and x= b respectively), is
required. Let EM be a parallelo-
gram inscribed between A and B, as
rectangles were inscribed in the
figures in Arts. 165, 166.
Area of NM= y^x ■ sin <d.
Area APQB = limit of sum of all the parallelograms like
RM, infinite in number, that can be inscribed between AP and
BQ ; that is,
area APQB = ( y sin w • da; = gin u | y dx.
Unless otherwise specified, the axes used in. the examples in
this chapter are rectangular.
EXAMPLES.
1. Find the area between the line 2y— 5x — 7 =0, the «-axis, and the
ordinates for which x = 2 and z = 6.
The rectangle FM represents an element of area, y dx.
The area required is the limit of the sum of these element-
ary rectangles, infinite in number, from AB to DC.
That is,
area = J^^ y di = i j"^° (5 a; -)- 7) dx = i [^ + ^ *1 °
= 36} square units.
If the unit of length used in drawing the figure
were one inch, the figure would contain 36| square
inches.
Fio. 105.
181.]
AREAS OF CURVES.
315
8. Solve Ex. 1 without the calculus, and thus verify the result obtained by
the calculus.
L
3. (a) Find the area of the circle
x' + j/2 = 9 ; (6) find the area of the figure
bounded by this circle, and the chords for
which x~l and a = 2.
Let APB be the circle whose equation
is x^ + y'^ = 9. Take a rectangle I'M, sup-
posed to be infinitesimal, with a width dx,
for the element of area. Its area is ydx.
The area of the quadrant AOB is the limit
of the sum of all these elements of area,
infinite in number, between 0 and A.
Hence,
OAB=( ydx = C\/9 - x^dx - i fiVO - X'' + 9sin-i|1 =|ir8q.unit8.
.-. area circle = 4 • OAB = 9 x square units.
(6) Draw the ordinates TB and NL at the points T and N where x = l
and X = 2 respectively. The area of TRLN is equal to the limit of the sum
of all the elements of area, PM, that lie between TB and NL. That is,
area TBLN=^^ydx = Cy/9 - x^dx = \ \x-J<i - z" + 9 sin-i?!'
= i{(2V5 + 9sin-i|) - (V8 + 9sin-i})}
= V5 - v^-l- Ksin-if - sin-'i).
Here the radian measures of the angles are to be employed.
Now
V2 = 1.414 ; sin-'f = (41° 40.8') = .727 radians ; sin-i J = .340 radians.
.•• area required = 2 • TRLN= 5.126 square units.
Note 1. Other end-values of x may be used in finding the area of this
circle. Thus
areacircle = 2AiBA = 2 P ydx = 2 P V9-x:^dx = [xV9-x^ + 9sin-i?l
= 9 sin-i 1-9 sin-' (-1)= — -9[--] = 9ir square units.
Note 2. These problems may be stated thus : Find by the calculus (a) the
area of a circle of radius 3, (6) the area of a segment between two parallel
chords, distant 1 and 2 units, respectively, from the centre. In this case it
is necessary to choose axes (as conveniently as possible), to find the equation
of the circle, and then to proceed as above.
316
INTEGRAL CALCULUS.
[Ch. XX.
4. Find the area between the curve y = 2 a^, the y-aiis, and the lines
y = 2 and y = i.
The area is represented by ABLB. At any point
P{x, y) on the arc RL talce for the element of area an
infinitesimal rectangle MP. Its area is x dy.
xdy =-i- ( y^
=^[!''->s-i<*'-^'
= 5.1.. 2^(2^-1) = -(v/l6-l) =2.2797.
Fig. 107
* 2i
Note 3. The definite integral which gives the area may also be expressed
in terms of x. For, since y = 2 a;', dy = Qx^dx ; also, when y = 2, x = 1,
and when y = i, x = y/i.
2797.
ABLB= ^'~*xdy= (""^ 62'da: = l(\/IO - 1) = 2.
6. (a) Find the area of the figure bounded by the parabola y' = 4 ax,
the z-axis, and the ordinate for which x = x\. Show that this area is equal
to two-thirds of the rectangle circumscribing the figure. (6) Find the area
bounded by the parabola y'^ = 9x, and the .chords for which a; = 4 and
1 = 9.
6. Find the area between the curve y^ = 4 x, the axis of y, and the line
whose equation is y = 6.
7. Find the area included between the parabolas whose equations are
2/2 = 8x and x2 = 81/.
The parabolas are OML and OBL ; the area of
ORLMO is required. To find the points of inter-
section of the curve, solve these equations simul-
taneously. This gives (0, 0) the point O, which
is otherwise apparent, and (8, 8) the point L-
Area ORLMO = area ORLK - area OMLN
= V8 Tz^dx-; fVdx
= ^f — ^ = 21 J square units.
8. Find the area included between the parabolas whose equations are
3 «2 = 25 X and 5 x2 = 9 y.
181.]
AREAS OF CURVES.
317
9. Find the area included between the parabola (y — x — 3)' =i x, the
axes of coordinates, and the line a; = 9. Figure 52 shows that lliis problem
is ambiguous, for OTGML and OTKNL are each
bounded as described. On solving the equation of
the curve for y,
y = X ±Vx -\- Z.
Thus if OQ = x, QO = x + \/x-\-Z,
and QK = a; - Vk + 3.
.•. area OTGML
T
T
i
/
/
L
V
— «-.-
X
= i (x + Vx + 3)dx = 85^ square units ;
and area OTKNL
'9
Fig. 109.
= I {x -Vx + Z)dx = 49^ square units.
Also, the area MTN (the figure bounded by the
curve and the chord for which jc = 9) = area OTQML — area OTKNL
= 36 square units.
The area of MTN can also be found as follows :
Area MTN= limit of sum of infinite number of infinitesimal strips, like
KG, lying between T and MN.
Now strip KG = {QG- QK) dx = 2Vxdx.
area MTN
2 Vx (?x = 36.
10. Apply the second method used in finding area MTN in Ex. 9 to find-
ing the areas in Exs. 7 and 8.
11. Find in two ways the area between the parabola {y — x — by = x and
the chord for which 2 = 5.
12. Find the area between the parabola 3/ = x'^ — 8 x
+ 12, the X-axis, and the ordinates at x = 1 and x = 9.
Area =i'^^y dx = f'(a;^ - 8 x + 12) dx
= 18^ square units. (1)
The parabola crosses the x-axis at B and C where
X = 2 and x = 6.
Area APR = (*'"% dx = 2^ ;
-6
area BEC = Ty dx = - lOf ;
CQD=^^
I dx = 27.
Fig. 110.
318
INTEGRAL CALCULUS.
[Ch. XX.
Area required = area APB + area BEC + area CQD
= 2i - 10| + 27 = 18f, as in (1).
The sign of the area BEC comes out negative, because the element of area,
y dx, is negative as x increases from OB to OC ; for dx is then positive and y
is negative. On the other hand as x proceeds from Ato B and from C to Z>,
y dxis positive. The actual area shaded in the figure is 2^ + lOf + 27, i.e.
40 square units.
N.B. It should be carefully observed, as illustrated in this example, that
in the calculus method of finding areas bounded by a curve, the z-axis, and
a pair of ordinates, ai'eas above the s-axis come out with a positive, and areas
below the x-axis come out with a negative sign. Accordingly, the calculus
gives the algebraic sum of these areas ; and this is really the difference between
the areas above the x-axis and the areas below it.
13. (o) Find the area bounded by the z-axis and a semi-undulation of
the sine curve y = sin 2 x. (6) Find the area bounded by the z-axis and a
complete undulation of the same curve, (c) Explain the result zero which
the calculus gives for (6). (d) What is the number of square units bounded
as in (6) ?
14. Construct the figure, and show that, according to the calculus method
of computing areas, the area between the curve whose equation is 12 y = (a— 1)
(x — 3) (z — 6), the z-aiis, and the ordinates for which x— — 2 and z = 7, is
— fl square units ; but that the
actual number of square units in
the figure thus bounded is 12||.
16. Find the area between the
line 2y — 5z — 7 = 0, the z-axis,
and the ordinates for which z = 2
and z = 6, the axes being inclined
at an angle 60°.
Area APQB = ('^y sin 60° • dx
= BiTt60° C(5x+'!)dx
= 63.65 square units.
Note 4. In the light of the
preceding examples attention may
be again directed to the N.B.
above. These examples also show : (1) the element of area may be
chosen in various ways (compare Exs. 1, 4, 7, 9, 11) ; (2) the end values
used in a problem may be chosen in different ways (see Ex. 3, Note 1) ;
(3) the calculus method of computing areas should not be employed in a rule
of thoiab way, but with understanding and discretion (see Exs. 12, 13, 14).
181.] AREAS OF CURVES. 319
Note 5. Precautions to be taken in finding areas and computing
integrals. Suppose that the area bounded by the curve y =f(x), the x-
axis, and the ordinates at A and B for which x = a and x= b respectively,
is required. If the curve has an infinite ordinate between A and B, or if
the ordinate is infinite at A or J5, or at both A and B, or if either or both
the end values a and b are infinite, the area may be finite or it may be infinite.
It all depends on the curve ; in one curve the area may be finite, in another
curve it may be infinite. When infinite ordinates occur, either within or
bounding the area whose measure is required, and also when the end-values
are infinite, special care is necessary in applying the calculus to compute the
area. The calculus method for finding areas and evaluating definite integrals
can be u-sed immediately with full confidence, only when the end values a
and 6 are finite and when there is no infinite ordinate for any value of x from
a to 6 inclusive. For illustrations showing the necessity for caution and
special investigation in other cases see Murray's Integral Calculus, Art. 28,
Exs. .3, 4, 5, 6, Art. 29 ; Gibson, Calculus, § 126 ; Snyder and Hutchinson,
Calculus, Arts. 152, 155.
Note 6. For the determination of the areas of curves whose equations
are given in polar coordinates, see Art. 208. The beginner is able to proceed
to Art. 208 now.
EXAMPLES.
16. Calculate the actual Increases in area described in the Note and in
Exs. 2, 4, Art. 67.
17. Find the areas of the figures which have the following boundaries :
(1) The curve y = 3? and the line iy = x. (2) The parabola y^ + 8z and
the line 2 + ^ = 0. (3) The semi-cubical parabola y'^ = 7? and the line
J/ = 2 z. (4) The curves ^^ = 7? and i'' = 4 ?/. (5) The axes and the parab-
ola V^-(-\/^ = Va. (6) The curve a;^ ^. 6y = 0 and the line 2/ -|- 3 = 0.
(7) The curve (j/ -I- 4)' -f (z -|- 3)^ = 0 and the line z -I- 6 = 0. (8) The
hyperbola zy = 1 and the ordinates : (a) at z = 1, z = 7 ; (6) at z = 1,
z = 15 ; (c) at z = 1 and z = n. (d) The hyperbola xy = fc^ and the ordi-
nates at 2 = a and x = b. (And the z-axis in each case.)
18. Find the area of the loop of the curve 8 y^ = x*(Z -|- z).
19. Show that the area of the figure bounded by an arc of a parabola and
its chord is two-thirds the area of a parallelogram, two of whose opposite
sides are the chord and a segment of a tangent to the parabola.
[ScGGESTioN : First take a parallelogram whose other sides are parallel to
the axis of the parabola.]
Ex. 20. Prove that the area of a closed curve is represented by
i,^[x^-y^yt[.ox\^{xdy-yix)-^
taken round the curve. (See 'Williamson, Integral Calculus, Art 139 ;
Gibson, Calculus, § 128.)
320
INTEGRAL CALCULUS.
[Ch. XX.
182. Volumes of solids of revolution.
of the curve
Suppose that the arc PQ
Fio. 112.
revolves about the a>-axis. It is required to find the volume
enclosed by the surface generated by PQ in its revolution and
the circular ends generated by the y
ordinates AP and BQ. (This is put
briefly : the volvme generated by PQ-)
Let OA = a and OB = 6.
Suppose that AB is divided into
any number of parts, say n, each equal
to Ax. On any one of these parts, say
LE, construct an "inner" and an
" outer " rectangle, as shown in Fig.
112. Let (r be the point (x, li/), and .ff"
be the point (x + Ax, y + Ay). When
PQ revolves about the a^axis, the inner rectangle GR describes a
cylinder of radius GL (i.e. y), and thickness Aa;. At the same
time the outer rectangle KL describes a cylinder of radius KR
(i.e. y + ^y), and thickness Ax. It is evident that the volume
PQST is greater than the sum of the cylinders described by the
inner rectangles, and is less than the sum of the cylinders described
by the outer rectangles. That is,
sum of outer cylinders > vol. PQST > sum of inner cylinders.
The difference between the volume of the outer cylinders and
the volume of the inner cylinders approaches zero when Ax
approaches zero. Hence,
vol. PQST= lim^,^ {sum of inner (or outer) cylinders}.
That is,
vol. PQST = lim^jio f sum of cylinders like that generated
by GR when x increases from atob\
= lim^i^ / (vLG^ • Ax) = ir j i
(See Art. Ififi.)
182.]
VOLUMES OF REVOLUTION.
321
The infinitesimal differential wi/'dx,
which is the volume of an infinitesimal
cylinder of radius y and infinitesimal thick-
ness dx, is called an element of volume.
When PQ revolves about the jz-axis the
element of volume is evidently Tr3?dy. If
the ordinates of P and Q are c and d respec-
tively, the volume generated,
vol. PQTr= ir^^^^x^dv.
Note 1. It is almost self-evident that the volume of the inner cylinders
and the volume of the outer cylinders (Fig. 112), approach equality when
their thickness Ax approaches zero.
Note 2. See Art. 67(e).
EXAMPLES.
1. Find the volume generated by the revolution, about the x-axis, of the
part of the line .3 x + 10 y = 30 intercepted between
the axes.
The given line is AB. The element of volume
is iri/^ dx. At B, X = 0 ; at A, X = 10. Accord-
ingly, the end-values of x are 0 and 10. Hence,
vol. cone ABC= . j^'y^dx^r ^V^^Ydx
= 94.248 cubic inches.
2. Verify the result in Ex. 1 by finding the volume of the cone In the
ordinary way.
3. Derive by the calculus the ordinary formula for finding the volume of
a right circular cone having height h and base of radius a. (See Ex. 8.)
4. (a) Find the volume generated by the revolution of the ellipse
9x2^.16 2/2 = 144 about the i-axis. (6)
Find the volume bounded by a zone of the
surface and the planes for which x = 2 and
x = 3.
The element of volume is iry' dx.
(a) Vol. ellipsoid
= 2 vol. ABB' = 2 T C'^y^dx
=:?jr p(i44 _ 9xi)dx = 48t
= 150.8 cubic units.
FlO. 114.
322
Or,
INTEGRAL CALCULUS.
[Ch. XX.
vol. ellipsoid = t\ y^dx , = 160.8 cubic units.
(6) Vol. segment P.QQ'F' = tt ( y^^dx ■
87
= 16'
17.08 cubic units.
8. Find the volume generated by revolving the arc of the curve 2/ = x'
between the points (0, 0) and (2, 8), about the !^-axis.
The arc is OA. The element of volume, taking any
point P(x, y) on OA, is ira;^ dy. Hence,
vol. OAB = T ("' \^ dy ■■
rCy^dy = \tir
= 60.32 cubic units.
The integral may also be expressed in terms of x.
Thus, -^2
vol. OAB = IT j ^ a;2 ^y_
Since
y = x', (Zy = 3 z' dx.
.-. vol. OAB = 3 IT Cx* dx = >^ir = 60.32, as above.
6. Find the volume generated by revolving about the y-axis the arc of
the catenary ^ ^
between the lines a; = o and a; = — a. AC A' is the catenary ; A and A' are
the points whose abscissas are a and — a respec-
y tively. The volume generated by revolving ACA'
about 0 T is evidently the same as the volume gener-
-i" ^A ated by revolving CA. The element of volume is
wx^ dy. -
.-. vol. ACA'G = tJ "^a:^ dy. (1)
In this case it is easier to express the differential
-* and the end-values in terms of x than in terms of
FiQ. 117. y_ From the equation of the curve it follows that
X I
dy = I (eo — e~») dx. ■
Hence (1) becomes vol. ACA'G = - I (a;^ ea _ x'' e ») dx. (2)
Integration (by parts) of the terms in (2) gives
vol. ACA'G=^le + --4\= .878 a'.
182.]
EXAMPLES.
323
7. Find, by the calculus, the volume of the ring generated by revolv-
ing a circle of radius 5 inches about a line distant 7 inches from tlie centre of
the circle.
Let C be the circle and ST the line. Choose
for the X-axis the line passing through the centre
0 at right angles to ST, and take OY for the
^-axis. Then
the equation of the circle is x'^ + y^ = 25,
and the equation of the line is x = 7.
Through any point P(x, y) on the circle, draw
PPM parallel to the x-axis. Suppose that PO,
at right angles to PP, is of infinitesimal length
dy. Then the rectangle PG, on revolving about ST, generates an inflni-
tesiinal part of the volume of the ring. The limit of the sum of these parts
as y changes from B' to B, is the volume required.
The volume generated by PO = ir (_PM^ — PM^) dy.
Now PM=7 -PB = 7 - V25 - y^,
and PM= 7 + BP = 7 + V25 - y^.
. • . vol. generated by PG= 28 ir V'25 — y'' • dy.
vol. of ring = 2 f*" 28 irV25^^d!/=360ir2 cubic units.
|0r, vol. of ring = I 28 ir V25 — y'^ dj/ =350 t' cubic units,
as in Ex. 4 (o).]
8. Find the volume of a cone in which the base is any plane figure of
area A, and the perpendicular from the vertex to the base is h.
9. Find the volume generated by revolving the arc BEC (Fig. 110)
about the x-axis.
10. Find the volume generated by the revolution of MTKN (Fig. 109)
about the x-axis,
11. Find the volume generated by the revolution of ORLM (Fig. 108)
about the y-axis.
12. Find the volume generated by the revolution of ABLB (Fig. 107):
(a) about the y-axis ; (6) about the x-axis.
13. Find the volume generated by revolving the loop in Ex. 18, Art. 181,
about the x-axis.
324 INTEGRAL CALCULUS. [Ch. XX.
14. Find, by the calculus, the volume generated by the revolution about
the 2-axis, of the part of each of the following lines that is intercepted between
the axes, and verify the results by the ordinary rule for finding the volume
of a cone :
(1) 3a; + 42/ = 2; (3) 7a: + 3!/ + 20 = 0;
(2) 2a;-52/ = 7; {i) 3x - iy + 10 = ii. ^
16. Find the volume generated by the revolution about the j/-axis, of
each of the intercepts in Ex. 14, and verify the result by the usual method
of computation.
16. Find the volume generated when each of the figures described in
Ex. 17, (l)-(9), Art. 181, revolves about the x-axis.
17. Find the volume generated when each of the figures in Ex. 16
revolves about the j-axis.
18. The figures bounded by a quadrant of an ellipse of semi-axes 9
and 5 inches and the tangents at its extremities revolves about each tangent
in turn : find the volumes of each of the solids thus generated.
19. Find the volume of a sphere of radius a, considering the sphere
as generated by the revolution of a circle about one of its diameters.
Note 3. The volume of a sphere may also be obtained by considering the
sphere as made up of concentric spherical shells of infinitesimal thickness.
The volume of a shell whose inner radius is r and whose thickness is an infini-
tesimal dr is (to within an infinitesimal of lower order) 4 irt" dr. Accordingly,
volume of sphere = i'iirr'dr = J to'.
20. Find the volume generated by the revolution o£ the hypocycloid
xi+ yt = as about the x-axis. (^Ans. iVV'^'''-)
183. Derivation of the equations of curves. The equation of a
curve or family of curves can be found when a geometrical prop-
erty of a curve is known. Exercises of this kind constitute an
important part of analytic geometry. For instance, the equation
of a circle can be derived from the property that the points on
the circle are at a given common distance from a fixed point.
The statement of a geometrical property possessed by a curve
may involve derivatives or differentials. To derive the equation
of the curve from this statement is, quite frequently, a diiRcult
problem. There are a few simple cases, however, in which it is
possible to find the equation of the curve by means of a knowl-
edge of the preceding articles. A few very simple examples
have been given in Art. 167.
182, 183.] EQUATIOSS OF CUliVES. 325
Note 1. It may be worth while merely to glance at more difficult prob-
lems of this kind and at the text relating thereto, in Chapter XXVII. and in
Murray's Introductory Course in Differential Equations, Chaps. V. and X.
Also see Cajori, History of Mathematics, pp. 207-208, " Much greater than
. . . integral of it."
NotS 2. It has been shown in Arts. 59, 62, that for the curve whose
equation is /(a;, y) = 0, rectangular coordinates, if (x, y) denotes any point
on the curve and m is the slope of the tangent at (x, y), then
B! = ^ ; subtangent = y — ; subnormal = y J--
dx dy dx
Note 3. It has been shown in Arts. 63, 64, that for the curve whose
equation is /(r, 8) = 0, if (r, 9) denotes any point on the curve, f the angle
between the radius vector and the tangent at this point, and ^ the angle
which the tangent makes with the initial line, then
tany(' = r^; 4, = ^ + e;
dr
polar subtangent = r^ — ; polar subnormal = — •
NaB> Draw the curves iti the following examples.
EXAMPLES.
1. A curve has a constant subnormal 4 and passes through the point
(8, 6) : what is its equation ?
Here the subnormal, y-^ = 4.
dx
On using differentials, ydy = i dx.
Integration gives ^ + Ci = 4 a; + Cj ;
2,2
whence ^ = ix -k- k, m which * = C2 — ci.
Since (3, 6) is on the curve, ^ = 12 + *, whence *; = J.
Accordingly, ^ = ix + -, i.e. !/2 = 8 a; + 1, is the equation.
Note 4. In working these examples it is enlivening
and helpful, to express the given conditions by means
of a figure. This tentative figure can be corrected
when fuller information is derived. Thus, for Ex. 1
draw a curve passing through (3, 6), and at any point
P(x, y) on this curve make the construction in
Fia."ll9. F'g- 11^ showing the subnormal 4. Here Z MPN
= /LHPT. Now tan JlfPiV=-, i.e. ^ = -- Then proceed as above.
y dx y
326 INTEGBAL CALCULUS. [Ch. XX.
2. A curve has a constant subnormal and passes through the points
(2, 4), (3, 8) : find its equation and the length of the constant subnormal.
3. A curve has a constant subtangent 2, and passes through the point
(4, 1) : find its equation.
4. Determine the curve which has a constant subtangent and passes
through the points (4, 1), (8, e) : find its equation and the length of the
subtangent.
6. Find the curve in which the length of the subtangent for any point
is twice the length of the abscissa, and which passes through (3, 4).
6. In what curves does the subnormal vary as the abscissa ? Deter-
mine the curve in which the length of the subnormal for any point is pro-
portional to the length of the abscissa, and which passes through the points
(2, 4), (3, 8).
7. In what curves does the slope vary as the abscissa ? Determine
the curve in which the slope at any point is proportional to the length of the
abscissa, and which passes through the points (0, 2), (3, 5).
8. In what curves does the slope vary inversely as the ordinate ?
Determine the curve in which the slope at any point is inversely proportional
to the length of the ordinate and which passes through the points named in
Ex.7.
9. Determine the polar curves in which the tangent at any point
makes with the initial line an angle equal to twice the vectorial angle. Which
of these curves passes through the point (4, - j ?
10. Determine the polar curves in which the subtangent is twice the
radius vector. Which of these curves passes through the point (2, 0") 1
11. Determine the polar curves in which the subnormal varies as the sine
of the vectorial angle, and which pass through the pole.
CHAPTER XXI.
INTEGRATION OF IRRATIONAL AND TRIGONOMETRIC
FUNCTIONS.
184. The integration of differential expressions involving irra-
tional quantities and trigonometric quantities will now be con-
sidered. Examples of this kind and methods of treating them
have already been given in preceding articles. (See Art. 174,
Art. 175, Exs. 10-18.) Only a few very special forms are dis-
cussed in this book.
Note. Chapter XIX. provides a good part of the knowledge of formal inte-
gration sufficient for elementary work in physics and mechanics and for the
ordinary problems in engineering. Accordingly, this chapter may be merely
glanced at by those who have only a very short time to give to the study of
the calculus and thus find it necessary to take on faith the results given in
tables of integrals.
INTEGRATION OF IRRATIONAL FUNCTIONS.
185. The reciprocal substitution. This substitution, which some-
times leads to an easily integrable form, has been shown in Art.
177, Ex 6. Additional exercises are here appended.
Ex. 1. Find f ^
xWx^ - a»
Put x = -- Then dx = --dt; and
f <^^ = - f "^' = i f (1 - a2ta)"*<J(l - a'fi)
= i(l_a»t«)4 = ^^^'.
Exs. a-9. Derive integrals 23, 26, 27, 39, 42, 43, 54 o, 59 a, 61 a, pages
453-456.
327
328 INTEGRAL CALCULUS. [Ch. XXI.
Note. Trigonometric substitutions. Examples of a useful trigonometric
substitution liave been given in Art. 175, Exs. 4, 5. A differential expression
in which Va'^ + x^ occurs may sometimes be simplified for purposes of inte-
gration by substituting a tan d for x, and expressions containing Va;^ — a^
by substituting a sec 8 for x.
For instance, in Ex. 1 put a; = a sec 0. Then dx = asecff tan 6 d6 ; and
dx 1 C „j^ !„:„<. Va:2 - a'^
f ^^ = i f cos 9 de = — sin e
<xhL
186. Differential expressions involving Va + bx. By this is
meant differentials in which the irrational terms or factors are
fractional powers of a single form, a -)- 6x. (In particular cases o
may be 0 and 6 may be 1 ; the irrational terms or factors are then
fractional powers of x.) For preceding instances see Art. 175,
Ex. 3, and Exs. 4, 10 at the end of Chapter XIX.
If n is the least common denominator of the fractional indices
of a + hx, the expression reduces to the form
2^(x, -v''a + 6a!)da;, (1)
This can be rationalised by putting
a + 6a: = «".
For then x = ^ ~^ and dx = ^ i^^''^ dz ; and, accordingly, ex-
h . h
pression (1) becomes
^F\^^,zY-^dz.
This is rational in z, and accordingly may be integrated by the
preceding articles.
Ex, - ■ ^*
;. 1. f ^_^. Ex. 4. ("(3 + a;) V(2 + i)'dx.
1 + x'
Ex.2.
f ^^. Ex. S. (■ ^
•'ai + l •' V2^rx(7-l-5V2-x)
C xdx . Ex. 6. f-
Ex. 8. I ^''^ ■ Ex. 6. f^^^HdL^ dx.
\/(3x-2)< •' Vx + 1 - 1
180, 187.] IRRATIONAL FUNCTIONS. 329
187 A. Expressions of the form F(x, Vjt^ + ax + b) dx. B. Ex-
pressions of the form F{x, y/ — x^ + ax + b)dx ; F(u, v) being a
rational integral function of u and v.
A. The first expression can be rationalised by putting
Va^ + ax + b = z — x, (1)
and changing the variable from x to z.
For, on squaring and solving Equation (1) for x,
- = ^- (2)
From this, dx = gi"' + ""+/) d^. (3)
On substituting the value of x in (2) in the second member
of(l), ,
a -\-'Zz
Accordingly,
F{x, V^?+^^&) dx becomes 2 Ff^, ^_!±^y;±^d^.
^ ' ^ ^ ' \a^-2z a+2z ) {a+2zf
This is rational in z, and, accordingly, may be integrated by
preceding articles.
Ex. 1. Find
C xdx
Assume Vx^ — x + 1 = z — x.
_ 2^-1
From this,
2«-l
«" - z + 1
and Vxii - x + 1 = « - a; 2 z - 1
330 INTEGRAL CALCULUS. [Ch. XXI.
On substitution of these values in the given integral,
(See Art. 108.)
_ X + Vz^ — a; + 1 .
4(2i- 1 + 2 Vaj^-i + i)
+ J log (2 a; - 1 + 2 Vx''' - a + 1) + c
= J log (2 K - 1 + 2 Vx^ - X + 1) + Va" -x-^\+k.
(fc = i + c.)
It happens that this is not the shortest way of working this particular
example ; but the above serves to show the substitution described in this
article. The integral may also be obtained in the following way ; this
method is applicable to many integrals.
C xdz ^r/1. 2.-1 ^1 1 N^^
J ^3fi-x + \ -^ \2 y/x^-x+\ 2 Va'^ - a: + 1 /
= f i (X— x + l)-id(x2 _ a; + 1) + J r "^^
= \/a;2 _ a + 1 + J log (a; - J + Va;^ - x + 1) + c
= Vx" -x + l + ilog(2x-l + 2 Vx2 - X + 1) + ci.
Ex.2. C (a:-5)jx_^r/ .-3 ^ \ dx
•'Vx2-6x + 25 -' \ Vx-' -6x + 25 VCx - 3)^ + 16/
= Vx2-6x + 25 - 2 log (x - 3 + >/x'^ - 6 x + 25).
B. Suppose that —3?-\-ax + h = {x — p)(ij — x).
The second expression at the head of this article can be rational-
ised by putting
V— x^-'rax+h, i.e. -y/ (x — p){q — x) = (x—p)z, (3)
and changing the variable from x to z.
On squaring in (3), q — x = (x—p)z^;
on solving for x, x = ^ ^ ; (4)
whence, on differentiation, dx = — rp^- — ^ dz.
' ' (1 + zy
187.] IRRATIONAL FUNCTIONS. 331
Substitution of the value of x in (4) in the second member of
1 +z'
Accordingly,
F(x, ■y/-3:'+ax+b)dx becomes 2 (p-.q)FfS^^, iSfZElA^^^—
This is rational in z, and, accordingly, may be integrated by
preceding articles.
Note 1. Instead of (3) the relation
V(i-j))(?-a;) = (9-a:)z
may be used.
Note 2. If -s/±px- + qx + r occurs, it may be reduced to form A or
JS; thus, Vp-J±x^ + ix+--
' BO
P P
EXAMPLES.
3. Find f -^'^
Via -x-x^
Put Vl2 - a; - x-^ = V(a; + 4) (.3 - x) = (x + 4)z.
From this, on squaring, 3 — x = (x + 4)z\
.3 - 4 z2
On solving for x, x =
1 + 2^
Accordingly, dx = 7-^f^, Vl2 - K - a:-^ =(x + 4)z = — ^•
(1 + 2-)- 1 +Z''
' ■ >' X V12 - a; - a;'^ J4z^-3 2 V3 2z + -
.2>/3-x-V3(x+4),
2 V3 '^2 V3^ + V3(x+4)
lo!
4. Solve Ex. 3, using the substitution Vl2 - x - x^ = (3 - z) z.
J V'4 x-^ + 6 X +"n •' V'12 - 4 X - x2
7. ( ^^
•' X V12 - 4 X - x^
g r (-3x-4)^x_. r3x-4^3_4-[
' •'xVl2-4x-X'' L X xj
332 INTEGRAL CALCULUS. [Ch. XXI.
9. f ^ 10. CC3:' + 2x-3)j^.
•^ K V^M^T+T -^ a; Vx' + X + 1
11. f ^^ (Putx + 2 = z.)
-' (I + 2) Vx- + 4 X - 12
Note 3. The integrands in integrals of the form | xp(a + fta;^) ^ dx in
which ni is any integer and p is an odd integer, positive or negative, can
be rationalised by means of the substitution a + bx^ = z^. Thus :
2. r ^'<^ .
12.
Put a;2 _ (j2 _ 22_
Then xdx = zdz ;
and r^^^=r(3^ + a=)d2 = 5(2^ + 3n=)=^i±2«!^^?3^.
-^ Vx^ - a- •' 3 3
IS. Find f ^2 (see Formula XXI., Art. 177): (1) Using the
•^ X Vx^ _ ai
substitution x = a sec S ; (2) using the substitution x = - ; (3) using the
substitution x^ — a^ = z\ (Show the equivalence of the various forms of
the integral.)
14. Show the truth of the statement in Note 3.
188. To find j jr'"(o + bx^ydx. Here m, n, and p are constants,
positive or negative, integral, or fractional. The given integral,
as will be shown in the working of examples, can be connected
with simpler integrals in a particular way. By "a simpler inte-
gral" is meant one that is simpler from the point of view of
integration. For instance, if m = 5, the integral in which m = 3,
other things being the same, is simpler ; if 2' = — f > the integral
in which p = — ^, other things being the same, is simpler. It will
be found that the given integral can he connected with an integral in
tvhich the m is increased or decreased by n, or with an integral in
which the p is increased or decreased by 1 ; i.e., with one or other
of the four integrals :
r a-"-*" {a + botf)" dx, Cx^in + ft.T")"*' dx.
Cx^-^a + bary dx, C3r(a + bxf -^dx.
(«)
187,188.] IRRATIONAL FUNCTIONS. 333
When one of these four integrals is chosen, a relation between
it and the required integral can be expressed in the following
way:
Form a function of x in which the x outside the bracket has an
index one greater than the least index of the corresponding x in the
required and the chosen integrals, and in which the bracket has an
index one greater than the least index of the bracket in those integrals.
Give the function thus formed an arbitrary constant coefficient and
give the chosen integral an arbitrai-y constant coefficient ; equate the
sum of these quantities to the required integral. The value of the
arbitrary coefficients can then be determined.
For example, let
I 7f{a-\-b3fydx be connected with | a;''(a+6x")''"'da!.
The function formed by the rule is a;""'"'(a + bx")'. Put
Cafia + bafydx = Aaf+\a + bx'y + B Cx''{a + baf)"-^ dx, (1)
in which A and B are arbitrary constants.
It is now necessary to find such values for A and B as will
make (1) an identical equation.
In order to determine A and B, take the derivatives of both
members of (1), simplify, and then equate coefficients of like
powers of x. Thus, on differentiating the members of (1),
af (a + 6x-"y = A{m + l)«"(a + 6af )^ + ^x^+Xa + 63;»y-'«6af ">
+ Bx^'ia + &af )'-'.
On division by af (a + 6a?')''"*, and simplification,
a + 6a;" = Ah{m + »ip+l)a;» + Aa{m + 1) + B.
On equating coefficients of like powers of x and solving for A
and B,
m + np + l' m-rnp + \
334 INTEGRAL CALCULUS. [Ch. XXI.
The substitution of these values in (1) gives
f-
x'^ia + bx")Pax = ■
m + np + 1
"^^ Cx'^ia + boC^^P-^ dx.
m + tip
On connecting the required integral ■jvith each of the other
integrals in (a) and proceeding in a similar manner, the results
(1), (2), (4), page 451, are obtained. The deduction of them is
left as an exercise for the student.
Note 1. Formulas 1-4, page 451, are examples of what are usually termed
Formnlas of Reduction, Frequently integrals are obtained by substituting
the particular values of m, n, p in these formulas of reduction. To memorize
such formulas is, however, a waste of energy ; it is better, at least for
beginners, to integrate by the method whereby these formulas have been
obtained.
Note 2. It will be observed that some of these formulas fail for certain
values of m, n, p; viz., when m + np + 1 = 0, when m = — 1, and when
p = - 1. Other formulas or other methods may be applied in each of these
cases.
Note 3. Its success may be regarded as one proof of the above method.
In the large majority of text-books on calculus, formulas 1, 2, 3, 4, page 451,
are derived in a straightforward way by integration by parts. For this
derivation see almost any calculus, e.g. Murray, Integral Calculus, Appendix,
Note B. For other formulas of reduction for jx"'(a-|- bx'')pdx, obtained
by the method of "connection" or "arbitrary coefficients," see Edwards,
Integral Calculus, Art. 82, and integrals 5, 6, page 462.
EXAMPLES.
1. Find (" ^ i.e. ( x-^ (x'^ - a^)'^ dx. (See Ex. 1, Art. 185.)
'' xWx^ - a^ •'
Here m=— 2, n = 2, p = — i. The best integral to connect with is
obviously the integral in which the m is raised by 2, viz. f -. On
making the connection according to the directions given above,
(1) (x-^ (x^ - 0=)"^ dx = Ax-^x^ - 0^2)* + b((x^- a=)"J dx.
It is now necessary to find such values for A and B as will make this
equation an identical equation.
188.] IRRATIONAL FUNCTIONS. 335
On differentiation, and equating the deriyatives,
x-2 (z-i - a^yi = - Ax-^ (x2 - a2) i + 4 («2 _ a'^yi + 5 (^2 _ a^yi,
Ob simplifying, by multiplying through by z'^ (z^ _ a^y,
1 =- -4(z2 - a2) + At!^ + B3?.
On equating coefficients of like powers of z,
£ = 0 and Aa' = 1 ; whence A=—.
On substitution of these values of A and .B in (1),
2. Find f_^^— -, t.e. Cz«-(z2 - oS)"^ (fa. (See Ex. 12, Art. 187.)
Here m = 3, n = 2, j) = — J. It will obviously be an advantage to lessen
m. Accordingly, let connection be made with \x{x^ — a^y^dx. On doing
this in the way described,
(1 ) (x' (z2 - a2)~- dx = Ax'^ (z2 - n^) ^ + .B f z (3? - a^'^ dx.
It is now necessary to find such values for A and B as will make this an
identical equation.
On taking the derivatives and equating them,
2« (x' - o2)"^ = 2Az(x^-a^)^ + Az'i (z^ - a^yi + Bx (z^ - o')"*.
On simplifying, by dividing through by x (jfi — cC-y^
z2 = 2 ^ («2 - a2) JrAx^ + B.
On equating coefficients of like powers of x, and solving for A and £, it is
found that A-\, B = ^a\
Substitution in (1) gives
J VP"^^ •' Vz2 - a2
= i z^ v;^^3^ + i aH^^::^ = ^+l^?l>^HiF.
Here m = 0, n = 2, p = - A;. In this case i
•oceeding according to the rule,
(1) ("(z^ + a^)-* dx = AxCx^ + a2)-*+i + b((x'' + a2)-»+i dz.
Here m — 0,n = 2,p=—k. In this case it is better to increase p. On
proceeding according to the rule,
336 INTEGRAL CALCULUS. [Ch. XXI.
On difierentiation, simplification of the resulting equation by division by
(x^ + a^)-*, equating coefficients of lilie powers, solving for A and B, and
substitution of tlieir values in (1), it will be found that
C dx _ 1 f X \(2k 3)( '^ ]
J {,x' + a2)* 2 a\k - 1) I (a^ + a^)*-' ^ ^ J (a;^ + a^)'-! /
4. Derive f Va^ — z''' (it by this method. (See Ex. 5, Art. 175, Ex. 6,
Art. 176.) -^
6. Do Ex. 16, Art. 177, by this method.
Note 4. It is sometimes necessary to repeat the operation of reduction
two or more times.
6. Derive integrals 21, 22, 23, 28, 30, 35, 40, 41, 42, 44, pages 453-454,
and others of the collection.
7. Derive integrals 48, 53, 54, 65, 57, pages 455-456 [ \'2 ax ± x- —
xi(2o ± xfy. (Compare Exs. 6, 7, and Exs. 2-9, Art. 185.)
8. Derire formnlas 1-6, page 461.
9. Find C^^^^^.^.=-K-^')S
f ^ (te = - 1 3 a* sin-i^ - x(2 z^ + S a^)Va^-x^\.
•' Va^ - x' 8 1 a ^ ^ i
10. Using integrals 1-4 as formulas of substitution for the values of m, n,
p, M, 6, derive some of the integrals 21-30, 37^6, 53-61, pages 453-456.
Note 5> On the integration of irrational expressions also see Snyder and
Hutchinson, Calculus, Arts. 129-131, 139, 140. These articles convey valu-
able additional information, and, In particular, Art. 139 gives an interesting
geometrical interpretation concerning the rationalisation of the square root
of a quadratic expression. Also see the references given in Art. 192, Note 2.
INTEGRATION OF TRIGONOMETRIC FUNCTIONS.
N.B. On account of the numerous relations between the trigonometric
ratios, the indefinite integral of a trigonometric differential can take many
forms.
189. Algebraic transfonnations. A differential expression in-
volving only trigonometric ratios can be transformed into an
algebraic differential by substituting a variable, t say, for one of
the trigonometric ratios. The algebraic differential thus obtained
may possibly be integrated by some method shown in the preced-
ing articles. Knowledge as to what substitution wili be the most
189, 190.] TRIGONOMETRIC FUNCTIONS. 337
convenient one to make in a given case can best be acquired by
trial and experience. Illustrations of this article have already
been met in Art. 176, Exs. 10, 11, 16, 17.
Ex. 1. See exercises just referred to.
Ex. 2. Do Exs. 1-5, 7-9, Art. 190, making algebraic transformations.
190. Integrals reducible to | F(u) du, in which u is one of the
trigonometric ratios.
(a) I sin" X dx and | cos" x dx are thus reducible when n is an
odd positive integer. For
I sin" xdx= I sin"-^x ■ sin a; da; = — ( (1 — cos'a;)~ d (cos a;).
„ -I
The latter form can be expanded in a finite number of terms,
being an integer, and then integrated term by term. | cos" a; da;
can be treated similarly.
EXAMPLES.
1. j cos' xclx= f cos* x.cosxdx= )(!— sin'' x)^ d (sin x)
= 1(1-2 sin' X + sin* x) d (sin x) = sin a; - | sin' x + J sin' z + c.
2. i sin' X dx, I cos" a; da;, ( sin' x dx.
(6) I 8in"xeos"'xda5 is thus reducible when either n or m is a
positive odd integer.
3. ( sin» X cos' X di = ( sin^ x cos'^ x sin x dx
= — I (1 — cos2 x) cos^ X d (cos x) = — J (cos'^ X — cos^ x) d (cos x)
= — ^oos'x + T«rCos^a: + c-
cos' X dx
4. (1) f ""' ^ dx. (2) fcos'xsin^xdi, (3) ("
-' v'cosx "^ ^ Vsinx
(4) j cos' X sin' X dx.
NoTB. Case (a) is a special case of (6).
338 INTEGRAL CALCULUS. [Ch. XXI.
(c) I sec" X doD and | cosec" as dx are thus reducible when n is
a positive even integer.
6. j coaec'xdx = i cosec'a; • cosec^xda; = — j (1 + co\?xyd(fiotx)
= — cota;(l + Jcot^K + |cot<a;).
6. Show the truth of statement (c).
7. (1) jsec^xciK, (2) I cosec* a; dx, (3) \st<fixdx.
(d) j tan»»» X sec" x dx and | cot»» x cosec" x dx are thus reduci-
ble when n is a positive even integer, or when m is a positive odd
integer.
8. Show the truth of statement (d).
9. (1) i tan^ X sec* a; dx, (2) j sec' x Vtan x dx, (3) i tan'xsec^xdx,
(4) ( tan'xsec'xdx, (5) 1 cot' x Vcosec x dx, (6) cot^ x cosec' x dx.
19L Integration aided by multiple angles. It is shown in
trigonometry that
sin u cos u = ^ sin 2 u,
sin" M = J- (1 — cos 2 m),
cos- M = ^ (1 4- cos 2 u).
Accordingly, if n and m are positive even integers, sin" x, cos" x,
and sin" a; cos" a; can be transformed into expressions which are
rational trigonometric functions of 2 a;. Dififerential expressions
involving the latter are, in general, more easily integrable than
the original differential expressions in x.
Ex. 1. j cos* X dx = ("{^1 + cos 2 x)P dx = J f (1 + 2 cos 2 x + cos" 2 x) dx.
Now I 2cos2xdx = sin2x, and 4 cos"2xdx = ^ I (1 + cos4x) dx =
J(x + Jsin4x). .-. ( cos*xdx = f X + Jsin2x + ^sin4x + c.
Ex. 2. f sin" X cos" x dx = J Jsin" 2 x dx = } f (1 — cos 4 x) dx
= ix- j<jsin4x+ c.
Ex.3. (1) |sin*xdx, (2) (cos^xdx, (3) | sin*xcos'''xdx,
(4) j sin'xcos'xdi, (5) Tsin* x cos* x dx.
190, 192.] TRIGONOMETRIC FUNCTIONS. 339
192. Reduction formulas. There are several formulas which
are useful in integrating trigonometric differentials. A few of
them are deduced here ; the deduction of the others is left as an
exercise for the student.
