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THE MODERN MATHEMATICAL SERIES
LUCIEN AUGUSTUS WAIT . . . General Editor
(SBNIOB PEOFE880K OF MATHEMATICS IN OOENELL UNIVEKSITT)
The Modern Mathematical Series,
lucien augustus wait,
(Senior Professor of Matbematics in Cornell University,)
GENERAL EDITOR.
This series includes the following works :
ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen.
DIFFERENTIAL CALCULUS. By James McMahon and Virgil Snyj)er.
INTEGRAL CALCULUS. By D. A. Murray.
DIFFERENTIAL AND INTEGRAL CALCULUS. By Virgil Snyder and J. I
Hutchinson.
ELEMENTARY ALGEBRA. By J. H. Tanner.
ELEMENTARY GEOMETRY. By James McMahon.
The Analytic Geometry, Dififerential Calculus, and Integral Calculus (pub-
lished in September of 1898) were written primarily to meet the needs of college
students pursuing courses in Engineering and Architecture ; accordingly, prac-
tical problems, in illustration of general principles under discussion, play an
important part in each book.
These three books, treating their subjects in a way that is simple and practi-
cal, yet thoroughly rigorous, and attractive to both teacher and student, received
such general and hearty approval of teachers, and have been so widely adopted
in the best colleges and universities of the country, that other books, written on
the same general plan, are being added to the series.
The Differential and Integral Calculus in one volume was written especially
for those institutions where the time given to these subjects is not sufficient to
use advantageously the two separate books.
The more elementary books of this series are designed to implant the spirit of
the other books into the secondary schools. This will make the work, from the
schools up through the university, continuous and harmonious, and free from
the abrupt transition which the student so often experiences in changing from
his preparatory to his college mathematics.
DIFFERENTIAL AND INTEGRAL
CALCULUS
BY
VIRGIL SNYDER, Ph.D. (gottingen)
AN©
JOHN IRWIN HUTCHINSON, Ph.D. (chicago)
OF CORNELL UNIVERSITY
3jO<C
NEW YORK .:• CINCINNATI •:• CHICAGO
AMERICAN BOOK COMPANY
COPTEIGHT, 1902, BY
VIRGIL SNYDER and JOHN I. HUTCHINSON
EXTSEED AT StATIONKRS' HaLL, LONDON.
DiF. nrr. oal.
w. p. 5
QA303
SGI
PREFACE
The favorable reception accorded the two volumes on the
Calculus in this series shows that they have been serviceable
in supplying a real need. A general demand has arisen for
a similar treatment of the subjects in briefer form, suitable
for use in shorter and more elementary courses. Accord-
ingly, in response to numerous requests and suggestions, the
present volume has been prepared.
The part on the Differential Calculus is of essentially the
same character as the former separate volume (which will
be referred to in the text as D. C), but the range of topics
is restricted ; various theorems have been put in less ab-
stract form, and fewer alternative proofs have been given.
The chapter on the expansion of functions has been so
arranged that the remainder theorem may be omitted with-
out marring the continuity of the subject. In the treatment
of functions of two independent variables no use is made of
an auxiliary variable.
The characteristic features of the larger book are retained.
Some of these are as follows : —
1. The derivative is presented rigorously as a limit.
2. The process of differentiation is so arranged as to give
the a;-derivative of a function of u^ in which t^ is a function
of X ; the resulting type forms being printed in full-face
letters in the text and collected for reference at the end of
the chapter.
mstts^s
VI PREFACE
3. Maxima and minima are discussed as the turning
values in the variation of a function, with complete graphi-
cal representation.
4. The notions of rates and differentials are so presented
as to grow naturally out of the idea of a derivative, and are
not introduced until the student has become familiar with
the process of finding the derivative and with its use in
studying the variation of a function.
5. The related theories of inflexions, curvature, and
asymptotes receive direct and comprehensive treatment.
The part on the Integral Calculus has been written, en-
tirely anew.
The first five chapters discuss the ordinary methods of
integration. The aim has been to make clear the rationale
of each process, and to encourage the students to become
independent of formulas.
The method of reduction has been put in the simplest
possible form ; in the solution of problems students need
make no use of formulas of reduction.
In the resolution of rational fractions into simpler ones,
care has been taken to show the logical basis of the usual
assumptions.
The rationalization of a differential containing the square
root of a quadratic expression has been treated much more
fully than usual. The problem is interpreted geometrically
as equivalent to the rational expression of the coordinates of
a variable point on a conic in terms of a varying parameter.
This makes clear how the required transformations are sug-
gested and puts the subject in a more attractive form.
Special care has been taken in presenting the subject of
integration regarded as a summation so as to combine rigor
and simplicity. The ordinary cases of discontinuity, either
PREFACE Vll
of the integrand or of the variable of integration, are in-
cluded in the discussion.
In deriving the formula for length of arc, the definition
of such length is given as the limit of the sum of chords, a
definition which readily expresses itself, by the use of the
mean- value theorem, in the form of a definite integral.
The exercises, which are new throughout the book, are
carefully graded. Numerous illustrative examples are worked
out in the text, and are accompanied by various suggestions
and remarks relating to both theory and practice.
The authors gratefully acknowledge their indebtedness to
their colleague, Professor James McMahon, for permission
to make free use of McMahon and Snyder's Differential
Calculus, for a number of valuable suggestions, and for as-
sistance in reading portions of the manuscript and proof.