(a) To find A: isin''xdx, and B: lcos''xdx, when n is any
integer.
A. Integrate by parts, putting
dv = smxdx ; then u = sin"~' x,
V = — cos X, du = (n — 1) sin""'' a; cos x dx.
.: I sin" a; da; = — sin"~' x cos a; + («■ — 1) ( sin'^^'ajcos^xda;
= — sin"~'x cos a; + (n — 1) I sin"~^a; (1 — sin'' a;) dx
= — sin"~' X cos X + (n — 1) I sin"~^ a; da; — (n. — 1) | sin" x dx.
From this, on transposition and division by n,
/• n J sin""' X cos x , n — 1 f ■ „._» ■, /i ^
sin"a;da5 = 1 I sm" ^xdx. (1)
n n J
This is a useful formula of reduction when n is a positive
integer. From it can be deduced a formula which is useful when
the index is a negative integer. For, on transposition and division
by "' ~ , formula (1) becomes
n
fsin-^xda; = ^in-a^cosx^ n ^i^n^^^.
J « — 1 n — lJ
This result is true for all values of n, and, accordingly, for
n = N-^2. On putting iV + 2 for n, this becomes
/. V J sin*'+' a; cos X , N+2 C„i^N+i^.i^ ro\
%urxdx= f- ^^ ' I sin^+'^ajda;. (i;
N+1 N+IJ
If .y is a negative integer, say — m, (2) may be written
/dx _ _ 1 cos a; , m — 2 C dx ,^s
sin'"a; "" m — I sin"—' x m — lJ sin"'-^a;
340 INTEGRAL CALCULUS. [Ch. XXI.
In the above way calculate the following integrals :
Ex.1. (1) jsin'^xAE, (2) Jsin'sc*;, (3) \sm*xdx, (4) XBirfixdx.
Ex. 2. (1) f-^, (2) f '^, (3) f-^.
J sin-' a; J sin'x Jsin*x
Ex. 3. Compare the results in Exs. 1, 2, with those obtained for these
integrals by methods of the preceding articles.
B. Similarly to A there can be deduced results 69, 71, page 457,
for B. Formula 69 is useful for positive indices, and 71 for
negative indices.
Ex. 4. Deduce formulas 69 and 71.
Ex. 6. (1) Ccos*xdx, (2) (cos^xdx, (3) f-^, (4) f-^.
J J J cos*x J cos^x
Compare results with those obtained by methods of preceding articles.
(6) To find I sec" x dx when n is a positive integer greater than 1.
Put sec' xdx = dv; then sec"~^x = ?t,
tan x = v, (n — 2) sec"~^ x tan xdx= dxi.
.: I sec" xdx — sec"~^ x tan x — (n — 2) | sec"~^ x tan' x dx.
Prom this, on substituting sec'a; — 1 for tan' a;, and solving
for I sec" X da;,
/„„ n J sec"~'a;tana; , n — 2 f „-■,-,
sec"a!Ox = \-- — I sec" ^xdx.
n — 1 n — U
Similarly, result 73 for | Gc>sec''xdx can be obtained.
Ex. 6. (1) Csec'xdx, (2) isec*xdx, (3) sec^rdx.
Ex. 7. (1) (csc^xdx, (2) (csc*xdx, (3) CBC^xdx.
Ex. 8. Derive formula 73.
Ex.9. From formulas 72 and 73 derive formulas for I sec" a; da; and
I cosec"a;da; which are applicable when n is a negative integer.
[SuooESTiow : Use method employed in deducing formulas 70 and 71.]
192.] TRIGONOMETRIC FUNCTIONS. 341
(c) To find I tmV'xdx, in which n is a positive integer greater
than 1.
I tan" a; da; = jtaii--^ x tan'' xdxz= ftan-^^ x (sec= a; — 1) da?
= I tan"~^ X d (tan a;) — j tan"~^ a; dx
tan"->a; /*. „_» ,
= I tan" "a; da;.
n — 1 J
Similarly can be shown result 75 for i cot" a; da;.
When n is negative, say — m, then j tan" a; da; = jcofxda;,
and I tau"a;da; can be expressed in cotangents by formula 75.
Formulas applicable to cases in which n is negative, can be
deduced from formulas 74 and 75, by the method used in
deducing formulas 70 and 71.
Ex. 10. Deduce Formula 75, and formulas applicable to xta.wxdx and
( coV^xdx when n is negative.
Ex.11. (1) j"tan»X(ia;, (2) cot<xdx, (3) ftan^xdx, (4) (taa^xdx.
(d) I sln'"a;co8"a;da;. When m and n are integers, reduction
formulas can be derived for this integral in a manner similar to
that used in Art. 188 ; that is, by
(i) Connecting it with each of the four integrals in turn, viz. :
I sin"~^ a; cos" a; dx, j sin" a; cos""" x dx,
I sin^+^x cos"a;da;, | sin"a; cos"+*a;da; ;
(ii) Forming a new function by giving sin x and cos x each an
index one greater than the lesser of its indices in the required
integral and the integral with which it is connected, and taking
the product ;
(iii) Giving the connected integral and this newly formed
function each an arbitrary coefiBcient, and equating their sum to
the required integral;
342 INTEGRAL CALCULUS. [Ch. XXI
(iv) Determining the value of these coefficients by proceeding
as in Art.. 188.
The derivation of these reduction formulas is left as an exercise
for the student ; they are given in the set of integrals, Nos. 76-79.*
Ex. 12. Deduce formulas Nos. 76-79 by the methods outlined aboye.
Ex. 13. Deduce the formulas in Ex. 12 by integratiog by parts.
Ex. 14. Apply these formulas to finding the following integrals :
(1) Jsins X cos2 X dx ; (2) f cos* x sin^ x ; (3) j"^^ cbi.
Ex. 16. Deduce the integrals in Ex. 14 by the method outlined in (d).
Note 1. When m + n is a negative even integer, the above integral can
be expressed in the form I /(tan a;)d(tan x).
Ex.16. C5i5!£dx=r?i2!£._J_.a:=ftan'xsec««(te
J cos' X J cos* X cos* X J
= ( tan» X (1 + tan^ x)d tan x = ^(6 + 4 tan^ x) tan* x.
Ex.17. (1) f52!l^(ix, (2) f^il^dx, (3) {'-^^dx.
^^Jsin8z ^Wcoe'x ^^Jsin«x
Note 2. Special forms. Integrals 80-87 are occasionally required. For
their deduction see Murray, Integral Calculus, Arts. 64-57, or other texts.
It will be a good exercise for the student to try to deduce these integrals him-
self. For a fuller discussion of the integration of irrational and trigonometric
functions see the article Infinitesimal Calculus (Ency. Brit., 9th edition),
§§ 124 on ; also see Echols, Calculus, Chap. XVIII.
Note 3. On integration by infinite series. See Art. 197.
Note 4. Elliptic integrals. Elliptic functions. The algebraic inte-
grands considered in this book give rise only to the ordinary algebraic,
circular, and hyperbolic t functions. (The two last named are singly periodic
functions.) Certain irrational integrands give rise to a class of functions
treated in higher mathematics, viz. the elliptic (or doubly periodic) functions.
The term elliptic functions is somewhat of a misnomer; for the elliptic
functions are not connected with an ellipse in the same Vfay as the circular
functions are connected with the circle, and the hyperbolic functions with
the hyperbola. The elliptic integrals derived their name from the fact that
an integral of this kind appeared in the determination of the length of an
arc of the ellipse. Out of the study of the elliptic integrals arose the modern
•These formulas are derived in Murray, Integral Calculus, Art. 51, and
Appendix, Note C. Also see Edwards, Integral Calculus, Art. 83.
t See Appendix, Note A.
192.] TRIGONOMETRIC FUNCTIONS. 343
extensive and important subject of elliptic functions; this accounts for the
term elliptic in the name of these functions. The student may take a glance
forward and extend his mathematical outlook by inspecting Art. 174, Note 4 ;
Cajori, History of Mathematics, pages 279, 347-354 ; the section on elliptic
integrals in the article mentioned in Note 2, in particular, §§ 191, 192, 204,
205, 206, 219, 220 ; W. B. Smith, Infinitesimal Analysis, Vol. I., Arts. 123-125 ;
Glaisher, Elliptic Functions, pages 6, 175, etc.
EXAMPLES.
1. Derive integrals Nos. 80-82, 85-87.
2. Derive several of the integi-als 18-30, 36-46, 53-66.
8. (1) CJ^dt. (2) (^^ (.3) C ^
^'-1 •'(2« + i)i J(i+a;2)Vr^:T^
(4) C ""^ (5) ^^^-^'da:. (6) f (2^+1)'?^ .
•^ (1 + j;2) Vl - 4 x2 ^^ '' Vi^ + 3 X + 5
,7 f (2x + l)dx __ .g, r dv ,9. r xdz —
J^Vx-^ + 3x + 5 ^ ^•'(^ + l)VSf+T "- ^ Jjx^- 16)'
(10) C—^ — (11) f — ^2 (12) r^"^--^'dx.
*- ''J(:c2 + 4)8 ^ ^-'(l + x^)Vr=^ ^ J =^^
4. Derive the following integrals :
mJ^/!
+ ^ j_ .• -1 ^
(ii; = o sin"' — Va' — x^.
■ X a
(2) (J?^^ dx = - Via + z) (6 - 2) - (a + 6)sin-i J^*
6
(3) ("a/^^^ dx = V(a - X) (6 + X) + (a + 6) sin-i yj^^ ■
J ^b + X 'a + 6
(4) C J«Jl5 dx = 1/(0 + I) (6 + I) + (a - 6) log ( Va + x + V6 + x).
^ '6 + X
(5) f ^ '^^ ^2sin-iJiHg.
6. Show that, if /(m, v) is a rational function of « and », and m and n are
m
integers, then f[x-, (a + hx^)"}xdx can be rationalised by means of the sub-
stitution a + bx^ = z". (Ex. 14, or Note 3, Art. 187, is a particular case of this
theorem.) „
6. Show that (1) f"sina-»(fa = 1 -3 •5- (2ct- 1) x
^ ^ Jo 2 . 4 . 6 .-. 2 m 2
(2) Jo'
sin2m+ia; dx = — - -4 -e-- -TO — .^ being an integer).
3.5.7-(2to + 1)
CHAPTER XXII.
APPROXIMATE INTEGRATION.
INTEGRATION.
MECHANICAL
193. Approximate integration of definite integrals. It has been
shown in Arts. 165, 166, 168, that: (a) the definite integral
I f(x)dx may be evaluated by finding the anti-differential of
f(x)dx, <l>{x) say, and calculating 4>(b) — <^(a) ; (6) this last num-
ber is also the measure of the area of the figure bounded by the
curve y =f(x), the a>axis, and the two ordinates for which x = a
and x = b. In only a few cases, however, can the anti-differential
of f{x)dx be found; in other cases an approximate value of the
definite integral can be obtained by making use of fact (&). Thus,
on the one hand the evaluation of a definite integral serves to
give the measurement of an area ; on the other hand the accurate
measurement of a certain area will give the exact value of a defi-
nite integral, and an approximate determination of this area will
give an approximate value of the integral. The area described
above may be found approximately by one of several methods ;
two of these methods are explained in Arts. 194 and 195.
194. Trapezoidal rule for measuring areas (and evaluating definite
integrals).
Let the value of the definite integral | f{x)dx be
required. Plot the curve
y =f{x) from a; = a to x =b.
Let OA = a, OB=b, and draw
the ordinates AP and BQ. By
Art. 166, the measure of the
area APQB is the value of the
required integral. An approxi-
mate value of the area APQB
f JO, 12U. ^^^ ^ found in the following
c X
344
193,194.] APPROXIMATE INTEGRATION. 345
way. Divide the base AB into n intervals each equal to Ax, and
at the points of division Ai, A^, A^, •••, erect ordinates AiP^,
A^P^ AiPi, •■•. Draw the chords PP^, P1P2, AA> •"? thus
forming the trapezoids AP^, A^Pi, A^Pg, •■■. The sum of the
areas of these trapezoids will give an approximate value of
the area of APQB.
Area AP^ = | (AP + A^P^) Ax,
area A^P^ = \ {A^P^ + A^P^ Ax,
area A^P^ = \ (^2^2 + ^a-Ps) ^^t
)
area ^„_iQ = V (^„_,P„_i + BQ) Ax.
.: area of trapezoids = (^ AP + A^Py + A.2P2 + •■• + A„-iPn-i
+ ^BQ)Ax.
This result may be indicated thus :
area trapezoids = (| + 1 + 1 + -.. + 1+|) Ax,
in which the numbers in the brackets are to be taken with the
successive ordinates beginning with AP and ending with BQ.
Note. It is evident tliat the greater the number of intervals into which
6 — a is divided, the more nearly will the total area of the trapezoids come
to the actual area between the curve and the z-axis, and, accordingly, the
more nearly to the value of the integral. See Exs. 1, 2.
EXAMPLES.
1. Find ( a;2(ii;, dividing 12 — 1 into 11 equal intervals.
Here each interval. Ax, is 1. Hence, approximate value
= Ci • 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92 + 102 + 112 + J . 12J) = 5771.
The value of f "a;2 (Je = ["— + cl = 575f. The error in the result ob-
tained by the trapezoidal method is thus, in this instance, less than one-
third of one per cent.
2. Show that if 22 equal intervals be taken in the above integral, the
approximate value found is 576.125.
3. Show that on using the trapezoidal rule for evaluating i x'dx,
if 10 intervals be taken, the result is If units more than the true value,
and if 20 intervals be taken, the result is ^ of a unit more than the true
value.
346
INTEGRAL CALCULUS.
[Ch. XXII.
4. Explain why the approximate values found for the integrals in
Exs. 1, 2, 3, are greater than the true values.
"820
(•32°
6. Evaluate ( cos x dx by the trapezoidal rule, taking 10' intervals.
'° {Ans. .0148. The calculus method gives .0149.)
6. Evaluate ( sin x dx, taking 30' intervals.
Js,° , A... i\ni\a
JS,°
"SSO
{Ans. .0506. Calculus gives .0508.)
/•350
7. Evaluate I cos x dx, taking 1° intervals.
•^''^° {Ans. .1509. Calculus gives. 1510.)
195. Parabolic rule*" for measuring areas and evaluating definite
integrals. Let the area and the integral be as specified in Art.
194. For the application of the parabolic rule, the interval AB
is divided into an even
number of equal intervals
each equal to Aa;, say. The
ordinates are drawn at tlie
points of division. Through
each successive set of three
points (P, P„ K), (A, P„
Pi), •■■, are drawn arcs of
parabolas whose axes are
parallel to the ordinates. The area between these parabolic arcs
and the x-axis will be approximately equal to the area between
the given curve and the avaxis. The area bounded by one of these
parabolic arcs and the a>axis, and a pair of ordinates, say the
area of the parabolic strip APPiP^A^, will now be found.
Parabolic strip APP1P2A2 ■■
■■ trapezoid APP^A^ + parabolic
segment PP^P^. (1)
Now the parabolic segment PP1P2
= two-thirds of its circumscribing
parallelogram PPP^PiA (2)
• This rule, which is much used by engineers for measuring areas, is also
known as Simpson's one-third rule, from its inventor, Thomas Simpson
(1710-1761), Professor of Mathematics at Woolwich.
t See Art. 181, Ex. 19.
l!ir..J I'AHABOLIC RULE. 347
Area trapezoid APP^A^ = ^ AA.i{AP + A^Pi) ;
area PPP.,P, = area APP^A^ - area APP^A^
= 2 . ^ AA^ ■ AyP^ - \ AA^iAP
+ A,P,). (3)
Hence, by (1), (2), and (3), area parabolic strip APP1P2A2
= (AP+4:A,P, + A,P,)^.
Similarly, area of next parabolic strip AiP^PjPiAi
= (A,P, + iA,P, + A,P,)^;
and so on. Addition of the successive areas gives total area of
parabolic strip =(AP + 4. A,P, + 2 A.P, + 4 ^,^3
+ 2A^P,+ ...+BQ)^.
This result may be indicated thus :
Total parabolic area = (1+4+2 + 4 + . .. + 2 + 4 + 1) — , (4)
3
in which the numbers in the brackets are understood to be taken
with the successive ordinates beginning with AP and ending
with BQ.
EXAMPLES.
1. Find I x^dx, taking 10 equal intervals.
Here, each Interval = 1. Hence, the result by (4)
= (1 • 0» + 4 • 1' + 2 . 2» + 4 . 3' + 2 . 48 + 4 ■ 53 + 2 . 63 + 4 . 7«
+ 2 . 88 + 4 . 95 + 1 . 103) >< J = 2500.
_ . rr< TO
True value = — + c = 2500.
2. Calculate the above integral, using the trapezoidal rule and taking
10 equal intervals.
/•ii
3. Evaluate t x'^dx, both by the trapezoidal and the parabolic rules,
taking 10 equal intervals.
4. Evaluate Ex. 1, Art. 194, by the parabolic rule. Why is the result
the true value of the integral ?
6. Show that there is only an error of li in 20,000 made in evaluating
10
X* dx by the parabolic method, when 10 intervals are taken.
s.
348 INTEGRAL CALCULUS. [Ch. XXII.
6. Find the error in the evaluation of the integral in Ex. 6 by the trape-
zoidal method, when 10 intervals are taken.
7. Evaluate the integrals in Exs. 6, 7, Art. 194, by the parabolic rule.
Note. For a comparison between the trapezoidal and parabolic rules, for
a statement of Dnrand's rale, which is an empirical deduction from these
two rules, for a statement of other mles for approximate integration, and
for a note on the outside limits of error in the case of the trapezoidal and
parabolic rules, see Murray, Integral Calculus, Arts. 86, 87, Appendix, Note
E, and foot-note, page 186.
196. Mechanical devices for integration. The value of a definite
integral may be determined by various instruments. Accordingly,
they may be called mechanical integrators. Of these there are
three classes, viz. planimeters, integrators, and integraphs. These
instruments are a great aid to civil, mechanical, and marine
engineers. The area of any plane figure can be easily and accu-
rately calculated by each of these mechanisms. Their right to be
termed mechanical integrators depends on the facts emphasised
in Arts. 166, 168, 193-195 ; the facts, namely, that a definite inte-
gral can be represented by a plane area such that the number of
square units in the area is the same as the number of units in the
integral, and hence that one way of calculating a definite integral
is to make a proper areal representation of the integral and then
measure this area.
Planimeters, which are of two kinds, viz. polar planimeters and
rolling planimeters, are designed for finding the area of any plane
surface represented by a figure drawn to any scale. The first
planimeter was devised in 1814 by J. M. Hermann, a Bavarian
engineer. A polar planimeter, which is a development of the
planimeter invented by Jacob Amsler at Konigsberg in 1854, is
the one most extensively used. By it the area of any figure is
obtained by going around the boundary line of the figure with
a tracing point and noting the numbers that are indicated on a
measuring wheel when the operation of tracing begins and ends.
Integrators and integraphs also -serve for the measurement of
areas ; they are adapted, moreover, for making far greater compu-
tations and solving more complicated problems, such as the calcu-
lation of moments of inertia, centres of gravity, etc. The integraph
(see Art. 170, Notes 2, 3) is the superior instrument, for it directly
196.] PLANIMETERS, INTEGRAPHS. 349
and automatically draws the successive integral curves. These
give a graphic representation of the integration, and are of great
service, especially to naval architects. The measure of an ordi-
nate of the first integral curve, when multiplied by a constant
belonging to the instrument, gives a certain area associated with
that ordinate (see Art. 170).
Note 1. A bicycle with a cyclometer attached may be regarded as a
mechanical integrator of a certain kind ; for by means of a self-recording
apparatus it gives the length of the path passed over by the bicycle.
Note 2. Planimeters and integrators are simple, and it is easy to learn
to use them.
Note 3. A brief account of the planimete.r, references to the literature on
the subject, and a note on the fundamental theory, will be found in Murray,
Integral Calculus, Art. 88, and Appendix, Note F. Also see Lamb, Cal-
culus, Art. 102 ; Gibson, Calculus, § 1.30. For a fuller account see Henrici,
Report on Planimeters (Report of Brit. Assoc, for Advancement of Science,
1894, pages 496-523) ; Hele Shaw, Mechanical Integrators (Proe. Institution
of Civil Engineers, Vol. 82, 1885, pages 76-143). Por references concerning
the integraph see Art. 170, Note 3.
If.B. Interesting information concerning planimeters, integrators, and
the integraph, with good cuts and descriptions, are given in the catalogues of
dealers in drawing materials and surveying instruments.
Ifote 4. For approximate integration by means of series see Art. 199.
CHAPTER XXIII.
INTEGRATION OF INFINITE SERIES.
197. Integration of infinite series term by term. It is beyond
the limits of a short course in calculus to investigate the condi-
tions under which an infinite series can properly be integrated
term by term ; in other words, to determine what conditions
must be satisfied in order that equation (3) Art. 143 (e) may be
true.*
It must suffice here merely to state the theorem that applies
to most of the series that are ordinarily met in elementary mathe-
matics ; viz. :
A power series (Art. 145) can be mtegratecl term by term through-
out any interval contained in the interval of convergence and not
reaching out to the extremities of this interval. (For proof see
Osgood, Infinite Series, Art. 40.) The next two articles give
applications of this theorem.
198. Expansions obtained by integration of known series. Three
important examples of the development of functions into infinite
series by the aid of integration will now be given.
The three expansions for tan-' x, sin~* x, log (1 + a;), in Exs. 1, 2,
3, can also be derived by means of Maclaurin's theorem. (See
Art. 152, Ex. 10 (3).)
EXAMPLES.
Ex. 1. For -l<a;<l
1
l + a:2
= l-a;2 + a^ . (1)
r'-^^= Cdx- ('x^dx+ ('x*dx , (Art. 197)
Jo 1 + x'- Jo Jo Jll
tan-la; = 35-^ + ^ . (2)
3 o
• See Art. 147 and Infinitesimal Calculus, Arts. 172, 173.
.S50
197, 198.] INTEGRATION OF INFINITE SERIES. 351
This is known as Gregory's series.* (For complete generality the term
± nir, (n = 0, 1, 2, •••), should be in the second member.) Series (1) oscil-
lates when z = 1 ; but by a theorem on series (see Chrystal, Algebra,
Vol. II., Chap. XXVI., § 20) series (2) is convergent and represents tan-'x
even when x = 1.
Note 1. Series (2) can be used to calculate r. On putting x = l (2),
there is obtained
(a) ^ = l_l + l_l + ....
^ ^ 4 3 5 7
This is a very slowly convergent series. More rapidly convergent series
for calculating it are the following :
(6) 1=4 tan-i - - tan-' -^ ; (Machin's Series t)
4 o 2o9
(c) ^ = ton-1 ^ + tan-i i • (Euler's Series J)
ExEHCiSEs. Show by elementary trigonometry that formulas (6) and (c)
are true. Compute the value of ir correctly to four places of decimals:
(1) by using formula (6) and Gregory's series; (2) by using formula (c)
and Gregory's series. (The correct value of x to ten places of decimals is
3.14159265.S6.)
Ex. 2. For - 1 < a: < 1
Vrr^ 2-4 2.4-6
On integrating between the end values 0 and 1, as in Ex. 1, there resultB
This series is due to Newton, and was used by him in computing the value
of IT. When X = ^ this series gives
IT _ 1 1 1.3 , 1.3-5
6 2 2. 3- 23 2. 4.5- 26 2- 4. 6- 7- 2'
Exercise. Using the last result calculate r correctly to four places of
decimals.
* Discovered in 1670 by James Gregory (1638-1675), professor of mathe-
matics at St. Andrews and later at Edinburgh. It was also found by Leibnitz
(1646-1716). This series can also be derived independently of the calculus
(see texts on Analytical Trigonometry).
t John Macliin, died 1751, was professor of astronomy at Gresham College,
London. t Leonhai-d Euler, 1707-1783.
352 INTEGRAL CALCULUS. [Ch. XXIII.
Note 2. For historical information concerning trigonometry and the
computation of ir, see Murray, Plane Trigonometru, Appendix, Note A, and
Note C (Art. 6) ; Hobsoii, article " Trigonometry " {Eiicy. Brit., 9th edition);
also article "Squaring the Circle" {Ency. Brit., 9th edition).
Ex. 3. For-l<a;<l
-J— = l-a; + i2_a;8+ .... (i)
l + x
On integi-ating between the end values 0 and x, as in Exs. 1, 2, there results
log(l+x)=x-iaj2 + ia;«-ja;*+ .... (2)
This is called the logarithmic series.* (-Her? the base is e.)
The members of (2) are equal for values of a; as near 1 as one pleases. It
is also easily shown that they are finite and continuous for x = 1. Accord-
ingly, formula (2) is true also when x = 1.
On putting « = 1 in (2), log2 = 1 — ^ + J — J 4 — , a very slowly conver-
gent series.
On putting x=- 1 in (2), logO =- {i + i + i+i + ■■■) = - oo. (See
Art. 146.)
Note 3. Except for small values of x series (2) is very slowly convergent.
A more rapidly convergent, and thus more useful, seriis for the computation
of logarithms can be derived from (2) , as follows. On putting — a; for a; in (2),
log(l-a;) = -i- Ja;2_ij.3_ jj;4 . (3)
.■Aos}^^ = 2(x+ix^ + lx^ + ...). (4)
1 — X
On substituting for x this becomes
^ 2 )i« -I- 1
logr«L±l=2[—^ + ^ + ^ +-1. (5)
m L2W + 1 3(2»»-H)8 6(2m-H)B J
If™ = l, log2 = 2(l + 3_L. + ^+...) = .693.
Ifm = 2,log3-log2=.2(l + ^ + ^^+...]=.406.
.-.log 3= 1.099.
Exercises. (1) Find log4 to base e, by putting m = 3 in (5), assuming
the value of log 3. (2) Find the logarithms (to base e) of 5, 6, 7, 8, 9, 10, in
a similar way. (The logarithms of 4, 5, 6, 7, 8, 9, 10, to base e, to three
places of decimals, are respectively 1.386, 1.609, 1.V92, 1.946, 2.079, 2.197,
2.303.)
• Apparently first obtained in 1668 by Nicolaus Mercator of Holstein.
198,199.] INTEGRATION OF INFINITE SERIES. 353
199. Approximate integration by means of series. The methods
described or referred to in Arts. 194^196 for evaluating a deiinite
integral
(1)
j f(x)dx
yield a numerical result only. They do not give any information
as to the anti-differential of f(x)dx.
Some information, however, about the anti-differential of /(x) dx
can be obtained in certain cases (see Art. 197) by expanding /(x)
in a series in ascending or descending powers in x and then inte-
grating this series term by term. The new series thus obtained
represents the anti-differential of f(x)dx for values of x in some
particular interval of convergence. From this series an approxi-
mate value of (1) can be obtained, if the end-values a and h are
in the interval of convergence.
Instances have been given in Art. 198, thus
■ -n- 1 r'^ dx ^ 1 , 1
^ dx
in Ex. 2, C^-^^ = l + —^+ 1-3 +.
Jo vn:^ 2 2-3.2' 2.4.5.2'
EXAMPLES.
1. Given that C = 1 + x + ^ + ^ + - (Art. 1.52, Ex. 7), show that
21 o !
I e* dx = e* -f- c, in which c is a constant.
2. Given that C08x = l — — -1-^^ — •••, and that sin x = x — ^ -t- —
2 14! o ! 61
(Art. 152, Exs. 2, 5), show that j cosxdx = sini + c, and that | sinrdx
= — cos X + c.
S. Find an approximate value of the area of the four-cusped hypocycloid
inscribed in a circle of radius 8 inches. (This area can also be found exactly.
See Art. 209, Note 5, Ex. 1.)
4. Find an approximate value of the length of the ellipse x = asin^,
y = 6 cos (p. [Here 0 is the complement of the eccentric angle for the point
(.X, y).]
354 INTEGRAL CALCULUS. [Ch. XXIII.
It will be found (Art. 209) that
IT
length s = 4 o j VI — e"^ sin" 0 d<l>. (a)
On expanding the radical by the binomial theorem and taking the term
by term integral of the resulting convergent series it will be found that
-'• ['-(l)"f-(H)'?-(li!il)"f--J- ">
6. Apply result (6) of Ex. 6 to find the length of the ellipse whose semi-
axes are 5 and 4. (To three places of decimals. )
6. The time of a complete oscillation of a simple pendulum of length I,
oscillating through an angle a « ir) on each side of the vertical,vis
■VIX'
^ - , in which k = &\a\ a. (c)
'? Vl-i^sin^i^
Show that this time
Note 4. Integrals (c) and {a) in Exs. 6 and 4 are known respectively as
"elliptic integrals of the first and the second kind." The symbols F{k, 0),
E{e, if) are usually employed to denote these integrals (the upper end-value
here being ip"). Knowledge of these integrals was specially advanced by
Adrien Marie Legendre (1752-1833). See Art. 192, Note 4.
T. Show that:
1-3.5 J_
2.4-6 ' 13'
(1) r <^^ =1 + 1.1+1^.14.
•''' Vl"-^'« 2 5 2-5 9
m C—Jl— = \-\ 2 + 1 2jj5 1_J.
•'" v'(l + a;3)' 4 3 7 1-2 3^ 10
(3) r d^ =1 + 1.1+1.1-4.1+1
J" vl^^ 6 3 11 1-2 32 16
)'
2.
1.
5.
2-
1.1 +
3 33^
1
1
•4
-2
-7
• 3
i-
CHAPTER XXIV.
SUCCESSIVE INTEGRATION. MULTIPLE INTEGRALS.
APPLICATIONS.
200. In Chapter VI. (see Arts. 68, 69, 70), successive derivar
tives and differentials of functions of a single variable were
obtained. In Chapter VIII. (see Arts. 79, 80, 82), successive par-
tial derivatives and partial differentials of functions of several
variables were discussed. In this chapter processes which are the
reverse of the above are performed and are employed in practical
applications.
201. Successive integration : One variable. Applications.
Suppose that Cf{x)dx=fi{x), (1)
jflx)dx=f,{x), (2)
Jf,(x)dx=Mx). (3)
Then, by (3) and (2), Mx)=f[fMx)dx']dx; (4)
By (4) and (1), A(x) = f^ ffff(x)dx)dx dx. (6)
This is written /3(a;) = f f ff(x){dx)',
or, more usually, fjix) = I j I f(x) da^. (6)
The second member of (6) is called a triple integral. Similarly,
the second member in (4) is usually written j | fi{x)d3?, and is
called a double integral.
In general, ( I I "• I f{x)dx'^ denotes the result obtained by
355
356 INTEGRAL CALCULUS. [Ch. XXIV.
integrating f(x)dx n times in succession. This integral is indefi-
nite unless end values of the variable be assigned for each of the
successive integrations. This integral and the integrals in (4) and
(5) are called multiple integrals.
Note. It should be observed that here da?* denotes dxdxdx.-.U> n factors,
i.e. (dx)", and not d ■ x» (i.e. wji^-^dx). [Compare Art. 70.]
EXAMPLES.
1. Find \ \ I x'-difi.
J-j-pda-=J{J[pd.]dx}da=
for, since cj is an arbitrary constant, ^ may be denoted by an arbitrary con-
stant k\.
8. Determine the curves for every point of which -=-^ = 0. Which of
these carves goes through the points (1, 2), (0, 3) ? Which of these curves
has the slope 2 at the point (3, 6) 1
g = 0.
On integrating, ^ = Ci.
ox
On integrating again, y = cii + c^,
which represents all straight lines.
For the line going through (1, 2) and (0, .3), 2 = Ci + C2 and 3 = 0 + e^ ;
whence ci = — 1, c^ = 3. Hence the line is x + y = 3.
For the line having the slope 2 at (3, 5), ci = 2 and 5 = 3 ci + C2, whence
C2 = — 1. Hence the line is y=2x — \.
4. Determine the curves for every point of which the second derivative
of the ordinate with respect to the abscissa is 6. Which of these curves
goes through the points (1, 2), (- 3, 4) ? Which of them has the slope 3 at
the point ( - 2, 4) ?
201, 202."] SUCCESSIVE INTEGRATION. 357
N.B. Tlie student is recommended to write sets of data like those in
Exs. 0-7, and determine tlie particular curves that satisfy them. He is also
recommended to draw the curves appearing in these examples.
6. Determine the curves for every point of which the second deriva-
tive of the ordinate with respect to the abscissa is 6 times the number of
units in the abscissa. Which of these curves goes through the points (0, 0)
(1, 2) ? Which of them has the slope 2 at (1, 4) ?
6. Determine the curves in which the second derivatives -r-^ from point
to point vary as the abscissas. Find the equation of that one of these curves
which passes through (0, 0), (1,2), (2, 5). Find the equation of that one of
these curves which passes through (1, 1), and has the slope 2 at the point
(2, 4).
7. Determine the curves in which the second derivative of the abscissa
with respect to the ordinate varies as the ordinate. Which of these curves
passes through (0, 1), (2, 0), (0, 5) ? Which of them has the slope J at
(1, 2), and passes through (—1,3)?
8. A body is projected vertically upward with an initial velocity of 1000
feet per second. Neglecting the resistance of the air and taking the accelera^
tlon due to gravitation as 32.2 feet per second, calculate the height to which
the body will rise, and the time until it again reaches the ground.
9. Do Ex. 20, Art. 68.
10. When the brakes are put on a train, its velocity suffers a constant
retardation. It is found that when a certain train is running 30 miles an
hour the brakes will bring it to a dead stop in 2 minutes. If the train is to
stop at a station, at what distance from the station should the engineer
whistle "down brakes" ? (Byerly, Problems in Differential Calculus.)
202. Successive integration : several variables. Suppose that
Cf(x, y, z) dz =Mx, y, z), (1)
J7i(a', y, 2) ^y = /zCa!. y> ')> (2)
jfii^, y, 2) d^ =fi{^> y> ")■ (3)
The integration indicated in (1) is performed as if y and x were
constant; the integration in (2) as if x and z were constant; the
integration in (3) as if z and y were constant. (Compare Arts. 79,
80.)
358 INTEGRAL CALCULUS. [Ch. XXIV.
From (3) and (2), f^(x, y, z) =j \ J7,(«, y,z)dy\dx; (4)
from (4) and (1), =j \ /[//(«. V, ^) c'^J'^y \ d^- (5)
The second member in (4) is often written
fjfi{x,y,«)dydx; (6)
the second member in (5) is often written
\f{x,y,z)dzdydx. (7)
////
The integral in (6) is called a double integral, and the integral
in (7) a triple integral.
Note 1. It should be observed that according to (2), (3), and (4), inte-
gral (6) is obtained by first integrating /i(x, y, z) with respect to y, and then
integrating the result with respect to a; ; in (7), according to (1), (2), (3),
and (5), the first integration is to be made with respect to z, the second with
respect to y, and the third with respect to x. That is, the^rst integration sign
on the right is taken loith the first differential on the left, the second integra-
tion sign from the right with the second differential from the left, and so on.
When end-values are assigned to the variables, careful attention must be paid
to the order in which the successive integrations are performed.
Note 2. The notation used above for indicating the order of the variables
with respect to which the successive integrations are to be performed, is not
universally adopted. Oftentimes, as may be seen by examining various texts
on calculus and works which contain applications of the calculus, integrals (6)
and (7) are written
rj/i(a;, y, z)dxdy, J J \f{x, y, z) dx dy dz respectively.
In this notation the first integration sign on the right belongs to the first
differential on the right, the second integration sign from the right to the
second differential from the right, and so on ; and the integrations are to be
made, first with respect to z, then with respect to y, and then with respect to x.
In particular instances, the context will show what notation is employed.
EXAMPLES.
1. ( ( (x''yz'dzdydx=( (x^y(- + ci\dydx
202,203.] SUCCESSIVE INTEGRATION.
2. (*CCxh/z> dz dy dx (i.e. J '"* f"^^ Cl^ ^'^V^ ^^ ^V *")
8. Ci'^'^f dydx =('''' I ^"^ xYdy\dx=CxAy^+cY dx
4. Evaluate the following integrals : (1) i ( ( xy^zdzdydx.
(2) r^ r " (3 w - 2 r) dwdi). (3) C C"' \/ st - f' ds dt.
359
r2 cos tfdr dS.
<:^) jo jo "Jo °' ''dzdydx.
^^> Jo Jo
■tr IT
rdr de.
(9) J^' j]"°"'V5^^^.nJr<W.
203. Application of successive integration to finding areas: rec-
tangular coordinates.
EXAMPLES.
1. Find the area between the curve y^ = 8x, the x-axis, and the ordinate
„, for which r = 3.
At P, any point within the figure 0W3f
whose area is required, suppose that a rectan-
gle PQ having infinitesimal sides dx and dy
parallel to the axis is constructed. The area
0 WM is the limit of the sum of all rectangles
such as PQ which can be constructed side by
side in 0 WM. Let one of the vertical sides of
the rectangle be produced both ways until it
meets the curve and the x-axis in T and S;
complete the rectangle TV as in the figure.
First, find the area of the rectangular strip TV by finding the limit of the
sum of the rectangles PQ inscribed in it from S to T; then find the limit
of the sum of the strips like TV which can be inserted between 0 Y and MW.
r
J
1^
41,
£7
h
1
e
0
f ^
^ 8 >
af X
Fig. 122.
3G0 INTEGRAL CALCULUS. ' [Ch. XXIV.
Area rF=limX (rectangles P§) = J dydx,=y/Wxdx. (1)
y at5
Area Oilf JT = lim 2^ (strips 7T) = ^^^ [ j^ dyj dx (2)
eatO
= 2v'2 { X* dx = iV6 square units.
The last expression in (2) is usually written II dydx.
The area of 0 iVM may also be found by finding tlie limit of the sum of the
rectangles PQ which may be inserted between S and U, and then finding
the limit of the sum of the strips like iJi which may be inserted between
oaf and W. Thus,
area BL = P'^ dxdy = idxdy = (3 -^]dy; (3)
T
area OMP = ^[^(.S -t)dy = {^\s - 1') dy = 4v^. (4)
From (3) and (4), areaOJ!fP=l ( dxdy.
8
Note 1. The last expression in (1) is ydx, the element of area employed
in Art. 166.
Note 2. Ex. 1 has been solved as above merely in order to give a prac-
tical application of double integration.
Note 3. For finding areas by double integration in the case of polar
coordinates, see Art. 208, Note 3.
2. Express some of the areas in Art. 181 by double integrals, and per-
form the integration.s.
3. Find by double integration the area included between the parabolas
Zy^ = 25x and &x^ = 9y. [See Murray, Integral Calculus, Art. 61, Ex. 1.]
204. Application of successive integration to finding yolumes :
rectangular coordinates.
EXAMPLES.
1. Find the volume bounded by the surface whose equation is
a2 -•- 62 -t- ci - ^■
Fig. 0-ABC represents one-eighth of the volume required. Suppose that
an infinitesimal parallelopiped P1Q3 is taken at Pi{x, y, 0), having infinitesi-
2:3, 204.]
S UCCESSIVE INTEGBA TION.
361
mal sides da;, dy, dz, parallel to the x-, y-, and z-axes, respectively. The
volume of 0-ABC is the limit of the sum of all infinitesimal parallelepipeds
such as Pi §8 which can be enclosed by OB A, OAC, OCB, and the curvi-
FlQ. 123.
linear surface ABC. Construct a parallelopiped PQi by producing the
vertical faces of PiQ, to the height PiP. (The point F(x, y, z) is taken on
the surface ABC.)
Vol. P§i=l dzdydx^l i " "' dz dydx. (1)
Note 1. The numbers x and y are constant along FiP, and, accordingly,
in the integration of (1) x and y are treated as constants.
Now take a slice BGL the planes of whose faces coincide with two faces
of PQu as shown in the figure.
Vol. slice BPGLS = limit of sum of parallelopiped s PQi from StoG.
rvtitol /•*=evl- — — — I
That is, voL slice .BG = t | "' '''dz\dy-dx
Jy at S \_Jx=a J
_| r»=*^ a» j '*' .. h^az\dy\dX. (2)
Note 2. The number x is constant along 8G, and, accordingly, in the
integration of (2) x is treated as a constant.
Now find the limit of the sum of all infinitesimal slices like BGL from
OCB to A ; I.e. from z = 0 to a: = a. This limit is the volume of 0-ABC,
362 INTEGRAL CALCULUS. [Ch. XXIV.
f
.-. vol. O-ABC = j^^^^ Ij »' J " "'dzUyjax
= j ( °° jo "' ""dzdydx. (3)
On performing the integrations indicated in (3) (see Ex. 4 (5), Art. 202), it
will be found that
vol. O-ABC = \irabc. Hence vol. ellipsoid = | itdbc.
Note 3. Result (3) may be written I I I dxdydz.
JxntO JyulS J'HtPi
Note 4. The initial element of volume PiQi, i.e. dxdydz, is an infinitesi-
mal of the third order ; the parallelepiped PQi is an infinitesimal of the
second order ; the slice "RGL is an infinitesimal of the first order.
Note 5. Equally well, slices may be taken which are parallel to the
xz-plane or to the ya-plane.
Note 6. Instead of the parallelopiped PQi, equally well, a similar paral-
lelepiped can be taken whose finite edges are parallel to the y-&xis, or to the
X-axis.
2. Perform the integrations indicated in Ex. 1.
3. Do Ex. 1 by taking the elements in the ways indicated in Notes 6
and 6.
4. From the result in Ex. 1 deduce the volume of a sphere of radius
a. Also deduce the volume of this sphere by the method used in Ex. 1.
(Compare with the methods used in Art. 182, Ex. 19 and Note 3.)
5. Two cuts are made across a circular cylindrical log which is 20 inches
in diameter ; one cut is at right angles to the axis of the cylinder, the
other cut makes an angle of 60° with the first cut, and both cuts intersect
the axis of the cylinder at the same point. Find the volume of each of the
wedges thus obtained.
6. As in Ex. 5, for the general case in which the radius of the log is a
and the angle between the cuts is «. Thence deduce the result in Ex. 5.
7. The centre of a sphere of radius a is on the surface of a right cyl-
inder the radius of whose base is -. Find the volume of the part of the
cylinder intercepted by the sphere.
8. Taking the same conditions as in Exs. 5, 6, excepting that the cuts
intersect on the surface of the log, find the volume intercepted between the
cuts.
204, 205.]
SUCCESSIVE INTEGRATION.
363
205. Application of successive integration to finding volumes ;
polar coordinates.
A. The use of polar coordinates in finding volumes sometimes
leads to easier integrations than does the use of rectangular
coordinates.
Let 0, the origin, be taken
as pole. The infinitesimal ele-
ment of volume is formed
as follows : Take any point
P(r, 0, <l>). [Here r=OP,0 =
angle POZ, <^ = angle XOM,
Oilf being the projection of OP
on XOY. In other words,
tj> = the angle between the
plane XOZ and the vertical
plane in which OP lies.] Pro-
duce OP an infinitesimal dis-
tance dr to Pi, and revolve
OPPy through an infinitesimal
angle dO in the plane ZOP to the position OQ. Now revolve
OPPxQ about OZ through an infinitesimal angle d^, keeping
6 constant. The solid PP^QR is thus generated. Its edges
PPi, PQ, PR are respectively dr, rdB, r sin $d<f>; its volume (to
within an infinitesimal of an order lower than the third) is
r^ sin 6 drd<t> dd. On determining the proper limits for r, <^, 6, and
integrating, the volume required is obtained.
Ex. 1. Find the vohime of a sphere of radius a, using polar coordinates
and taking O on the surface of the sphere and OZ on the diameter through O.
(It will be found that the volume is given by the integral in Art. 202, Ex. 4,
(6). See Murray, Integral Calculus, Art. 63, Ex. 1.)
B. The element of volume can be chosen in another way, which
sometimes leads to simpler integrations than are otherwise obtain-
able. An instance is given in Ex. 2 below.
Fig. 134.
EXAMPLES.
S. Another way of doing Ex. 7, Art. 204.
In the figure, 0-ABC is one-eighth the sphere, and the solid bounded by
the plane faces ALBO, AKO, the spherical face ALBVA, and the cyliiidriciil
364
INTEGRAL CALCULUS.
[Ch. XXIV.
face AVBOKA is one-fourth of the part of the cylinder intercepted by the
sphere.
In AOK take any point P.
Let OP = r, and angle AOP=e.
Produce OP an infinitesimal dis-
tance dr to Pi, and revolve OPPi
through an infinitesimal angle d9.
Then PPi generates a figure, two
of whose sides are dr and rd8.
Its area (to within an infinitesimal
of an order lower than the second)
is rdi-de. (See Art. 208, Note 3,
Ex. 8.)
On this infinitesimal area as
a base, erect a vertical column
to meet the sphere in M. Then
PM = Va^ — r', and the volume
Fig. 125.
of the column is Va'^ — r^-rdrd9.
This is taken as the element of
volume ; the limit of the sum of these columns standing on AOK is the vol-
ume required. Keeping S constant, first find the limit of the sum of the
columns standing on the sector extending from 0 to .K" whose angle is dB.
Since OK = a cos 8, this limit is t ~°' -Va^ — i^ • r dr d6. This gives the
Jr=B
volume of a wedge-shaped slice whose thin edge is OB. One-fourth of the
volume required is the limit of the sum of all the wedge-shaped slices of this
kind that can be inserted between AOB and COB; that is, from fl = 0 to
» = -;
2
vol. required = 41 \ Va^
Je=a Jn=o
■i^-rdrdB = \Tra^-\a*.
[See Art. 202, Ex. 4 (9).]
In this instance this is a very much shorter way of deriving the volume
than by starting with the element dx dy dz, as in Art. 204.
3. Find the volume of a sphere of radius a, taking O at the centre :
(1) choosing the element of volume as in .4 ; (2) choosing it as in B.
4. The axis of a right circular cylinder of radius & passes through the
centre of a sphere of radius a {a> b). Find the volume of that portion of
the sphere which is external to the cylinder.
CHAPTER XXV.
FURTHER GEOMETRICAL APPLICATIONS OF
INTEGRATION.
206. In this chapter the calculus is used for finding volumes
in a particular case, for finding areas of curves whose equations
are given in polar coordinates, for finding the lengths of curves
whose equations are given either in rectangular or in polar coordi-
nates, for finding the areas of surfaces in two special cases, and
for finding mean values of variable quantities.
N.B. Many of the problems in this chapter are presented in a general
form. In such cases the student is recommended, when he obtains the
general result, to make immediate application of it to particular concrete
cases.
207. Volumes of solids the areas of whose cross-sections can be
expressed in terms of one variable. In Art. 182 the volumes of
solids of revolution were found by making cross-sections of the
solid at right angles to the axis of revolution, taking these cross-
sections an infinitesimal distance apart, and finding the limit of
the sum of the infinitesimal slices into which the solid is thus
divided. This method of finding the volume of a solid can some-
times be easily applied in the case of solids which are not soiids
of revolution. The general method is : (a) to take a cross-section
in some convenient way ; (6) to express the area of this cross-
section in terms of some variable ; (c) to take a parallel cross-sec-
tion at an infinitesimal distance from the first cross-section ; (d) to
express the volume of the infinitesimal slice thus formed, in terms
of the variable used in (b) ; (e) to find the limit of the sum of the
infinite number of like parallel slices into which the solid can
thus be divided. There is often occasion for the exercise of judg-
ment in taking the cross-sections conveniently.
366
INTEGRAL CALCULUS.
[Ch. XXV.
EXAMPLES.
1. Find the volume of a right conoid with a circular base of radius a and
an altitude k.
Note 1. A conoid is a surface which may be generated by a straight line
which moves in such a manner as to intersect a given straight line aud a given
curve and always be parallel to a
given plane. In the conoid in this
example the given plane is at right
angles to the given straight line, and
the perpendicular erected at the
centre of the circle to the plane of
the base intersects the given straight
line.
Let LM be the fixed line and ARB
the fixed circle having its centre at
C. Take a cross-section PQR at
right angles to LM, and, accordingly,
at right angles to a diameter AB.
Let it intersect AB in D, and denote
CD by X.
Area
PQB = \PDQR = PD- QD.
Now PD = h, and, by elementary geometry.
I,
P
a M
\
\
\
I
\
1
1
J"
1
.At
A
4
V
1 \
B
/
D /
c
Fig. 126.
QD = VAD ■ DB = V(a - a;) (a + z) = Va" - a:*.
.'. area PQR = hVa^ -.x^.
Now take a cross-section parallel to PQR at an infinitesimal distance from
it. Since CD has been denoted by x, this infinitesimal distance may be
denoted by dx.
Vol. LM-BQABB = 2 vol. LG-TSAT
• = 2 Urn (sum of slices PQR)
inlC
= 2 ft f" Va2 - a;-^ dx = i ira^ft.
That is, the volume of the conoid is one-half the volume of a cylinder of
radius a and height ft. (See Echols, Calculus, Ex. 3, p. 266.)
Note 2. As already observed, finding the volumes of solids of revolution
is a special case under this article.
Note S. Tuio general methods of finding volumes have now been shown,
namely, the method shown in Arts. 204, 205, and the method shown in this
article.
207, 208.]
ABE AS: POLAR COORDINATES.
367
2. Do Ex. 1, denoting AD by x.
3. Do Ex. 8, Art. 182 and Ex. 1, Art. 204 by method of this article.
4. Find the volume of a right conoid of height 8 which has an elliptic
base liaving semi-axes 6 and 4, and in which the fixed line is parallel to the
major axis. Find the volume in the general case in which the height is h,
the semi-major axis a, and the semi-minor axis 6.
6. A rectangle moves from a fixed point, one side varying as the dis-
tance from the point, and the other side as the square of this distance. At
the distance of 3 feet the rectangle is a square whose side is 6 feet. What
is the volume generated when the rectangle moves from the distance 2 feet
to the distance 4 feet ?
6. On the double ordinates of the ellipse bH^ -f oV = o^^'^i ^"d in planes
perpendicular to that of the ellipse, isosceles triangles having vertical angles
2 a are erected. Find the volume of the surface thus generated.
7. A circle of radius a moves with its centre on the circumference of an
equal circle, and keeps parallel to a given plane which is perpendicular to the
plane of the given circle : find the volume of the solid thus generated.
8. Two cylinders of equal altitude ft have a circle of radius a for their
common upper base. Their lower bases are tangent to each other. Find the
volume common to the two cylinders.
208. Areas: polar coordinates. Suppose there is required the
area of the figure bounded by the curve whose equation is
f(r, ff) = 0, and the radii vectores drawn to two assigned points
on this curve.
(iVi.Bii
Let LG be the curve
f{r, e) = 0, and F and
Q the points (r„ ^i)
and (r,, 62) respectively;
it is required to find
the area POQ. Sup-
pose that the angle POQ
is divided into n equal
angles each equal to A6,
and let VOW be one of
these angles. Denote Fas
the point (r, 0). Through
V, about 0 as a centre,
draw a circular arc intersecting OW in M.
368 INTEGRAL CALCULUS. [Ch. XXV.
Through W, about 0 as a centre, draw a circular arc intersectiug
OV in N. Denote MWh^ A;-.
Then, area OVM=\'r£^e (PL Trig., p. 175), and area ONW
Let "inner" and "outer" circular sectors, like FO-Sf and NOW
in the case of VW, be formed for each of the arcs like FTT which
are subtended by angles equal to A6 and lie between P and Q. It
is evident that
total area of inner sectors <area POQ<total area of outer sectors.
(1)
In the case of the arc VW the difference between the inner and
outer sectors is VMWN. On noting this difference for each arc
and transferring it to the radius vector OPS, as indicated in the
figure, it is apparent that the total difference between the areas
of the inner and outer sectors is PBCS. Now
area PBCS = area OSC - area OPB = i {OS" - OP') \6 ;
and this approaches zero when A6 approaches zero.
From these facts and relation (1) it follows that
Area POQ = limit of area of inner sectors (or outer sectors)
when A0 approaches zero, that is, when the number of these
sectors becomes infinitely great. That is,
Area POQ — limit of sum of areas of sectors VOM from OP
to OQ when A^ approaches zero
= lim^^io X i '""^^ = i r ^ d9. (See Art. 166. )
Note 1. The element of area in polar coordinates is thus ^r^dO; this i.s
the area of an infinitesimal circular sector, of which the radius is r and the
.angle is an infinitesimal, dS. The differential of the area also has the same
form J r^dS. In the element of area d9 must be infinitesimal, in the differen-
tial dd need not be infinitesimal. (See Art. 67 ft.)
Note 2. It is not necessary that the angles A9 be all equal. (See Art. 166,
Note 3.)
208.] AREAS: POLAR COORVINATEH. 369
EXAMPLES.
1. Find the area of a loop of the curve r = a sin 2 8.
It is first necessaiy to find tlie values of B at the beginning and at the end
of a loop. At 0 (see Fig., page 464) r = 0 ; hence, sin 2 9 = 0 at 0. If
sin 29 = 0, then 2 9 = 0, t, 2 tt, •••, and, accordingly, 9 = 0, -, tt, •■■.
Any pair of consecutive values, say 0 and ^, are values of 9 at 0 at the
beginning and end of a loop.
.-. area of a loop = J T ^r^ d9 = ^ (" ^sin^ 29 = 5^^^(1- cos 4 e)de
= «!r9-5i£l^l^=i.a^.
4 L 4 Jo «
2. Find the area of one of the loops of the curve r= a sin 3 9.
3. Find (1) the area of a loop of the lemniscate r^ = a'^ cos 2 6 ; (2) the
area of a loop of the curve r^ — a^ cos nB.
4. Show that (1) the area included between the hyperbolic spiral rS = a
and any two radii vectores is proportional to the difference between the
lengths of these radii vectores ; (2) the area included between the logarithmic
spiral r = e"' and any two radii vectores is proportional to the difference
between the squares on these radii vectores.
6. Find the area enclosed by the cardioid r^ = a' cos -•
2
6. Find the area of the oval r = 3 + 2 cos 9.
7. Compute the area of the loop of the folium of Descartes ^s + y* = 3 a zy.
Suggestion for Ex. 7 : Change to polar coordinates, and then use the
substitution z = tan 9.
Note 3. On finding areas of curves by double integration. For the sake
of illustration an example will be shown in which areas, in polar coordinates,
are found by double integration.
8. Find the area of the circle
r = 2 a cos 9.
Take any point P in ODA.
Let 0P= r, angle AOP=e. Pro-
duce OPs. distance Arto Q ; revolve
OPQ through an angle A9. Then
PQ sweeps over the area PQRS.
Area PQBS
= i 6Q^ • A9 - J OP^ ■ A0
Fig. 128. = r ■ Ar • A9 + ^ (Ar)2 . A9.
370 INTEGRAL CALCULUS. [Ch. XXV.
One can proceed to find the limit of the sum of the areas like PQBS in
ODA, in either of the two following ways (o) and (6).
(o) Starting with PQBS as an element of area, find the area of the
sector BOC; then, using BOC as an element of area, derive therefrom
the area of ODA. Thus,
r=OB
V^ r r=2 a cofl 0
sucfsa. BOC =\\m:^r^2Li PQBS =\ rdr-AB;
area Oi>^ = lim A«=y) 2/ -BOC =j j rclrde = ^^.
e=o
(b) Starting with PQBS as an element of area, find the area of the
circular strip ODF ; then using GDF as an element of area, derive there-
from the area of ODA. Thus,
«=co.-'(5»') ^ ^
area GDF = limAOio 2^ PQBS = J ^"'^ rd0 ■ Ar;
(fa),
area ODA = liniArioX (?Z)F =C' C"' " ^J ,-dfl dr = ^l
/. area of circle = 2 area ODA = ira\ [Ex. 4 (7), Art. 202.]
In this method of computing areas the infinitesimal element of area is
thus rdrdS.
Note 4. For discussions on the sign to be given to an area, on the areas
of closed curves, and on the area swept over by a moving line, see Lamb,
Calculus, Arts. 99, 101 ; Gibson, Calculus, §§ 128, 129 ; Echols, Calculus,
Arts. 163, 164.
209. Lengths of curves: rectangular coordinates. Let it be re-
quired to find the length of an arc „, .
PQ of the curve whose equation is
y =f(x), or Fix, y) = 0. Let P, Q
be the points (xj, y,), (x^, y^) respec-
tively, and denote the length of PQ
by s.
Suppose that chords like VW
are inscribed in the arc from P to
Q. Through V draw VN parallel
to the o^axis, and through W draw " fig. I2ii.
' AX
208,209.] LENGTHS OF CURVES. 371
WN parallel to the y-axis. Let V be (x, y) and W be (x + Ax,
y + Ay). Then VN^Ax, WN=Ay, and
chord VW=: y/(Axy + (%/ (1)
Now suppose that Ax, and consequently Ay, approach zero;
then the arc VW and the chord VW both become infinitesimal.
The smaller the chords VW from P to Q are taken, the more
nearly will their sum approach to the length of the arc PQ. The
difference between their sum and the length of PQ can be made
as small as one pleases, simply by decreasing the arcs. Thus :
8 = limit of sum of chords VW when these chords become
infinitesimal *
■'^0 ^ \-
'+'r3-'
= JJ*V*+(^)"^ • dx. (Definitions, Arts. 22, 23, 166.) (4)
Similarly, from form (3),
Note 1. The quantities under the integration sign in (4) and (5) are the
infinitesimal elements of length in rectangular coordinates. The difterential
of the arc also has the same forms (Art. 67 c) ; see Note 1, Art. 208.
Note 2. In (4) the integrand must be expressed in terms of a; ; in (5) in
terms of y.
Note 3. The process of finding the length of a curve is often called the
rectification of the curve ; for it is equivalent to getting a straight line of the
same length as the curve, t
• For rigorous proof of this, depending on elementary algebra and geom-
etry, see Rouch^ et Comberousse, Traite de Geometric (1891), Part I., § 291.
For a proof of the same principle and for interesting remarks on the length
and rectification of a curve, see Echols, Calculus, Arts. 165, 172.
t The semi-cubical parabola was the first curve that was ever rectified
absolutely. William Neil (1637-1670), a pupil of Wallis at Oxford, found
the length of any arc of this curve in 1657. This was also accomplished
372 INTEGRAL CALCULUS. [Oh. XXV.
Note 4. It can be shown : (a) that the difference between an infinitesimal
arc and its chord is an infinitesimal of an order at lea.st three lower ; (6) that
the limit of the sum of an infinite number of infinitesimal arcs is the same
as the limit of the .sum of the chords of these arcs. (See Infinitesimal Cal-
culus, Art. 19, Ex. 6, Note, and Art. 21, Theorems A and B.)
EXAMPLES.
1. Find the length of the four-cusped hypocycloid x» + y* = a*
Length of a quadrant = (' ° -Jl + (^ j dx. (1)
On differentiation, -x~^ + -y i^ = 0; whence ^ =
3" '3" dx ' ' dx \x
. quadrant = r^il^d^= C'^^l+Adx= r^-dx = \
Jo j;l Jo j;I Jo ^ •
.: length of hypocycloid = 4x|a = 6a.
Note 5. The hypocycloid, sometimes called the astroid, may also be
represented by the equations x — a cos' e, y = a sin' 6. (This may be veri-
fied by substitution.) Orf using these equations it follows that
dx= — 3a cos" 0 sin 6 dB, dy = 3a sin" e cos $ dB,
Thence (1) becomes :
whence -r- = — tan
dx
re=o
length of quadrant = — 4 ^ Vl + tan" B • 3 a cos" 6 sin B dB
n
(•2 3 „
= 3 a i sin e cos BdB = — , as before.
Jo 2
(Ex. Show that the area of the hypocycloid x = a cos' 8, y = a sin' 6
is J ira" ; and that the volume generated by its revolution about the i-axis is
-^ ira', as obtained otherwise in Art. 182, Ex. 20.)
2. Find the lengths of the following :
(1) The circle a;" + j/" = a". (2) The arc of the parabola y" = 4 ax, (a) from
the vertex to the point (zi, y{); (6) from the vertex to the end of the latus
independently by Heinrich van Heuraet in Holland. The second curve to
be rectified was the cycloid. This was effected by the famous architect,
Sir Christopher Wren (1632-1723), in 1673, and also by the French mathe-
matician, Pierre de Fermat (1601-1665).
209, 210.]
LENGTHS OF CURVES.
373
rectum. (3) (a) The arc of the cycloid a; = a (S — sin 9), y = a (1 — cos 6)
from e = $0 to e = Bi; (_b) a, complete arch of this cycloid. (4) The arc of
X X
the catenary y = "' {e' ■{- e "), {a) from the vertex to {xi, yi) ; (b) from the
vertex to the point for which x = a.
2 2
3. Find the whole length of the curve (-J +(r) =1. Thence
deduce the length of the hypocycloid.
4. Show that in the ellipse i = a sin 0, y = b cos tp, <j> being the com-
plement of the eccentric angle, the arc s measured from the extremity of the
minor axis is a I Vl — e^sin^ d0, e being the eccentricity. (This integral is
called "the elliptic integral of the second kind.") Then show that the perim-
eter of an ellipse of small eccentricity e is approximately 2jra( 1 — — )•
210. Lengths of curves : polar coordinates. Let it be required to
find the length of an
arc PQ of the curve
/(r, 6) = 0. Let P and
Q be the points (Vi, 0{),
0'2! ^2)) respectively, and
denote the length of arc
PQ by s. Suppose that
chords like VW are in-
scribed in the arc from
P to Q. Let V and W
be denoted as the points
(r, e), (r + Ar, $ + ^6),
Q(»-3,ej)
respectively
chord VW=yj{
Fig. 130.
Then, from Eq. (2) Art. 67 d,
sinA(9V ,
^H^i^.sin^Ae + ^Y-Afl. (1)
The length of the arc PQ (see Art. 209) is the limit of the sum
of the lengths of the chords FTTfrom Pto Q, when these chords
become infinitesimal, that is when A^ approaches zero. Hence,
from (1) and the definitions of a derivative and an integral,
-;>*©'•*
(2)
374 integhal calculus. [Ch. xxv.
It can also be shown [see the derivation of result (6), Art. 67 d],
that s = Q.p^[Vl.dr. (3)
Note 1. The quantities under the integration sign in (2) and (3) are the
infinitesimal elements of length in polar coordinates. The differential of the
arc also has the same forms, Art. 67 d ; see Note 1, Art. 209.
Note 2. In (2) the integrand must be expressed in terms oi 8 ; m (3),
in terms of r.
Note 3. The intriusic equation of a cnrve. See Appendix, Note B.
EXAMPLES.
1. Find the length of the cardioid r = a(l — cos 0).
— £7V"+(a'*
The substitution of the value of r and — in the integrand and simplifica-
tion, give
s = 2ov^ (""Vl —cosede = ia f" sin -de = 8 a.
Jo Jo 2
2. Find the lengths of the following :
(1) The circle r = a. (2) The circle r = 2asintf. (3) The curve
a
r = a sin' - ■ (4) The arc of the equiangular spiral r = ae« ■=■" », (a) from
o
e = 0 to 9 = 2 TT ; (6) from » = 2 ir to » = 4 ir. (5) The arc of the spiral of
Archimedes r = a0 from (?-i, di) to (vi, S^)- (6) The arc of the parabola
r = a sec2 ^, («) from 9 = 0 to » = 9i; (6) from e = --to«= + -.
2 2 2
211. Areas of surfaces of revolution.
Note 1. Geometrical Theorem. Let KL and B8 (Fig. 131 a) be in the
same plane. In eleraentai-y solid geometry it is shown that if a finite straight
line KL makes a complete revolution about BS, the surface thu-- generated by
KL is equal to 2 tTM ■ KL, in which TM is the length of tht perpendicular
let fall on RS from T, the middle point of KL.
Suppose that an arc PQ of a curve y =f(x) revolves about the
o^axis, and that the area of the surface thus generated is required.
210, 211.]
AREAS OF SURFACES.
375
Let P and Q be the points Car,, j/i) and (a^ y^) respectively. Sup-
pose that PQ is divided into small arcs such as KL, and denote
K and L as the points (x, y) and Cx -|- Ax, y + ^y) respectively.
Fig. 131a.
QU.,i;»i
SO
Fig. 131 6.
Draw the chord KL, and from T, the middle point of this chord,
draw TM at right angles to the x-axis. Then the area generated
by the chord KL when the arc PQ revolves about the x-axis
= 2nTM-KL
= 2;r(2/ +
i^^)\Ri
Ax. (Note 1.)
The smaller the chords KL are taken, the more nearly will the
surfaces generated by them approach coincidence with the surface
generated by the arc PQ, and the difference between area of the
latter surface and the sum of the areas of the former surfaces
can be made as small as one pleases by decreasing Ax. Accord-
ingly, the area of the surface generated by the arc PQ is the
limiting value of the sura of the areas of the surfaces generated
by the chords KL (from P to Q) when these chords become
infinitesimal. That is, area of snrCice generated by PQ
= lim^52 ^(2/ + i A2/)^l + (ff J^^
(Definitions of derivative
a C^ U . Idy\i-, (Definitions of d
=2,£^vVl+(^)'«^. 'and integral.)
(1)
(2)
376
INTEGRAL CALCULUS.
[Oil. XXV.
If the length of the chord 7i"/> be denoted by -» I
this integral takes the form
surface = 2 ,r C'y\^ + [^dy.
(3)
Note 2. Each of the expressions to be integrated in (2) and (3) may be
denoted by 2 try ds [Art. 67 /(9)], and is called an element of the surface
of revolution.
If PQ is revolved about the y-axis, the element of surface is 2 irx ds ;
and the surface
''-C:-i*m"-
(4)
The questions, whether to use form (2) or (.3), and whicb of (4) to employ,
are decided by convenience and ease of working. (See Art. 208, Note 1, and
Art. 67/.)
Note 3. In a similar manner it can be shown that the area of the surface
generated by the revolution of an arc of a curve about any straight line in
the plane of the arc, is .
2 TT \ Ids, (5)
in which ds denotes an infinitesimal arc of the curve, I the distance of this
infinitesimal arc from the straight line, and ei and eg are coordinates of some
kind that denote the ends of the revolving arc. An illustration is given in
Ex.4.
EXAMPLES.
1. Find the surface generated by the revolution of the hypocycloid
X* + y' = a* about the a;-axis.
Surface
= 2 2
Jx=0
ir-PN-.ds
:4ir I
-'ir-
(See Art. 209, Ex. 1.)
= - 6 Tra* (""(a* - x^)^d{a^ - x^)
n X
Fio. 132.
2n.j
AHEAS OF SURFACES.
377
In this case an easier integral is obtained by expressing the surface in
terms of y and dy, as in form (3). Thus,
Surface = 2 ■ 2n j'^ysjl +('{^Ydy = 4iraH°yhy = ^Ta\
2. Calculate the surface of the hypocycloid in Ex. 1, using the equations
x = a cos' e, y = a sin^ 6.
3. Derive formula (5).
4. The cardioid r = n(l — cosS) revolves about the initial line : find the
area of the surface generated.
Surface = 2 jrl FN ■ ds.
Je=o
Now FN = r smd = a(\ —cos ff)sme, and ds = oV2Vl — co&ede
Ex. 1, Art. 210).
»r^,
8-0
Fig. 13S.
.-. surface = 2\^ira- (""(1 - cos«)^ sin SdS = [-5^^02(1 - 008 9)^"
5. Find the area of the spherical surface generated by the revolution of a
circle of radius a about a diameter.
6. A quadrant of a circle of radius a revolves about the tangent at one
extremity. What is the area of the curved surface generated ?
7. Calculate the area of the surface of the prolate spheroid generated by
the revolution of the ellipse hV + a^?/^ = a^J^ about the a;-aiis.
8. In the case of an arch of the cycloid x = a{6—sm6), y=a(l— cosff),
compute : (1) the area between the cycloid and the a;-axis ; (2) the volume
and the surface generated by its revolution about the avaxis ; (3) the volume
and the surface generated by its revolution about the tangent at the vertex.
9. Find the volume and the surface generated by revolving the circle
r? f {y — hy = a^, (ft > a), about the i-axis.
378 INTEGRAL CALCULUS. [Ch. XXV.
10. Find the area of the surface generated by the revolution of the arc
of the catenary in Ex. 6, Art. 182.
11. The arc of the curve r = asm2 6, from S = 0 to 6 = - (i.e. the
first half of the loop in the first quadrant), revolves about the initial line :
find the area of the surface generated. What is the area of the surface
generated by the revolution of the second half of the same loop about the
same line ?
12. A circle is circumscribed about a square whose side is a. The smaller
segment between the circle and one side of the square is revolved about
the opposite side of the square. Find the volume and the surface of the
solid ring thus generated.
212. Areas of surfaces whose equations hare the form z =f(x, y)
or F(x, y, z) =0. It is shown in solid geometry that:
(a) The cosine of the angle between the ly-plane and the tangent plane
at any point (a;, y, z) on such a surface, supposed to be continuous, is
{-(ir-(i)T
(1)
(6) The area of the projection of a segment of a plane upon a second
plane is obtained by multiplying the area of the segment by the cosine of
the angle between the planes.
It follows from (a) and (6) that :
(c) If there be an area on the xy-plane equal to A, then A is the area
that would be projected on the ly-plane by an area on the tangent plane at
(a;, y, z) which is equal to
w-(ir-(i)'
(See C. Smith, Solid Geometry, Arts. 206, 20, 31 ; Murray, Integral Calcu-
lus, Art. 75.)
Let z =f(x, y) be the equation of a surface BFCRAGB [Fig. 123] whose
area is required. Let P{x, y, z) be any point on tliis surface, and Pi the
point (x, y, 0) vertically below P. Let PiQi be a rectangle in the xy-p\a.ne
having its sides equal to Ax and Ay respectively, and parallel to the x- and
j/-axes. Through the sides of this rectangle pass planes perpendicular to the
xj/-plane, and let these planes make with the surface the section PQ, and
with the tangent plane at P the section PQ^. {QiQ produced is supposed
to meet in §2 the tangent plane at P.)
Then, area Pi Qi = A.r • Ay.
Hence, by (2), area PQ^ = yjl + (MX+ (fY ■ ^V • ^-
^12] AREAS OF SURFACES. 379
Now the smaller Ax and Ay become, the more nearly will the section PQ^
on the tangent plane at P coincide with the section PQ on the surface.
Accordingly, the more nearly will the sum of the areas of sections like PQ2
on the tangent planes at points taken close together on the surface, become
equal to the area of the surface ; moreover, the difference between this sum
and the area of the surface can be made as small as one pleases. Con-
sequently, the area of the surface is the limit of the sum of the areas of
these sections on the tangent planes when these sections become infinitesimal.
Tliat is,
area BFCBAGB = f ^=*'^' f "="'' Jl + (-)'+ (PY ' ^V dx.
Jjc^o Jy^O ^ \dxl \dVI
Note. The integral \ y A/' 1 + [—\ + { \ dy\dx gives the area
of the strip or zone RGL, and the integral ( RGLdx gives the sum of
Jz=0
these zones from BOC to A.
EXAMPLES.
1. Find the area of the portion of the surface of the sphere in Ex. 7,
Art. 204, that is intercepted by the cylinder.
The area required = 4 area A VBLA (Fig. 125). In this figure, the equation
of the sphere is x^ + y'^ + z'^ = a^,
and the equation of the cylinder is a;^ + j/^ _ ax.
The area of a strip L V, two of whose sides are parallel to the 3!/-plane, will
first be found ; then the sum of all such strips in the spherical surface
AVBLA will be determined.
A„..™..=j-rc"['+(i)"-(in'*-
Since the required surface is on the sphere, the partial derivatives must be
derived from the equation of the sphere.
Accordingly, §£ = _2, |f = _!?,.
dx z dy z
hence, 1 + (f^V + (f£y= 1 +^ + ^ = ^ = -
Also, RK=Vax - x'^.
■ x^-y^
area^FjBi^= i \ dy dx
Jo Jo .^^2 _ a;2 _ 2,2
= afTsin-i y Y^'dx
= a\ sin-iA/ — ^— dx.
Jo y a + X
380 INTEGRAL CALCULUS. [Ch. XXV.
This integral can be evaluated by integrating by parts. The integration
can be simplified by means of the substitution sin z =-J — - — It will be
' a + I
found that area required = 4 area A VBLA = 2 (jr — 2)0^ = 2.28.32 a^.
2. Find the area of the surface of the cylinder intei-cepted by the sphere
in Ex. 7, Art, 204.
3. By the method of this article, find the surface of the sphere x^ + r/^
+ 22 = «2.
4. A square hole is cut through a sphere of radius a, the axis of the
hole coinciding with a diameter of the sphere : find the volume removed and
the area of the surface cut out, the side of a cross-section of the hole being 2 b.
6. Find the area of that portion of the surface of the sphere inter-
cepted by the cylinder in Ex. 4, Art. 205.
213. Mean values. In Art. 168 it has been stated that if the
curve 2/=/(x) be drawn (Fig. 101), and li OA = a and OB = b,
then, of all the ordinates from A to B,
^x. 1 Sivea. APQB J„ •^'^'^^''^ ,,,
the mean value = ^^— — ^^ ( 1^
AB b-a ^ ^
Result (1) can be derived in the following way which has
also the advantage of being adapted for leading up to a more
general notion of mean value. The mean value of a set of quan-
tities is defined as
the sum of the values of the quantities
the number of the quantities
For instance, if a variable quantity takes the values 2, 5, 7, 9,
the mean of these values is -^ — — — ^^^- or 54.
4 ^
Now take any variable, say x, and suppose that f(x) is a con-
tinuous function, and let the interval from x = a to x = b be
divided into n parts each equal to Ax, so that n Ax = b — a. Let
the mean of the values of the function for the n successive values
of a;,
a, a + Ax, a -I- 2 Ax, •••, a-f-n — lAx,
be required. The corresponding n successive values of the func-
tion are „, , , , ,
f(a), f{a + Ax), /(a -f- 2 Ax), ..., /(a -|- n - 1 ■ Ax).
213.] MEAN VALUES.
Heace, mean value of function
. f(a) +f(a + Ax) +/(ffl + 2 Ax) + • • • +f(a + n-l-Ax)
381
(2)
Now n Ax = b — a, whence n =
mean value
Ax
Substitution in (2) gives
^/(a)Ax+/(a+Ax)Ax+/(a+2Aa;)Aa;-| \-f(a+n—l Ax)Ax
b-a ^3)
Finally, let the mean of all the values that f(x) takes as x varies
from a to 6 be required. In this case n becomes intinitely great
and Ax becomes infinitesimal; accordingly [Art. 166 (2), (3)]
(3) becomes
mean value =
J>1
b-a
dx
(4)
as already represented geometrically in Art. 168.
Note 1. Reference for collateral reading. Echols, Calculus, Arts.
150-152.
EXAMPLES.
1. Find the mean length of the ordinates of a semicircle (radius a),
the ordinates being erected at equidistant intervals on the diameter.
Choose the axes as in Fig. 134. Then the equation of the circle is
J.2 -I- 1/2 _ ^2_ Let PN denote any of the ordi-
nates drawn as directed.
Mean value =
T"^" PN ■ dx f ° Va- - 2'^ dx
{-a)
2a
2.2a
.7854 a.
2. Find the mean length of the ordinates of a
semicircle (radius a), the ordinates being drawn at
equidistant intervals on the arc.
Let PN be any of the ordinates drawn at equi-
distant intervals on the arc, that is, at equal incre-
ments of the angle 6.
re=n
Mean value = =^fe^
PN-de
ir -0
=t
sin e de
:2a =.6366 a.
382 INTEGRAL CALCULUS. [Ch. XXV.
Note 2. A slight inspection will show that it is reasonable to expect the
results in Exs. 1,2, to differ from each other.
Suggestion : Draw a number of ordinates, say 4 or 6 or 8, as specified
in Ex. 1, and compare them with the ordinates of equal number drawn as
specified in Ex. 2.
3. Find the average value of the following functions: (1) 1 x'^-\-ix — ^
as X varies continuously from 2 to 6 ; (2) ofi — Zx'^ + ix-'rllasx varies from
— 2 to 3. Draw graphs of these functions.
4. Find the average length of the ordinates to the parabola y^ = 8 x
erected at equidistant intervals from the vertex to the line i = 6.
6. (1) In Fig. 108 find the mean length of the ordinates drawn from
OiV to the arc OML, and the mean length of the ordinates drawn from OiVto
the arc ORL. (2) In Fig. 107 find the mean length of the abscissas drawn
from 0\', (a) to the arc OR; (6) to the arc RL; (c) to the arc ORL.
(3j In Fig. 109 find the mean ordinate from OL, (a) to the arc TKN ; (6) to
the arc TOM.
6. (1) In the ellipse whose semiaxes are 6 and 10, chords parallel to
the minor axis are drawn at equidistant intervals : find their mean length.
(2) In the ellipse in (1) find the mean length of the equidistant chords that
are parallel to the major axis. (3) Do as in (1) and (2) for the general case
in which the major and minor axes are respectively 2 a and 2 6.
7. On the ellipse in Ex. 6, (3), successive points are taken whose eccen-
tric angles differ by equal amounts : find the mean length of the perpen-
diculars from these points, (1) to the major axis ; (2) to the minor axis.
8. In the case of a body falling vertically from rest, show that (1) the
mean of the velocities at the ends of successive equal intervals of time, is one-
half the final velocity ; (2) the mean of the velocities at the ends of succes-
sive intervals of space, is two-thirds the final velocity. (The velocity at the
end of t seconds is yt feet per second ; the velocity after falling a distance
s feet is V2 gs feet per second.)
9. A number n is divided at random into two parts : find the mean value
of their product.
10. Find the mean distance of the points on a circle of radius a from
a fixed point on the circle.
The interval 6 — a in (1) and (4) through which the variable x
passes is called the range of the variable, and dx is an infini-
tesimal element of the range. In (1) and Ex. 1 the range is a
particular interval on the a^axis. In Ex. 2 the range is a certain
angle, namely t ; in Ex. 8 (2) the range is a vertical distance ; in
B
c
^
A»
c
v
3
O
A 2:
213.] MEAN VALUES. 383
Ex. 8 (1) the range is an interval of time. There are various
other ranges at (or for) whose component parts a function may
take different values. For instance, a curved line as in Ex. 10, a
plane area as in Exs. 11, 13 ; a curved surface as in Ex. 15 (1) ; a
solid as in Exs. 16, 17. The definition of mean value [or result
(4)] may be extended to include such cases, thus :
liin 2 {(value of function at each infini-
tesimal element of the range) x (this
the mean value of a func- \ _ infinitesimal element)} ^
tioaover a certain range/ the range
11. Find the mean square of the distance of a point within a square
(side = a) from a corner of the square.
In this case "tlie range" extends over a square.
Choose the axes as shown in Fig. 136. Take any point
P (cc, J/) in the range, and let its distance from O be
d. At P let an infinitesimal element of the range
be taken, viz. an element in the shape of a rectangle
whose area is dy dx. Now d^ = x'^ -i- y^. .•. mean
value of (P for all points in
Fig. 136.
(" (''{x^+y'')dydx
OACB = J^-J^ = I (i2.
area of square
12. Find (1) the mean distance, and (2) the mean square of the distance,
of a fixed point on the circumference of a circle of radius a from all points
within the circle. (Suggestion : use polar coordinates.)
13. Find (1) the mean distance, and (2) the mean square of the distance,
of all the points within a circle of radius a from the centre.
14. Find the mean latitude of all places north of the equator.
15. For a closed hemispherical shell of radius a calculate (1) the mean
distance of the points on the curved surface from the plane surface ; (2) the
mean distance of the points on the plane surface from the curved surface,
distances being measured along lines perpendicular to the plane surface.
16. Calculate (1) the mean distance, and (2) the mean square of the dis-
tance, of all points within a sphere of radius a, from a fixed point on the
surface.
17. Calculate (1) the mean distance, and (2) the mean square of the dis-
tance, of all points within a sphere of radius a, from the centre.
384
INTEGRAL CALCULUS.
[Ch. XXV
18. Find (1) the mean distance, and (2) the mean square of the distance, of
all points on the surface of a sphere of radius a, from a fixed point on the surface.
19. Find (1) the mean distance, and (2) the mean square of the distance, of
all points on a semi-undulation of the sine curve 2/ = a sin x, from the a;-axis.
214. Note to Art. 104. Proof of (6). Let ^ be the given curve
jj- 2/ = f(x), and E its evolute.
Let Ci be the centre of curva-
ture for Ai, and C2 for A^. Denote
any point in K by (x, y), the radius
'A^[Xi,Vi) of curvature there by R, the cor-
responding centre of curvature in
E by (a, j8), the points A^, A2, Ci,
C2, by (xi, ?/,), (a-2, 2/i), («„ ySj), («„
P2), respectively, the radii of cur-
vature AiCi and A2C2 by B^ and B2.
It will now be shown that
Fig. 137.
length of arc C1C2 = Bi - Bu
ATcCA=r''^l+(f'j-dp. (See Art. 209.) (1)
.^j8.
da
On substituting the value of — from (3) Art. 104, and the
d/3
value of d/3 derived from (1) Art. 104, and noting that
x = Xi when /3 = /81, and x = X2 when /8 = ySj,
Equation (1) becomes
•ocAc.=£^;Vi+(;
dx
dx\dx^
1 +
daf
dj^
dx. (2)
d/?
Differentiation of B in Art. 101, Eq. (1), will snow that ^^ is
dx
the same as the integrand in (2). Then, since B — Bi when
X = Xi, and B = Bn when x = X2, and — dx = dB (Art. 27), Equa-
d.i;
tion (2) becomes
arc CiC.,= f'~'' — dx= r^''dB= C^"" dR =^ B2- B,.
J i=x^ dx i/x=x, Jk=k^
N.B. On lengths of curves in space see Appendix, Note C.
CHAPTER XXVI.
NOTE ON CENTRE OF MASS AND MOMENT OF
INERTIA.
N.B. For a full explanation and discussion of the mechanical terms in
this note, see textbooks on Mechanics.
215. Mass, density, centre of mass. For this note the following
definition of mass may serve : The mass of a body is the quantity
of matter which the body contains* The principal standards of
mass are two particular platinum bars; the one is the "imperial
standard pound avoirdupois," which is kept in London, and the
other is the " kilogramme des archives," which is kept in Paris.
Note. The weight of a body is the measure of the earth's attraction upon
the body, and depends both on tlie mass of the body and its distance from
the centre of the earth. The same body, while its mass remains constant,
has different weights according to the different positions it takes with respect
to the centre of the earth.
The density of a body is the ratio of the measure of its mass to the measure
of its volume ; that is, the density is the number of units of mass in a unit of
volume. The density at a point is the limiting value of the ratio of (the
measure of) the mass of an infinitesimal volume about the point to (the
measure of) the infinitesimal volume. A body is said to be homogeneous when
the density is the same at all points. If a body is not homogeneous, tlie " den-
sity of a body," defined above, is the average or mean density of the body.
Centre of mass. Suppose there are particles whose masses are m^,
»i2, jftj, ■■•, and whose distances from any plane are, respectively,
dj, di, dj, •••. Let a number D be calculated such that
^ _ midi + m-A + m^dj ••• . .^ let D = ^ "^'^-
wii + ?7i2 + wij + • • • ' 2m
For any given plane, D evidently has a definite value.
» A real definition of mass, one that is strictly logical and fully satisfac-
tory, is explained in good text-books on dynamics and mechanics. (For
example, see MacGregor, Kinematics and Mechanics, 2d ed.. Art. 289.)
385
386 INTEGRAL CALCULUS. [Ch. XXVI.
If (aa, Vi, 2i), fe 3/2, Z2), (2:3, ^3, 23), ■••, respectively, be the coordi-
nates of these particles with respect to three coordinate planes at
right angles to one another, then the point (x, y, z), such that
x = ^, y=^, z=^, (1)
Swi 2m 2m
is called the centre of mass of the set of particles.
If the matter " be distributed continuously " (as along a line,
straight or curved, or over a surface, or throughout a volume), and
if Am denote the element of mass about any point (x, y, 2), then,
on taking all the points into consideration, equations (1) may be
written :
X = l'm^»^2a:-Am ^^^ similarly for y and 2. (2)
lim^,^„2Am' -^ ^ ^^
From (2), by the definition of an integral,
i X dm \ y dm, \ z dm,
^ = —C ,V = —. ,3 = ^7 (3)
I dm, \ dm, 1 dm,
If p denote density of an infinitesimal dv about a point, then
dm = pdv (4); and, on integration, tn=\pdv, (5)
Ex. Write formulas (3), pntting p dv for dtn.
Suppose that the body is not homogeneous; that is, suppose
that the density of the body varies from point to point. Let p
denote the density at any point (x, y, z), let dv denote an infini-
tesimal volume about that point, and let p denote the average or
mean density of the body. Then
mass
P
of body Jp*^^
vol. of body C^v
Note. The term centre of mass is used also in cases in which matter is
supposed to he concentrated along a line or curve, or on a surface. In these
cases the terms line-density and surface-density are used.
215.]
CENTRE OF MASS.
387
EXAMPLES.
1. In a quadrant of a thin elliptical plate whose semi-axes are a and b,
the density varies from point to point as the product of the distances of each
point from the axes. Find the mass,
the mean density, and the position of
the centre of mass, of the quadrant.
Choose rectangular axes as in the figure.
At any point P(x, y), let p denote the
density and dm denote the mass of a
rectangular bit of the plate, say, dx ■ dy.
Let M denote the mass, p the mean
density, and (x, y) the centre of mass,
of the quadrant.
Now dm = pdx dy. But ptxxy ; i.e.
Fia. 138.
kxy, in which k denotes some constant
Accordingly, M= \dm = i i
-' •'1=0 •'y=0
kxy dy dx = ^k a^V^.
Also,
Here
Similarly,
- _ mass of quadrant _ \ k a^b'^ _ kah
volume of quadrant J vab 2 ir
p . X ■ dv A- 1 I " x-y dy dx
^ ka^b" _
y — ^^b. Hence, centre of mass is (^5 a, {^ 6).
= Ao-
2. Find the centre of mass of a solid
hemisphere, radius o, in which the density
varies as the distance from the diametral
plane. Also find the mean density.
Symmetry shows that the centre of mass
is in OT.
Take a section parallel to the diametral
plane and at a distance y from it.
The area of this section
= ir.CP'^ = '.r(a2-y2).
For this section, p coy, i.e. p = ky, say.
; /a \
0 ^
Fio. 139.
Then
Also
J.'
p.y.ir(a2-!/2)(i!/ fcT J !/5(a2 - y2)dy
f
^°P^(a^ - y')dy
kir
|;y(a^
■ y^)dy
-_ M _\kira* _
\ka.
vol. \ ira^
This is the density at a distance | a from the diametral plane.
388
INTEGRAL CALCULUS.
[Ch. XXVI.
8. The quadrant of a circle of radius a revolves about the tangent
at one extremity. Find the position of the mass-centre of the surface
thus generated. In this case let the
"surface-density" be denoted by p.
Symmetry shows that the mass-centre
is in the line PL. Let y denote the
distance of the mass-centre from OX.
In PL take any point N, at a dis-
tance y, say, from OX. Through JV
pass a plane at right angles to PL,
and pass another parallel plane at an
infinitesimal distance dy from the first
plane. These planes intercept an infini-
tesimal zone, of breadth ds say, on the
surface generated.
1'
[,
M
^„
■LY
1
.1
/^
0
P
X
Fig. 140.
Area of this zone = 2 ir • CiV • ds = 2 v^MN— MC)ds.
Now, at C {x, y)
x2 -I- j/2 = a2
Accordingly, ds = -W 1 4- 1 -? j ■ dy =
Va2-
-dy.
Hence, area zone = 2 7r(a — Vo'^— 2/2)_^^___di;=:2ira(— — ^::::^— \\dy.
y/a^ - y^ \ Va' - y'^ /
^j:%y.(2..CN.ds) ^ 2'a,§;y{-^^^_-l)dy
p ■ area zone 2ir ap\ I — " — 1 ] du
.■.y=-
ir -2
= .876 a.
4. In the following lines, curves, surfaces, and solids, find the posi-
tion of the centre of mass ; and, in cases in which the matter is not dis-
tributed homogeneously, also find the total mass and the mean density
("line-density," "surface-density," or "density," as the case may be).
(The density is uniform, unless otherwise specified.)
(1) A straight line of length I in which the line-density varies as (is k
times), (a) the distance from one end ; (6) the square of this distance; (c) the
square root of this distance.
(2) An arc of a circle, radius r, subtending an angle 2 « at the centre.
(3) A fine uniform wire forming three sides of a square of side a.
(4) A quadrantal arc of the four-cu.sped hypocycloid.
(5) A plane quadrant of an ellipse, semi-axes a and 6.
215.] CENTRE OF MASS. 389
(6) The area bounded by a semicircle of radius r and its diameter,
(a) when the surface density is uniform ; (6) wlien tlie surface density at
any point varies as (is k times) its distance from the diameter.
(7) The area bounded by the parabola Vx + Vy = Va and the axes.
(8) The cardioid r = 2 (z(I - cos e).
(9) A circular sector having radius r and angle 2 a.
(10) The segment bounded by the arc of tlie sector in Ex. (9) and its chord.
(11) The crescent or lune bounded by two circles which touch each other
internally, their diameters being d and |<Z, respectively.
(12) The curved surface of a right circular cone of height h, (a) when
the surface density at a point varies as its distance from a plane which passes
through the vertex and is at right angles to the axis of the cone ; (6) when
the surface density is uniform.
(13) A thin hemispherical shell of radius a, in which the surface density
varies as the distance from the plane of the rim.
(14) A right circular cone of height h in wliioh, (a) the density of each
infinitely thin cross-section varies as its distance from the vertex ; (6) the
density is uniform.
(15) Show that the mass-centre of a solid paraboloid generated by revolving
a parabola about its axis, is on the axis of revolution at a point two-thirds the
distance of the base from the vertex.
(16) A solid hemisphere of radius r, (a) when the density is uniform ;
(6) when tlie density varies as the distance from the centre.
(17) Show that the mass-centre of the solid generated by the revolution
of the cardioid in Ex. (8) about its axis, is on this axis at a distance f a from
the cusp.
(18) If the density p at a distance r from the centre of the earth is given
by the formula p = po^'" , in which po and k are constants, show that the
kv
earth's mean density is sin kE - kli cos kE
^^ ¥r^ '
in which B denotes the earth's radius. (Lamb's Calculus.)
[Answers : (1) f « from that end, M=i kP, -p = \kl; (h)\l,M=\ kP,
p = \kl^; (c) ^l, M=l kl^, p -^kli. (2) On radius bisecting the arc at dis-
tance r — - — from centre. (3) At a distance J a from the centre of the
square. (4) Point distant | a from each axis. (5) Point distant -— from
axis 2 a, — from axis 2 b. (6) (a) On middle radius, at point distant —
Ztt 3ir
from the diameter ; (6) On middle radius, at point .589 a from the diameter,
mean density = .4244 maximum density. (7) Point distant I a from each
axis. (8) The point (ir, | a). (9) On middle radius of sector, at distance
^ sinjt fjQjj^ jjjg centre. (10) On the bisector of the chord, at distance
390 INTEGRAL CALCULUS. [Ch. XXVI.
2 J. sin_« f^oai the centre. (11) On the diameter through the point
a — sin a cos
of contact and distant \l d from that point. (12) (a) On the axis, at distance
J h from the vertex ; (6) on axis, at distance f h from vertex. (13) On the
radius perpendicular to the plane of the rim, at a distance § a from the centre.
(14) (a) On the axis, f A from the vertex; the mean density is the same as
the density at the cross-section distant f h from the vertex ; (6) on the axis,
at a distance | h from the vertex. (16) (a) On a radius perpendicular to the
base, at a distance .375 r from it; (6) on radius as in (a), at distance Ar
from the base.]
216. Moment of inertia. Radius of gyration. These quantities are
of immense importance in mechanics and its practical applications.
Moment of inertia. Let there be a set of particles whose masses
are, respectively, Wj, m.j, jjig, ••■, and whose distances from a chosen
fixed line are, respectively, r-i, j-j, r^, •••■ The quantity
wiir,^ + m2?'2^ + m^r^^ -\ , i.e. 2 wr2 (1)
is called the moment of inertia of the set of particles with respect
to the fixed line, or axis, as it is often called. It is evident that
for any chosen line and system of particles the moment of inertia
has a definite value. In what follows, the moment of inertia will
be denoted by I,
It can be shown, by the same reasoning as in Art. 215, that
definition (1) can be extended to the case of any continuous dis-
tribution of matter (whether along a line or curve, or over a sur-
face, or throughout a solid) and any chosen axis; thus.
= {r^dm,
in which r denotes the distance of any point from the axis, and
dm an infinitesimal element of mass about that point.
Radius of gyration. In the case of any distribution of matter
and a fixed line, or axis, the number k, which is such that
j^2 _ the moment of inertia _ J ^^^^"^
the mass ~ C^^ '
is called the (length of the) radios of gyration about that axis.
216.]
MOMENT OF INERTIA.
391
EXAMPLES.
1. Find the radius of gyration about its line of symmetry of an isosceles
triangle of base 2 a and altitude h.
The density per unit of area will be denoted by p.
Fig. 141.
Let P be any point in the triangle, and make the construction shown in
the figure. Denote NO by y.
ry=h rx=LN
,1 -Z. PN'' ■ p -dx dy over AOC J^''}y^)^ ^ '^^^^
Sp ■ dxdy over ABC p ah
Then k^ -.
Now
LN_
AO'
CO ah h^ "-^
.-. /fc2=i^ = -a2; whence k = ~
ah 6 v'6
In this example, the moment of inertia is ^ a^h.
2. Show that the moment of inertia of a homogeneous thin circular plate
about an axis through its centre and perpendicular to its plane is Jpira*, in
which p denotes the surface density, and that its radius of gyration is J a\/2.
I On using polar coordinates, I = \r'^- dm = j i"^ ■ p ■ dA = p I 1 r^-rdrddA
Y 3. Find the moment of inertia of a solid
homogeneous sphere of radius a about a
diameter, m being the mass per unit of
volume. Suppose that the sphere is gener-
ated by the revolution of the semicircle APB
about the diameter AB. Let rectangular
axes be chosen as in the figure. At any
point P{x, y) on the semicircle take a thin
rectangular strip PN at right angles to AB
392 INTEGRAL CALCULUS. [Cii. XXVI.
and having a width Ax. This strip, on the revolution, generates a thin circu-
lar plate. It follows from Ex. 2, since m is the mass per unit of volume, that
/ of this plate about AB = -Tr ■ PX* ■ Ax.
.: I of sphere = 2 — tt • FN* Ax from ^ to B
2
= 2 . ^ r " (a2 - x^ydx = ^ mwaK
2 Jo
Here, on denoting the mass of the sphere by M,
M = \ mira^ ;
hence, I=i ^<i^ >
accordingly, fc'' = | o^ ;
and thus, k — .632 a.
4. Find the moment of inertia and the square of the radius of gyration
in each of the following cases ;
(Unless otherwise specified, the density in each case is uniform. The
mass per unit of length, surface, or volume is denoted by m, and the total
mass by M. )
(1) A thin straight rod of length I, about an axis perpendicular to its
length : (a) through' one end point, (6) through its middle point.
(2) A fine circular wire of radius a, about a diameter.
(.3) A rectangle whose sides are 2 a. 2 i ; (a) about the side 2 ft,
(ft) about a line bisecting the rectangle and parallel to the side 2 ft.
(4) A circular disc of radius a : (a) about a diameter, (6) about an
axis through a point on the circumference, perpendicular to the plane of
the disc, (c) about a tangent.
(5) An ellipse whose semi-axes are a and 6 : (a) about the major axis,
(6) about the minor axis, (c) about the line through the centre at right
angles to the plane of the ellipse.
(6) A semicircular area of radius a, about the diameter, the density
varying as the distance from the diameter.
(7) A semicircular area of radius a, about an axis through its centre of
mass, perpendicular to its plane.
(8) A rectangular parallelepiped, sides 2 a, 2 6, 2 c, about an edge 2 c.
(9) A rijjht circular cone (height = U, radius of ba.se = ft), about its axis.
(10) A thin spherical shell of radius a, about a diameter.
(11) A sphere of radius a, about a tangent line.
(12) A right circular cylinder (length = ?, radius = iJ) : (a) about its
axis, (6) about a diameter of one end.
21ii.] MOMENT OF INERTIA. 393
(13) A circular arc of radius a and angle 2 o : (a) about the middle
radius, (6) about an axis through tlie centre of mass, perpendicular to the
plane of the arc, (c) about an axis through the middle point of the arc,
perpendicular to the plane of the arc [Lamb's Calculus, Exs., XXXIX.].
[Answers: (1) (a) Iml^ ^P; (6) ^^mP, j'j f. (2) }. Ma\ J a^.
(3) (a) r- = ia^; (b) A-2=ia2. (4) (a) k^ = \a'; (6) A;^ = f a' ; (c) k^
= I a^ (5) (a) i Mb-^ ; (6) J itfa^ ; (c) J .V((7-^ + ft^). (6) \ Ma^, | a^.
(7) k^ = f i - ^\ a\ (8) A;2 = Ka' + &')■ (») A '"'^ &**, ,% b'. (10) A;^
= §a2. (11) k^ = iaK (12) («) 7=iJtfB-i; (6) 1= Mi', R' + ^P).
(13) (a) A2 = j„2^_!iBJ^\; (6) k^ = a'U_^a\., (c) A^ =
-(-^•>J ^ "' ^ ''
Note. For interesting examples on centres of gravity and moments of
inertia, see Campbell, Calculus, Chaps. XXXVI., XXXVII, Chandler, Cal-
culus, Chaps. XXXIII, XXXIV. For discussions on mechanics and exam-
ples, see Osgood, Calculus, Chap. X., and Campbell, Calculus, Chaps. XXX.-
XXXV,
CHAPTER XXVII.
DIFFERENTIAL EQUATIONS.
If.B. The references made in this chapter are to Murray, Differential
Equations.
217. Definitions. Classifications. Solutions. This chapter is
concerned with showing how to obtain solutions of a few differen-
tial equations which the student is likely to meet in elementary
work in mechanics and physics.
Differential equations are equations that involve derivatives or
difierentials. Such equations have often appeared in the preced-
ing part of this book.
Thus, in Art. 37, Exs. 2, 11, 13, differential equations appear ; Equations
(1), Art. 63, (2)-(5), Art. 67 (a), (2)-(5), Art. 67 (c), C3)-(6), Art. 67 (d),
are differential equations ; so also, in Art. 68, are (1) and (2), Ex. 6 ; equa-
tions in Exs. 13, 14, and some of the equations in Exs. 10, 11 ; several equa^
tions in Ex. 1, Art. 69; Equations (2)-(4), Ex. 1, Art. 7."; the answers to
Exs. 2-4, Art. 73 ; in Ex. 4, Art. 79 ; in Exs. 5-8, Art. 80 ; Equation (8),
Art. 96 ; etc., etc.
Differential equations are classified in the following ways, A
and B:
A. Differential equations are classified as ordinary differential
equations and partial differential equations, according as one, or
more than one, independent variable is involved. Thus, the equa-
tions in Ex. 4, Art. 79, and in Exs. 5-8, Art. 80, are partial differen-
tial equations; the other equations mentioned above are ordinary
differential equations. (Only ordinary differential equations are
discussed in this chapter.)
B. Differential equations are classified as to the order of the
highest derivative appearing in an equation. Thus, of the exam-
ples cited above, Equations (2)-(5), Art. 67 (a), are equations of
the first order; Equations (2), Ex. 5, Art. 68, and (8), Art. 96, are
394
217-219.] DIFFERENTIAL EQUATIONS. 395
equations of the second order; the last equation but one in Ex. 1,
Art. 69, is an equation of the nth order.
A solution (or integral) of a differential equation is a relation
between the variables which satisfies the equation. Thus, in
Art. 73, Ex. 1, relation (1) satisfies Equation (4), and, accordingly,
is a solution of (4).
Ex. 1. Show that relation (1) satisfies Equation (4) in Art. 73, Ex. 1.
Ex. 2. See Ex. 4, Art. 79, and Exs. 5-8, Art. 80. In these examples the
equations in the ordinary functions are solutions of the differential equations
associated with them.
Ex. 3. Show that the relations in Exs. 2-5, Art. 73, are solutions of the
differential equations obtained in these respective exercises.
218. Constants of integration. General solution. Particular solu-
tions. It has been seen in Art. 73, Ex. 6, that the elimination of
n arbitrary constants from a relation between two variables gives
rise to a differential equation of the nth order. This suggests the
inference that the most general solution of a differential equation
of the nth order must contain n arbitrary constants. For a proof
of this, see Diff. Eq., Art. 3, and Appendix, Note C. Simple
instances of this principle have appeared in Art. 73, Exs. 1-5.
A general gelation of an ordinary differential equation is a solu-
tion involving n arbitrary constants. These n constants are called
constants of integration. Particular solutions are obtained from the
general solution by giving the arbitrary constants of integration
particular values. The solutions of only a few forms of differential
equations, even of equations of the first order, can be obtained.
N.B. For a fuller treatment of the topics in Arts. 217, 218, see Diff. Eq.,
Chap. I.
EQUATIONS OF THE FIRST ORDER.
219. Equations of the form f{x) dx + F{y) <fy = 0. Sometimes
equations present themselves in this simple form, or are readily
transformable into it; that is, to use the expression commonly
used, " the variables are separable." The solution is evidently
j'fix)dx+j'F(y)dy = c.
39G INTEGRAL CALCULUS. [Ch. XXVII.
Ex.1. Solve ydx + xdy = 0. (1)
On separating the variables, h — = 0,
X y
and integrating, log x + logy = logo ;
whence xy = c. (2)
Solution (2) can be obtained directly from (1) on noting that ydx + xdy
is d (xy).
Ex.2. Vl - x:'dy + v^l - y'^dx = 0. Ex.3. n{x + a}dy + m(y + b)dx = 0.
220. Homogeneous equations. These are equations of the form
Pdx+ Qdy = (), in which P and Q are homogeneous functions
of the same degree in x and ij. Tfie substitution of vx for y
leads to an equation in v and x in which the variables are easily
separable.
Ex.1, {y'^ -x^)dy + 2xydx-0. Ex.3, y'^dx ^ixy + x'^)dy = (i.
Ex. 2. (z2 + j/2) (te + xy dy = 0. Ex. 4. (t/^ -2xy)dz = {x^ - 2 xy) dy.
221. Exact difierential equations. These are equations of the
form
Pdx+Qdy==0, (1)
in which the first member is an exact differential (see Art. 179).
If P and Q satisfy test (2), Art. 179, then (1) is an exact differ-
ential equation, and its solution is
C{Pdx+Qdy) = c.
Ex. 1. xdy + ydx = 0. (See Ex. 1, Art. 186.)
Ex.2. i2xy + Z)dx+(x-^ + iy)dy = 0.
Ex.3, (e' sin y + 2x)dx + e' cosy dy = 0.
Ex. 4. (ox - y2) dy = (a:2 - ay) dx.
Inte^ating factors. Equations that are not exact can be made
exact by means of what are termed integrating} factors. In some
cases these factors are easily discoverable.
220-222.] DIFFEHEyriAL EQUATIONS. 397
EXAMPLES.
5. Solve xd'j -ydz = 0. (1)
The first member does not satisfy tlie test in Art. 179 ; thus (1) is not an
exact differential equation. Multiplication by 1 -i- xy gives
y X
whence logy — logx = logc, and, accordingly, y= ex.
Multiplication by 1 -e- x^ gives
xdy -ydz_n.
X^
whence - = c, i.e. y = ex.
X ' "
Similarly, multiplication by 1 -^ y- makes (1) integrable.
The multipliers used above are called integrating factors. In the follow-
ing examples these factors can be obtained by inspection.
6. Solve (i/2 - X-) dy-i-2xydx = 0. (See Ex. 1, An. 220.)
On rearranging, y- dy + 2xy dx — x-dy = 0,
and using the factor 1 h- y", dy + ^ j-y rfa: - x^ dy ^ q_
T.T, - • x^
Whence, on integration, y -\ — = c ;
y
i.e. x^ + y'^ — cy = 0.
7. 2aydx = x(y- a) dy. 8. (y + xy-)dx = (x'h/ - x)dy.
Note. On Integrating Factors see Diff. Eq., Arts. 14-10.
222. The linear equation ^ 4 J*y = 0, (1)
in which P and Q do not involve y. (It is called linear because
the dependent variable and its derivative appear only in the first
degree.) This is, perhaps, the most important equation of the
first order.
It has been discovered that er^^ is an integrating factor for
this equation. On using this factor,
e^''"(|+/'j/) = QeI'-; (2)
whence, on integration.
Note. For the discovery of the integrating factor, see Diff. Eq., Art. 20.
398 INTEGRAL CALCULUS. [Ch. XXVII.
EXAMPLES.
1. Show that (2) is an exact differential equation.
2. x^-ay = x+l.
dx
On using form (1), -^ -- y = \ + x''^.
dx X
Here P =
■-■ .-. (pdx = -a\ogx = iosx-'. .-. eJ-^'^ = x-».
X J
On using this factor, x-''{dy — ax-' dx) = x-''(l + x"') dx ;
and integrating, jix-" = 1 1- c,
\ — a —a
X 1
whence y = — '■ h ex".
1 — a a
3. (1 -x2)^-xy= 1. 4. cos2x-^+ j; = tanx.
dx dx
dx x^
Some equations are reducible to form (1). For example,
f^+Py^Qyn. (3)
On division by y", y-" ^ + Pw'-» = Q.
dx
On putting y'-" = u, it will be found that (3) takes the linear form
|^ + (l-«)Pr =(!-»)§. (4)
6. Derive (4) from (.S).
1.'^ + -^ = xyi. 8. f? = xV-xy.
dx \ — x^ dx
223. Equations not of the first degree in the derivative. Three
types of these equations will be considered here, viz. A, B, 0, that
follow. (Let -^ be denoted by p.)
A. Equations reducible to the form as = f{y, p). (1)
On taking the ^-derivatives, -= <t>( Vt P> 1 say. (2)
P \ dyj
Possibly, (2) may be solvable and give a relation, say,
Fip, y, c) = 0. (3)
222,223.] DIFFERENTIAL EQUATIONS. 399
The p-eliminant between (1) and (3) is the solution. If this
eliminant is not easily obtainable, Equations (1) and (3), taken
together, may be regarded as the solution, since particular corre-
sponding values of x and y can be obtained by giving p particular
values.
Ex. 1. x=:y + a logp.
On taking the ^-derivative, - = 1 + 2 ffi ; whence 1 — p = a ^ •
p p dy ay
On integrating, y = c — a log (p — 1);
and thence x = c + a log ^-^ —
P
Ex. 2. p'^y + 2px = y. Ex. 8. x = y+p^.
B. Equations reducible to the form y =Kx,p). (4)
On taking the x-derivative, p = Mx, p, —] say. (5)
Possibly, (5) may be solvable and give a relation, say,
F(p, X, c) = 0. (6)
The p-eliminant between (4) and (6) is the required solution.
If this eliminant is not easily obtainable. Equations (4) and (6),
taken together, may be regarded as the solution, since they suffice
for the determination of x and y by assigning values to a param- '
eter p.
Ex.4. iy = x^-\-p^. Ex. 6. 2 y + p" = 2 a;«.
C. Clairant'B equation, viz. y=px + fip). (7)
In this case y = cx+ f(c) (8)
is obviously a solution.
This solution can be obtained on treating (7) like (4), of which it is a
special case.
Thus, on taking the x-derivatives in (7),
p=p + [x+/'(p)]|.
From this, a;+/'(p) = 0 (9), or ^ = 0. (10)
Equation (10) gives p = c.
Substitution of this in (7) gives (8).
As to the part played by (9) see Diff. Eq., Art. 34.
400 INTEGRAL CALCULUS. [Ch. XXVII.
EXAMPLES.
6. y=px + -- 't.y=px + ay/\+p^.
8. x\y —px) = HP^. [Suggestion: Put x^ = u, y^ = ».]
Note 1. Sometimes the first member of an equation f(x, y, p) = 0 is
resolvable into factors. In such a case equate each factor to zero, and solve
the equation thus made. (This is analogous to the method pursued in solv-
ing rational algebraic equations involving one unknown.)
9. Solve p» - p2(z + 2, + 2) + p{xy + 2x + 2y) -2xy = 0.
On factoring, (p — x) = 0, p - y = 0, p — 2 = 0.
On solving, 2 y = i^ + c, y = ce', y = 2x + c.
These solutions may be combined together,
(2y -x^-c){y - ce'){y - 2 jc - c) = 0.
Note 2. On Equations of the first order which are not of the first degree
see Diff. Eq., Chap. III.
224. Singular solutions. Let a differential equation f{x, y,p)=0
have a solution f(x, y, c) = 0. The latter is geometrically repre-
sented by a family of curves. The equation of the envelope of
this family (Art. 120) is termed the singular solution of the differ-
ential equation. That the equation of the envelope is a solution
is evident from the definition of an envelope (see Art. 120) and
this fact, viz. that at any point on any one of the curves of the
family the coordinates of the point and the slope of the curve
satisfy the differential equation. The singular solution is obviously
distinct from the general solution and from any particular solution.
For example, the general solution [(8), Art. 223] of Olairaut's equation
is, geometrically, a family of straight lines. The envelope of this family of
lines is the singular solution of (7). The envelope of (8) may be obtained
by the method shovrn in Art. 123. Differentiation of the members of (8)
vfitb respect to c gives „ , j^i/ ^
0 = x+f'{c).
The envelope is the c-eliminant between this equation and (8).
EXAMPLES.
1. Show that the singular solution of Ex. 6, Art. 190, is j/' = 4 ax.
2. Find the singular solutions of the equations in Exs. 7, 8, Art. 223.
2J4, 225.]
DIFFERENTIAL EQi'ATIONS.
401
3. Find the general solution and the singalar solution of :
(l)y=px+p^. (2)p2a; = y. (S) 8u{l + pY = 27(x + y)(.\ -p)'.
Note 1. The singular solution can also be derived directly from the dif-
ferential equation, without finding the general solution ; see reference below.
Note 2. On Singular Solutions see Diff. Eq., Chap. IV., pages 40-49.
225. Orthogonal Trajectories. Associated with a family of curves
(Art. 120), there may be another family whose members intersect
the members of the first family at right angles. An instance is
given in Ex. 1. The members of the one family are said to be
orthogonal trajectories of the other family.
For example, the orthogonal trajectories of a family of concentric circles
are the straight lines passing through the common centre of the circles.
A. To find the orthogonal trajectories of the famiif
/(OS, y, a)=0, (1)
in which a is the arbitrary parameter. Let the differential
equation of this family, which is obtained by the elimination of
o (see Art. 73), be , , ,>
Fio. 143.
Fig. 144.
Let P be any point, through which pass a curve of the family
and an orthogonal trajectory of the family, as shown in Fig. 143.
For the moment, for the sake of distinction, let (x, y) denote the
coordinates of P regarded as a point on the given curve, and let
402 INTEGRAL CALCULUS. [Ch. XXVIL
(X, T) denote the coordinates of P regarded as a point on the
trajectory. At P the slope of the tangent to the curve and the
slope of the tangent to the trajectory are respectively — and -— -.
dx dX
Since these tangents are at right angles to each other,
dy^_dX
dx dr'
Also x= X, and y=Y.
Substitution in (2) gives
<^(^> Y, -ff) = 0- (3)
But P{X, Y) is any point on any trajectory. Accordingly, (3)
or, what is the same equation,
is the differential equation of the orthogonal trajectories of the
curves (1) or (2).
Hence : To find the differential equation of the family of orthog-
onal trajectories of a given family of curves substitute for —
in the differential equation of the given family. ^ ^
EXAMPLES.
1. Find the orthogonal trajectories of the family of circles which pass
through the origin and have their centres on the as-axis.
The equation of these circles is
a;2 + !/2 = 2 ax, (4)
in which a is the arbitrary parameter.
On differentiation and the elimination of a (Art. 73), there is obtained
the differential equation of the family, viz.
2,2_a:2-2a;i/^ = 0. (5)
dx
The substitution of — — for ® gives the differential equation of the
orthogonal curves, viz. ^
y^-x^ + 2xy^ = 0. (6)
(ly
225.]
DIFFERENTIAL EQUATIONS.
r
403
Fio. 145.
Integration of (6) [see Art. 221, Ex. 0] gives
a;2 + 2/2 = cy,
(7)
the orthogonal family, viz. a family of circles passing through the origin and
having their centres on the y-axis. (See Fig. 145.)
2. Obtain the orthogonal trajectories of the circles (7), viz. the circles (4).
3. Derive the equation of the orthogonal trajectories of the family of
lines y = mx.
4. Derive the equation of the family of concentric circles whose centre
is at the origin.
B. To find the orthogonal trajectories of the family
f{r, e, c)=0,
(8)
in which c is the arbitrary parameter. Let the differential equa-
tion of this family, which is obtained by the elimination of c, be
i^(.,e,^;-) = o.
(9)
404 INTEGRAL CALCULUS. [Ch. XXVII.
Let P be any point through which pass a curve of the given
family and an orthogonal trajectory of the family, as shown in Fig.
144. For the moment, for the sake of distinction, let {r, ff) denote
the coordinates of P regarded as a point on the given curve, and
let {R, ©) denote the coordinates of P regarded as a point on the
trajectory. At P (see Art. 63) the tangent to the given curve and
the tangent to the trajectory make with the radius vector angles
whose tangents are respectively r — and R ——■
Since these tangent lines are at right angles to each other,
r^ = L-; whence^ = -ri?^ = -i?^i?®.
dr j^d®' dd dR dR
dR
Accordingly (9) may be written
But P{R, 0) is any point on any trajectory. Accordingly (10),
or the same expression in the usual symbols r and B,
i) = «' (!«')
is the differential equation of the orthogonal trajectories of the
curves (8) or (9).
Hence : To find the differential equation of the family of orthogo-
nal t7-ajectories of a given family of curves, substitute — 7-^ — for —
in tlie differential equation of the given family.
EXAMPLES.
6. Find the orthogonal trajectories of the set of circles r = acose, a
being the parameter.
Differentiation and the elimination of a gives the differential equation of
these circles, viz. j_
— + r tan fl = 0.
de
On substituting, as directed above, there is obtained
r^ = tane,
dr
the differential equation of the orthogonal trajectories. Integration gives
another family of circles r = c sin 9. (11)
Fir, 9, -r^*^^'
225] DIFFERENTIAL EQUATIONS. 405
6. Sketch the families of circles in Ex. 5, and show that the problem
and result in Ex. 5 are practically the same as the problem and result in Ex. 1.
7. Find the orthogonal trajectories of circles (11), viz. the circles in
Ex.5.
N.B. Various geometrical problems requiring differential equations are
given in the following examples.
XoTE 1. On applications of differential equations of the first order, see
Diff. Eq., Chap. V.
8. Find the curves re.spectively orthogonal to each of the following
families of curves (^sketch the curves and their trajectories) : (1) the parabolas
y- = iax; (2) the hyperbolas xy = k- ; (3) the curves a"-^y = x" ; interpret
the cases n = 0, 1, — 1, 2, - 2, ± ^, ± |, respectively; (4) the hypocycloids
jJ + y* = a' ; (5) the parabolas y = ax^ ; (6) the cardioids r = a(l — cos S) ;
(7) the curves r" sin nd = a"; (8) the curves r* zz a" cos n$ ; (9) the lemnis-
cates 7^ = 0^0082 S; (10) the confocal and coaxial parabolas r = ;
(11) the circles x^ + y^ + 2my = a^, in which m is the parameter. ''"
9. (a) Show that the differential equation of the confocal parabolas
y- = 4 a(x + a) is the same as the differential equation of the orthogonal
curves, and interpret the result. (6) Show that the differential equation of
the confocal conies — 1 ^ — = 1 is the same as the differential equation
a^+l b^+l
of the orthogonal curves, and interpret the result.
10. Find the curve such that the product of the lengths of the perpen-
diculars drawn from two fixed points to any tangent is constant.
11. Find the curve such that the product of the lengths of the perpen-
diculars drawn from two fixed points to any normal is constant.
12. Find the curve such that the tangent intercepts on the perpendiculars
to the axis of x at the points (o, 0), (-a, 0), lengths whose product is V.
13. Find the curve such that the product of the lengths of the intercepts
made by any tangent on the coordinate axes, is equal to a constant a^.
14. Find the curve such that the sum of the intercepts made by any
tangent on the coordinate axes is equal to a constant a.
EQUATIONS OF THE SECOND AND HIGHER ORDERS.
Only a very iew classes of these equations will be solved here ;
namely, simple forms of linear equations with constant coefBcients
and homogeneous linear equations. Three special equations of
the second order will also be briefly discussed.
406 INTEGBAL CALCULUS. [Ch. XXVH.
226. Linear Equations. Linear equations are those which are
of the first degree in the dependent variable and its derivatives.
The general type of these equations is
in which P^ P^, ■■•, P„, X, do not involve y or its derivatives.
(For some general properties of these equations see Mun-ay, Integral
Calculus, Art. 113, Diff. Eq., Art. 49.)
A. The linear equation^^+l'i^^^+JP2^^^^+'"+Pnl/=0,(l)
in which the coefficients P^Pj, ■••, P„, are constants.
The substitution of e"" for y in the first member, gives
(m» + Pi?ft"-' + PiW,'-^ -\ h P„)e~
This expression is zero for all values of m that satisfy the
equation ^„ ^ p^^n-i ^ p^^»-2 + . . . + p_, = Q ; (2)
and, accordingly, for each of these values oi m, y = e°" is a solu-
tion of (1). Equation (2) is called the auxiliary equation. Let
mj, mj, •••, m„, be its roots. Substitution will show that y = CiC"!',
y = Cje"!*, .", y = 0^6""", and also
y = ce"^* + c^e'^ + ... + c„e">>% (3)
in which the c's are arbitrary constants, are solutions of (1).
Solution (3) contains n arbitrary constants and, accordingly, is the
general solution.
Note 1. If two roots of (2) are imaginary, say o + tj3 and a — 1/3, t
denoting V— 1; the corresponding solution is
According to Art. 179 this may be put in the form
y = ei"(c,e'?* + de-'^")
= «"{ci(cos j3x + i sin px) + ci(cos px — i sin j3x)},
= «"{(ci + C2) cos px + i'(ci — Cj) sin /3a:},
= e^C^l cos /3a; + JB sin ;3x) ,
in which A and B are arbitrary constants, since ci and cj are arbitrary
constants.
226.] DIFFERENTIAL EQUATIONS. 407
Note 2. If two roots of (2) are equal, say mi and rtii each equal to a, the
corresponding solution, viz.
j/i = Cie"i* + C2e'"2*,
becomes y — (,ci + Oi)e'", i.e. y = ce"",
which does not involve two arbitrary constants. Put m2 = a + h; then the
solution takes the form
y = cie" + 026'"+*'',
On expanding e** in the exponential series (Art. 162, Ex. 7), this equation
becomes
y = e'"(A + Bx + \ c^liV + terms in ascending powers of K), (4)
in which ^ = ci + Ca and B = c^h. On letting h approach zero in (4), the
latter becomes , , „ ,
y = e"{A + Bx).
(The numbers ci and cj can always be chosen so that ci + c^ and Cjft are
finite.)
If a root a of (2) is repeated r times, the corresponding solution is
2/ = (ci + c-iX + cax^ + ••• + CrZ'-'^ye'".
Note 3. On Equation (1), see Diff. Eq., Arts. 50-55.
EXAMPLES.
1. Solve ^-3^+22, = 0.
da;' dx
The auxiliary equation is m' — 3m + 2=0;
its roots are — 2, 1, 1.
Accordingly, the solution is y = Cie-^ + (cj + Cix)^,
2. Solve ^ + a^y = 0.
The auxiliary equation is m^ + a^ = 0;
its roots are ai, — ai.
Accordingly, its solution is y = cie'" + c-^e-'"
= Acosax + B sin ax. (See Ex. 1, Art. 73.)
3. Solve the following differential equations :
(1) i>2y - 4 Dy + 13 y = 0. (2) D'y - 7 I>v + 6 y = 0.
(3) ^_12^-16w = 0. (4) *^-l0^ + 02^-160^ + 136y=r0.
^ ' cb? dx * ^ <lx*^ dx'> dx'' dx
408 INTEGRAL CALCULUS. [Ch. XXVII.
B. The "homogeneous" linear equation
a,»d!Ly_ + p,a,»-if^ + i>2^»-2f^+...+i,„y = 0, (5)
in which Pu p^ •••, Pm ^^^ constants.
First method of sohition. If the independent variable x be
changed to z by means of the relation
z = log X, i.e. X = e*,
the equation will be transformed into an equation with constant
coefiBcients. (For examples, see Art. 92 and Exs. 3 (i), (v), (vi),
page 147.)
4. Show the truth of the statement last made.
6. Solve Exs. 7 below by this method.
Second method of solution. The substitution of a^ for y in the
first member of equation (5) gives
[m(m — l)'"(m — n + 1) +pim(m — !)••• (m — n + 2)H \-p„]3f.
This is zero for all values of m that satisfy the equation
m(m— 1)"-(to— 7i+1)+Pito(??i— l)-"(m— 7i+2)^ \-p„=0. (6)
Let the roots of (6) be wij, m^ •■■, m„; then it can be shown,
as in the case of solution (3) and equation (1), that
y = CiJfi + c,af^ + ••• + c.af^
is the general solution of equation (5).
The forms of this solution, when the auxiliary equation (6)
has repeated roots or imaginary roots, will become apparent on
solving equation (5) by the first method.
EXAMPLES.
6. Show that the solution of (5) corresponding to an r-tuple root m of
(6), is 2/ = i™[ci + C2loga; + C8(logx)2+ ... +Cr(logi)'— 1] ; and show that
the solution of (5) corresponding to two imaginary roots a + ip, a — ijS, of (6) , is
y = x^[ci cos (p log i) + C2 sin (/3 log z)].
226, 227.] DIFFERENTIAL EQUATIONS. 409
7. Solve the following equations :
(1) x^I/h/ -xDy + 2y = 0. (2) x'^D'y - xDy -\- y = 0.
(3) x'^D'y - 3 xDy + iy = 0. (4) xW^y + 2 x'D^y + 2y = 0.
Note 3. Equations of the form
(« + 6^)"§ +i'i(« + ''^)"-'£^ + P2(a + 6x)"-^2^2 + - +Pny = 0
are reducible to the homogeneous linear form, by putting a ■\- hx = z.
8. Show the truth of the last statement.
9. Solve (5 + 2 z)^^, _ C(5 + 2 a;) $^ + 8 y = 0.
ox- dx
Note 4. On Equation (5), see Diff. Eq., Arts. 65, 66, 71.
227. Special equations of the second order.
dht
A. Equations of the form -j^^fiy).
For these equations 2-^ is an integrating factor.
EXAMPLES.
1. ^+a2j/ = 0. (See Ex. 2, Art. 226.)
On using the factor 2^, 2 ^-^ f^ = - 2 a^^.
dx dx dx' dx
On integrating, i^j =— d'y^ + k
= a2(c2 - y'i), on putting a^'c^ for k.
On separating the variables, ' = adx,
and integrating, sm-'- = ax + a.
This result may be written y = c sin (ax + a),
or y — Asinax + B cos ax.
2. Show the equivalence of the last two forms. Express A and B in
terms of c and a, and express c and a in terms of A and B.
3. Show that 2 ^^ is an integrating factor in case A.
ax
4. Solve the following equations :
(l)g = a^.. (2)3=e..
(S^ If ^ = ^, find «, given that -^ = 0 and x = a, when t = 0.
410 INTEGRAL CALCULUS. [Ch. XXVII.
B. Equations of the form f{^^, ||, a;] = 0. (1)
On letting p denote -i, this may be written /( ^, p, x\= 0. (2)
(XtC \(tX J
Integration of (2) may give <^(/), x, c) = 0,
and this may happen to be iutegrable.
EXAMPLES.
6. Find the curve whose radius of curvature is constant and equal to a.
(This example is the converse of Art. 99.)
6. Solve the following equations :
(2) xDhj + i>i/ = 0. (4) (1 + a:)Z)2j/ + i)j/ + a; = 0.
C. Eqnations of the form /(|^j ^, v) =0. (1)
This (see Art. 90) may be written
<^|'^'^) = «- <2)
Integration of (2) may give
F{tp, y, c) = 0,
and this may happen to be integrable.
EXAMPLES.
7. Solve
^, + a-iy = 0. (See Ex.1.)
This is
dp
P^y = -a^y.
1.
Now proceed
as in Ex.
8. Solve the
following equations :
m.S
-O''
W'S-©"-'""-
(3) y'D^y + 1 = 0.
(4) Z)2y+(Z)2/)2 + l=0.
Note 5. For the solution of equations in the form I>^y=f{x), see
Art. 201.
^^7.] DIFFERENTIAL EQUATIONS. 411
Note 6. On forms like A, B, C, see Diff. Eq., Arts. 77, 78, 79, respectively.
Note 7. References for collateral reading. For a brief treatment of
differential equations and for Interesting practical examples, see Lamb, Cal-
culus, Chaps. XI., XII. (pp. 456-640) ; also see F. G. Taylor, Calculus,
Chaps. XXIX.-XXXIV. (pp. 493-564), and Gibson, Calculus, Chap. XX.
(pp. 424-441).
EXAMPLES.
Solve the following equations :
(1) rde = tan 6 dr. (2) (1 + y)dx + x{x + y)dy = 0.
(3) (4y + Sx)dy+(y-2x)dx=0. {i) x^-y=Vifi+^'. (5) ^+ytana: = l.
(6) x~-2y = xWl +a;2. (7) (dx + iy + 5)dx +(ilO y + i x + l)dy = 0.
dy 4x 1
(8) yiydx-x dy) + x Vx^ + y' dy = 0. (9);£+-5— ^y
(^•^^ ^ S + JTT ^ = P^- (11)^-2'^ = "I^- (12) y= = a2(H- ;,^).
dx ' X^+1" (X2 + 1)8
'. (12) y2 = a2(H-;>2).
(1.3) (px- !/)(pj/ + x)= A2p. (14) J52a;3 + a;2j5)/ = I. (15) k = 2 y - 3f>2.
(10) p^ + 2 py cot x = y^. (17) i/Vl+p^ = a; also find the singular solution.
(18) y — px = Vh'^ + a-p^ ; also find the singular solution. (19) xp'' = {x — ay,
and also find the singular solution. (20) y^ - o'y = 0. (21) -t-| + 4 y = 0-
<->^S- (->{S)'=i- <«)-S+l-- <-'(S)=«(2)'
APPENDIX.
NOTE A.
ON HYPERBOLIC FUNCTIONS.
1. This note gives a short account of hyperbolic functions and
their properties. The student will probably meet these functions
in his reading ; for many results in pure and applied mathematics
can be expressed in terms of them, and their values are tabulated
for certain ranges of numbers.* There are close analogies between
the hyperbolic functions and the circular (or trigonometric) func-
tions (a) in their algebraic definitions, (6) in their connection with
certain integrals, (c) in their respective relations to the rectangular
hyperbola and the circle.
2. Names, symbols, and algebraic definitions of the hyperbolic
functions. The hyperbolic functions of a number x are its hyper-
bolic sine, hyperbolic cosine, hyperbolic tangent, •••, hyperbolic
cosecant, and the corresponding six inverse functions. These func-
tions have been respectively denoted by the symbols siuh x, cosh x,
tanh X, cotli X, seek x, cosech x, sinh~^ x, etc. These are the symbols
in common use. As to symbols for the hyperbolic functions, the
following suggestion has been made by Professor George M.
Minchin in Nature, Vol. 65 (April 10, 1902), page 531 : " If the
prefix hy were put to each of the trigonometrical functions, all the
names would be pronounceable and not too long. Thus, hysin x,
hytanx, etc., would at once be pronounceable and indicate the
• See tables of the hyperbolic functions of numbers in Peirce, Short Table
of Integrals (revised edition, 1902), pages 120-12.3 ; Lamb, Calculus, Table
E, page 611 ; Merriman and Woodward, Higher Mathematics, pages 162-168.
41.3
414 ISTEGHAL CALCULUS.
hyperbolic nature of the functions." This notation will be
adopted in this note.*
The direct hyperbolic functions are algebraically defined as follows :
hysin x = £f_ZLf_?, hycos x = ^-— ^^— »
hytanx=»^ = ^l^*^, hycotx = |?i:gg^ = *'" + ^-% -(1)
hycos X e-' + e-" hysmx e" — e-" '
hysec x = ; — = — j hycosec x = , — = — •
hycos X hysin x
There is evidently a close analogy between these definitions
and the definitions and properties of the circular functions. [See
the exponential expressions (or definitions) for sin a; and cos a; in
Art. 153.]
From the definitions for hysin x and hycos x can be deduced, by
means of the expansions for e' and e~^ (see Art. 152, Ex. 7), the
following serits, which are analogous to the series for sinx and
cos a; (Art. 162, Exs. 2, 5) :
hysinx = x + ^ + ^ + ....
' " (2)
hycosx = l+|| + |:^ + ...;
The second members in equations (2) may be regarded as defi-
nitions of hysin x and hycos x.
EXAMPLES.
1. Derive the following relations, both from the exponential defini-
tions of sin z, cos z, hysin a;, hycos a;, and from the expansions of these func-
tions in series : (1) cos x = hycos {ix) ; (2) i sinx = hysin (la;) ; (3) cos (tx)
= hycos X ; (4) sin (ix) = % hysin x.
2. (a) Show that e' = hycos x -I- hysin x, e-' = hycos x — hysin x.
[Compare Art. 179 (1), (2).] (6) Show that hysin 0 = 0, hycosO = l,
hytanO = 0, hysin co = oo, hycosao = x, hylan oo = 1, hysin (—x) =
— hysin x, hycos ( — x) = hycos x, hytan ( - x) = — hy tan x.
* The symbols used in W. B. Smith's Infinitesimal Analysis are hs, he,
ht, hot, hsr, hcsc.
APPENDIX. 415
3. Show that the following relations exist between the hyjierbolic
functions :
(1) hycos2 X - hysin^ 2 = 1; (2) hysec^ x + hytan^ 2 = 1;
(3) hysin (x ±y)= hysin x ■ hycos y ± hycos x • hysin y ;
(4) hycos {x ±y')= hycos j; ■ hy cos 2/ ± hysin x • hysin y ;
(5) hytan (2 ± y) = (hy tan x ± hytan |/) h- (1 ± hytan z . hytan y) ;
(6) hysiu 2 z = 2 hysin x ■ hycos 2 ;
(7) hycos 2 2= hycos^z + liysin22 = 2 hyco822 -1 = 1+2 hysin^z ;
(8) hytan 2 z = 2 hy tan 2 h- (1 + liytan-2 z).
Compare these relations with the corresponding relations between the
circular functions.
4. Show the following: (1) ^ihS^iJUH = hycos x ; (2) <^(hycos 2) ^
dx dx
hysinz; (3) ^(MHI£l=hysec2z ; (4) ^(hycotz)^_ ,^ ^(hysecz^
<i2 di (22
= -hysec2.hytan2; (6) ^OlZ55££l = _ hycscz • hycotz ; (7) ("hysinzdz
(Jz J
= hycosz; (8) j hycos z dz = hysin z ; (9) j hytan z dz = log (hycos 2) ;
(10) J hycotz<22 = log(hysin2) ; (11) Jhysec zdz = 2 tan-'e* ;
(12) J hycsczdz = log/hytan? j. Compare these relations with the cor-
responding relations between the circular functions.
5. Malce graphs of the functions hysin 2, hycos 2, hytan z. (See Lamb,
Calculus, pp. 42, 43.)
V X X
6. Show that the slope of the catenary - = hycos - is hysin — Sketch
this curve.
Inverse hyperbolic fnnctions. The statement "the hyperbolic
sine of y is x" is equivalent to the statement "y is a number
whose hyperbolic sine is x." These statements are expressed in
mathematical shorthand,
hysin y = x, y = hysin-i x. (3)
The last symbol is read " the inverse hyperbolic sine of x," or
"the anti-hyperbolic sine of x." The other inverse hyperbolic
functions are defined and symbolised in a similar manner.
The inverse hyperbolic functions can also be expressed in terms
of logarithmic functions, and thus they may be given logarithmic
definitions. (This might have been expected, for the direct hyper-
bolic functions are defined in terms of exponential functions, and
the logarithm is the inverse of the exponential.)
416 INTEGRAL CALCULUS.
Let hysini/ = a;; then a; = |(e* — e"*).
This equation reduces to e^* — 2 xe" — 1 = 0.
On solving for e", e» = a; + Vaf^ + 1. (4)
(For real values of y, e* being positive, the positive sign of the
radical must be taken.)
From (4) y = hysin-i x = log(x + Vx2 + 1). (5)
N.B. The base of the logarithms in this note is e.
In a similar manner, on putting
X = hycos y = ^ (e* + e"'),
and solving for e*.
^ = x±Vx'-l. (6)
For real values of y, x is greater than 1 and both signs of the
radical can be taken.
From (6) and the fact that (a; + Var* — 1) (x — Va;^ — 1)= 1, and
thus log (a; — Va^ — 1) = — log (a; + Va.-^ — 1), it follows that
y = hycos-i x = ± log(a; + VasS _ i). (7)
In a similar manner it can be shown that
hytai.-ia;=llogrl±^. (8)
where a^ < 1 for real values of hytan~' x ; and that
hycot-ix=llog|±l, (9)
where a^ > 1 for real values of hycot"' x.
EXAMPLES.
7. Derive the relations (7), (8), (9).
8. Solve equations (5), (7), (8), (9), for x in terms of y, and thus
obtain the definitions of the direct hyperbolic functions.
9. Show that the differentials of hysin-i x, hycos"' x, hytan-i x, hycot"' x,
are respectively '^^ , ± '^-^ . -^^ for a;^ <; ], ^_ for a;2 > 1.
Vi^ + 1 Va:-^ - 1 1 - a;^ x^-l
Compare these with the differentials of sin~i x, cos"' x, tan-' x, cot"' x.
APPENDIX. 417
10. Following the method by which relations (6)-(9) have been derived,
show that :
hysin-i ^ = log =^ + ^=^^ + <^^ . hyc«8-i ^ = ± log ^+^^'-«' ;
a a a a
hytan-l - = ^ log ^^^^ for z^ < a^ ; hycot-i - = ^ log ?^±^ for a;' > a^.
a 2 a-x a 2 x - a
11. Assuming the relations in Ex. 10, show that the z-difierentials are :
dfhTsin->gU /^ : d/'bycos-ig^ = ± '^^ ;
V a/ y/x^ + a2 \ a] y/-j? - a?-
d ( by tan-i -] = -^^^ for z^ < a^ ; d (' by cot-> -\ = - _2J?5_ for x^ > a^.
\ a/ a — x^ \ a/ x- — a'^
Compare these differentials with the differentials of sin-' -, cos*' -, tan"' -,
a a a
12. Assuming the relations in Ex. 10 as definitions of the inverse hyper-
bolic functions, derive the definitions of the corresponding direct hyperbolic
functions. (Soggestion. Follow the plan outlined in Ex. 8.)
3. Inverse hyperbolic fanctions defined as integrals. It follows
from Ex. 11. Art. 2, that
f '^^ = hysin-i^ + c ; f ^"^ = ± hycos-' - + c ;
r-^ = lhytan-?+c,(x^<a^; f^ = -ihyoot-^^ +c,
J ar—sr a a J or— a a a
Accordingly, these inverse hyperbolic functions can be ex-
pressed in terms of certain definite integrals, viz. :
f" a^ .=:l0g" + ^"+"=hT8in-l «.
Jo Va52 + o2 a a
p_^_ ^ log «+Vi?HZ = ± hyeos-i «.
Jo Vx2-a2 a a
Jo o,i -^i 2 a a — u a a
r«^^=_ilog£±^ =-lhycot-i«,«2>«2.
418 INTEGRAL CALCULUS.
These relations between definite integrals and inverse hyperbolic
functions may be taken as definitions of the functions.
The inverse circular functions can be defined by integrals which
are very similar to the integrals appearing in the definitions of the
hyperbolic functions. Thus :
f '^^ ==sin-'^. f
X
Va- — x^
dx _l,,^-iM r dx
= ^t&n-^-, -J^::^=-icot-'-.
'o a^+x^ a a J^ a' + 3? a a
EXAMPLES.
1. Find the area of the sector AOP of the hyperbola x'' — y^ = a'^
(Fig. 147), P being the point for which x = u. Thence show, from the
definition above, that hycos"' - is the ratio of twice the sector AOP to the
square whose side is a.
2. Find the area of the sector BOP bounded by the j/-axis, the arc
BP of the hyperbola i/^ — x^ = a^ (the conjugate of the hyperbola in Ex. 1),
and the line OP drawn from the origin to the point P , P" being the point for
which x = u. Then show, from the definition above, that hysin'^ - is
the ratio of tioice the sector BOP to the square whose side is a. "'
3. Sketch the curve y(^a^—x^) = a^. Calculate the area between this curve,
the axes, and the ordinate for which x=u{u''<ia!'). Show that hytanr^ — is the
ratio of this area to the area of the square whose side is a.
4. Sketch the curve y(x^ — a^) = a'. Calculate the area bounded by this
curve, the x-axis, and the ordinate at x = u(u^>a'^). Show that hycot'^ - is
the ratio of this area to the area of the square whose side is a.
4. Geometrical relations and definitions of the hyp^rbolic functions.
In Fig. 146 P is any point (x, y) on a circle 3^ + y^ = a". Let the
area of the sector AOP be denoted by u and the angle AOP by 6.
Then, by plane trigonometry,
^a^e; whence, 0 = ^- (1)
i 1
In Fig. 147 P is any point on a rectangular hyperbola 3?—y^=a''.
(The a of the hyperbola bears no relation whatever to the a of
APPJESDIX.
419
the circle.) Let the area of the sector AOP be denoted by u.
Then
« = area 0PM— area APM = \xy — l ^ .r — d- dx ;
iT + y t
whence, « = -r log
«*i__a- + \'.i-'- a'-* a
= -J log ■
(2)
From (2), log-^^±l = '^; whence, •I±I = e°'.
a a- a
^-f = a\
.r — V
•-= e
(3)
T
^
A
/
Op —
-^1-
-H-^
il X
'
•'^ '
Fig. 146. Fig. 147,
From equations (3), on addition and subtraction,
?^=i(e'' + e'''); y=l(e''-e'''); .". ^ =
a ' a ' X
(4)
• That is, u = J a' hycos-i - ; whence, hycos-' - = — .
a a a-
t If a = 1 , log ( jr + y) = 2 M = twice area A OP. On account of the relation
between natural logarithms {i.e. logaritlims to base e) and the areas of hyper-
bolic sectors, natural logarithms came to be called hyperbolic logaritlims.
The connection between these logarithms and sectors was discovered by
Gregory St. Vincent (1584-1667) in 1647
420
INTEGRAL CALCULUS.
Relations (4) lead to geometrical definitions of the hyperbolic func-
tions. These definitions are given in the following scheme. This
scheme, supplemented by relation (1), also shows the close geo-
metrical analogies existing between the hyperbolic and the circular
functions.
N.B. In Figs. 146, 147 the a and u of the circle are not related
in any way to the a and u of the hyperbola.
In a circle ar' + y' = a" (Fig.
146), if P is any point (a;, y)
and u = area sector AOP,
In a hyperbola a;^ — t/' = a'
(Fig. 147), if P is any point
(x, y) and u = area sector AOP,
then
y ■ 2m
^ = sin — ,
a a^
- = cos -J-,
a a'
V i 2m
^ = tan— ;
X a'
whence,
2m
=^=sin
a"
.->^ = cos-'5
a a
then
^=hyslii2^,
a a'^
a
hycos
2u
= tan-'^.
whence,
2u
X
^ = hy8in-i^ = hyco8-l
= hytan-i K
These results may be expressed in words :
The circular functions may he defined by means of the relations
connecting a point (x, y) on the circle x' + y'^a' and a certain cor-
responding circular sector; and the hyperbolic fnnctions may be
defined by means of the relations connecting a point {x, y) on the
rectangular hyperbola 3? — y^=(^ and a certain corresponding hyper-
bolic sector.
Each of the Inverse circnlar fonctlons may be expressed as the ratio
of tivice the area of a certain sector of a circle of radius a to the
square described on the radius of the circle, and each of the Inverse
hyperbolic fnnctions may be expressed as the ratio of twice the area of
a certain sector of a rectangular hyperbola of semi-axis a to the square
described on this .temi-axis.
(For a more general notion see Ex. 3 following.)
The term hyperbolic arose out of the connection of these func-
tions with the hyperbola.
APPENDIX. 421
EXAMPLES.
and thence show that
1. Show that hysin-i J = hycos-i | = liytan"' ^. Represent each of these
functions geometrically. Compute hysin-' |. \_Ans. 1.099.]
2. Show that hysin*' J = hycos*^ J = hytan"' |. Represent each of these
functions geometrically. Compute hysin"' |. \^Ans. .693.]
3. Show that, if AP (Fig. 146) is an arc of an ellipse hH'^ + a^j/S = a'^ifta,
and u denote the area of the elliptic sector AOP, it is possible to write
- = cos — , "- = sm — •
a ab b ab
Also show that, if .4P (Fig. 147) is an arc of a hyperbola -; — r^ = !> and
u denote the area of the hyperbolic sector AOP, then
? = hycos?«, y = hysin^-^.
a ab b ab
(Williamson, Integral Calculus, Arts. 130, 130 a.)
X- tfi
4. Show that a point P{x, y) on the ellipse -j + rj = 1 •" E^- 3 may be
represented as (acosfl, 6 sin 9), and show that e(= eccentric angle of P)
= (2 area sector AOP -h ab).
Show that a point P(x, y) on the hyperbola -^^ — ^ = 1 in Ex. 3 may be
represented as (ohycosr, fthysint)), and show that c =(2 area sector
AOP-i-ab).
5. The Gudennannian. Suppose that
sec <l> + tan <t> = hycos v + hysin v. (1)
From (1) and the identities sec^ <^ — tan^ <^ = 1, hycos'v —
hysin^v = l, it follows that
sec<^ = hycos'y, (2) tan <^ = hysin v. (3)
Since [see Art. 2, Ex. 2 (a)] log (hycos v + hysin v) = v, relation
(1) may be written ^ ^ j^^ ^^^^ + + tan +) ; (4)
that is, by trigonometry,
V = log tan (j + J) = 2.302585 log,„ tan (^^ + f\- (5)
422 INTEGRAL CALCULUS.
Wften any one of the relations (l)-(5) holds between two numbers
V and <^, <^ is said to be the Gudermannian of v.* This is expressed
by this notation : . j /c\
•' + = ga V. {p)
In accordance with the usual style of inverse notation each of
the relations (4), (6), (6) is expressed
V = gd-i +. (7)
The second members of (4) and (5) are more frequently denoted
by the symbol X(<t>), which is read " lambda </>," than by gd'^ <^.
Geometrical representation of A.(<^) or gdr^ <^. If at P{x, y) in
Fig. 147, a; = a sec <i>, then y = a tan <^, since x' — y'= a\ On mak-
ing this substitution for x and y, it can be deduced that
area sector AOP = ^ a^ log (sec </> + tan <^). (8)
From this,
log(sec</, + tan,^),».e. XW (or ^d"^ <^) = ?^^5£t^^0P, ^g^
(X
From (4), (6), (8), i>.= ^^^^ • sector^ OPX ^^^^
If the area of sector AOP be denoted by u, relations (9) and
(10) may be expressed
a^ a'
To constriwt an angle whose radian measure is <^. In Fig. 147,
about 0 as a centre with a radius a describe a circle. From M
draw a tangent to this circle, and let the point of contact be at P*
in the first quadrant ; and draw OP'. Now 0M= OP' sec MOP ;
i.e. X = a sec MOP. But, according to the hypothesis in the last
paragraph, a; = a sec <t>. Hence, angle MOP' = 4>.
If ii point P{x, y) on the hyperbola x^ — y^ = a^ (see Ex. 4, Art. 4) be
denoted as (aseo0, atan^), ij> is the angle which has just now been con-
structed.
• This name was given by the great English mathematician Arthur Cayley
(1821-1895) "in honour of the German mathematician Gudermann (1798-
1852), to whom the introduotinn of the hyperbolic functions into modern
analytical practice is largely due." (Chrystal, Algebra, Vol. II., page 288.)
APPENDIX. 423
EXAMPLES.
1. Denve result (8).
2. (a) Show that, 0 and v being as in equations (l)-(7),
gdv = sec-i (hycos v) = tan-i (hysin v) = cos-' (hyseo v) = sin-^ (hytan v)
= cot"! (hycosec r) = cosec-i (hy cot v) ; hytan | « = tan | <|>.
(6) Show that grd-i,^=hycos-i(seo0)=hysin-i(tan^); pda;=2tan-ie^- -•
3. (a) Show that the derivative of X(0)(i.e. gd-^ip') is sec0. (6) Show
that X(-<^) = -X(0). [Suggestion. Show that X(- i^)+ X(0)= logl.]
(c) Sketch the graph of X(i^).
4. Show that | hysec udu = gdu; j sec u du = gd-^ u.
Note. Keferences for collateral reading on ftyperftoZic/anction*. Gib-
son, Calculus, §S 60, 111, 110 ; Lauib, Calculus, Arts. 19, 23, 40, 44, 72, 98,
Exs. 2, 3 ; F. G. Taylor, Calculus, Arts. 62-80, 439 ; W. B. Smith, Infinitesi-
mal Analysis, VoL I., Arts. 99-113 ; McMahon and Snyder, Diff. Cal,
pp. 320-325. For further information see Chrystal, Algebra (ed. 1889),
Vol. II., Chap. XXIX., §§ 24-31 (pages 276-291) ; the notes on pages 288,
289 contain interesting information about the history and literature of the
subject. Also see Hobson, Treatise on Plane Trigonometry, Chap. XVI.
An excellent account of hyperbolic functions, starting from the geometrical
standpoint and showing practical applications, is given in McMahon, Hyper-
bolic Functions {i.e. Merriman and Woodward, Higher Mathematics, Chap.
IV., pages 107-168).
NOTE B.
INTRINSIC EQUATION OF A CURVE.
1. The intrinsic equation of a curve. Usually the equation of a
curve involves either the Cartesian coordinates x and y or the
polar coordinates r and 6. In some cases the intrinsic equation
is especially useful. In the intrinsic equation of a curve the
coordinates chosen for any point P are (a) the distance of P from
a chosen fixed point on the curve, this distance being measured
along the curve, and (h) the angle made by the tangent at P with a
chosen fixed tangent of the curve. These coordinates are denoted
respectively by s and <j>. The relation connecting them, f{s, <t>)=0
say, is called the intrinsic equation of the curve. The term
intrinsic is used because the coordinates s and 4> are independent
of all points or lines of reference other than the points and
tangents of the curve itself.
424
INTEG RAL CA L CUL US.
EXAMPLES.
1. Derive the intrinsic equation of a straight line. Let AB be any
straight line. Let O be the chosen
I I I I fixed point, and P(«, 0) be any point
on the line. It is required to find the
equation which is satisfied by s and <t>.
The direction of the line at P is the same as the direction at 0 ; hence the
intrinsic equation is ^ = 0.
2. Derive the intrinsic equa- P(8,W ■I'y^
tion of a circle of radius o.
Talce (Fig. 107) O for the fixed
point, and the tangent at 0 for
the tangent of reference. Let
P(s, <p) be any point on the
circle. Then s = arc OP and
<t> = angle TBP. Now arc OP
= a ■ angle <p ; i.e. s = a<p.
Fig. 148.
2. Derivation of the intrinsic equation of a curve. The intrinsic
equation of a curve is usually derived from its equation in
Cartesian coordinates or from its
equation in polar coordinates. The
general method of doing this will
now be shown.
Let the equation of the curve be
f{x,y) = 0. (1)
Take Q for the iixed point, and
the tangent at 0 for the tangent of
reference. Take any point P on the
curve; let its Cartesian coordinates
be X, y, and its intrinsic coordinates be s, <^.
Express s in terms oi x, y; suppose that
s=Mx,y). (2)
Also express <^ in terms oi x, y; suppose that
<i> =/2(p, y)- (3)
The elimination of x and y between equations (1), (2), (3), will
give the required equation between s and </>.
Fig. 149.
APPENDIX. 425
Similarly, let the polar coordinates of P be r and 0, and let
the polar equation of the curve be
F{r,e) = 0. (4)
Express s in terms of r, 0; suppose that
s = F,{r,6). (6)
Also express <^ in terms of r, 6 ; suppose that
^ = F,(r,e). (6)
The elimination of r and 6 between equations (4), (5), (6), will
give the required equation between s and 4>.
Note. A tangent parallel to the z-axis is usually chosen for the tangent
of reference.
EXAMPLES.
1. Derive the intrinsic equation of the hypocycloid
x^ + y^ = ai (1)
Take the cusp on the positive part of the i-axis for the fixed point, and
the tangent there for the tangent of reference. Then at any point P(x, y)
on the arc in the first quadrant
Un<^=-(?/*-=-a;^), (2)
and s = iJ(J-x^). (3)
From (1) and (2), sec^ <j, = tan^ tf, + 1 = a^ -i- x^.
Substitution for z^ in (3) gives 2 « = 3 a siu^ <f,.
2. If in Ex. 1 the chosen fixed point O be at a distance 6 along the
curve from the cusp and the chosen fixed tangent (not necessarily at O)
make an angle u with the tangent at the cusp, show that the intrinsic
equation of the hypocycloid is
2 (s + 6) = 3 a sin" (^ + o).
3. Find the intrinsic equation of the caidioid r = o(l — cos 8).
Let the polar origin be chosen for the fixed point, and the tangent there
be chosen for the tangent of reference. Let P(x, y) be any point on the
cardioid. Then s= J^*-y|r2 + /^Vdfl = 4 afl - cos|y (1)
426 INTEGRAL CALCULUS.
Also, (Art. 63) , ^ = 0 + ten-i ^=e + tan-i f tan -"^ = 1 9. (2)
dr \ 2/
On substituting in (1) the value of S from (2),
s = 4af 1 - cos* )•
4. If in Ex. 3 the chosen fixed point be at a distance 6 from the polar
origin and the chosen tangent of reference make an angle a with the tan-
gent at the polar origin, show that the intrinsic equation of the cardioid is
6. Derive the intrinsic equation of each of the following curves, the
fixed point and the fixed tangent being as indicated: (1) the catenary
X X
y = - (e« + e"»), the vertex and tangent thereat ; (2) the parabola y^ = i ax,
8
the vertex and tangent thereat ; (.S) the parabola r = a sec^ -, as in (2) ;
(4) the cycloid x = a(ff - sin ff), y = a(l— cose), with reference to (o)
the origin and tangent thereat, (ft) the vertex and tangent thereat ; (5) the
logarithmic spiral r = ce"* ; (6) the semi-cubical parabola 3 ay''' = 2 x', the
origin and tangent thereat; (7) the curve y = alogsec-, the origin;
a
(8) the semi-cubical parabola y^ = ax'^ ; (9) the tractrix x = Vc' — y'' +
clog^t — " — y ^ tjjg point (0, c). (For an account of the tractrix and
y
for various problems which reveal its properties, see the text-books of
Williamson, Byerly, Lamb, and F. G. Taylor, on the calculus.)
[Answers: Ex. 5. (1) s = a tan 0, (2) s = a tan ^ sec^ + a log tan
(i + -], (3) as in (2), (4) (a) s = 4 a(l - cos0), (6) s = 4asln0,
\2 4/ , ,
(5) s = c(e«*-l), (6) 9s = 4a(sec8 0- 1), (7) s = alogtan (5^- 1),
(8) 27s=8n(sec8 0 - 1), (9) s = clogsec^.] ^^ *'
3. Radius of curvature derived from the intrinsic equation. The
radius of curvature at a point on a curve can very easily be
deduced from the intrinsic equation. For, according to Arts. 98,
99, the radius of curvature being denoted by B,
R = ^.
APPENDIX. 427
EXAMPLES.
1. In Art. 2, Ex. 5 (1), E = a sec^ <t>.
2. Find the radius of curvature for each of the curves in Art. 2, Ex. 1,
Ex.3, Ex. 5 (4), (6), (6), (9).
{Answers : Ex. 1. f a sin 2 <^ ; Ex. 3. | a sin 5^ ; Ex. 5 (4). (a) 4 a sin (p,
(6) 4 a cos <^ ; (5) a ce'^ ; (6) f a sec' ^ tan 0 ; (9) c tan 0.]
Note. On the intrinsic equation of a ciirve, see Todhunter, Integral
Calculus, Arts. 103-119; Byerly, Integral Calculus, Arts. 114-123.
NOTE C.
LENGTH OF A CURVE IN SPACE.
(This note is supplementary to Arts. 209, 210.)
The lengths of plane curves have been derived in Arts. 209,
210. ' The principle used there is that the length of an arc is the
limit of the sum of the lengths of infinitesimal chords inscribed
in the arc. The same principle is employed in finding the lengths
of curves in space.
Thus in Fig. 93 or Fig. 95,
1 f >i f d R _ ( ^'"^it of the sum of chords PQ, inscribed from
{AioB, when the chords approach zero.
Now length of chord PQ = V{Axf + (^yy + {i^zf (1)
-4
^-(HT^diJ- <^'
Hence, by tlie definitions in Arts. 22^ 166,
xat J
length of arc ^^=/Vl+(|)^ + (|J^- (3)
Similarly there can be derived from (1),
length Of arc ^B=/Vl + (g)V(fJ<^.
ya\A
2 at 5
-H
^-(ITHIT- <'>
428
INTEGRAL CALCULUS.
If the coordinates {x, y, z) are expressed in terms of a third
variable, t say {e.g. see Arts. 158, 159), relations (1), (2) can be
expressed thus :
length of chord
PQ=^4f;) ^m +(^\ ^i;
At
At
(5)
whence, length of arc AB= Cyjf'^Y+f^'+f^\dt. (6)
EXAMPLES.
1. Find the length of the helix, a curve traced on a right circular cylinder
80 as to cut all the generating lines (elements) of the cylinder at the same
angle.
The equations of the helix, as derived below, are
a; = a cos 9, y = asin9, 2 = afltana, (1)
in which a is the radius of the right circular cylinder x^ + y^ = a', and a is
the angle at which the helix cuts the elements of the cylinder.
Equations (1) may be written
X = a cos e, y = asm 0, z = c9, (2)
in which c ~ a tan u.
z]
Fig. 151.
In Fig. 150 P{x, y, z) is any point on the helix.
APPENDIX. 429
Fig. 151 shows the cylindrical surface AGB "unwound" and laid out as
a plane surface. AtP:
X— On = a cos 6,
y = nm = a sin 5,
z = Pm = Am tan a (Figs. 150, 151),
= ad tan a.
Tlie length of the arc APB (Fig. 150)= length of the straight line APB
(Fig. 151) = .^mC sec a = jra sec a.
Accordingly, the length of the arc which encircles the cylinder = 2 ira sec a.
This length s will now also be derived by the calculus method shown in this
article.
From equations (1) on differentiation,
— = -asin«, ^=acose, — = atana.
de dd de
= 21" y/a'- siif^ 8 + a- cos^ 0 + a* tan^ a d6
Jo
= 2 a Vl + tan^ a. ( " dS = 2 ira sec «.
Jo
Thus, if a = 10 inches and a = 30=', the length of an arc encircling the
cylinder is 72.6 inches.
2. Show that the length of the arc of the helix in Ex. 1 from B-Bi to
fi = flj is 2 a{92 — 9i) sec a. Hence find that the length of the arc on a cylin-
der of radius 4 inches from S = 25° to S = 75° when « = 35° is 8.6 inches.
3. Show that the equations (2) of the helix in Ex. 1 can be transformed
into a;2 + y^ = (fi, y = a: tan -•
c
Then, using these equations, find the length of the arc encircling the
cylinder.
4. Show that the equations (2) of the helix in Ex. 1 can be transformed
into x = a cos - , y = a sin - •
c c
Then, using these equations, show that the length of the arc measured
from the point where z = 0 to the point where z = zi is
' c-
430 INTEGRAL CALCULUS.
6. Show that the length of the arc of the curve
I = 2 o cos t, y = 2asmt, z = bfi,
from the point at which « = 0 to the point at which t = ti \s
2 26 a
Sketch the curve.
6. Show that the length of the arc from the point on the xy-plane to the
point where z = ii on the curve
r2 ?/2
Make a figure showing this curve.
7. Sliow that the length of an arc of the curve
X = 4a cos' 6, y = ia sin" $, z = 3c cos 2 6,
from the point at which e = a to the point at which B = p is
3 Va^ + c-(cos 2 a - cos 2 (3) .
2 2 1
Show that this is a curve encircling the cylindrical surface x^ + y^ =(4 a) \
Make a figure with a sketch of the curve, and show that its length is 24 Va^ + dK
QUESTIONS AND EXERCISES FOR PRACTICE
AND REVIEW.
o»:o
A large number of examples are contained in several works on
calculus, in particular in those of Todhunter, Williamson, Lamb,
Gibson, F. G. Taylor, and Echols. Special mention may also be
made of Byerly's Problems in Differential Calculus (Ginn & Co.).
Exercises of a practical and technical character, which are con-
cerned with mechanics, electricity, physics, and chemisti-y, will
be found in Perry, Calculus for Engineers (E. Arnold) ; Young
and Linebarger, Eleinents of the Differential and Integral Calculus
(D. Appleton & Co.) ; Mellor, Higher Mathematics for Students of
Chemistry and Physics (Longmans, Green & Co.). Many of the
following examples have been taken from the examination papers
of various colleges and universities.
CHAPTERS II., III., IV.
1. Explain what is meant by a continuous function.
2. Explain what is meant by a discontinuous function. Give examples.
3. (1) Given that /(x) = a;2 + 2 and i^(x)= 4 +a/x, calculate /{F(r)}
and nm}. (2) K /(x) = |^, show that J^lg^^ = ff±. (3) If
y =f(x)= + ^^" and z=f{y), calculate z as a function of x. (4) If
4 — 7 I
y = 0(z) ■= . ~ , show that x = <t>(,y), and show that x = <t>^(x), in which
02(a;) is used to denote (^{^(x)}, not {<t>(.x)f. (5) If /(^) = ^3T> show
tha.tp(x) = x,f\x) = f(x),f*(x)=x. (6) If j^=/(x)=2^^,showthat
X = f(y). (7) If f(x, y) = ax^ + bxy + cy\ write f{y, i) , /(x, x) , and f{y, y).
4. Define the differential coefficient of a function of x with regard to x.
State what is the interpretation of the differential coefficient being po.sitive
or negative.
431
432 DIFFERENTIAL CALCULUS.
6. Give a geometrical interpretation of -" when x and y are connected
by the relation /(a;, y)=0 or y = ^(x).
6. Show that the derivative of a function with respect to the variable
measures the rate of increase of the function as compared with the rate of
increase of the variable.
7. Find geometrically the differential coefficients of cos x and sin x.
8. Deduce from first principles the first derivatives of x", sin x, tan x,
tan-1 x, logo X, a', a'"**, log sin —
9. Find the derivatives of - and uv, with respect to x, where u and v
are functions of x.
10. Investigate a method of finding the derivative with respect to x of a
function of the form {/(x) }*('', and apply it to differentiate x*^i+*^.
r2i»
__ log_(cosxl ^„ ^^^, 1 6 + acosx^
(1 + X2)n j; O+ftcOSX
e""-' ", tan- 1 e', x»e" sin« x, log /^asinlogxX
12. Show that (1) i)sin-i-vP^=i)cos-i-vl^^^: (2) Z)sin-i^^-±-^
+ Jsin-i^(°-^-^')(^-»')=0.
a + 6x
18. If x'2^' + cos X — sin X tan y — sin y = 0, show that
dy _ ( — 2 xy^ + sin x) cos' y + cos x sin y cos y
dx 3 x'y- cos' y — sin x — cos^ y
14. Differentiate: (1) ^ ^'""' '^ + log VHTp ; (2) tan-' ^''' " °' "" =';
Vl — x' a + 6 cos X
(3) cos-i*+A525^; (4) 3in-i& + asinx „. ^^^^.i Va' - 6' sinx
a + 6 cos X a + 6 sin X 6 + a cos x
(6) Vmsin'x+ncos'x; (7) (2 a* + x*) ^a* + x* ; (8) ("nnx)".
/- /- (cos mx)»
(9) (cosx)""; (10) ta.n-^ Yl+^^i±XL^.
VI + x' - Vl - X2
("Answers to Ex. 14 : (1) ^'""'^ ; (2) ^^' " °' ; (3) ^«' " ''' ;
CI— x'")^ 6 + ocosx o + ficosx
W.4l5?= w^S^; (6)k™-n) «™2x .
a + 6smx a + 6cosx 2 Vm sin' x + n cos^ x
f7) 4v/^ + 3-v/x . ,g- mn (sin nx)"-' cos (mx - ivt) . .qn (-cosx>1"»-»
. ^V"* *' (cosmx)»+i > \ 1 \ J
(cos' X log cos X — sin' x); (lO) —^ "1
QUESTIONS AND EXERCISES. 433
CHAPTER V.
1. If the equation of aplane curve be y = 0(a;), find the equations of the
tangent and the normal at any point, and find the lengths of the tangent,
normal, subtangent, and subnormal.
2. Deduce the equation of the tangent at the point (z, y) on the curve
y =/(«), vfhen the curve is given by the equations x = it>{t), y = ^(0-
X
Prove that - + ^ = 1 touches y = be ' a,t the point where the latter crosses
the y-axis. "
3. Find an equation for the normal at any point on the curve whose
equation is f(x, y) = 0.
4. At what angle do the hyperbolas 3? — y^ = a^ and xy = b intersect ?
Draw sets of these curves, assigning various values to a and 6.
5. Find the angle of intersection between the parabolas y'^ = iax and
x^ = i ay.
6. Find an expression for the angle between the tangent at any point of
a, curve and the radius vector to that point. Show that in the cardioid
r = a(l + cos 0) this angle is — H —
7. Determine the lengths of the tangent, normal, subtangent, and sub-
normal, respectively, at any point of each of the following curves : (1) the
X X
hyperbola b'h:^ - aV = a'^b^ ; (2) the catenary y = - (e° + e~°) ; (3) the
parabola t/2 = 9 a;. r^^,. (1) -1 Via'' - a;2)(a» - e^z^), \ Va* - e'x",
ax a^
^'-"N ^; (2) y' , yi, -^y—, y-y/f^^^-, (3) io,7i,8,4j.]
X a^ ^yi _ a^ a y/yi _ gi a
8. Show that all the points of the curve y'^ = ^ a(x ■'r a sin -J at which
the tangent is parallel to the axis of x lie on a certain parabola.
9. (1) In the curve r=a sin' |, show that <t> = i^. (2) In the lemnis-
cate r2 = a' sin 2 d, show that f = 2 fl, ^ = 3 «, subtangent = a tan^ 0 Vsin 2 S.
10. Solve the following equations : (i) 4 a^ + 48 a;^ + 165 a; + 176 = 0 ;
(ii) 9a;* + 6z'-92a;2 + 1041-32 = 0; (iii) 16 «« + 104 z* + 73 z' - 277 «=
- 161 z + 245 = 0.
11. Show that the condition that az» + 3 ftz' + 3 cz + d = 0 may have
two roots equal is (6c — ady = 4 (ac — 6') (fed— c^).
434 DIFFERENTIAL CALCULUS.
12. Prove, geometrically or otherwise, that provided f(x) satisfies a
certain condition which is to be staled
f(x + h) -fix) = hf'(,x + eh),
where d is a proper fraction. Show that it is possible that in this relation 6
may have more values than one.
13. If A is the area between the graph of fix), the z-axis, a fixed ordi-
dA
nate, and the variable ordinate f(x), show that — =/(*)•
dx
CHAPTEE VI.
1. Find the nth derivative of the product of two functions of z in terms
of the derivatives of the separate functions.
2. Find the fourth derivative of x^ cos' x and the nth derivatives of
1 X"^
(i) X? cos ax ; (ii) x* cos* x ; (iii) tan-' - ; (iv) sin' x cos' x ; (v) — — - ;
(vi) e" sin 6a;. " ^ ~
3. Show that
(i) Z)"(^U(-l)-»"(" + ^) - ("+"'-l)«; (ii)i,.(x-ilog2)=^^^:ilI';
\x»/ X"+" X
(iii) i)-/Lij'\-2(-l)''«! (i^N j9Ve.iDi) = _e"nicosa[;sina;(sinx + 3).
4. If 2 = a(l - cos 0, y = a{nt + sin t), then ^ = - IL£211±1.
dx^ a sin' t
6. Derive the following : (i) Ii e>+xy—e=0, D' y = y • (^ - y)e'' + 2x
(ii) If I* + y' + 4 ah:y = 0, (yS + a=a;)'|^ = 2 a^yix^y^ + .3 a«). (iii) If
aa;= + 2Aiy + 6y2 + 2s2+2/!/ + c = 0, g^ = °ftc + 2/.^^ - «/^- ftg' - c/,'_
(tc'^ (hx + by +f)^
6. Prove the following: (i) If y = sin (m tan-' a;), (l + x^)'' ^ +
2x(l + a;2)^+m2y = 0. (ii) If y = (x +\/a;2 - 1)", (x^ - 1)^+ x^?^ -
(ix „ dx2 dx
n^=Q. (iii) Ify2=sec2i, y+|-|=3s/s. (iv) If y=(l+x2)'^sin (mtan-'x),
(1 + x2) ^ - 2(7» - l)x^ + m(m - \)y = 0.
7. If ae* + be-y + ce* — «-* = 0, determine a relation connecting the first,
second, and third derivatives of y.
CHAPTER VII.
1. Write a note on the turning values of functions of one variable.
2. Assuming/(x) and its derivatives to be continuous functions, investigate
the conditions that /(a) should be a maximum or a minimum value of /(x).
QUESTIONS AND EXERCISES. 435
3. Show how you would proceed to find the maximum and minimum
values o£ a single variable, and to discriminate between them.
4. If f(x) have a maximum or minimum value when x = a, and f(x) be
continuous at x = a, prove that /'(x) must vanish when r = a. Show by
means of a diaaram that the converse is not necessarily true. Examine the
case in which /(a;) has a maximum or minimum value when x = a, and/'(i)
is discontinuous when x = a.
5. If i' + 3 x-y + 4 y' = 1, show that v^J is the maximum and that J is
the minimum value of y, where x can have all possible values.
6. ABCD is a rectangular ploughed field. A person wishes to go from
.4 to C in the shortest possible time. He may wallc across the field, or take
the path along ABC ; but his rate of walking on the path is double his rate of
walking on the field. Show that he should make through the field for a point
on BC distant 6 — ^ from C, a and 6 being the length of AB and BC
respectively. "^"^
7. Prove that the greatest distance of the tangent to the cardioid
r = o(l + cos B) from the middle point of its axis is aV2.
8. AB is a fixed diameter of a circle of radius a and PQ is a chord per-
pendicular to AB ; find the maximum value of the difference between the two
triangles APQ, BPQ for different positions of the chord PQ.
9. Show that the point on the curve iay = z', which is nearest the point
(a, 2 a), is the point (2 a, a).
10. Show that the minimum value at which a normal chord of the ellipse
ab
?- + 2. = 1 recuts the curve is tan"'
a2 62 a^- b^
11. Prove that the greatest value of the area of the triangle subtended at
the centre of a circle by a chord, is half the square on the radius of the circle.
12. A slip noose in a rope is thrown around a square post and the rope is
drawn tight by a person standing directly before the vertical middle line of
one side of the post. Show that the rope leaves the post at the angle 30°.
13. Show that the maximum and minimum values of integral algebraic
functions occur alternately.
14. (i) Show that the points of inflexion on a cubical parabola y'' =
(x - a)\x - b) lie on a line 3 x + a = 4 6. (ii) Show that the curve
y(x2 + a^) = a^{a - x) has three points of inflexion on a straight line,
(iii) Show that the curve x' -axy + b^ = 0 has a minimum ordinate at
X = —^ , and a point of inflexion at (— 6, 0).
-^2
436 DIFFERENTIAL CALCULUS.
15. Find where the following curves have maximum or minimum ordi-
nates and points of inflexion respectively : (i) y = x< — 4 x' — 2 x'^ + 12 a; + 4 ;
(ii)y = xe»; (iii) 2; = xe-' ; (iv) a/ = xe-»". yAns. (i)x=-l, 1, 3,
l±lV3; (ii)x = -2; (iii) x = 1, x = 2 ; (iv)x = ±^, x = 0, x = ± Vf. I
v2 -■
16. Find the inflexional tangent of the curve 2/ = x — x^ + x'. \Aiis. 27 y
= 18x + l.]
17. Show that : (i) The cone of maximum volume for a given slant side
has its semi-vertical angle = tau-i v^; (ii) The cone of maximum volume
for a given total surface has its semi-vertical angle = sin-i ^.
18. Show the march of each of the following functions: (i) sin^xcosx;
(ii) sin 2 X - X ; (iii) x(a + xy^a - x)'.
19. Examine the following functions for maxima and minima :
^(xi-il „)X-^ + 2x+ll (iii) l-x + x\ (iv)l + x-fx^
^''x*-xMJ_ ^ '^ X2-I-4X + 10' ^ M+x-x'''' M-x + x^'
(v; X Vox - x2 ; (vi) (x-l)*(a; + 2)3; (vii) (1 + x)^ - (x - x^) ;
(viii) secx-x; (ix) sin x(l + cos x) ; (x) asinx + 6cosx; (xi) x' ;
(xii) — ^. Ans. (i) Two max., each = \; two min., each =-\;
(ii) max. = 2, min. = f ; (iii) min. =|; (iv) max. =3, min. = J; (v) min.
_3V3^2. (vi) min. = 0, max. = 124 . 93 H- 7' ; (vii) max. = 0, min. = 8 ;
10
(viii) sin x = ^-; (ix) max. = 1.299; (x) max. = \/a°+ 6', min. =
— Va-' + 6* ; (xi) min. for x = - ; (xii) min. = e.
e J
CHAPTERS VIII., IX.
1. What is meant by partial differentiation ?
2. State precisely the restrictions as to the function /(x, y) so that the
theorem " ^ = "■ may hold, and prove the theorem.
dxdy dydx j_ ^
Show that if /(x, y) =xy^ — ='-, the theorem does not hold forx=0, y=0,
and explain why. x + y
3. Explain the meaning of a partial derivative. In what sense may we
logically speak of the partial derivative of c with respect to a, when c is a
function of a and b, and a and 6 are both functions of x ?
4. Prove Euler's theorem for a homogeneous function 0 of x, y, 2 :
xS± + y^ + z^=n4,.
dx dy dz
QUESTIONS AND EXERCISES. 437
6. If u be a homogeneous function of the nth degree in any number of
variables x, y, z, ■■■, then x^ + y^ + z^+ - = nu.
ox. dy dz
6. Verify that J^ (5-) = T" (ff) '° ^^ <=ase of each of the foUowing
functions: sin (s^j/), cus /-l^). 'og [^' + y\ ^/^V
7. Verify the followmg : (i) If u = sin-i ? + tan-i t^ x^ +y^=Q
y X dx dy
(ii) Ifv=(4a6-cT*,f^ = T^- (iii)If«=2»tan-i2^-ynan-"?,-i!^
dc' dadb x y Qx dy
= ^?TT/ ('^) " y=f{y + aa;) +i>{y- ax), in general ^, = a^ ^.
X + y Qx" Qy^
(V) If u = log ^ + 2 tan-i ?, du = -i-^ (y dx-i dj;). (vi) If u=tan-" ^,
°^""=ijf?' S + S + S = °- (^"^ If « = Bin(,^ + ^x + .,),
^^^nbSgS+^(^+^+^)"=°- (^"0 If «=.^^.
e^a "dxdy dy' 4
8. Verify the followmg: (i) If (s 0^ + 2] (^\^ - ( a^A- l\^ ^
\ dx.^ )W) \ dx^ Jdxdx^'
©)"=(|-)&'- « "<'-«(g-')-(l)'-<->|g
= W«L=-5\. (V) If ^-sec»oosec»^ + yn»tan»« = 0andz=logsec9,
„ \ 0 / off* a8
rt2w
g+„^ = 0
CHAPTER X.
1. Define curvature of a curve. Find an expression for the radios of
curvature of a curve whose equation is in the form y =/(i).
2. Show that the curvature at any point of the curve given by x = 0(«),
y = ^ (t) is ^Jl — ~^ '^ , where accents denote differentiations with respect
to t. (<p'^ + vl'")*
3. roranycurve/(r, ^)=0 show that radius of curvature = — r^
in which ^=tan-i?:^. ^^'I'V+D
438 DIFFERENTIAL CALCULUS.
4. Find the coordinates of the point on the parabola x^ = iay for which
the radiiis of curvature is equal to the latus rectum.
6. Show that at a point of undulation the tangent has contact of at least
the third order.
6. Show that the circle (4 a; -3 a)^ + (4 r/ - 3 a)2= 8 a^ and the parabola
\/x+Vy=-\/a have contact of the third order at the point (-, -). Find
the order of contact of the curves y — x^ and y=3x^ — 3x + l.
7. Show that the circles of curvature of the parabola y^ = 4ax for the ends
of the latus rectum have for their equations x^+y^ — 10 ax ±i ay — 3 a^=:0,
and that they cut the curve again in the points (9 a, T 6 a).
8. Find the radius of curvature of each of the following curves :
(i) The cardioid r^ = a^ cos J e. (ii) y = 2x + 3x'^-2xy + y^ at (0, 0).
(iii) xy^ = a''(a + x) at ( — a, 0). (iv) The tractrix x = a log cot — a cos $,
y = asiB8. {\) y = x — sin x at the origin, and where a = - • (vi) The expo-
z 2
nential curve y = ae°. (vii) r" = a" cos ni9. (viii) r = osinn9 at (0,0).
(ix)r^=a'oos3 9. lAns. (i) f-i/or. (ii) J\/5. (iii) i a. (iv)— ocotS.
(v)0.2V2. (vi)i^. (Vii) ^^;^^:^,. (viii)i«a. (ix)^.]
CHAPTER XIII.
1. Define an asymptote to a curve. Derive a method of finding the
asymptotes of an algebraic curve whose equation in Cartesian coordinates is
of the nth degree.
2. Show that the asymptotes of the cubic x^y — xy^ + y'^+xy + x — y = 0
cut the curve again in three points which lie on the line x + y = 0.
3. Find the asymptotes of the curve xy'^ - x^ + 2x'' ■{■ 3y + x— I = 0.
Show that the points at a finite distance from the origin in which the
asymptotes cut the curve lie on the line 3y + 2x — l=0.
4. Draw the curve x^y = x' — n'. Show that it has an asymptote which
crosses the x-axis at an angle tan"' 3.
6. Find the asymptotes of the following curves: (i)xy''—x'hi=a^(x+y') + b^.
{i\)l + y = e'. (_\\i) x' - xy'^ + ay'^ - a'^y = 0. (iv) (x'^ + j/2)(j/2 _ 4a;2)
+ 42;2(x-l) + x2(4x + 3) = 0. (v) (x-2a)j/2 = a:3_a3. (vi)x» + 3i/3
=a^(y—x). (vii) x'+2x2!/ + x?/2-x2-xy+2 = 0. (viii) r sin 2 9 = a cos 3 9.
(ix) y' = x2(2 a - x).
6. Find the asymptotes of the curve x^y — xy^ + 6 a'^xy + a'h/ — 16 ah^ = 0.
Show that the origin is a point of inflexion.
QUESTIONS AND EXERCISES. 439
7. Define a family of (plane) curves, and the variable parameter of the
family. Define the envelope of a family of curves. Define an ultimate
intersection of a family of curves. Define the locus of the ultimate intersec-
tions of a family of curves. Illustrate the definitions by concrete examples
and diagrams, and furnish any explanations you may think necessary.
8. Show that in general the locus of ultimate intersections of the family
touches each member of the family. Show that this locus is, in general, the
envelope of the family. Explain the necessity of the qualifying phrase "in
general."
9. Explain the method of finding the envelopes of the curves /(a:, y, t)=0,
where { is a variable parameter.
10. Write a note on "singular points of curves," explaining what they
are, giving illustrations, and showing how to find them.
11. Ellipses of equal area are described with their axes along fixed straight
lines. Show that the envelope consists of two equilateral hyperbolas.
12. Prove that the circles which pass through the origin and have their
centres on the equilateral hyperbola x^ — y^ = a^ envelop the lemniscate
(x2-)-2/2)2 = 4a2(i2-y2).
13. P is a point on a parabola of which A is the vertex. Find the equa-
tion of the curve touched by all circles described on AP as diameter.
14. A circle passes through the origin, and its centre lies on the parabola
^^ =: 4 ax. Show that the envelope of all such circles is a cissoid.
15. A straight line moves so that the product of the perpendiculars on it
from two fixed points (± c, 0) is constant (= ft^). Show that its envelope is
the ellipse ^^^ + 1 = 1, or the hyperbola -^^ - g = 1.
16. Find the envelope of circles passing through the centre of an ellipse
a^yi ^ i)ix^ — 0^1)2 and having centres on the circumference of the ellipse.
lAns. {7? + 2/2)2 = 4(323.2 + 62j,2).]
17. Ellipses are described having their axes coincident in direction with
those of a given ellipse, and lengths of axes proportional to the coordinates of
a variable point on the given ellipse. Show that the ellipses all touch four
straight lines.
18. Find the equation of the envelope of the line zsina + ycosa =
a sin a cos «.
19. From a fixed point on the circumference of a circle chords are drawn,
and on these as diameters circles are drawn. Show that the envelope of the
series of circles is a cardioid.
20. If a cannon is fired at an elevation 6, and the projectile has an initial
velocity equal to that attained by a body in falling h feet, the equation of the
parabolic path, referred to horizontal and vertical axes through the point of
440 DIFFERENTIAL CALCULUS.
projection, ia y = x tan 6 — — sec'' $. Find the envelope of the paths for
diflerent elevations.
CHAPTERS XV., XVI.
n=co
1. A function f(x) is defined by an infinite series f(x) = ^ <t>n(x,') ; state
and prove a sufiBoient condition that the equation —f(x) = ^ — 0n(a;) may
. . ax ^i, ax
be true. n=i
2. Write a note on the conditions under which (1) the integral, (2) the
differential coefficient of an infinite series, may be obtained by integrating or
differentiating the series t«rm by term.
3. Prove that if /(z) be a continuous function of x, then
f{x + h)=f{x-)+ hf>{x + eh),
where 0 < 9 < 1.
Show clearly how this proposition may be applied to prove Taylor's theo-
rem, and specify the circumstances in which the theorem as you state it is true.
4. Prove Taylor's theorem for the expansion of f{x + h) in ascending
powers of h, carefully specifying the conditions which f(x) must satisfy.
Find an expression for the remainder after n terms of the series have been
written diwn.
5. State Maclaurin's theorem, and give the conditions under which it is
applicable to the expansion of functions. Derive the theorem.
6. Expand in series of ascending powers of x the functions : (i) cos mx.
(ii) tan-i(a + a:). (iii) sin (m sin-^ g). (iv) (1 + y)', where j/ < 1.
(v) e"" + e""". (vi) 6*^'+', 4 terms.
7. Expand the following functions in powers of x : (i) e''° '. (ii) tan-i x.
(iii) cot-i X. [Ans. (i) l + x + ix^-ix^--^x^+ —. (ii) For
values of x from x = — 1 to x = 1, x — ^ 3? + i ifi — } x'' + •■■ ; for |a;|>l,
|-^3-^-5ii+-- ('")^°' l*'<^' l- + i^-i=^+-: for
8. Calculate the values of the following :
(i) t I'Vl -x'^dx. (ii) i'xcotxdx. (iii) f e''dx. (iv) fVsinxdx.
W f ^ dx. ^Ans. (i)ixkl-ix''-^\x^-.^xfi + ...).
(ii) x-^-^-A^.... (iii) 2/l+l+-J_+_J_ + I + ...V
9 225 6615 ^ ^ \ 3 1-2.5 1.2-3.7 I.2.3.4. 9 )
(iv) ^ + 2i'-t-2x^-2i^_2i^_2!j!4.... rv^x ^ + _^f 1
'•'^21 31 41 61 71 81 ^^ 3-31 5.51 "J
QUESTIONS AND EXERCISES. 441
CHAPTERS XVIII.-XXII.
1. Explain and illustrate the meaning of integration.
2. If f(x) be finite and continuous for all values of x between a and 6,
prove that lim„^ A {/(n) + f(a + h) + f{a + 2h)+ — + /(a + n - 1 A)} ia
0(6) -0(o), where h=^^^ and — 0(x) = /(z).
n dx
3. Explain fully how it is that the area included between a curve, the
axis of X, and two ordinates corresponding to the values xq and xi of x is
represented by the definite integral v'^ydx.
4. Give an outline of the reasoning by which it is shown that the area
bounded by the two curves y = (p{x) and y = ^(x), and the two ordinates
x = aandx = 6, is j {^(x)— ^(x)}dx.
5. Prove Simpson's or Poncelet's rule for measuring a rectangular field,
one of whose sides is replaced by a curved line.
The graph of y = x^ is traced on a diagram. If O be the point (0, 0) on
it, P the point (10, 100), and PM the ordinate from P, find the area of OMP
cut off between OJf, MP, and the curve, by taking all the ordinates corre-
sponding to integral values of the abscissas, and applying the rule you adopt.
Tell exactly by how much your calculation is wrong.
6. Show how to find the volume of the surface generated by the revolu-
tion of a given curve about an axis in its plane.
7. Find the area cut off between the parabola y = x^ and the circle
z2 + S/2 _ 2.
8. Trace the curve whose equation is a*y'^ = ii^{a^ — x^), and find the
whole area enclosed by it.
9. Show that the area included between the curve ^^(2 a — x) = x' and
its asymptote is 3 Tra-.
10. Determine the amount of area cut oft from the circle whose equation
is x^ + !/^ = 5 by a branch of the hyperbola whose equation is xy = 2.
11. Trace the curve ay + 2 x(x — a) = 0. Find the area of the closed por-
tion contained between the curve and the axis of x. If this portion revolves
round the axis of x, find the volume generated.
12. A curved quadrilateral figure is formed by the three parabolas
y2 _ 9 ax + 81 cfi = 0, y^ - 4 ax + 16 a^ = 0, !/2 - ax + a^ = 0, the other boun-
dary being the axis of x. Find the area of the quadrilateral.
13. Show that the volume of the solid generated by revolving about the
X-axis, an arc of a parabola extending from the vertex to any point on the
curve, is one-half the volume of the circumscribing cylinder.
442 DIFFERENTIAL CALCULUS.
14. Determine the curve for any point of which the subtangent is twice
the abscissa and which passes through the point (8, 4).
IB. Write the equation including all curves that have a constant sub-
normal. Determine the curve which has a constant subnormal and which
passes through the points (0, ft), (6, k), and find what is the length of its
constant subnormal. [Ahs. by^ = {k^ - K'yx + bh^ ; ^'~^'-1
16. In what curve is the slope at any point inversely proportional to the
square of the length of the abscissa ? Determine the curve which has this
property and passes through (2, 5), (3, 1).
17. State and derive the rule knovm as "integration by parts." Apply
it to find I K» log r dx.
18. Show that if the integral of /(x) is known, the integral of f~^ix), the
function inverse to f(x) , can be found.
19. Show how to integrate I=i-^, where /(x) and <t>{x) are rational
4>{x)
integral functions of x, and give some of the standard types for the integrals
on which the value of / may be made to depend. Show how to integrate the
fraction when the equation ^(x) = 0 has repeated imaginary roots.
20. Show that if /(u, v) is a rational function of u and v,flx, ■x}"^ \dx
arA-h ^ ^cx + dj
can be rationalised by means of the substitution ^ = £".
ex + d
21. What is meant by a formula of reduction for an Integral ?
Investigate formulas of reduction for the following : (i) ( sin" 0 dff
JC x"
sin" e cos" e dd ; (iii) I . -dx ;
(iv) ( 2» sin a; dx.
22. Explain how it is that C' 00$'"+^ e de = 0.
dx
-p)Vax'' + 2bx + c
23. Evaluate \ , by means of the substitution
y(x —p) = Vax^ + 2bx + c.
24. Evaluate the following integrals, and verify the results by differentia^
tion: r^°"""'''^^ f-sin-iJI^dx, f^ ^' , C^^dS,
•^ (1 + x^)5 •'o ^ « + ^ •'i ""^ * '=°* * ■'f cos' e
f de r x^ dx C dx C dx
J a^ gos2 9 + 62 sin 2 e' Jx'^-l' Jx(.3 + 4x5)8' J 3 sin a; + sin 2 x'
(x^{a + x)hx, j* ^^^"^^ — dx, Jx^ tan-i X dx, (e'^sin^xdx,
QUESTIONS AND EXERCISES. 443
Jitaig^^^^ J^,,^, Plog(^ + aS)<to, ^Q2M)^, j;i2g
zda:
r da: /• (a: + l)dx "^
J X \/-i2+ 5 3: - e' J Vx^ + X + 1
CHAPTERS XXIV., XXV.
1. Find an expression for the area bounded by a curve given in polar
coordinates and two straight lines drawn from the pole.
2. Show how to find the length of the arc of a plane curve whose equa-
tion is given (i) in rectangular Cartesian coordinates, (ii) in oblique Carte-
sian coordinates, (iii) in polar coordinates.
3. Investigate a formula for finding the superficial area of a surface of
revolution about the axis of x.
4. Trace the curve r- = a- cos 3 6, and find the area of one of its loops.
5. Show that in the logarithmic spiral, r = a', the length of any arc is
proportional to the difference between the vectors of its extremities.
6. Find the area of the curve r Va^ + 6^ = (a^ + 6^) cos S ■+■ a^.
7. Find the surface of a spherical cup of height h, the radius of the
sphere being S.
8. Find the average value of sin x sin (« — x) between the values 0 and
a of the variable x.
9. Find the volume bounded by the surface ■\- + -\- + -\J- = l and the
coordinate planes. a o c
10. The axis of a cone is the diameter of a sphere through its vertex ;
find, in terms of its vertical angle, the volume included between the sphere
and the cone, and examine for what angle it is greatest.
11. Determine the areas of each of the following figures ; (i) The segment
cut off from the parabola y'^ = 4 ax by the line 2x — 3y-|-4a = 0. (ii) The
curve (-\ + (^\ =1- (iii) The evolute of the ellipse (ax)^+ (6y)^ =
(a2 - 62) i (iv) The figure bounded by the ellipse 16 x" + 25 y' = 400, the
lines X = 2, X = 4, and 2 y + x = 8. (v) The curve (x^ + y^)^ = a^x^ + h^-
(vi) The oval y = x'^ -I- V(x — 1)C2 — x). (vii) The loops of the curve
aV = x2(a2 - x2). (viii) The segment of the circle x^ + y^ = 25 cut off by
the line x -|- y = 7. (ix) The area common to the ellipses bhfl + a'y^ = a^V^,
a^x^ -I- 62y2 = oSfts. \ Ans. (i) \ a\ (ii) \ xa6. (iii) \ v ("' -V^^^,
(v) '•(o'-H&')_ ^^j^ »_ ^yj;^ j.^gjj ^^2. (viii) j^i sin-i j^s - f
2 4
i6
(ix) 4a6tan-i5."|
444 DIFFERENTIAL CALCULUS.
12. Find the volume and the area of the surface generated by the reTolu-
tion of the cardioid r = a(l — cos B) about the initial line. [Area = ^ jra-.]
13. Show that the volume enclosed by two right circular cylinders of
equal radius a whose axes intersect at right angles is -i^ a', and the surface
of one intercepted by the other is 8 a^.
14. Show that the volume included between the surfaces generated by
the revolution of a hyperbola and its asymptotes about the transverse axis
and two planes cutting this axis at right angles is the same, no matter where
the sections are made, provided that the distance between the planes is kept
constant.
15. The parabola y^ = &x intersects the circle z^ + ya _ jg. Show that
if the larger area intercepted between the curves revolves about the a;-axis,
the volume generated is 60 tt cubic units ; and show that if the smaller area
intercepted revolves about the y-axis the volume generated is 2|i Vs x cubic
units.
16. An arc of a circle of radius a revolves about its chord. Show that if
the length of the chord is 2 a«, volume of the solid = 2 ira'(sin a — \ sin' «
— « cos «), surface of the solid = 4 TO'^(sin a — a cos a).
17. I'ind the area of the segment cut ofl from the semi-cubical parabola
27 ai/2 = 4 (i — 2 ay by the line x=ba. Also find the volume and the area
of the surface generated by the revolution of this segment about the x-axis.
^Ans. %>■ a% T-a^ I ^ + ! log (>^ + 1) | .]
18. A number n is divided at random into two parts. Show that the
mean value of the sum of their squares is | n-.
19. Show that the mean of the squares on the diameters of an ellipse, that
are drawn at points on the curve whose eccentric angles differ successively
by equal amounts, is equal to one-half the sum of the squares on the major
and minor axes.
20. Prove that the mean distance of the points of a spherical surface of
radius a from a point P at a distance c from the centre is c -I- — or a + — ,
according as P is external or internal.
CHAPTER XXVII.
1. Solve the following equations :
(1) x^y dx-{v? + f)dy = 0. (2) 3 e* ton j/ dx + (1 - e^) sec" y dy = 0.
(3) (x2-4a:2/-2!/2)(te + (j,2_4a:!/-2 3;2)dy = o. (4) xDy~y=xV¥+^.
(5) {xi + y'^){xdx-\-ydy)=a\xay-ydx). (6) {x^ + \)Dy + 'ixy = ixK
(7) 6(x + \)Dy = y- y*. (8) p^-ixyp + Sy^ = 0, in which p = D^y.
(9)$+y=x^y^. (iO)^ + ^-=|^y=l. iU)y=x^-ip^. (12)x+2pj/=p^
dx X ax X'
QUESTIONS ANT) EXERCISES. 445
(13)2),33, + 2i>.^ + 2),2, = 0. (14) g - 3 g + 4 | _ 2 j, = 0.
[Solutions : (1) 3 2/^ log y = x^ + c. (2) tan j/ = c(l - 6=^)3. (3) x8 - 6 ar^y
- 6 zy2 + y3 = c. (4) 2 j^ = z(ce' - ce-'). (5) x^ + y^ = 2 a^ tan-i ^ + c.
(6) 3(x2 + 1)2, = 4 z' + c. (7) VxTT(l - 2/') = cyK (8) 2/ = c(a; - c)^.
(9) 2 2^-6 = 03^ + 5 25. (10) y = e2(1 + ce'). (11) (Z^ + y)2(22 _ 2 y)
+ 2 x(z2 - 3 2^)c = c2. (12) 1 + 2 C2/ = c^z^. (13) y = Ci + e-«(c2 + CsZ).
(14) 2/ = e»(ci + C2 cos z + cs sin z). (15) y = a + c^ + e'(C8 + qz).
(16) xy = ci logz - log (z - 1) + C2. (17) 2^ = z (ci + C2 log z) + cjz-i.
(18) sm(ci-2V2 2/)=C2e-2». (19) x=-y/cy^-y+ — i— hycos-i(2c2/-l) + Ci.
c 2cVc
(20) 2 z = log(2,2 + ci) + Ci. (21) 15 Ci^ = 4(ciz + a^)^ + CjZ + Cs-J
2. Find the singular solutions of :
(1) z2p2-3z2(p+2y2+z8=0. (2)zp2-2 2/p+az=0. (3) Solve equation (2).
r/SToiations; (1) z2(2/2-4zS)=0. (2) 2/2 = az^. (3) 2 2/ = cz^ + ?•]
MISCELLANEOUS.
1. How far does the symbol — obey the fundamental laws of algebra ?
dx
2. Prove that if D denote — , and f(D) be any rational algebraic func-
dx fj2
tion of D, then f{D)uv = uf(D)v + Dnf'{D)v + i^-^ .f"{D)v + ....
5. If 0 denote any function of z, prove that ^"(^'<>) = n ^^ + z^-
dz» dz"-i dz»
By this theorem or otherwise find the value of D^{x sin mx).
4. If z = e», provethat A('i._iyA_2V"('-^-n+lV = *^— ,
d9\de j\de 1 \de j dz"'
(d d\" I d\'' 't d\^
— X — u= — z — «.
dz dz/ \dz/ \dxl
6. If ^(z) is a function involving positive integral powers of z, prove the
symbolic equation <t> f— ( e" • a jl = &"<)>( a + — ja.
6. Show how to find the values of -^ and ^ when z and y are con-
dx dx
nected by the equation /(z, y) = 0.
446 DIFFERENTIAL CALCULUS.
7. If u =/(x, y) and if x = <t>(f), y = f («), state and prove the rule for
obtaining the total derivative of u with respect to t.
If X = r cos e, « = r sin e, transform (x^ — y"-) " ^ + xy I ^ — ~-^ into
dxdy V3x* dy^J
an expression in which r and fl are the independent variables.
8. Calculate the rath derivative of (sin~i x)^. Show by the use of Mac-
laurin's theorem that (sin-ix)2 = 2/'— + ? ^ + ?-ll^+...Y
^ ^ U 3-4 3.5.6 I
9. The curves « = 0, «' = 0 intersect at (x, y") at an angle a. Show that
dx dy dx dy
tan«:
dx dy dx dy
Show that the curves — i- ■'- = 1 and \- S- = \ intersect at right angles
10. Show that the total surface of a cylinder inscribed in a right circular
cone cannot have a luaximum value if the semi-angle of the cone exceeds
tan-i i, i.e. 26° 34'.
11. Through a diameter of the base of a right circular cone are drawn two
planes cutting the cone in parabolas. Show that the volume included between
these planes and the vertex is — of the volume of the cone.
3 w
12. Calculate the area common to the cardioid r = a (1 — cos $) and the
circle of radius | a whose centre is at the pole.
13. Find the area and the perimeter of the smaller quadrilateral bounded
by the circles x^ + y^ = 25, x^ + y' = 144, and the parabolas, y^ = S x,
y"- + 12 (X + 2) = 0.
14. Given the cardioid r = 4 (1 — cos e) and the circle of radius 6 whose
centre is at the cusp, find the length of the circular arc inside the cardioid
and the lengths of the arcs of the cardioid which are respectively outside the
circle and inside the circle.
16. If a curve be defined by the equations -^ = -^ = — =— , find an ex-
■^(0 iKO /W
pression for the radius of curvature at a point whose parameter is (.
16. Expand (by any method) i^ cosec' x in a series of powers of x as far
as the term in x*. At what place of decimals may error come in by stopping
at this term, when x is less than a right angle ?
17. Trace the curve x* + y* = a^xy, and find the points at which the tan-
gent is parallel to an axis of coordinates. Find the area of the loop.
18. Trace the curve x = a sin 2 9 (1 + cos 2 9), j/ = a cos 2 9 (1 — cos 2 9).
(a) Prove that 6 is the angle which the tangent makes with the axis of x, and
obtain the equation of the tangent to the curve. (6) Find the length of the
radius of curvature in terms of e.
QUESTIONS AND EXERCISES. 447
19. Find 2l? under each of the following conditions : (i) x^ = e**" \ '' )■
dx
(ii) y = e^ tan-' x. (iii) e' + 2 = e" + y. (iv) y = • (v) sin (x^)
- €^ - x^j, = 0. x+y/l -x^
20. Four circles z^ + ^,2 = 2 ax, x" + j^^ = 2 ay, x^ + 2/2 = 2 6z, x^ + y^ = 2 62/,
form by their intersections in the first quadrant a quadrilateral ; prove that
the area of this is (a^ + 6^) cot-» _?-«L_ _ (a - 6)2.
o^ — 6^
21. Prove that the area of a sector of an ellipse of semi-axes o and 6 be-
tvfeen the major axis and a radius vector from the focus is — (^ — e sin 0),
vrhere ^ is the eccentric angle of the point to which the radius vector is
drawn.
22. Trace the curve xy* = a* ; and find whether the area between it, a
given ordinate, and the coordinate axes is finite.
Show also that if the tangent at P meet the axis of x in T, then MT = 3 OM.,
where M is the foot of the ordinate at P, and 0 is the origin.
23. If u be a homogeneous function of n dimensions in x and y, show that :
(i)x25!« + 2xy-^-|-!^^=n(n-l)«. (ii)x^+ s/-^^ = (n-l)^".
(Hi) x_a!!L^„5!«_,„_i,aM. f.,^ f^Aj.,.d_\2„-
dxdy 'dy^ By \ dx " dyj
24. Prove the following : (i) If u = sin-i Cxyz), 3a 5« 5" _ ^^^2 „ ggg ^
ax dy dz
(ii) If M = log(tanx + tan« + tan2), sin2x^ + sin2!/^ + sin22^ = 2.
dx dy d^
(iii) Ifu = log(x» + j/» + z»-3xj/z), |!^ + |» + ^ = _J—- . (iv) If
dx By dz x + y + z
u = tan2 X tan^ y tan^ z, du = iu l-r^ — I- . :( + . ^ )■
\sm2x am2y sin2z/
25. If b be the radius of the middle section of a cask, a the radius of either
end, and h its length, show that the volume of the cask is ^T(3a^ + iab
+ 8 b'^)h, assuming that the generating curve is an arc of a parabola.
26. OM is the abscissa, MP the ordinate of a point P(xi, j/i) on the
hyperbola — — ^ =1, (xi, yu both being positive). If A is the vertex nearest
a^ b^ / \
P, show that area AMP = I xm - i a6 log ( ^ + ^ ) , and area sector OAP
/ \ \a 0 J
= JaMog(|^ + |).
27. Show that the mean of the squares on the diameters of an ellipse that
are drawn at equal angular intervals is equal to the rectangle contained by
the major and minor axes.
448 DIFFERENTIAL CALCULUS.
28. Find the mean square of the distance of a point within a square from
the centre of the square.
29. Through a diameter of one end of a right circular cylinder of altitude
h and radius a two planes are passed touching the other end on opposite sides.
Show that the volume included between the planes is (tt — f)o%.
30. Show that the integration of the expression f(x, y)dxdy may be per-
formed in any order, provided the limits of x and y are independent of each
other.
31. Evaluate ( ( \ x<^y?zy dx dy dz taken throughout the space bounded
by the coordinate planes and the plane x + y + z = 1.
32. Prove geometrically or otherwise that xdy—ydx=r^ de, and show that
the area of a closed curve is represented hy \\ (xdy — y dx).
33. The equation to a curve being written in terms of the polar coSrdi-
nates r and 9, p being the perpendicular from the pole to the tangent and
u = -, show that, - = a^ + f— V-
34. If a is a first approximation to a root of the equation /(i) = 0, deter-
mine graphically or otherwise the conditions under which a — =2^ is a valid
second approximation. jW
36. If /(«) be a finite and continuous function of x between x = a and
x = b, show that a value xi of x, lying between a and 6, may be found such
that/'(a;,) = {/(6) -/(a)} - (6 - a).
If the function be x'+cx, find the point in question when a = a and 6=2 a,
and thence show that in this case Xi is such that ° ~ ^' is constant for all
values of a. ^^
36. Find the radius of curvature of the curves: (i) lima^on r=acose+6,
wherer- = -; (ii) ay2 =(a;-a)(x-6)2 at (a, 0). Trace the curves. ^Ans.
ia' — 0^ 2 a J
37. (1) Trace the curve r =0+6 cos e, a>6>0 ; find its area. (2) Find
the area of the loop of j/^ = (x - 1) (a; - .3)2. (3) Find the area between the
i-axis and one arch of the harmonic curve j/=6 sin -• \Am. i(2 a'^+lfi)r,
•^,2a6.1
15 J
38. Trace the curve 9 y^ = (i + 7) (x + 4)2. Find the area and the length
of the loop, and the volume and area of the surface generated by the revolu-
tion of the loop about the x-axis. [^ns. |\/3, 4\/3, J tt, 3 ir.]
QUESTIONS AND EXERCISES. 449
39. Find the limiting values of: (i) log-I^^iS-^, when e=T; (ii) /l2g?V
when I = 00 ; (iii) ; — x" — x — ^j^^^^ x = l; (iv) -i , when
l-x + logx ' "• '^ 2x2 2ztan7rx
X =
1
0; (v) ^5i5J=\x2, whenx = 0; (vi) 2^^^-^, when x = 0 ; (vii)
x2-o2'
when x = a.
40. Find the mass of an elliptic plate of semi-axes a and 6, the density
varying directly as the distance from the centre and also as the distances from
the principal axes.
41. From a fixed point A on the circumference of a circle of radius a, the
perpendicular j1 y is let fall on the tangent at P. Prove that the greatest
3"\/3
area AFY can have is — — o".
8
42. A rectangular sheet of metal has four equal square portions removed
at the corners, and the sides are then turned up so as to form an open rec-
tangular box. Show that the box has a maximum volume when its depth is
i(a -I- 6 — Va' — ab + b'), a and 6 being the sides of the original rectangle.
43. Two ships are sailing uniformly with velocities «, v, along straight lines
inclined zX an angle 9 : show that if a, b, be their distances at one time from the
point of intersection of the courses, the least distance of the ships is equal to
(ai) — 6m) sin 8
(a" -I- »2 - 2 MO cos e)i
44. A right circular conical vessel 12 inches deep and 6 inches in diameter
at the top is filled with water : calculate the diameter of a spherical ball which,
on being put into the vessel, will expel the most water.
45. A statue a feet high is on a pedestal whose top is 6 feet above the level
of the observer's eyes. How far from the pedestal should the observer stand
in order to get the best view of the statue ? [Ans. y/b^a + b) feet.]
46. The lower corner of a leaf, whose width is a, is folded over so as just
to reach the inner edge of the page : find the width of the part folded over
when (1) the length of the crease is a minimum, (2) when the area of the tri-
angle folded over is a minimum. [Ans. (1) Ja; (2) Ja.J
47. (1) Show that the cylinder of greatest volume for a given surface has
its height equal to the diameter of the base, and its volume equal to .8165 of
that of the sphere of equal surface.
(2) Show that the cylinder of least surface for a given volume has its
height equal to its diameter, and its surface equal to 1.1447 of that of the
sphere of equal volume.
450 DIFFERENTIAL CALCULUS.
48. Trace the graph of y = sm 2 z - sin a: ^^^^ ^^^ angles at which it
cos X
crosses the a>-axis, and show that its finite maximum distance from the a>axis
is (2? - 1)1
49. An ellipse, whose centre is at the origin and whose principal axes coin-
cide with the axes of x and y, touches the straight line qx+py=pq ; find the
semi-axes when the area of the ellipse is a maximum, and also the coordinates
of its point of contact with the given line.
60. Find the volume of the greatest parcel of square cross-section which
can be sent by parcel post, the Post-office regulations being that the length
plus girth must not exceed 6 feet, while the length must not exceed 3 feet
6 inches.
INTEGRALS.
FOR EXERCISE AND REVIEW.
The following list of integrals provides useful exercises in
formal differentiation and integration. It will also afford some
assistance in the solution of practical problems as a table of refer-
ence. Those who have to make considerable use of the calculus
will find it a great advantage to have at hand Peirce's Short Table
of Integrals* (Ginn & Co.).
GENERAL FORMULAS OF INTEGRATION.
Formulas A, B, C, pages 294, 205; formula for integration by parts,
page 298.
FUNDAMENTAL ELEMENTARY INTEGRALS.
Formulas I.-XXVI., pages 293, 294, 301, 302. (These should be mem-
orised.)
REDUCTION FORMULAS FOR ( x*''(^a + bx'^yp dx.
[Here Xdenotes (a + 6x").]
1. (x"'XPdx='''^-"^'^''^' _a{m-n + l) U^nnXPax.
J b(np + ni + l) b(,np + ni+ l)J
2 fx^XPdx =^"^'^V' _ 6(m + n + np + 1) Ur^^n^Pax.
J a{m+l) aim+l) J
S. fx"»XP dx = a;"^' XP ^ anp__^ f x™XP-i dx.
J m + np+l m -It np + 1 J
4. f x" JP dx = - «'"';'^^;; ^ m + n + np + 1 C^^xp^x ax.
J an(p + 1) an(,p +1) J
* There are two editions, the briefer edition of 32 pages and the revised
edition of 134 pages.
451
452 DIFFERENTIAL CALCULUS.
5. ix^XP dx = '"-''^'^^^' - '"-"+^ ra^-»X.+i dx.
J m + 1 TO + 1 J
r (fi 1 {m — n + np — V)bC dx
J x" J> ~ (to - l)oa;'"->X''-i (to — l)o J 2'"-"Xp
g r_*L- = 1 m-n + np-\C dx
J x^X" an(p- 1)x"'-'Xp-i on(p - 1) J x"'X^-^
g rXPda:, Xy+i &(m - )t - np - 1) TX^dx
J 2" a(m — l)!™-! a{m — 1) J a;"-"
10 C^'dx X' . anp CXp-^dx
J x" (np — TO + l)!"-' np — m + \J a?»
,, fz-'da; »"■-"+' a(TO - >i + 1) rz^-'tfe
J X? 6(TO-np+ l)X''-i 6(m-jjp + l)J Xf
J Xr ~an(p -l)XP-i an(p - 1) J X?-i"
18. f ^^ = 1 r 5 + (2„_3)C ^ 1
J (a + 6z'')» 2(» - \)a L(a + 6x^)"-i J (a + 6x2)"-iJ
Put cfi for a, 6 = 1, and compare with Ex. 3, Art. 118.
14 f g'tfa ^ zL^ L 1 C dx
J(o + 6x2)" 2 6(71- l)(a + 6x2)»-i 2 6(re - 1) J (a + 6x»)»-''
J K2(a + fcx^-jn a J x2(a + 6x2)n-i a J (a + bx^)" '
EXPRESSIONS CONTAINING Va + bx.
Also see Ex. 10, page 312.
C dx _ V a + 6a; 6 C dx
^ x^y/a + bx a* ^aJj-V^
bx
17. fy£L+^^^^2VHT6i + ar-^
•' ^ "^ x Va + 6x
INmGRALS. 453
EXPRESSIONS CONTAINING Vx^ ± a".
Also see Ex. 7, page 312.
18. f _^_=log(' + ^^^-«!V See XXIV., XXV., page 181.
n
19. f (x» ± a^)'dx = 5i2i±^±-!!^ ("(i" ± a»)'''dx.
J n + 1 n + 1 J
20. f (i= ± o»)idx = ? Vx" ± a-J ± ^ log (x + Vz^ ± a^).
ai. f (li" ± a2)idx =1(2 x» ± 5 o^) Vx^ ± a^ + §^*iog(x + Vi^ ± a').
■Jo 8
22. fxaCx^ ± 0=)* (ix = I (2 x2 ± a») Vx'' ± a^ - ^ log (x + Vx^ ± o") . "
•7 8 8
23. f_i^ = ±.-
84. f^!^_
:2\/xa -t a"
= - Vx^ia" T — log (I + ViS ± a«).
+ log(x+Vx2-a«).
S6. C ''^ = »
(J^ ± a»)i Vx" - a^
26. f^ = hog ? ; (• ^ =lsec-ig.
•' xCx' + a!-)* " o + V^T^ •'x(x»-as)i " "
27. C tlx ^ -^ Vx" ± «'
88. a. ( ^ ^_VFTT' _Lj g+VFT?.
6. f ^ =^^^^^^ + J-sec-ig.
■'x»(x^-a»)i 2a^ 2a3 a
^ I I
. f(i5-o2)*(ii /-5 -„ ,a
6. I -^^ = Vx' — a^ — a cos-' - •
J X X
30 r (x^ ± g^) * dx ^ _ VFT^i^ ^ t ^ ^,^^^
J X^ X
454 DIFFERENTIAL CALCULUS.
EXPRESSIONS CONTAINING Va^ - x\
Also see Ex. 7, page 312.
n
31. f (a'' - z'')^dg = ^<^°' ~ fy +J^ ({a^ - x-^y'^dx.
J n + 1 n-^\J
r x^dx _ _ a^-'Va'^-a:' (?n - l)a' C x^-^d
33. t ar" V a' — x'dx = 1 I ^— ^^
J m + 2 'm + 2J y/^fZTi
Va^ — a;2 , m — 2 f da;
34. C ^? rf:. = V^'-^" + m-2 r
Jx^Va^-i'^ (m - l)a'''a;"'-i (TO-l)a2J
a;"'-^ Va2 - x^
J x" (m — 2)a!'»-i TO — 2 J x" Va'^ — «'
36. C(a2-x')idx = -Va2^^ + 2!siiri?.
J^ ^ 2 2a
37. r (a2 - x2) t (Ja; = 5 (5 a2 - 2 x2) Va" - x^ + ?-"- gin-i -•
Jo b fl
38. far^Crt^ _ x^)* cZx = - (2 a;^ - a2) Va^ _ x2 + ^ sin-i ^.
J 8 8 a
39. f ^'^^ ^_gV5a^r^ + g!sin-ig.
40.
r ^— ^= — ^ • 41. r_^!^= — ?^_-.„i'5.
(a^ — x'')' '^^ ^"^'^ ~ ^^ (a^ — x'^)^ Va''' — x'^ "
42. f ^5 = -^"'-^1 43. (■ ^5 =liog. »:
x2(a2-x2)^ ""^ -'x(a2-x2)i " a + V«r.2-x2
•^x3(32_a;2)^ ^ a^x^ 2 a' a + Va" - x^
5. ga' - x^)^ ^^ ^ V^^r^. _ 3 log «±vgZg.
-/ X X
;. (•i«izi^cZx = -:
^* .VaiE^^_ . ,x.
46. tV" -■" '-dx = -"" - =-sin-
INTEGRALS. 455
EXPRESSIONS CONTAINING V2 ax - x\ V2 ax + x\
[Here X denotes V2 ox — a;^, and iJ denotes V2 as + «».]
47. a. j"| = sin-i^. 6. JJ = log (x + a + ZJ.
48. a. rXdx=^^l^X+«-%m-i^^:i«.
J 2 2 o
6. JZeix = ^ Z - 1 log (x + a + Z).
49. a. fx'»Xdx = -?!^I^ + i2™±iL« fx-iXcix.
J TO + 2 m, + 2 J
h. rx".zdx=^::^ii^°-(^'» + ^)« fx^-zdx.
J »i + 2 TO + 2J
60. a. f-^ = ^ + '"-^ f '^ .
J x-'X (2 j» - l)ox'» (2 TO - l)a J x^-'X
. C dx _ — Z m — 1 C dx
' J x<'Z (2»n — l)ax"' (2 m — '\.)a J x"-^ Z
61 a C'^^^- g"~'X . (2TO-l)a fx"— 'dx
J X TO m J X
J rx"dx_x"-'Z (2m-l)g fx"*-' dx
' J Z TO m J Z
Jx™ (2TO-3)ax'» (2 m -3)a^x»'-i
6. CLax = ?^ "'-g f^<fa.
J x™ (2 TO - 3)ax'» (2 TO - 3)a J x"-'
a. fxXrfx ^ _ 3 °' + «^ - 2 ^' x+ g-' Bin-' ^^.
J 6 2a
6. rxZ(fa = -^''''-'f-^^'z+g^log(x + a + Z).
64. a. C-^ = -^. 6.f^ = _^.
J xX ax -' xZ ax
68. a. Ct^ = _X+asin-i5^^. 6. r^ = Z - alog(x+ a + Z).
66. a. r?!^=-^+-3«X + 5a2sin-i^^^.
J X 2 2 a
6. f 5l^ = 5_zl« Z + - a2 log (,x+a + Z).
J Z 2 2
53
456 DIFFERENTIAL CALCULUS.
67. a. ('^^ = X+asin-i5.=^. ft. (^^ = Z + a\og(x + a + Z).
J X a J X
68. o. C^dx^-^-sin-^^-:^- b. f4*K = - — + log(» + « + 2)-
J x' X a J x^ X
J x' 3 aa^ J x^ 3 aa?
60. a. f^ = ^^. 6. r^ = -^±«.
., Cxdx X 1 rxdx_ X
EXPRESSIONS CONTAINING a + bx± ex''.
■ bx + cx2 v* ac - 62 V4 ac - 6"'
a. f ^ 2 tan-i ^'^ + f> , for 62<4ac
J a + 6i + cx2 v* ac - 6^ V4 ac - 6"
1 ,„g 2ca; + 6-V&^^::i^^ f^, j, ^ 4 ^^
Vb'^ -iac 2cx + b + Vb'^ -4ac
dx _ 1 ,„„ \/62 + 4 ac + 2 ex - 6
5. f_^_ = ^^log:
^ a + 6x - cx2 Vft2 4- 4 or.
; + 6x - cx2 V62 + 4 oc Vb^ + 4kac -2cx+ b
63. a. C '^^ = — log (2 cz + 6 + 2 Vc Va + 6x + ca;").
-' Vo + 6x + cx'^ Vc
6. f^
dx _ J_ . 1 2 ex - 6
bx — cx^ Vc v'6'' + 4 ac
64, a. (Va + bx + cx''dx = ?J^L±A Va + 6x + ex'
J 4c
6» - 4 oc ,
-log (2 ex + 6 + 2\/c Va + 6x + cx2).
8c«
6. ('Va + 6x-cx^dx = ^^^^Va+6x-cx' + ^!±i°g3in-i4^=|:
•' 4 c Q * ■v/h2-U4/i
8 c* V62+4ac
„, „ I X dx Va + bx + ex"
a. f— ^
•' VST
•^ Va +
6x + ci'^ ''
^ log (2 ex + 6 + 2 Vc Va + fix + ex").
2e^^
xdx _ _ Va + 6x — ex" , 6 ain-i 2cx — 5
6x — ex" " 2 e^ "^S" + 4 ac
N.B. Other algebraic integrals that are occasionally useful are given
in Exs. 7-10, page 312, and in Exs. 4, 6, page 343.
66
INTEGRALS. 457
EXPONENTIAL AND TRIGONOMETRIC EXPRESSIONS.
The most elementary of these are given in the integrals on pages 293, 301.
J/inen+l<|. /■ sin"'*"^ic
sinicos»xdx = — ^^i^ =■ 6. lsin»a;cosx= -•
n + 1 J « + 1
67. a. Csin^xda; =5- Jsin2a:. 6. fcos^xdz =|+ Jsin 2i.
ca f » J sin»-'a:cosx , n — 1 f ■ „ ., ,
J n n J
ca C ^ J cos»-'a;sini n-1 f „ ., ,
69. I cos"a;da; = 1 I cos" ■'xaa;.
J n n J
70 C dx _ 1 cos a: n — 2 C dx
J sin"z n — 1 sin»-'x n — \J sln»-2x
71 C (to _ 1 sin X n — 2 r dx
J cos"x n — 1 cos"-'x re — 1 J cos"-2x
72. f sec-xdr = ^'^ ^ "^"'"'^ + ^^ Csee-^xdx. (Cf. 71.)
J n — 1 n — lJ
73. Ceosec»xdx ^ _ ""^ ^ cosec-'x ^ n^ (cosec-'xda;. (Cf. 70.)
J n — 1 n — 1.'
74. ('tan''xdx = '^°"''^- ftan'-^xdx.
J re — 1 V
75. Ccot" K di = - ££^2:^ _ fcot-Sx da;.
J re — 1 -/
76. jsin"»x COS" 05^05 = -"'" "^"^ -
»M + n
+ ^^: — i r8in»»»-*a5c«B"«dx.
77. Jsm">x COS" X dx = «fa'"'"^«' '^^""^ ""
»w + 1
+
m+ji+^ C ^lum+i X COS" X dx.
»» + 1 J
"J
sin*" a; cos" x dx ■
sin"»+i X cos»-lx
m + n
79. I sin»» X COS" 35 dx = - -
+ ^ — i- Tsin'" X co8»-2 X dx.
i"+lX
+ m + n + 2 fgi^m pj (5ogn+2 X dx.
w + 1 ^
n+ 1
458
"I
82. fi
DIFFERENTIAL CALCULUS.
80. t sin mx sin nxdx=- "'"('" + "> + ^'" C"* " ")'^.
2 (m + n) 2 (m - n)
oosm*cosnx(ii= sin (m + n)x sin (m - n)x.
2 (m + 11) 2 (m - n)
■„ „„„ J., cos (m + n)z cos(m — n)x
sin mi cos na ox = ^^ ^ ^^ ^ —
2 (m + Ji) 2 (771 - n)
83.
f ^ = _^__tan-i('J?^^ tan?V whena>6
Ja+6cosi Va- — 6'^ \ 'a + 6 2^
V6 + o + Vft — a tan -
■log
V¥
Vb + a — Vb — a tan -
-, when a < 6.
84. f—
(fc
6sinx Va^-fta
1
atan-+ 6
tan-i — , when a>6
atan-+t-V62_a2
log , when o<5.
86. f <^ _ 1 tan-' /j_tanx\
J a^ cos2 X + 6^ sin2 X a6 V a /'
86. f e- sin nx dx = ^C" "'° f - " ""^ "^^ (See Ex. 19, Art. 176.)
87. f e" cos nx dx = ^"(" ^'" "? + ° °°^ '^). (See Ex. 6, Art. 176.)
•' a- -(- «^
y-ginz
y-coix
■y -. Blll-\l y -COS' *«
459
y
;
/
/
/
/
2
0/
77
2
y-ta.
Y
3S-
n a;
/%'f\ r>".
A'
27r
;r
\^^^
^__,,,--^
Js:
^^^
3/= tan
'a?
460
461
The Parabola x'+y' =a^
Tbe Semi-Cubical Parabola,
r
The Cubical Parabola a' y=-x'
Y
The Astroid or Four-Cusped
Hypocy cloid, x '-*- y'^a ^
Asymptote
The CiSBOld of Diocles
u2=-=
The Witch of Agnesi
462
^^ "
Q
The Folium of Descartes
0 X
The Catenary
Aitffmptote ^0 ^
The Exponential Curve
y.e'
The Cycloid
x-a (8-siae),y-al.l-cosd)
The Logarithmic Curve
U-loff .1
Pai'abohi
The CardioiU
»-a(l -cos 6)
403
The Lemniscate, r!.a'cos 2ff, The Curve, r-a sin 2e The Parabolic Spiral
Asymvtott
The Spiral of Aruhlniedes, r^^ag
The Hyperbolic or Beciprocal
Spiral, r g — a
The Lituus or Trumpet,
T'e-a^
The Lo^rithmlc or Equiangular
Spiral, r-e"'^ ovlosr-ad
404
ANSWERS TO THE EXAMPLES.
CHAPTEE I.
Art. 4. 1. 45°, 0°, 63^26' 4", 71° 33' 54", 75° 57' 49", 78° 41' 24",
80°32'16", 82°52'3U", 104°2'11", 99°27'44", 135°, 126°52'.2, n0°33'.3,
2. (.18, .033), (.29, .083), (.5, .25), (.87,-.75), (5.72,32.66), (-1.07,1.15)
(- .35, .12), (- .18, .033), (- .09, .008). 3. [The latter part. ] (a) - -
(6) 21 + 1; (c) 3z=; (d) ^ (e) ^; (/) 1^; {g) ^P; (ft) -^
!/ \by lb!/ 2^ o^
(i) — . 4. a. oc, ± .5774, ± .2582, 0, ± .4045, ± 1.8074 ; 90°, 30° and
150°, 14°28'.7 and 165°31'.3, 0°, 22° 1'.4 and 157°58.'6, 61°2'.7 and 118°57'.3.
6. 27, 12, 3, 0, 6.75, 18.75; 87°52'.7, 85° 14'. 2, 71°3b'.9, 0°, 81° 34'.4.
86°5«'.8. c. », ±1.4142, ± 1, ± .8165, ± .5774, ± .5; 90°, 54°44'.l and
125° 15'.9, 45° and 135°, 39° 14' and 140° 46', 30° and 160°, 26° 34' and 153° 26'.
d. 0, ±.1937, ±.4330, x., ±.0945, ±.3034; 0°, 10°57'.7 and 169°2'.3,
23°24'.8 and 150°3r>'.2, 90°, 5° 24' and 174° 36', 16° 52'.7 and 163°7'.3.
e. 00, ±.8661, ±.8183, ±1.25, ±.9139; 90°, 40°53'.8 and 139°6'.2,
39°17.'6 and 140° 42. '4, 51°20'.4 and 128° 39'.6, 42°25'.4 and 137°84'.6.
6. Where z = ± 2.57 ; where x=± 2.78.
CHAPTER II.
Art. 12. 1. 35.2426 or 26.7574, 23.0186 or 21.1214, 3VsIn5e + —
+ 7 8in2z + 2. 2. 68, 28, 14, 3sin2z - 5slnz + 21. 8. — — 2^. 4. 18 +
2 — 49 a;
SVx + z, 4 + \^2T2. 6. oj^ + bxy + cx^, (a + 6+ c)x\ (o + 6 + c)y^.
CHAPTER III.
Art. 20. 1. (a) 22.977 ; (6) - 4.448. 2. (a) 21.22 ; (6) 40.42 ;
(c) 161.58. 3. (o) .0047 ; (6) - .014. 4. (a) - .0035 ; (6) .0104.
Art. 21. 3. 76.59, 22.24. 4. 212.2, 404.2, 538.6. S. .80756,
- .8023, - .60137, .5959.
Art. 22. 4. (o) 2z, 2x, 2z; (6) Zx^ Sx% 3x^. 6. 4z», 2x + 4,
-i, -■^-3 + 4z. 6. 6«, 12t2-8-| 7. 6y6 3j,_8 + I.
Art. 26. 2. 2 irr times, r being the measure of the radius ; 1.51 sq. in.
per second ; 2.83 sq. in. per second. 3. .866rt times, o being the measure
of the side ; 25.98 and 51.96 sq. in. per second. 4. 4 tt- times, r being the
measure of the radius ; 9.425 and 37.7 cu. in. per second. 6. 5|} mi. per hour.
465
466 DIFFERENTIAL CALCULUS.
Art. 27. 8. Sx^dx, dx, 2 dx, 3 dx, a dx, 2 zdx, Uxdx, etc. 4. 1.6;
1.681. 6. 42^; 43.696. £'!. 5.03 and 9.425 sq. in. £x. 1.3 and 2.6 sq. in.
CHAPTER IV.
Art. 31. 6*2 + 142-10, 2 a; -17, -2 a; + 21.
Art. 32. 4. (5x*-8x3 + 21a;2 + 2x-2)(fcc, •".
^^ 33 J 3 z* - 14 x3 + 6 a:'' 16 a: - 21a' - x<^ - 2 a:' + 44 a: - 96
(3 x2 - 7 X + 2)2 ' (x' + 8)2 ' (2 x2 - 9 X + 3)2 '
(3x<-14x° + 6x2)dx _ 2 a)-—.—
(3x2-7x + 2)2 ' ■■■■ ■ ' 640' 245"
Art. 35. 2. tlCiiillii. 8.
14x3
4 t + 17 3 X + 7
Art. 37. 1. 2 «^, 12 u8^, 63 u^^, 8 x', 12 ifi, 84 x", 27 x^ - 34 x + 10.
dx dx dx
8. 240 x(5 x2 - 10)23, 120 x\3 x« + 2)9, (432 x^ + 300 x> - 168 x2 + 448 x - 50)
(4x2+ 5)'(3x*-2x+ 7)*. 4. - 2«-3u', - 7 «-««', - 11 u'I'm', - 7 X"',
-15x-«, -170X-", -8x(x2-3)-6, -60x8(3x' + 7)-8, 15x4-21x2 +
72-|5 + 5i- 6. i«-i2)«,-f«-ii)«,f«f2)«,Jx4,jxt,|xt, ^ ^^ .
X X- o X V 3 x2 — 6
i^(2x2 + 7^_3)-|, ^ . -|(3x-7)-^, 6x-5x-i-x-^-
** V2X + 7
2x"*+i<Ex"^. 6. V2u^2-ia/^ \/3x^3-i, 5V7x^7-i, 2V5(2x + 5)*'5-i,
V3(6 X + 7) (3 z2 _)- 7 a; _ 4)*^3-'. 7. — + c, and give c any three particular
4
constant values. 8. (In eacli of these expressions ft is to be given any three
a^ 1 2^ 2^ 62
particular constant values.) —+k, — - + k, -x^ + i, -x^ + &, -x5 + -
6 X 3 6 5 X
-2Vx + ft. 12. 6x2 + 34x-61, max™-'- ?i6x-»-i, *^ ~^"
(1 -x2)2' (a + x)
Vl + x2 a:' 3 x2vTT^ (a -6x2)^ (l-x2)^
. mnx»-i(l + X")"-', 12 6x2(a + 6x')', i"-i(l-x)»-»
(1 - x) Vl - x2
[m-(m + n)x], _«Zl1^. 14. «. «1L^, 4 2(^' + ay),
2V'^^ri^ 2/^-« o(3t/2-2x2)
9x2y-8x-14xy2_2y8 -(x + a)y2 _x ft'x.
14x22/ + 6xi/2-3x«-16y' (a + 2/)(62_aj,_2j/2) + 2,(a; + a)2' y a^y
b, — J, I, J, — f . 17. y = x2 + ft, in which ft is an arbitrary constant ;
y = x2 + 1. 18. 5 ft. per second. 19. 10 mi. an hour ; 8j ft. per second.
20. (4,8). 21. 3hr. ;60nii. 22. f ft. per second. 23. 36°62'.2.
24. 36°62'.2.
ANSWEBS. 467
^^ 33 , (6g + 4)log.e 6a; + 4 .434 (Ox + 4) 11
■ ■ 3x-^ + 4x-7' Sx'^ + ix-f S'ji^ + ix-f lUlog.a'
a, .29858. 2. J, .144765. 3. -^, -^—, ^-— , ^ ,
1 - x2 1 _ a;^ (1 - x) vx Vx^ + a-^
— i— , 1 + log X. 4. log (x2 + 3 s + 5) + c, log c (x« - 7 a; - 1), log Vkx,
xlogx
in which c and A; are arbitrary constants. {Ex. Write each of these anti-
derivatives with the arbitrary constant involved in other ways.)
g ,. - (2167 + 1877 X -t- 228 x^) Vx -f 2 ,j, 6(x° - 2)
30(4x-7)T(3x-l-5)? («+ l)2(x + 2)2
91 x' + 475 X + 450
(0
15 (2 X + 6)^ (7 X - 5)5 (x + 3)^
Art. 40. 1. 2xe^, 2..303(10'), 2.303 (6 x ■ lO^"), -i^e*^- 2. 2ea,
2v'x
2.303 (2 «. lO"'), 2«e''+3, 4.606 (lO^f+O. 3. e' x"-! (x + m) , na'" • x»-i log a,
e'(l -x)-l (i_3.)g-x i e'''('2--lV 4. le^ + c, le^' + c,
I 6*"+' -V c., c being an arbitrary constant.
Art. 41. 2. (3x-)-7)"r2xlog(3x-|-7)-|- "^'^^ j, (3x+7)2'riog(3z+7)»
+ -^1, as last, ^(i=M^\x'".x»-H«logx+l),e''.e', - V-*Vlogx,
ox+YJ \ x-" / eVx/
- - log a.
Art. 42. 1. — sin 2 K = cos 2 « • — (2 «) = 2 cos 2 « . — , 3 cos 3 « • Bu,
dx dx dx
J cos ^ M • a', f cos i u — , V cos Y « ■ -D"- 2. Z) sin 2 x = cos 2 x • D (2 x)
= 2 cos 2 X, 3 cos 3 X, J cos ^ x, 6 x cos 3 x^, 3 sin 6 x, 20 x* cos 4 x',
„. . . . . „ , c. . T .t A 2cos2xsin 3x— 3sin 2xcos3x
20 sin* 4 X cos 4 X. 3. 5 cos 6 «, « cos J i^. 4. — — ,
sin 2 X + 2 X cos 2 X, 2 x sin ^x + ^] + x^ cos (a: + j] • 6- 45° and 136°.
6. Where x = mtt ± • 9553, in which n is any integer. 7. 63° 26' and 116°
34'. 8. Where x = nr — -, in which n is any integer ; 54° 44'. 1 and 126°
4
15'. 9 ; where x = nir + -, n being any integer. 9. n cos nx, nx"-' cos x»,
4
n sin"-i X cos X, 2xcos(l+x2), ncos(nx + a), n6x"-'cos(a + tx"),
12sinMxcos4x, ^°°^^-^'"=', £2ill^^^, cotx, e'cos(e') . logx + ^i!^^
X2 X X
10. (a) sin x+c, J sin 3 x + c, ^ sin (2 x + 5) + c, ^ sin (x^ - 1) + c, in
which c is an arbitrary constant. (6) ^ sin 2 x + c, | sin (3 x — 7) + c,
J sin a^ + c, in which c is any constant.
468 DIFFERENTIAL CALCULUS.
Art. 43. 3. Where z = nr, n being an integer; where a;=(4» — 1)-
± . 485, 2 nir - . 485. 6 - - cot ». 6. cot-; 60°. 7. - 2 sin (2 a: + 5),
a 2
- 15 cos2 5 a; sin 6x, 2 x cos x — i^^ sin a;, — ' ^"'^ „, - (m cos nx sin ma;
(1 + cosa;)^
+ ncos7na;sinnz), e™"(l— xsina;), e'"(acos'/na;— msinmx). 8. — cosa;+c,
- 2 cos J a; + c, - ^ cos (3 a; - 2) + c, - ^ cos (a:^ + 4) + c ; c being an arbi-
trary constant.
Art. 44. 3. 2 sec'' 2 u ■ Du, 3 sec^ 3 u ■ Du, m sec^ mu-u',2,nu sec^ na^ • u',
2 sec^ 2 a;, i sec'' ^ x, m sec" mx, 6 x sec" 3 1", 12 x" sec" 4 x^, nmx"-' sec" mx»,
6 tan 3 X sec" 3 X, 12 tan" 4 x sec" 4 x, nm tan»-i mx sec" mx, |tan (|x + 3)sec"
(*x + 3), -i-orcosecx. 4. tanx + c, Jtan-2x + c, Jtan(3x + a) + c
sin X
6. When x is an odd multiple of - and dx is finite.
Art. 48. 1. -2csc2(2x+3), Jsec(Jx+3) tan Qx+3), -3csc(3x-7)
cot(3x-7), 58in(5x + 2), nsec»xtanx. 2. -6cot (3«+ l)csc"(3« + 1),
sec'(J«- l)tan(|«- 1), - Jcsc"| (« + 6) cot§(<-|- 5), -18fcsc"9«",
14(7 «- 2) sec (7 «- 2)" tan (7 -2)".
'-1 1 2 2
Art. 49. 2.
Vl-x"» Vl-2x-a;2 \ + 'J? (1 - 12) Vl - 5 a;^
1 xsin-ix^ iVl + cscx. 4. sin-iz+a, sin-ix"+«,
Vl - x" Vl - x-"
I sin-i i' + a, in which a is an arbitrary constant.
■ 2 jix»-i 2 a
Art. 50. 3.
x"»+l ' l+x"' V2 ax - x-'
Art. 51. 1. ^ , 2 ^/, _l£_, J»L^. 2.
l + 4x"' H-42/"(fo;' l+x*' l + y«(ix l + lGt^'
6xdx 6^ 1-x" 1 1
l + «8' l+9x*dt ' l+x"' l+3x" + x«' Vl - x"' 2(l + x2)'
" ^ " 7. tan-i X + c, Un-i x" + c, i tan-i x< + c.
;" -I- fr"
2(a + 2x)v'x(a + x) a^ + a:-
Art. 52. 2.1^. Art. 53. 2. ^ ~^
Art. 55. 1.
x*-a« a:Vx*^ VT^ Vo^^^ a^Hl
2
l + x""
Art. 56. 2. (3x"!/2+3)dy+(2xy'+2)dx, 3(y2-ax)%+3(x"-ay)(?x, etc.
8 --J^ _-\/^ -f*^'"/'?^'""' y tan X + log sin y _ dx dy
'x' 'x' \al \yl ' log cos X - X cot s/ ' " 2%/x 2v^'
i(S^^)' '"(^ + ^1' (!'tanx + logsin2,)dx-(logcosz
— X cot y) dy.
ANSWERS. 469
Page 77. 1. (i) 24 a;» + 15x^+ 124 x + 55, (ii) a + 6 + 2 x, (iii) (a + z)™"'
(6 + X)"-' [m(6 + X) + n(a + x)], (iv) ('"x - nx + m6 - «a)(^ + a)-'
(x+6)"+'
(y) {m + inx-nz)x'-^^ ^^i^ o^ ^ ^^ii^
(1 + *->""" (a-^ - x^)i (1 + xO^
(viii) ^(^-^) , rix^ -l(i + _J^V (X
2VxV'a + i(Va+ V'x)2 a;^ Vl-x'/
^^.^ a^ + a^.^-4xV 2. (i) ^g^^ + '-'^-l' , (ii) Zl^ifr., (iii) -"
Va^-x'-' 7x*-|-:ix'^-17x + 2 a'-x^ xVo'^-x'^
(iv)secx, (v) ^ ■■ 3. (i)20x*cos4x5, (ii)-78iiil4x, (iii)6sec23xtan3x,
vT+x2
(iv) 8 8ec2(8x + 5), (v) x'»-i(l + mlogx), (vi) ;)?x«-isin''-ix«cosx»
(vii) n(sinx)"-'sin(n + l)x, (viii) cos (sin x) • cos x, (ix) ^^ —
(X) ncotnx. 4. (i) — ^, (ii) — -^ , (iii) r-^- 6. (i) —
^ ■^ ^ x«-l^ "^ tan^x-l M-i* ^ e'+e-
X /■ ^ n /„^ — Va' -
(ii) - 1, (iii) ~-^ . (iv) ; " , . , , (v)
^ ■^ ^ ' .^j _ J.2 cos'x + li-sia-x a + 6cosx
(vi) e~ sin"-' n(a sin rx + mr cos rx), (vii) — log a -a'
(iii) j;.V^i^^5^, (iv) e'V(l + logx), (v)x('').x'{^ + logx+(logx)2}
2 XM* . , 1 , , ^ , , . COS a; (cos !/ + sin y)
(■") -4xY + cosy' Ov)-{"»«ec(x,)-y}, W - .^^ ^ ^.^^ ^ _ ,;„ y) _ i
(^'> eTT^' ^""^ x-^-xylogx- ^^"'^ x(l + nj)- ^^ (l + logx)^'
Vl - x-'
10. (i)2y-J, (ii)8(-ll, (iii) sec X, (iv)-cotz, (v) ^
11. (i) (12 x» + 18 X + 5) (6 x2 + 3), (ii) (e"" ' + 2 tan 0 sec' (, (iii) g,
(iv) f!^. 12. (i)00°, (ii)73°41'.2, (iii) 90°, (iv)2°21'.7, (v)70°31'.7.
14. Speed of Q in inches per second is 116.82, 225, 7, .^19. 18, 390.9, 436,
451.39, 390.9, 225.7, respectively. P- 419
CHAPTER V.
Art. 59. 2. See answer Art. 4, Ex. 3. 4. (i) ± 1, ± i, 45°, 135°,
26° 34', 153° 26'. (ii) 2, 63° 26'. (iii) - f, 146° 19'. (iv) - 1, 1.35°.
(v) IJ, 66°18'.6. (vi) IJ, 56°18'.6. (vii) IJ, 56°18'.6. (viii) -.6667,
146° 19'.
Art. 61. 2. y = x-12, 2y + x + Q = 0, x + y=0, y = 2x-18.
470 DIFFERENTIAL CALCULUS.
3. y + 2.0056 j; + 2.19... =0, !/=4.60o6x- 10.6—, 2.6056 !/= x + 14.6 •••,
X + 4.6056 y = 53.45 •••. 4. (i) Tangents : y = x + 2, x + y + 2 =0, 2y =
x + S, 2y + x+S = 0; normals : y + x = 6, y = x — G, y + 2x = 24:, y = 2 x
- 24 ; (ii) y=2x-S, 2y+x = 2i; (iii) Sy+2x = 13, 2y = Sx;
(iv)x + y = e,x = y; (v) 2 ;/= 3a; - 3, 3y + 2x = 15; (vi)2y =3x,3y
+ 2x = 13; (vii) 2i/=3a;-10,3!/ + 2x = 24; (viii) 3 y + 2 x = 24,
2 y = 3 X - 10.
Art. 62. 1. The lengths of the subnormal, subtangent, tangent, and
normal, are respectively : (1) 3, 5^, 6^, 5 ; (2) 4, 4, 5.60, 5.66; (3) -^'
- "" ~ ^'^ , — V(n--i-Xi2)(rt--e--^i^) , ^'^"•'-^•''^1^, e being the eccentricity ;
Xi Xi a
(4) sin xi, cos x,, tan Xi, tan Xi Vl + cos'^ Xi, sin Xi Vl + cos^ xi ; (5) j/i^, 1,
vT+l/I^, j/ivTTy?. 2. Where X is 7 ± 2V6. 3. Infinitely great.
6. xxi"^ + yyC^ = ai 7. xxi"^ + yyi"^ = a^. 8. a sin e,2a sin^- tan -'
2 a sin ^ , 2 a sin - tan - . 12. 90°, 0°, cot-i 4*, i.e. 32° 12'. 5.
2 2 2
Art. 64. 1. n) a,aff^,aVTTT^,rVTT^- (2) ^,2re,^Vrf+l=i
2 )• 2
aVe(T+T¥); (3)--, -a, - Va2 + r\ - VaT+T' ; (4) 7ia?»-i, ^^,
a a n
off"-' VnH^, -VnM^- 3. ar, - , r Vl + a-, - Vl + a-. 4. (a) V =
n a a
34°55'.2, 0 = 74°55'.2; i/- = 50°41'.9, 120°41'.9; (6) v^ = 26° 33'.9,
<p = 65° 12'.8.
Art. 65. 1. In feet per second: 0, 4; 2.828, 2.828; 3.5", 1.79; 3.7/,
1.33. Solution for x = 2 : Where x = 2, the tangent to the parabola has
a slope 1. Accordingly, the moving point is there going in a direction
which is at angle 45° to the z-axis. Hence, the speed of the x-coordinate
(i.e. — ^ = — X cos 45° = 4 X — ; also ^ = 4 x — .1 2. 20 and 22.36 ft.
\ dt) dt ^-2 dt V2 J
per sec. Suggestion .- Difierentiation with respect to the time gives
2«^2? = 4 — .1 3. .399 and - 9.97 ft. per sec. ; 9.7 and — 2.425 ft. per sec.
4. 442.82 and 161.2 ft. per sec. ; 199.15 and 427.08 ft. per sec. 6. (1) (2, 8"),
(-2, - 8); (2) (i, ,h)- (- h - ^y, (3) 300.
ANSWERS. 471
Art. 66. 3. 25.1 cu. in. ; ■^. 4. 4 jrr^ . Ar ; 50.3 sq. in., 502.7 cu. in. ;
jk- lb. i§5- 5. 1.35 sq. in. ; 7:5 approximately.
Art. 66a. 2. CI) - 1, - 1, i ; (2) - J, - §, - 1 ; (3) 2, 2, 3, 4;
(4) - J, - ■, 3, - f ; (5) 2, 2, - 3, - 3, 1. 3. n»j--^ = 4p«(« - 2)»-2.
Art. 67. 4. 1.6, .4. 6. J r^, i.e. 2 e" ; .0048, .035. 7. .0.349, 0, .0025.
9, J° + ^, > + »^; .(/? 10. 2.41, .1. 11. aV^+J','^^/'^^TVK
'X ^ a 'x o^
12. .078. 14. irx', irz2. 15. 5 03, 10.05. 18. 10.37, 5.06. 19. J^^^
' a- — x-'
^ Vo'^ - x^ '•'''''("' - ^^) . 2jr6 v^2313^, e being the eccentricity. 20. ^
a a^ a r <
a, r cosec a, V'2 ar.
CHAPTER VI.
Art. 68. 1. (i)_— 2--; (ii) 8+4 + ^=; 0") "'"* '
(1 + x2)2 ^ ' a;3 4 y-^ ' ^ ' (1 - sin x)^
00 -C-IJ5_L5^. ..«_il;ca).8..nx. e.(i)-^,
12. 24i. IS. ^ = f x2 ^. 2 a; + ci, y = ^x» + x^ + cix + cj, in which ci and
ax
Ci are arbitrary constants. 14. 8 «/ = x' — 9 x + 19. 16. y = 4 x' + x.
16. (2) '— I ft. per sec' per sec. 17. In ' in. per sec' per sec . (i) 1152 ir^,
(ii) 768 ir2, (ili) 384 tt^ (iv) 0. 18. s = J jri^ + Ci< + C2. 19. 15.5 sec,
3881.9 ft. 20. -^^ sec.
Art. 69. 2. e', a' (log a)", a"*", 6"a'' (log a)». 4. cos/^z + — V
• / , »t\ , / , n7r\ . (-l)"-'(n-l")! (-1)"-12 ■ (n -1)!
a''sin(ax+ — ), a»cos ax+— - )• 6. ^ ^ =^ —,^ ^- :^- >—
\ 2J V 2/ x» (a;-2)»
(- l)"n ! (- l)"7i! ^ 2- nl (- l)»ac"(OT + w -1)1
xn+i ' (1 .). a;)»+i' (3 _ x)»+i' (6 + fx)"+«
I (1 + x)"+» (1 - x)»+' / l(l-x)»+i (l+x)-+iJ
Ai+ "71 2 g + ft COS g 6 + a COS tf
■ ■ 6sine ' b^ain^e
Art. 12. 2. (x*-120x2 + 120)xsinx-20(x2-12)x2cosx. 3. (x+n)e',
2»-i(» + 2x)e2».
Art. 73. 3. (1) y' = X!^"; (2) xY' + 2y = 2xj/' ; (.3) y' + 2 zj/" = 0 ;
(4) (x2 -2 j/2)y2' - 4 xyy' - x2 = 0 ; (5) yy' = x(y!/" + y"-'). 4. (1) y' = 0 ;
(2) y = xy'; (3) y" = 0 ; (4) y" = j/ ; (5) y" = m^y ; (6) y" + m^j^ = 0 ;
(7) y" + m'j/ = 0. 5. y2(l + y^') = r^ ; i2(l + y^') = rV i {1 + y"}^ = ry".
472 DIFFERENTIAL CALCULUS.
CHAPTER VII
Art. 76. 4. A minimum ; neither a maximum nor a minimum. 8. See
Ex.3. 12. See Ex. .3. 13. (1) Min. for a; = J ; max. for a; =- 2. (2) Min.
at -1-"^^ ; max. at zd±J^, (3) Max. for a; = 0 : min. for a;= ^~ ;
6 6 ^ ^ ' 12 '
min. for x = "*" — : for z = 2, neither a max. nor a min. (4) Max. for
12 ^
x = — \; min. for x=\; neither a max. nor a min. for a; = 2. (5) Min. for
x = i. (6) Max. wlien a; =— 4, and when x = 3 ; min. when a; = — 3, and
when a; = 4. (7) Min. for a; = 16 ; max. for a: = 4 ; neither for x = 10.
(8) Max. for a; = — 10 ; min. for a; = — 2 ; neither for z = 2. (9) Min.
value is +-, i.e. +.3678. (10) Max. when z = e. (11) Max. value = 8;
e
min. value = 2. (12) Max. or min. when sin x = v'l according as the angle
z is in the first or the second quadrant. (13) Max. when z = cotz.
16. {ay/2,ay/i).
Art. 77. 7. Each factor = Vthe number. 8. -• 9. A square.
10. (i) {a^ + hh^; (ii) a + 2y/ab + b ; (iii) 2 ah. 11. Let the perpen-
diculars drawn from A and B to MN meet MN in B and S respectively ;
then (1) BC = CS; (2) BC = ^^^. 12. (i) fr; (ii) fr. 13. 19°28'.
14. (i) Vol.=.5773 vol. of sphere; (ii) height=r'y/2. 16. (i) Vol. =5VTa-6 ;
(ii) height = i 6. 16. 1. 17. 2s i.e. 114° 35' 29" .6. 18. Vfa. 19. 1:2.
22. IJ times the rate of the current. 23. —d,—d. 24. (a* + 6*)^.
« 3 3
26. -S-.
V2
Art. 78. 1. (1) (0, 0) ; (2) (3, - 3) ; (3) (f, W) ! W (2, f) ;
(5) ( ± -^' - ) ; (6) where x = 0, and where z = ± VS ; (7) where z = 0,
and where z = ±2-v/3. 2. (1) Where z = — ; (2) where x = — ; (3) where
5 4
sc = ±-^; (4) (c, 6); (5) (c, m); (6) (&, ^)-
CHAPTER VIII.
Art. 79. 2. 3z2+e'sii)y, 4;/ + e' cosy — cos 2 sin y, 6 z — sin 2 cos y.
8. (a) -Ili^ and ^|L; (6) ^li^ and ^^4 . ^^^ ^-2^ ^„^ - 4.3
4-\/ll9 5^119' 3VS9 5V89' 3V47 4V37'
respectively.
QQQ 1 CO
Art. 81. 3. Increasing :::^ units per second. 4. Decreasing
units per second.
20V119 5-\/89
ANSWERS. 473
Art. 82. 3. .030;. 036011. 4. (i) ^L!!l=l^- (ii) yz logy . dx+xy-^du;
x'^ + y-
(iii) yxy-^dx-\-x'\osx-dy; (iv) ^dx: + \osx ■ dy ; (v) uO-^^dx+^-^^^dy]-
^ \ X y j
6. .025. 6. 2.2; 2.37. 7. .0017. i. »"-\yzdx-\-zx\osx ■ dy + xy\ogx ■ dz).
Art. 83. 3. 4.72 sq. in.
CHAPTER IX.
Art. 90.
V \dy^l dydy>f ' U,./
Art. 93. \.{f'{f><t>"' - t>'f"')-f"{f'<t>" - 'P'f'<)\-^pi. 2. -4asin?.
.•) 2
3. - a. 4. - (o'^ siii^ 9 + 6'^ cos- e)'^ ^ ab.
Pagel47. i. (i) !^-2 2,^ = 0. (ii) '^'^ + ^'^ = 0. 2. ^-2('!^V
rfi/-' di/ di/'* dy^ dx'- \dxj
= cos2^. 3. (i) ^?'-+« = 0. (ii)^+2, = 0. (iii) ^=0. (iv) ^
dy^ dt- *2 dz^
+ a^y = 0. (v)^+y = 0. (vi) ^ + 2^'?(+ y = 2. 4. (i) tan « ;
. (ii) — 3sin*t cost; 3sin'<(4— 5sin-<").
at cos-' t
CHAPTER X.
Art. 95. 1. (1) First order at (1, 1); (2) second order at (2, 8).
2. y =60;"-- Ox + 3. 3.-1. 4. j/ = 3a;2-3i + 1. 5. y = a;^ - 3 x + .3.
Art. 96. 1. 5.27 and (- 4, f); 2.635 and (- ?, \\). 2. B = 145.5 ;
(-m.aoj'j).
Art. 100. 1. The curvature ot y =x^ is one-half that of y=6 x^ — 9x-|-4.
2. — ; iJ=-88.4; (-87.5, - 12.r)).
125 ^
3
Art. 101. 3. ^P + ^y'; f2p-|-8x, — y^-\; 2;) and (2p, 0).
pi \ 4PV
4 (6^x^+«VlI = (ai^W. Centre at f^^ll^x', -"i^^^V
^ -* aW ~ a6 ' la*' M ^ j"
(4) - 3(axy)i ; (x -I- 3 v^, y+Z i/n^y). (5) 3 a sin 9 cos e ; (n cos« «
3 1
-|-3acos«sin2«, asin3«+3asin<oos2 0- (6) C4a-f9x)^x- . / _^_,x^^
0 a \ a
474 DIFFERENTIAL CALCULUS.
Ij' + t— V (V) -2ai (a, -ia). (8) ± 4 a sin^ ■ (a- 9 + sine, - y).
6. (1) (£±lll. (2) (a* + 9^r. (3)csecg. (4)ia. (5) 2 a cosec^ ,//.
2 y^ C a'la; c
i.e. ?_ •
a
Art. 102. 1. (i)a; 6. (2)2^-^. (3) f vTSF. (4) -^. (5) ±|^ .
(61 .VIT^. (7) ±<1J^. (8) ±«»"-'^"^ + »')^
Art. 103. 3. (1) (axy-(by)i=(a^+b^)i (2) (k + y)^ -(a; -t/^i
= (4a)i [Suggestion: Show that a +;3 = ?(!? + ^y, a-p = -(---Y,
2\x aj '2\x a/
and deduce therefrom.] (3) (z + y)^ +(x — yy = 2a^.
CHAPTER XI.
Art. 108. 2. J, i. 3. The tangent at the middle point of a parabolic arc
is parallel to the chord of the arc. 4. — 3 ± V^ ; find the abscissa of the
point where the tangent is parallel to the chord joining the points whose
abscissas are 3 and 4.
Art. 109. 1. - 3.69 ■•■, .51 ••■, 3.18 •■-. 2. -4.03293,1.2556,1.77733.
3. 2.8.58,-3.907. 4. 2.34. 5. 2.046. 6. -3.806. 7. 2.129.
8. 2.216, - .5392, -1.676. 9.3.693. 10. 1.4231,-0.6696. 11. 2.195823.
CHAPTER XIII.
Art. 123. 6. (1) a;2 + j/2 = a2. (2) b^x'' + a'^y^ = a'^bK (3) iay'^ + bxy
+ cx2 = 4ac-62. (4) 4 ij/ + a^ = 0. (5) 4 1/'* = 27 a^i. (6) (x - o)^
+ (j,-6)2=r2. 6. (1) x2 + !/^ = n'^. (2)x^-y''=a\ (3) (ax)^ + (by)^
= (a2 _ ft2) |. 7. (i) The lines x±y = 0; (2) 27 cy'^ = 4 x'. 10. A parabola ;
y^ =4 ax if the fixed point be (a, 0) and the fixed line be the y-axis.
Art. 124. 3. 4xy = a'^ 4. x^ + y^ =aK 5. a;^ + ?/' = a^.
Art. 126. i. (1) x = a,y-b. (2)k = 2. (3) y + 3 = 0, 2x + 3 = 0.
(4) 2,+ 1=0. 6. (2) (2, f). (3) {-i, -3), (- f, -Y). (4) (-i, -1).
8. (1) x = 0, j/ = 0. (2)x = 2a. (3) !/ = 0. (i) x = ±a, y = ±b. (5) y=0,
x = a. (6) x = 0. (7) y = 0. (8) !/ = 0. (9) a; = (± 2 n + 1) ?, in which
n is any integer.
ANSWERS. 475
Alt. 127. 2. bx±ay = 0. 5.(l)y=x. (;i) x + y = l, x- y = 1.
(3) x=2, j/ + 3=0, 2(2^-a:)=5. (i) x=y±l, x+y -±1. {b)0y = 3x + 2.
Art. 128. 2. (1) Lines parallel to the initial line and at a distance
± nav from it, n being an integer. (2) The line perpendicular to the initial
line at a distance a to the left of the pole. (3) The two lines which are
parallel to the initial line and are at a distance 2 a from it. 4. r sin (9—1) = 1 ;
?•=!.
Art. 131. 3. (1) Node at origin; slopes there are ±1. (2) Cusp at
(— 3, 1) ; slope there is 0. (3) Cusp at (2, 1) ; tangent there is parallel to
the y-axis. (4) Double point at (0, 0) ; slopes of tangent there are 1, — f .
(5) Cusp at (1, 2) ; slope of tangent there is 1. (6; A conjugate point at
(3, 0).
CHAPTER XIV.
Art. 136. 1. 50 ft., N. 5.3° 8' E. 2. 51.96 ft., W. 8. 58.8 ft.,
N. 16° 3' E. 4. 9.39 ..., 3.42 ..., and 5.77 ... miles respectively.
Art. 137. 1. 9.83 ... and 6.88 ... ft. respectively. 2. 25.5 ft., 43° 17'
(nearly) to the horizon. 3. 228.8 ft., 377.3 ft. 4. 258.3 inclined at an
angle (— 63° 27') to the given displacement.
Art. 138. 1. 2.236 ft. per sec, E. 26° 34' S. 2. 20.47 miles per hour,
N. 42° 3' E. 3. At an angle 60° to the river bank. 4. 11.37 and 3.84
miles per hour, respectively . 5. 21.56 mi. per hour toward the south,
24.18 mi. per hour toward the west.
Art. 139. 2. 4 ft. per sec; 3.79 ft. per sec; 3.79 ft. per sec; 3.84 ft. per
sec; 21.21 ft. per sec. 3. Decreasing 9.7 ft. per sec; increasing 8.77 ft. per sec
4. (a) a sin 6 — (in which — is the rate at which the radius vector is re-
^ dt dt
/Iff ft fiff
volving) ; (6) a (1 — cos 9) — ; (p) 2 a sin - — in a direction making an
dt 2 dt
a Off
angle - with the radius vector (i.e. '■ — with the initial line). At the points
(1), (2), (3), (4), the values of (n), (6), (c) are respectively, in inches per
sec.: (1) 6.236,5.236,7.405; (2) 4.53, 2.62, 5.236; (3) 4.53, 7.85, 9.07;
(4) 0, 10.47, 10.47.
Art. 140. 1. 14.8 mi. per hour, N. 37° 50' E.; 4.9 mi. per hour, per
hourN. 37° 50' E.; 9.08, 11.7, 3, 3.87 mi. hr. units. 2. 98.9 ft. per sec.
per sec; 58.1 ft. per sec. per sec.
Art. 141. 2. — acos 9 (— ) , in which e denotes the angle from the
(f)''
horizontal diameter to the radius drawn to P. 3. (a) 7047.75 ir^ ft. per
sec. per sec. ; (6) 3169.5 r^ ft. per sec. per sec 4. 10.89 ft. per sec. per sec.
6. (a) 6 in. per sec. per sec; (6) 2 in. per sec. per sec; (c) 6.32 in. per
sec. per sec, in a direction inclined at 71° 34' to the normal. 6. (a) 2 in.
476 DIFFERENTIAL CALCULUS.
per sec. per sec; (6) 1.61 in. per sec. per sec. directed toward the centre of
curvature for P; (c) 2.56 in. per sec. per sec. in a direction making an
angle (— 38'^ 50') with the tangent at P. 7. Wholly normal, 1.61 in. per
sec. per sec. 9. Vel. =4.4 ft. per sec; at = 8.383 ft per sec. per sec. ;
tin = 4.84 ft. per sec. per sec. ; a = 9.68 ft. per sec. per sec
CHAPTER XV.
Art. 146. S. (1) Convergent. (2) Convergent. (3) Divergent.
(4) Divergent except when J) > 2. (5) Convergent if p > 2. 4. (1) a; < 1,
convergent; i>l, or 2 = 1, divergent. (2) Absolutely convergent if
22< 1, divergent if i^ = i^ divergent if x'' > 1. (3) Absolutely convergent
for all values of i. (4) i < 1, or j- = 1, convergent; x>l, divergent.
(5) Same as in Ex. (4). (6) Same as in Ex. (3).
CHAPTER XVI.
cos X +
2! 3!
^2 hZ
Art. ISO. 8. (a) cosa: — Asini cosa: + — sini + •••; (6) cos A
— I sin ft cos ft H — sin ft + •••.
2! 3!
Art. 151. 4. e + e(3;-l) +;^(x-l)2 + ■•..
Art. 152. 10. (1) 1+ — + — + 5il£!+ ...; (2) 5! + 5i + 5i+....
^ 2! 4!G! ^"^212 45
12. (1) c + x + f-|2.'-?if^-?^%?B_^+...; (2) log^ + (6-a)
2-2! 3.31 ^ 1-3 1 •2-5 1-2-3-7
CHAPTER XVII.
Art. 162. 2. (a) 6 X + 2/ + 3 3 = 19, ?-=-? = y-i= 5_=i ; (&) Zx-
O o
6y + 72 + 19=0, 2x + y + 2 = 0 and 7i-33 + 27=0; (c) 4a: + 8j;
-3z + 6 = 0, 2s- y = 20 and 3 a; + 43 = 32; (d) 4 x - 18 j( - z = 31,
9x + 2y = 12anda; + 4z = 126; (e) 3x + y + 2z = 0, x = 3yand2!/=2;
(/) 2x + y — 42=4, x = 2y and 4 j/ + 2 = 20.
Art. 163. 2. 2x+ 12y -92 + 48 =0, 6 x- j/ = 32 and 9x + 2 2 = 78;
(a) 2 1 + 12 y - 9 2 + 48 = 0 and 3 X - 2 y + 2 = 22, 6 X + 29 «/ + 40 z = 632 ;
(6) 2 X + 12 y - 9 2 + 48 = 0 and 4x + y-Sz + S = 0, 27x + 30j/ + 46 2
= 834. 3. x-2!/ — 2 + 5 = 0, x = 2 + 3 and y = 2z; (a) x — 2 y
-2 + 5 = 0 and 7x-2y-2 = 25, y = 2z; (6)x-2y-2 + 5=0 and
2x + Zy + z = 2i; x-3y + 72 = 7. 4. 8x - 27 y - 24 2 = 122,
ANSWERS. 477
3 z + 2 = 15 and 8 y - 9 z + 75 = 0 ; (a) 8 z - 27 j/ — 24 z = 122 and 3 x -
2y-Sz = 15, 332-48y + 65z = 61J; (6) 8z - 27 y - 24 z = 122 and
or + 2 y + 4 z = 4, 60 x + 56 y - 43 z + 225 = 0. 5. 13 x + 30 !/ = 198 and
32 y + 39 z = 696, 90x- 39y - 32 z = 0. 6. t/ + 4 z = 24 and 9 z = z + 43,
x—4y + 9z = 7.
CHAPTER XVIII.
Art. 167. 3. y =x'; y = z' - 347 ; y = x^ + 5li ; y - k = :i^ - h'.
4. y = i X + c ; y = i X ; y = i X — 5 ; y = 4 X + 29. 5. y = ix^ + c ; y = 4 x'^;
y = 4x'~2;y = 4x'i-13;y = ix-62. 8. 16 «2; 64 ; 256; 400 ; 16 «2 + lO,
etc. ; 16 12 + 20.
Art. 168. 3. |. 4. 2 ; 0. 5. 4 ; 0.
Art. 170. 4. (a) 2 2/ = z2, 6y = z^ 24 j/ = z*; (6) t/ = z2 + 5 a:, 63/=2z»
+ 15*2, 12;/= z^ + lOz'; (c) !/ = 1 — cosz, y =3; — sinz, 2y = x^-\-2 cosz— 2;
((Z)y = e='-l,!/ = e'-z-l,2y = 2e^-i2_2x-2. 6. y = l, !/=2,
y = cos a;, y = e*.
CHAPTER XIX.
Art. 174. 9. Jz« + c, ^jz'' + c, ^jX*^+c, -fz-i8 + c, -/^z-w+c,
-^+ c, -^ + c, Jz^ + c, — -L_z*^-+' + c, |z* + c, Jz* + c, 8 Vz + c,
2 z-! z* .y/2 + 1
-^ + c,-~ + c. 10. i„4+c, ^5'<5 + C, ^ + c, 12s* + c.
Vz 14 ^ 2 u*
m+n -, m+3 , 6+n ^
11. X " +c, 1 ' +k, V " +c he. 12. log CO,
n» + n m + 3 6 + 71 t + s
logc(s + 2)2, - Jlogc(7-z«), logc(4«'-3f + 11). 13. e' + c, fe^' + c,
2e«' + c, ^ l-c, )-c. 14. — icos3z+c, *sin7z + c, I tan5z + c,
log 4 2 log 10
— cos (z + a) + c, i sin (2 z + a) + c, f tan r— + - ] + c. 16. ^ sec 2 z +c,
Jsecfz + c, sin-'t + c, Jsin-iz^+c, ^sin-i5z+c, fsin-'z'' + c, log(»+ vT+c')
+ c, ^taxrH'^ + c, tan-i2z + c, sec^t + c, sec-i3z + c, Jsec~iz2+r,
^ vers-i3z + c or ^ sin-'(3z — 1) + c, Jveis-'4z + c or Jsin-i (4z — 1) + f
16.
•-f<3+16« + c, o^z + V-aV + ?a^z'« + A -'^+0, .?Lgi%o,
- sin az cos nx + c.
a n
[In the following integrals the arbitrary constant of integration is omitted.]
Art. 175. 11. isin^z, ^22L5 (3 + 2 tan^z), -jtan(4-7z), -\e-^.
12. log(z+l) + ^i|^^,|+3z+31ogz-l, 3 (z+2)*(z-8), t't(x-2)^
478
(2z + 3).
DIFFERENTIAL CALCULUS.
IS. i(x+a)i ii!?L±iEl!, -^Vs^ry^, K4 + 6t,)«.
o n
14. le»+««, -i^, log(tan-iz), -cos(logz). 16. A(< - 1)H3 « + 2),
n 3 log 4
— {a+by)^, ^(j»+2)^, f sin|i. 16. Jsmz(3-sin2x), |taiia;(tan2x+3),
5 h
^cosSx-cosz-^cos^z, ntan/^-V 17. - } log (3 + 7 cos z),
-ilog(9-2sinz), -|V4-3tanz, J-sin-i (^^'^"'V 18. VoM^^
5 3 1 \/3 \ V7 /
VO^^^
Art. 176. 7. — (az-1). 8. -(z + l)e-». 9. ae°(z2 -2ax + 2 a^).
10. zlogz-z. 11. iz2(logi-J). 12. ^z'(31ogz-l). IS. ztan-^z
— log Vl + z^. 14. J (1 + ^''') ta''"' z — J z. 16. 2 cos x + 2 z sin z — z^ cos z.
16. e'[z" — mz™-' + TO(m — l)z'"-^ — ••• + (— l)'"-'m(TO — 1) ••• 3 • 2 • z
+ (-l)"'-ml]. 17. - Jzcos2z + Jsin2z. 18. - Vl - z^ • sin-iz + z.
Art. 177. 7. ^- tan-i 5-±^ ; sin-> ?^ ; log (z + 3 + y/^fi+Wx+W).
2v^ 2v^ V2B
8. \\ogl±l- sin-i^i^; log(2z -5 + 2 Vx^ -5z + 7). 9. -^^
1-1 V53 V33
log
2Z + 5-V33. 1 J 2z + 5-\/6T. ,
2 z + 5 + VSS ' v^ 2 z + 5 + Vel
[ log (8 z - 3 + 4x^4 z2-3 z+ 5).
10. _2_j^„.,8z-5.
^/71 -v/tT
J sin
., 8 z + 5
13 '
1 log VT37 + 5 + 8z
V137
/l,37-5-8z
11. vers-i5 and sin-i 5^li ; 4 vers-' — and i sin-i ^^-=-^ ; J^sec-i^.
4 4^ 9 9^' 5
12. isec-i^jzi; >/'z\/9^^ + 9sin-if'); V^- ". zV9 - z2 + 9sin-i|;
|logUn^^+jV Jlogtani^^. 14. ilogsec(3z+«); ^ log sin (4 1^ +«'');
Art. 178. 3. log (z + 3)2 +
z-1
6z
5. log(z2 + 4)2(x-l)';
,^„^|^y_7^^^.,| g l^g|^«. 7. iog(2z + 5)(z-7)3.
8. iz'-2z + log(^ + V'- 9- Jz2 + log:5:^5izii. 10. log^ '^ ^ +
z — 1 z (z — 1)2
|log(2z+5). 11. log i2i:£K£±5l. 12. log(z-3)2(z+3)3(z-2)(z+2)«.
18. log (z-1) ? — 14. logV4z + 5+ ^ „. 18. logz +
z-1
4(4z + 6)
ANSWERS. 479
f log (2 z + 5) + ?. 16. log (z + 4y v'37+2 + "^ • 17. log (a; + 1)2
X o (o x-\-2j
+ -^rr&2 "• log »: - ^ tan-i-f . 19. f log (3 2 - 2) - J log (a;^ + 5)
_J_tan-i-^. 20. logs + 2tan->a;. 21. z + 4 log ?i±-5 - VS tan-i -^.
22. logi2 + V3tan-i-?- 23. log z3(a2 + 3)2. 24. 2 logi -?- 2 tan-i5.
V3 I 2
26. log „ ^~^ , + J tan-i 2^:^. 26. tan-i x + log vzMH ^ —
i2 — 2z+5 2 a:2 4-l
Art. 179. 4. e^ cos y ; x' + ix^y + ix — Q y. 6. cos z tan y — sin z ;
2/2
ze" — 2 zj/ + z2 ; 3 z — 2 z^ — zy — 2-.
Page 311. 1. -i^^ +C, i22<«+») + C, '^ + « gn-K+3 _!■ c,
y/2 + m + l TO + « + 3
-^r' + c, -12T?i + 291og|, ^ + 8»-flog(«2 + 3)-llv^Un-i-^ + c,
f -2z + |log(z2-2) --5_log2^i:!^+c, _I_tan-i -51-+C,
^ 2V2Z + V2 6-^/5 2\/5
-I^ log ^~^^ + c, 7 a* + if - a' + ^F. i sin-i ^ + c, i log (z-' +
4 V3 0 + 2 V 3 ^
Vz« - 9) + c, Jz+__i— -+Jlog(22-l) + c. 2. llogsecCmz
O (2 0 — 1) TO
+ n) + c, |tan3z + |log (sec3z + tanSx) +4z + c, co, 2.4288.
3. X cos-i X - Vl -x^ + c, a; sec-' x - log (x + Vx'^ - 1) + c, x cot-i x
z
+ i log (1 + z2) + c, z{(log z)2 - 2 logz + 2} + c, - oe"»(z2 + 2 az + 2 o2) + c,
-(z' + 3z2 + 6z + 6)e-' + c, cosz(l -logcosz) + c, ^""^' (logz- ^ \
m + l\ ,TO + 1/
+ c. 4. |z*-f\/z + c, 18(fz^ + ^z^ + ^zi + z^) + 91og?-^:ii + c,
, , qt 1 , , z* + l
4 (3* _ 2*) + 4 log 5_=-i, Vx2-1 + log (X + Vz2^n;) + c. 8. .206 (the
2^-1
base being 10), i(l-^). i(e'-l)' -t'A'^'- 6. -^log(m + ncose) + c,
1 f ■ a^ 9\ , 1 i„ *„ /z , 7r\ , secz , , secx-2
log sine tan- + c, — log tan [- + -] + c, — ; — A log
\ ^) V2 V2 8/ 8(sec2x-4) " secz + 2
+ c (see restUt in Ex. 3, Art. 118), sin"' (^:ELl\ + c, ~ log8 {mz + n) +c,
\ 2 / 3 TO
— 1— sec-i«^+c, tan-ie^ + c, Jlog^^^l^+c, J log f+^'li^ + c,
TO log a m *€' + €-" ' 1 - tan 2 9
4 \/2sln-i( V2sin5 )+ c, cos z cos ?/ — 2/2 + a; + c, cosx slny + z — !/ + c.
480 DIFFERENTIAL CALCULUS.
CHAPTER XX.
Art. 181. 6. (6) 76. 6. 18. 8. 5. 11. ^V5. 18. (a) 2; {d) 4.
16. .802025; 6.644025; 1.8564; .401. 17. (1) ,'i; (2)10|; (3) 3.2;
(4) 68^^; (5) \a^; (0) \2y/Z; (7) No area is bounded ; (8) (a) log 7, i.e.
1.946; log 15, i.e. 2.708; log re ; i^log-. 18. W^^l-
a
Art. 182. 9. W'f- 10. -'-V*T. 11. ip^-. 12. (a) f(2v'4-l))r;
(6)^4-2/2-1)^. 13. W- 18. 405^1 -|\x, 225(1 - 1) TT.
Art. 183. 2. y2 _ 48 a; _ 80 ; 24. 3. a; - 4 = 2 log !/. 4. a - 4 = 4 log !/ ;
4. 5. 3!/2 = i6 2. 6. 5!^2_48x2- 112; the conies ?/2 = fcr^ + c, ft and c
denoting arbitrary constants. 7. 3 y = z^ ^ 6 ; the parabolas y = kx^ + c,
ft and c being arbitrary constants. 8. y- = 1 x + i; the parabolas
y'^ = Ar + c, ft and c being any constants. 9. The circles r = c sin 8 ;
»• = 4 sin 9. 10. 1^ = ce' ; r'-^ = 4 e'. 11. r = a{\ — cos S), in which a is an
arbitrary constant.
CHAPTEE XXI.
Art.186. 1. |v^(VJ-3)+4tan-iv^+c. 2. 2(\/a-tan-iVx)+c.
3. j(3x-2)^- / +c. 4. ^(2 + x)^(5x + 17)+c.
3 V3 a; - 2
6. -|log(7 + 5\/2 -x)+c. 6. a;+ 1 +4Vx + 1 + 4 1og(Vm - l) + c.
Art. 187. 5. i V4a;2 + 6a; + 11 + J log (2 x + J + \/4ir'^ + 0z+ 11) + c.
1
8. 3sin i£±2_Alog^^S=V^^^_ 8_ log £3l±^^!±^ + c.
4 V3 VU-Sz+VO+x x+l + v^Hx+l
10. Vr^ + a: + 1 + 5 log(j; + J + VP"+TT1) - 3 log''" ^ + ^^'"'"^"'"^ + c.
r + 2 , ' ' 2+1+VsH^+l
11. ^sec-' 1- c.
Art. 190. 2 |cos'a-— COS! + c; sin a;— J sin' a; + c ; f cos' a; — J cos^ a;
— cosa; + c. 4. (1) f cos*a;(cos^a; — 4)+c; (2) 5sin*a;(J— ^sin2a;+j'jSin%) + c;
(3) 2Vsinz (1 — |sin2x + |sin*a;)+ c; (4) 3 cos^a;(j'r cos-x — i)+ e.
7. (1) Jtan^a;+tanx + f; (2) — Jcot'a; — cotx + c; (3) i tan^x+f tan'x+tanx + c.
9. (1) ^Jj tan' x(3tan2x + .'))+ c; (2) 2 tan^xC^ + ^ tan^x + -^ tan«x)+ c ;
(3) I tan^x(^ + j tan'x)+ c; (4) sec'x(^ sec'x - ^ sec^x + ^)+ c;
(5) § Vcscx(5 — esc2x)+ c; (6) — csc'x(} esc* x - « csc'^x + ^)+ c .
ANSWERS. 481
Art.191. 3. 0)}(^-sin2x + 5HLt?Ur; (2) A(5:c + 4 sin 2x
- J sin' 2 a; + } sin 4 I) + c ; (3) -^ - ^'"'*^ _ ^'"^ ^ ^ + c ■
16 64 48 '
(4) ,>,oos2x(cos»22-3) + c; (5) ^i,^3i - sin4z +?ilLi^W c.
Art. 192. 1, (i)_sinx^cosx^|_^^. ^^^ _ jsin^xcosx - |cosx + c ;
(S) -""''?''"' (sin°» + j) + |x + c; (4) -Jsin^xcosx- i^2S£(sin2z + 2) + c.
* 16
;;. >l)-cjtx + c;(2)iloglaa ?- J cot ar cscx + c; (3) -1.5°i^ - |cotx + c.
^ sin' I
6. (1) isinxcosx(2cos2x + 3) + tx+c; (2) ^ sin x(cos*x + J cos2x + 5) + c ;
^^^^^^-3 +l'^PX+c; Witanxsec3x + |secxtanx+|log(secx+tanx) + c.
COS X /I
6. (1) ^ tan X sec X + J log ton ( I + ? ) + c ; (2) i ton x (sec^x + 2) + c ;
f \4 2/ ^ , .^
(3) i ton xsecSx + f | tan x sec x + log tan ( ^ + 1 j j. + c. 7. (1) i log ton -
- J cot X cosec X + c ; (2) -J cot x (cosec- x + 2) + c ; (3) — J cot x cosec' i
- J /cot X cosec x - log tan I j + c. 11. (1) J tan!" x - log sec i + c ;
(2) -Jcot3x+cotx+x+c; (3) J ton' x-tan x+x+c; (4) J ton*x-i ton^x
+ logsecx + c. 14. (1) ^(sinxcosx + x) — Jsinxcos^x + c; (2) —J sin x cos' x
+ j'l sin X coss x+ -^^ sin i cos x + A-* + ^ ; (3) - - ^?^ (3 - cos^x) - ^ + c.
2sini 2
17. (1) -!cot'x-icot6i + ci (2) itan«H-c; (3) -x>sCot3x(3cot2x+5) + c.
Page 343. 3. (1) 3 «i + J log (A^lDl _ V3 tan-i (^A±l\ + c ;
«- 1 \_\/3 /
(o,«r2,:±^+,; (3)4.ton-(-^)+.; (4)-i^log^^l^igl^+c;
8v2r + l v^ \V1^4x=/ 2V5 Vl-ir'+VE
(51 _->■•* ■r-J^_yers-'|+c; (6)2v'33+3x + 5-21og(x+4 + V'xH3x + 5)+c;
(7)21og(x + f+Vx^+'3x + 6) + ^log^« + «'-^^^S('°±l^±iJ + c;
V^ 0+1 ^ • W Til \ ^jj4 _ jg^2 32(x«-16)
CHAPTER XXII.
Art. 195. 2. 2525. 3. 3690 ; 3660 ; (true value = 3060). 6. 333 in
20,000. 7. .06075 ; 1509.
482 DIFFERENTIAL CALCULUS.
CHAPTER XXIV.
Art. 201. 4. The parabolas ?/ = 3 x" + Cia: + C2, whose axes are parallel
to the y-axis ; 2 y = 6 k'^ + 11 a; — 1.3 ; y = iix'^ + \bx + 22. 6. The cubical
parabolas y = x^ -^ CiX ■\- c^; y = x^ + x ; y = x^ — x + 4. 6. The cubical
parabolas y = c3?+CiX + Ca, in which c, Ci, Cj are arbitrai-y constants ;
6y =x^ + llx; 5y + x'' + \6 =22x. 7. The cubical parabolas x = Cij/'
+ C2y+C3; 120K=ll!/3-251y + 240; 7 x + iy' =e2y-8o. 8. 15,528 ft.;
62.1 sec. 10. Half a mile.
Art 202. 4. (1) 37 ; (2) 385>5 a'' ; (3) 6 a^ ; (4) - | o'tt . (5) J irabc ;
(6) |,ra'; (7) ^'; (8) ^'; (9) i Tra' - i a'.
Art. 203. 3. 5.
Art. 204. 6. 1154.7 cu. in. 6. faHana. 7. f (ir - J)a'. 8. 5440.6
cu. in. ; ira'' tan a.
Art. 205. 4. I ir(a2 - 6«)5-
CHAPTER XXV.
Art. 207. 4. 301.6 ; i irafift. 5. 65f cu. ft. 6. faft^cota.
7. f (3 IT + 8) a'. 8. fa2/i.
Art.208. 2. ^. 3. 2?; 5:. 6. | ^a^. e. 11 ir. 7. 4 a^.
12 2 n ' ^
Art. 209. 2. (1) 2»-a; (2) (5) {^2 + log (v'2 + l)}a ;
(3) 4a/'cos*-"-cos^V8a;(4)5(e--e-»), Sfe-lV 3. -t («-^ + «& + fe-^) .
^ ^ \ 2 2/ '^ ^2^ ^ 2V e/ a + 6
Art. 210. 2. (3) ^.2^; (4) (a) i sec a, in which Z is the difference in
length of the radii vectores to the extremities of the arc ; (4) (6) like (4) (a) ;
(5) ? rSzvTTfi? - «i vTT«? + log ^2 + Vl±^1 . (6) a tan *1 sec ^ +
2 L *i + Vl + ^iJ ^ 2
a log tan ^|i + iV 2 a /sec ^ + log tan ^^^ •
Art. 211. 6.4^2. 6. ttCtt -2)a2. 7. 2 irft^ + 2 xaft^i^*".
8. (1) Sra^; (2) Sir^a', V'^a'^ ! (3) ^V. -»^ Jra^. 9. 2 7rV6, 4 Tr'^afi.
10. 2ira2fl-iy 12. j!jiraS(3T-2); -J_,ra2(ir + 4).
\ ej 2 -x/^
2V2
6
Art. 212. 2. 4 a2. 3. 4 ira". 4. Surface = 8 a [2 6 sin-i
.aam-^—!^—]-
2-62
ANSWEBS. 483
Art. 213. 3. 134J ; 9}. 4. 4.62. 6. (1) 21, 5J ; (2) J, 1.14, .94 ;
(3) 5i, 91. 6. (1) 9.425; (2) 15.71; (3) 1.671 6, 1.571 o. 7. ^, ^.
IT r
9. in^. 10.1.273 a. 12. 1.132 o, 1.5 a". 18. fo, ia^. 14.32.704°.
15. io, |a. 16. fa, |o2. 17. | a, | a^. 18. 1.273 a, 2 a^. 19. .6.366 0,^0'.
CHAPTER XXVII.
Art. 219. 2. 2^ Vl - a;^ + x\^l-y^= c. 3. (j/ + 6)"(a; + o)" = c.
Art. 220. 1. x^ + y'' = cy. 2. k2(^k2 + 2y^) = c*. 3. ij/^ = c'^Cx + 2 3^).
4. xy{x — y) = c.
Art. 221. 1. I!/ = c. 2. x'h/ + Zx + 2y'' = c. 3. e' sin y + z^ = c.
4. 3 oxy — y'' = a;3 4. <;. 7. a log (x'^y) - y = c. 8. log — = — •
y xy
Art. 222. 3. VI — z^ ■ y = sin"' a; + c. 4. j/ = tana: — 1 + ce-""'.
5. y = a:5(l + ce^). 7. Sj/^ = c(l - x^)*- 1 + a;2. 8. j/2(x2 + 1 + ce'') = 1.
Art. 223. a. !/2 = 2 ex + cK 3. y.= c - [p^ + 2^) + 2 log (p - 1)],
X = c - [2p + 2 log (i) - 1)]. 4. log ip-x) = —5— + c, with the given
p — x
relation. 5. (x' + !/)2 (x2 - 2 y) + 2 x(x2 - 3 y)e = c". 6. y = ci + ?.
7. y = ex + a Vl + c2. 8. j/^ = cx2 -|- c^. <'
Art. 224. 2. x2 + y2 = a' ; x2(;j;4 _ 4 j/2) = 0. 3. (1) y = ex +c\
a;2 + 4y = 0. (2) (y + x-c)2 = 4xy, xy =0. (3) (x -y + c)« = a(x + y)«,
1 + 3^ = 0.
Art. 225. 3. The concentric circles x'' + y^ = a'. 4. The lines y = mx.
8. (1) The ellipses y^ + 2 x' = c^ ; (2) tlie hyperbolas x^ - ys = c^ ; (3) the
conies X- + ny'^ = c ; (4) the curves y^ — x^ = c' ; (5) the ellipses x' +
2 y2 = c^ ; (6) tlie cardioids r = c(l + cos $) ; (7) the curves »•» cos 716 = 0";
(8) the curves j" = c sin n9 ; (9) the lemniscates r' = c^ sin 2 ff, vf hose axes
are inclined at an angle 45° to the axes of the given system ; (10) the con-
focal and coaxial parabolas r(l — cos ff) = 2 c ; (11) the circles i* + y'^ - 2 Ix
-)- a^ = 0, in which I is the parameter. 10. The conies that have the fixed
points for foci. 11. The conies that have the fixed points for foci. 12. The
eoincs bh:'' ± a'^y'^ = a^h^. 13. The hyperbola 4 xy = o''. 14. The parabola
(x - yy - 2 a(x + y) + a2 = 0.
Art. 226. 3. (1) y = e2'(acos3x-(-6sin3x). (2) y=cie2'+C2e'+C8e^.
(.3) y = cie*'+e '''(C2 + C3X). (4) y = 62^(01 + C2x) + e*'(C3 cos5x+C4sin 5x).
7. (1) y = x(a cos log x + 6 sin log x). (2) y = x(ci + c^ log x).
(3) y = x''(ci + C2 log x). (4) y = eix'i + x(c2 cos logz + ej sin log x).
9. y = (5 + 2 x)2{ci(5 + 2 x^i + C2(5 + 2 xy^i}.
484 DIFFERENTIAL CALCULUS.
Art. 227. 4. (1) y = Cie" + Cae"". (2) e^" + 2 Cfie"-» = Ci^.
(.3) t ^-^fr { I ( vers-i 1^~t\- Vox - a;^ 1 . 6. The circle of radius a.
6. (1) !/ = CiX+(ci-Hl)log(a;-ri)+c2. (2) j/ = ci log a + ca. (3) 2(j/-6)
= «' '> + e-(^ ">. (4) j/ = cilog(l+a:)+Ja:-Ja;2^(;2. g. (i^ yi^x'^+CiX + c^.
(2) log2^=:cie^+C2e-^ (3) (a;-ci)2 = c2(i/Hc2). (4) 2/ = logcos(ci-a:) + C2.
Page411. (l)r=osine. (2) a:ei' = c(l+a;+i/). (.3) c(2j/2 + 2i-s^-a2)=V3
_ (y3+l)^ + 2y. (•4>) 3.2-2 cy + cK (5) j/ sec x = log (sec k + tan a;) + c.
(l-V3)a; + 22/
(6) 3 y = 12(1 + a:2)l + ca;^. (7) Zx"^ + ixy + by"^ + bx-^ y = c.
(8) (z - 2c')y2 = c^a:. (9) y{x^ + l)'^ = tan;' x + c. (10) 60!/3(a: + 1)2 =
10a* + 24a;S + 15x' + c. (11) x= — ^ (c + g sin-'p). ;/ = - ap +
Vl - p-''
— ^^^ (c + a sin-ip). (12) a; + c = a log (p + Vl + p'^, y = a vT+i^
(13) 2/^ = cx2--?^. (14) I = cary + c2. (15) 2/=|(p2+jB)+i log (2i)-l).
c + 1
(16) !/(l±cosa;)=c. (M) y'^+{x + cy = a^;y'^ = a\ (18) y=ca;+ VftHa'^c^;
6%2 + aV = 0^62. (19) 9(y + c)'^ = 4a:(a: - 3 a)2 ; a; = 0. (20) y = Cie<"
+ f2e-" + Cssin(a2; + «). (21) !/ = (cie» + C2e-') cosa; + (C3e'4-C4«-^)sina;.
(22) 2/ = e2'(ci + cja;) + Cje-'. (23) y = c^x + dx-K (24) 2/ = -' +
xi|c2Cosf-^logxj + C8sin^^loga;H- (2a) y = Ci{x+ay + <^{x + ay.
(2B) (cix + Cif + a = Ciy\ (27) 3 x = 2 aJ(i/i - 2 Ci)(2^4 + Ci)i + Ca.
(28) !/ = ci log X + J x2 ^ C2. (29) e— » = c^x + a.
INDEX.
[The numbers refer to pages."]
Abdank-Abakanowicz, 290.
Absolute, constants, 16 ; valae, 14.
Acceleration, 105, 22^-229.
Adiabatic curves, 86.
Aldis, Solid Geometry, 212.
Algebra, Chrystal's, 62, 65, 181, etc.;
Hall and Knight's, 65, 233.
Algebraic equations, theorems, 94, 16S.
Algebraic functions, 17, 56, 93.
Allen, see 'Analyiic Geometry.'
Amsler's planimeter, 348.
Analytic Geometry, Aahton, 129; Candy,
5; Tanner and Allen, 129; Went-
worth, 129.
Analytical Society, 39.
Angles at which curves intersect, 81.
Anti-derivatives, 45, 48.
Anti-di£Eerentials, 45, 291, 292.
Anti-differentiation, 269, 291.
Anti-trigonometric functions, 17.
Applications: elimination. 111; equa-
tions, 93, 94, 171 ; geometrical, 79 ;
physical, 79; rates, 90; of inte-
gration, 313, etc. ; of successive
integration, 360, etc.; of integra-
tion in series, 350; of differentia-
tion in series, 240; of Taylor's
theorem, 244-248, 254-256; to mo-
tion, 214.
Approximate integration, 344 ; by means
of series, 353.
Approximations : to areas and integrals,
■ 278, 344, .353 ; to roots of equations,
171; to values of functions, 44; to
small errors and corrections, 92,
138.
Arbitrary constants, 16.
Arbogaste, 36.
Arc: derivative, 98,99; length, 370,375,
*27 ; Huygheus' approximation,
249.
485
Archimedes, see ' Spiral.'
Area, 10; approximation to, 3(4, ,146,
derivative, differential, 95, 97 ; me-
chanical measurement, 318, 3i\i;
of curves, 313, 367, 369; of a
closed curve, 319, 370 ; of surfaces
of revolution, 374; of other sur-
faces, 378; precautions in finding,
319; sign of, 318, 370; swept over
by a moving line, 370.
Argument, 142.
Ashton, see 'Analytic Geometry.'
Astroid, see 'Examples.'
Asymptotes, 199, 212, 213; circular, 205;
curvilinear, 204; oblique, 203; par-
allel to axes, 201 ; polar, 205 ; vari-
ous methods of finding, 2(H.
Asymptotic circle, 205.
Bernoulli, 271.
Binomial Theorem, 245.
Bitterli, 290.
Borel, divergent series, 235.
Burmann, 19.
Byerly, see ' Calculus.'
Cajori, History of Mathematics, 36, 40,
270, 325, 343,
Calculation of small corrections, 92.
Calculus, 1; differential, 11, 33, 270;
integral, 11, 33, 45, 270 ; invention,
1,270; notions of, 11.
references to works on: Byerly,
Problems, 108, etc.; Campbell,
225; Echols, 35, etc.; Edwards,
Integral, 334, etc.; Edwards,
Treatise, 127, etc.; Gibson, 41,
etc.; Harnack, 170, etc.; Lamb,
41, etc. ; McMahon and Snyder,
Dijr., 41, etc.; Murray, Integral,
284, etc. ; Osgood, 170, etc ; Perry,
486
INDEX.
12, 431, etc. ; Smith, W. B., 133,
343; Snyder and Hutchinson, 277,
etc. ; Taylor, 127, etc. ; Todlmnter,
Diff., 65, etc. ; Integral, 2U; Wil-
liamson, Diff., 65, etc.; Integral,
284, etc. ; Young and Linebarger,
431.
Campbell, see ' Calculus.'
Candy, see 'Analytic Geometry.'
Cardioid, see ' Examples.'
Catenary, see ' Examples.'
Cauchy, 234 ; form of remainder, 250.
Centre of curvature, 157, 158; of mass,
385.
Change of variable, in differentiation,
143; in integration, 296.
Changes in variable and function, 30, 31.
Chrystal, see 'Algebra.'
Circle, curvature of, 155 ; of curvature,
156; osculating, 152, 159; see 'Ex-
amples.'
Circular asymptotes, 205.
functions and exponential functions,
250.
Cissoid, see ' Examples.'
Clairaut's equation, 399.
Commutative property of derivatives,
131.
Comparison test for convergence, 237.
Complete differential, 134.
Compound interest law, 65.
Computation of tt, 351.
Concavity, 148.
Condition for total differential, 138.
Conjugate points, 208.
Conoids, 366.
Constant: absolute, 16; arbitrary, 16;
elimination of. 111; of integra-
tion, 281, 283, 287, 395.
Contact: of curves, 149; order of, 149;
of circle, 150; of straight line,
151.
Continuity, continuous function, see
' Function.'
Convergence : 234, 237 ; interval of, 237 ;
tests for, 237, 238; see 'Series,'
' Infinite Series.'
Convexity, 148.
Corrections, 92.
Cos z, derivative of, 69; expansion for,
245, 248.
Criterion of integrability, 309.
Critical point, critical value, 114, 116.
Crossing of curves, 81, 151, 255.
Cubical parabola, see ' Examples.'
Curvature : 153 ; average, 154 ; at a point,
154, 155; total, 154, centre of, 157,
158 ; of a circle, 155 ; circle of, 156 ;
radius of, 156, 159.
Curves: area of, 313, 367, 369; asymp-
totes, 199, 212, 213; contact of,
149; derived, 38; differential, 38;
envelope, 190; equations derived,
324; evolute, 160; family, 190;
integral, 289, 290; involutes, 164;
length, 370, 373, 427 ; locus of ul-
timate intersections, 191 ; Loria's
Special Plane, 212; of one pa-
rameter, 257, 260; parallel, 164;
twisted, skew, 258; see 'Exam-
ples.'
Curve tracing, 211.
Curvilinear asymptotes, 204.
Cusps, 193, 206, 207, 209, 210.
Cycloid, see ' Examples.'
Decreasing functions, 113.
Definite integral, see ' Integral.'
De Moivre's theorem, 251.
Density, 385.
Derivation of equation of curves, 324.
Derivative: definition, 32; notation, 35;
general meaning, 40; geometric
meaning, 37 ; physical meaning,
39; progressive, regressive, 167;
right and left hand, 167.
Derivatives ; of sum, product, quotient,
46, 48-52; of a constant, 47; of
elementary functions, 66-75 ; of a
function of a function, 54; of im-
plicit functions, 75, 137; of in-
verse functions, 56 ; special case,
55; geometric, 95-102 ; successive,
103, 108 ; meanings of second,
104, 105.
Derivatives, partial, 76, 128, 129; com-
mutative property of, 131 ; geo-
metrical representation, 130; il-
lustrations, 139-142; successive,
131.
Derivatives, total, 134; successive, 139.
Derived, curves, 38; functions, 32, 34.
Descartes, 270.
Difference-quotient, 32, 34.
Differentiable, 35.
Differential calculus, see ' Calculus.'
Differential coefficient, see ' Derivative.'
Differential, differentials, 42, 44; com-
INDEX.
487
plete, 134; exact, 138; geometric,
95-102; infinitesimal, 276; par-
tial, 134; successive, 109; total,
134, 135 ; illustrations, 139-142 ;
condition for total, 138; integra-
tiou of total, 309.
Differential equations, 112, 394; classifi-
cation, 394 ; Clairaut's, 399 ; exact,
39f>; homogeneous, 396, linear,
397, 406, 408; order, 394; ordi-
nary, 394; partial, 394; second
order, 40!) ; solutions, 112, Siin,
400 ; references to text-books, 112,
411, etc.
Differentiation, 33,291; general results,
46; logarithmic, 63 ; of series, 240;
successive, 103; see 'Derivative,'
' Derivatives.'
Direction cosines of a line, 258.
Discontinuity, discontinuous functions,
see ' Functions.'
Displacement, 214, 216, 218.
Divergent series, see 'Series.'
Double points, 193, 206, 207.
Doubly periodic functions, 342.
Durand's rule, 348.
Echols, see ' Calculus."
Edwards, see 'Calculus.'
Elementary integrals, 293, 301.
Elimination of constants, 111.
Ellipse, see ' Examples.'
Ellipsoid, 360.
Elliptic functions, 279, 342.
integrals, 279, 342, 354.
End-values, 276.
Envelopes, contact property, 193, 195;
definition, 191 ; derivation, 194,
197.
Equations, approximate solution of, ITl ;
derivation of, 324 ; graphical rep-
resentation, 19, 128 ; roots of, 94,
171 ; of tangent and normal, 83.
Equiangular spiral, see ' Examples.'
Errors, small, 92, 136; relative, 92, 136.
Euler, 139, 251, 351; theorem on homo-
geneous functions, 139.
Evolute, definitions, 160.
properties of, 161.
Evolute of the ellipse, see ' Examples.'
Exact differential, 138.
equations, 396.
Examples concerning :
adiabatic curves, 86.
astroid (or hypoeycloid) , 85,98, 158,
161, 319, 324, 376, 405, 425.
cardioid, 90, 97, 159, 369, 374, 377, 389,
405, 425, 433, 44li.
catenary, 322, 373, 378, 426, 433.
circle, 85, 159, 315, 369, 374, 377, 388,
389; 391, 404, 449.
cissoid,203.
cubical parabola, 91, 97, 98, 158, 279,
287, 316, 319, 322.
cycloid, 86, 158, 161, 373, 377, 426.
ellipse, 85, 102, 164,203, 321, 324, 373,
382, 387, 435, 447, 449, 450.
evolute of the ellipse, 161, 104.
exponential curve, 85.
folium of Descartes, 86, 203, 369.
harmonic curve, 448.
helix, 328, 329.
hyperbola, 86, 91, 158, 159, 161, 203,
204, 212, 405, 433.
hypoeycloid, see ' Astroid.'
lemniscate, 159, 369, 405, 433.
limaipon, 448.
parabola, 85, 86, 91, 98, 100, 158, 159,
161, 164, l!lfi, 197, 203, 213, 273,
280, 287, 316, 317, 319, 359, 374,
382, 389, 405, 426, 433.
probability curve, 203.
semi-cubical parabola, 85, 86, 158,
280, 319, 426.
sinusoid, 85, 280.
tractrix, 4'26.
the witch, 86, 159, 203.
Spirals :
Archimedes', 90, 97, 99, 159, 374.
equiangular (or logarithmic), 90,
159, 369, 374, 426.
general, 90, 159.
hyperbolic (or reciprocal), 90,
369.
logarithmic, see ' Equiangular.'
parabolic (or lituus), 90.
reciprocal, see ' Hyperbolic'
Expansion of :
cos X, 245, 248.
log {l + x), logarithmic series, 244,
352.
sin X, 245, 248.
sin-l I, 351.
e', exponential series, 249.
tan-ix, Gregory's series, 350.
Expansion of functions :
by algebraic methods, 249.
by differentiation, 240.
488
INDEX.
Expansion of functions :
by integration, 350.
by Maclaurin's series, 247-249.
by Taylor's series, 243-247.
Explicit function, 16.
Exponential curve, see ' Examples.'
function, 17; expansion of, 249 ; and
trigonometric, relations between,
250.
Extended Theorem of Mean Value, 177.
Family of curves, 100.
Fermat, 120, 270, 372.
Fluent, fluxion, 3!).
Folium of Descartes, see ' Examples.'
Forms, indeterminate, 180.
Formulas of reduction, 334, 339.
Fourier, 276.
Fractions, rational, integraticm of, 305.
Frost, Curve Tracing, 204, 200, 212.
Function, 14; algebraic, 17,56, 342; cir-
cular, 342; classificatiou , 16 ; con-
tinuous, 18, 25, 35, 129; derived,
32, 34; discontinuous, 18, 25, 27;
elliptic, 279, 342; explicit, 10;
exponential, 17, 61; graphical
representation, 19, 20, 128; homo-
geneous, Euler's theorem on, 139;
hyperbolic, 304, .342, 413; implicit,
16, 75, 137 ; increasing and decreas-
ing, 113; inverse, 15, 56, 71; irra-
tional, 17, 327 ; logarithmic, 17, 61 ;
many-valued, 15; march of a,
121; maximum and minimum val-
ues of, 114 ; notation for, 18 ; of a
function, 54, 55; of two variables,
16, 128; one-valued, 15; periodic,
312; rational, 17, transcendental,
17; trigonometric and anti-trigo-
nometric, 17, 66, 3.36; turning val-
ues of, 115; variation of, 115.
Gauss, 234.
General integral, see ' Integral.'
spiral, sie 'Examples.'
Generalized Theorem of Mean Value,
182.
Geometrical hiterpretation, a certain,
336.
Geometrical representation of :
derivatives, ordinary, 37.
derivatives, partial, 130.
fmictions of one variable, 19.
f unctious of two variables, 128.
function of a function, 55.
integrals, definite, 284.
integrals, indefinite, 287.
total differential, 135.
Geometric derivatives and differentials,
95-102.
Gibson, see ' Calculus.'
Glaisher, Elliptic Functions, 343.
Goursat-Hedrick, Mathematical Anal-
ysis, 170.
Graphical representation of functions,
19 ; of real numbers, 13.
Graphs, sketching of, 121.
Gregory, 235, 351.
Gregory's series, 351.
Gyration, radius of, 390.
Harkness and Morley, Analytic Func-
tions, 233, Theory of Functions, 35.
Harmonic curve, 448.
Harmonic motion, 78, 107.
Harmonic series, 234.
Harnack, see ' Calculus.'
Hele Shaw, Mechanical Integrators, 349.
Helix, 258, 428.
Henrici, Report on Planimeters, 349.
Herschel, 19, 40.
Hobson, Trigonometry, 233, 352, 423.
Homogeneous, differential equations,
396.
functions, Euler's theorem, 139.
linear equation, 397, 406, 408.
Homer, Horner's process, 247, 256.
Hutchinson, .s^e ' Calculus.'
Huyhen's rule for circular arcs, 249.
Hyperbola, see 'Examples.'
Hyperbolic functions, 304, 342, 413.
spiral, see 'Examples.'
Hypocycloid, see 'Examples.'
Implicit functions, 16; differentiation,
75, 137.
Increasing function, 113.
Increment, notation for, 4, 30, 31.
Indefinite integral, see ' Integral.'
Indeterminate forms, 180.
Inertia, centre of, 386.
moment of, 390.
Infinite numbers, 14, 28, 29.
orders of, 29.
Infinite series, 230; algebraic proper-
ties, 234; differentiation of, 232,
240; general theorems, 235; inte-
gration in, 353; integration of.
INDEX.
489
232, 350; limiting value of, 231;
questions concerning, 231 ; Osgood,
article iind piiraphlet, 233, 2;ili, 237 ;
remainder, 23U; study of, 233.
■Si'c ' Series.'
Iniinitesiiniil, 1,28, 43,45.
liitinitesimal difTerential, 2TG.
lufiuitesimals, 28; orders, 29; summa-
tion, 271.
Inflexion, points of, IKi, 125, 127.
Inflexioijal tanj;ent, 127.
Integral curves, 28SI, 2'JO.
Integral, definite, approximation, 344,
353; definition, representation of,
properties, 275-27!), 284, 285.
Integral: double, 355; element of, 27li;
elementary, 293, 301 ; elliptic, 27i),
342,354, 373; general, 283.
Integral, indefinite, 281, 283; represen-
tation of, 287.
Integral: multiple, 356; particular, 283;
precautious in finding, 319 ; triple,
355.
See * Calculus.'
Integrand, 271.
Integraph, 2<«, 348, 349.
Integrating factors, .196.
Integration, 2(10, 2«Jl ; as summation,
275, 291 ; as inverse of differentia-
tion, 281, 291; constant of, 281,
283, 287 ; general theorems in, 294 ;
successive, 355, 357.
Integration : by parts, 298 ; by substitu-
tion, 296, 304, 328, 336 ; by infinite
series, 350, 353; by mechanical
devices, 348.
Integration of: infinite series, 232, 350;
irrational functions, 327 ; rational
fractions, 305; total difTerential,
309 ; trigonometric functions, 336.
See ' Applications.'
Integrators, 318, 349.
Interpolation, 256.
Intrinsic equation, 374, 423.
Invention of the calculus, 1, 270.
Inverse functions, 15, 56, 71.
Involutes, 164.
Irrational functions, integration, 327.
Isolated points, 206, 208.
Jacobi, 131.
Kepler, 120,
Klein, 62.
Lagrange. 36, 249, 270.
Lagrange's form of remainder, 250.
Lamb, see ' Calculus.'
Laplace, 270.
Legendre, 3,")4.
Leibnitz, 36, 39, 195, 270, 271, 351.
theorem on derivative of product,
110.
Lemniscate, see 'Examples.'
Lengths of curves, 370, 373, 427 ; of tan-
gents and normals, 84, 88.
Limacj'on, see 'Examples.'
Limits, limiting value, 20, 23, 36; in in-
tegration, 276; of a series, 231.
Linear differential equations: of first
order, 397; with constant coefli-
cients, 406; homogeneous, 408.
Linebarger, see 'Calculus.'
Litnus, see 'Examples.'
Locus of ultimate intersections, 191.
Logarithmic, differentiation, 63.
function, 17,62.
series, 244, 332.
spiral, see 'Examples.'
Ii0ria> Special Plane Curves, 212.
Machin, Xil.
Maclaurin, 250.
theorem and series, 247, 252.
Magnitude, orders of, 29.
Mass, centre of, 385.
Mathews, G. B., 235.
Maxima and minima, 113; by calculus,
114-120; by other methods, 120;
of functions of several variables,
120; practical problems, 121.
McMahon, proof, 138.
See 'Calculus.'
Mean values, 380.
Mean value theorems :
differentiation, 1()4, 169, 174-179, 182.
integration, 286, 380.
Mechanical integrators, 348.
Mechanics, 385.
Mellor, Hif/her Mathematics, 431.
Mercator, 3,'i2.
Minima, see ' Maxima.'
Moment of inertia. 390.
Morley, see ' Harkness.'
Motion, applications to, 214.
Motion, simple harmonic, 78, 107.
Muir, on notation, 131.
Multiple, angles in integration, 338.
integrals, 356.
490
INDEX.
Multiple, points, 206, 209.
roots, 93.
Neil, 371.
Newton, 39, 171, 351.
Nodes, 207.
Normal, equation of, 83; length, 84, 88.
Notation for: absolute value, 14; de-
rivatives, 35, 39, 103; differentials,
42; functions, 18; increment, 4;
infinite numbers, 14; integration,
270, 283, 284, ,158; inverse func-
tions, 19, 5(); limits, 23: partial
derivatives, 7B, 129, 131, 135 ; sum-
mation, 270.
Notation, remark on, 'A&.
Numbers, 13 ; finite, infinite, infini-
tesimal, 14, 28 ; transcendental, 62.
e and t,, 62, 328.
graphical representation, 13, 14.
Oblique axes, 314.
Order of, contact, 149.
derivative, differential, 104, 256.
differential equation, 394.
infinite, 29.
infinitesimal, 29, 256.
magnitude, 29.
Orthogonal trajectories, 401, 403.
Oscillatory series, 234.
Osculating circle, 152, 159.
Osgood, W. F., pamphlet, 233, etc. ; see
' Calculus.'
Parabola, see ' Examples.'
Parabolic rule, 346.
spiral, see ' Examples.'
Parallel curves, 164.
Parameter, 190, 257.
Partial derivative, see ' Derivative.'
Partial fractions, 305.
Particular integral, see 'Integral.'
Pendulum time of oscillation, 354.
Periodic functions, 3^2.
Perry, on notation, 131.
See ' Calculus.'
Picard, 277.
Pierpont, Theory of Functions, 14, 27,
131, 167.
Planimeters, 348, 349.
Henrici, Report on, 349.
Points, see 'Critical,' 'Double,' 'Iso-
lated,' 'Multiple,' 'Salient,' ' Sin-
gular,' ' Stop,' ' Triple,' ' Turning.'
Power series, 237, 240, 350.
Precautions in integration, 319.
Probabilities, 249.
Probability curve, see ' Examples.'
Progressive derivative, 167.
Radius of curvature, 156, 159.
of gyration, 390.
Rate of change, 11, 39, 40, 41, 90.
variation, 132.
Rational fraction, integration, 305.
Reciprocal spiral, see ' Examples.'
Rectification of curves, 371.
Reduction formulas, 334, 339.
Regressive derivative, 167.
Relative error, 92.
Remainder after n terras, 236.
Remainders iu Taylor's and Maclaurin's
series, 243, 246, 250.
Right- and left-hand derivatives, 167.
Ring, 323.
Rolle, 169.
Rolle's theorem, 166, 168.
Roots of equations, 94, 171.
Rouche et Comberousse, 371.
Rules for approximate integration, 344,
34(), 348.
Salient points, 208.
Schlomilch-Roehe's form of remainder
250.
Second derivative:
geometrical meaning, 104.
physical meaning, 105.
Semi-cubical parabola, 371.
See 'Examples.'
Series, 65 ; absolutely convergent, 235 ;
conditionally convergent, 235 ;
convergent, 234; divergent, 234,
235; harmonic, 234; oscillatory,
234.
See ' Convergence,' ' Expansion,' ' In-
finite Series,' ' Power Series.'
Serret, 320.
Skew curves, tangent line, and normal
plane, 258-261, 266-268.
Sign of area, 318, 370.
Simpson, Simpson's rule, 346.
Sin X, sin-i I, expansions, 245, 248, 351.
Singly periodic functions, 342.
Singular points, 206, 208.
Singular solution, 400.
Sinusoid, see ' Examples.'
Slope, 5, 6, 11, 38, 79, 87.
INDEX.
491
Slopes, curve of, 38.
Smith, C, Solid Geometry, 212, 378.
Smith, W. B., Infinitesimal Analysis,
133.
Suyder, see 'Calculus.'
Solution, see ' Differential Equation.'
Speed, 2. ii, 4, 214.
Sphere, surface, 'ill, 3"',l.
volume, 324. 3C2, 3(j3.
Spiral, see ' Examples.'
Stationary tangent, 127.
Stirling, 2.W.
Stop points, 208.
Subnormal, rectangular, 84.
polar, 88.
Substitutions in integration, 296, 304,
328, 330.
Subtangent, rectangular, 84.
polar, 88,
Successive differentiation, 103.
derivatives, 103, 108.
differentials. IC).
integration, 355, 357.
of a product, 110.
total derivatives, 134.
Summation, examples, 271.
integration as, 27.">.
Surfaces, applications of differential
calculus, tangent lines, tangent
plane, normal, 21)2-205.
areas of, 374, 378.
volumes, 320, 3(50, 303, 3C5.
Tangent, 5 ; e<iuation of, 8;> ; inflexional,
127; length, 84, 88; stationarj-,
127 ; to twisted curve, 259, 206 ; to
surface, 202.
Tanner, see 'Analytic Geometry.'
Taylor, F. G.. see 'Calculus.'
Taylor's theorem and series:
applications ; to algebra, 256 ; to cal-
culation, 44, 135, 245, 24(!, 247 ; to
contact of curves, 255 ; to maxima
and minima, 254.
approximations by. 44, 135, 24.'<.
expansions by, 244-247.
for functions of one variable, 44, 242,
243, 246, 252.
for functions of several variables,
250.
forms of, 24;i, 2+4, 246.
historical note, 249.
Test-ratio, 2.(8.
Time-rate of change, 39, 133.
Todhunter, see ' Calculus.'
Total derivative, 134.
differential, KW.
rate of variation, 132.
Tractrix, see ' Examples.'
Trajectories, orthogonal, 401, 403.
Transcendental functions, 17.
numbers, 13.
Trapezoidal rule, 344.
Trigonometric functions, direct and in-
verse, 17, 71.
differentiation of? 00-75.
integration of, 3oO.
relations with exponential, 250.
substitutions by, 328.
Trigonometry, Hobson, 233, 352, 423.
Murray, 71, etc.
Triple points, 21_17.
Turning points, values, 115.
Twisted curves, see ' Skew curves.'
Undulation, points of, 126.
Value, see ■ .\verage,' 'Limits,' 'Maxi-
mum,' 'Mean,' 'Turning.'
Value of 7r, computation of, Xil, 352.
Van Vleck, E. B., '235.
Variable, dependent, independent, 13, 15.
change of, 143.
Variation, continuous variation, inter-
val of variation, 24.
Variation of functions, 113.
total rate of, l.'>2.
Veblen-Lennes, Inlinitesimal Analysis,
14, 27. etc.
Velocity, 91, 214-222.
Volumes, methods of finding, 320, 300,
,^3, ;v;."i.
W.illis, '270, 371.
Wentworth, see 'Analytic Geometry.'
■Whittivker, Modern Analysis, 234,
239.
Williamson, see 'Calculus.'
Witch of .\gnesi, see ' Examples.'
Wren, 372. ~
Younc. jice 'Calculus.'
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