CONTENTS
DIFFERENTIAL CALCULUS
CHAPTER I
Fundamental Principles
ARTICLE PAGE
1. Elementary definitions 1
2. Infinitesimals and infinites 2
3. Fundamental theorems concerning infinitesimals and limits in
general 4
4. Comparison of variables 5
5. Comparison of infinitesimals and of infinites. Orders of mag-
nitude 7
6. Useful illustrations of infinitesimals of different orders . . 9
7. Continuity of functions 13
8. Comparison of simultaneous infinitesimal increments of two
related variables 15
9. Definition of a derivative 20
10. Geometrical illustrations of a derivative 20
11. The operation of differentiation 24
12. Increasing and decreasing functions 25
13. Algebraic test of the intervals of increasing and decreasing . 27
14. Differentiation of a function of a function .... 28
CHAPTER II
Differentiation of the Elementary Forms
15. Differentiation of the product of a constant and a variable . 30
16. Differentiation of a sum 31
17. Differentiation of a product 32
18. Differentiation of a quotient 33
19. Differentiation of a commensurable power of a function . . 34
20. Elementary transcendental functions 38
21. Differentiation of log^ x and log,, u 39
ix
X CONTENTS
ARTICLE PAGS
22. Differentiation of the simple exponential function ... 41
23. Differentiation of the general exponential function ... 42
24. Differentiation of an incommensurable power .... 42
25. Differentiation of sin u ^ . 44
26. Differentiation of cos u . 44
27. Differentiation of tan u . ' 45
28. Differentiation of cot m 45
29. Differentiation of sec m 46
30. V Differentiation of esc m 46
31. Differentiation of vers u 46
32. Differentiation of sin-^ u 47
33. Differentiation of the remaining inverse trigonometric forms . 48
34. Table of fundamental forms . 49
CHAPTER III
Successive Differentiation
35. Definition of the wth derivative , 52
36. Expression for the nth derivative in certain cases ... 54
CHAPTER IV
Expansion of Functions
37. Convergence and divergence of series ..... 57
38. General test for interval of convergence 58
39. Remainder after n terms 61
40. Maclaurin's expansion of a function in power-series ... 62
41. Taylor's series 66
42. Rolle's theorem 67
43. Form of remainder in Maclaurin's series 68
44. Another expression for remainder 70
45. Theorem of mean value 75
CHAPTER V
Indeterminate Forms
46. Definition of an indeterminate form 77
47. Indeterminate forms may have determinate values ... 78
48. Evaluation by development 80
49. Evaluation by differentiation 81
CONTENTS XI
ARTICLE PAGE
50. Evaluation of the indeterminate form §- 84
51. Evaluation of the form co . 0 86
52. Evaluation of the form go — co. . • •• . .86
53. Evaluation of the form 1" 88
54. Evaluation of the forms 0^,00^ 88
CHAPTER VI
Mode of Variation of Functions of One Variable
55. Review of increasing and decreasing functions .... 91
56. Turning values of a function 91
57. Critical values of the variable 93
58. Method of determining whether </>' (x) changes its sign in
passing through zero or infinity 93
59. Second method of determining whether <^' {x) changes sign in
passing through zero 95
60. Conditions for maxima and minima derived from Taylor's
theorem 97
61. The maxima and minima of any continuous function occur
alternately 98
62. Simplifications that do not alter critical values ... 99
63. Greometric problems in maxima and minima .... 100
CHAPTER Vn
Rates and Differentials
64. Rates. Time as independent variable 105
65. Abbreviated notation for rates 108
66. Differentials often substituted for rates 110
CHAPTER Vni
Differentiation of Functions of Two Variables
67. Definition of continuity 113
68. Partial differentiation 114
69. Total differential 116
70. Language of differentials 119
71. Differentiation of implicit functions ...... 120
72. Successive partial differentiation ...... 121
73. Order of differentiation indifferent . . . • . .121
XU CONTENTS
CHAPTER IX
Change of the Variable
ARTIOLK PAOB
74. Interchange of dependent and independent variables . . 124
75. Change of the dependent variable 125
76. Change of the independent variable 126
APPLICATIONS TO GEOMETRY
CHAPTER X
Tangents and Normals
77. Geometric meaning oi-^ - % ^29
78. Equation of tangent and normal at a given point . . . 129
79. Length of tangent, normal, subtangent, subnormal . . . 130
Polar Coordinates
80. Meaning of p^ 133
81. Relation between -~ and p-r 134
ax ^ dp
82. Length of tangent, normal, polar subtangent, and polar sub-
normal 135
CHAPTER XI
Derivative of an Arc, Area, Volume, and Surface
OF Revolution
83. Derivative of an arc 138
84. Trigonometric meaning of — , -r- 139
85. Derivative of the volume of a solid of revolution . . . 140
86. Derivative of a surface of revolution 140
87. Derivative of arc in polar coordinates 141
88. Derivative of area in polar coordinates 142
CHAPTER XH
Asymptotes
89. Hyperbolic and parabolic branches 143
90. DefinitiGD of a rectilinear asymptote 143
C0:N TENTS XIU
Determination of Asymptotes
ARTICLE PAGE
91. Method of limiting intercepts 143
92. Method of inspection. Infinite oi'dinates, asymptotes parallel
to axes 144
93. Method of substitution. Oblique asymptotes .... 147
94. Number of asymptotes 149
95. Method of expansion. Explicit functions .... 150
CHAPTER Xin
Direction of Bending. Points of Inflexion
96. Concavity upward and downward . . . . . . 152
97. Algebraic test for positive and negative bending . . . 153
98. Analytical derivation of the test for the direction of bending . 156
99. Concavity and convexity towards the axis .... 157
CHAPTER XIV
Contact and Curvature
100. Order of contact .159
101. Number of conditions implied by contact .... 160
102. Contact of odd and of even order 161
103. Circle of curvature 163
104. Length of radius of curvature; coordinates of center of
curvature 163
105. Direction of radius of curvature 164
106. Total curvature of a given arc ; average curvature . . . 166
107. Measure of curvature at a given point 166
108. Curvature of osculating circle 167
109. Direct derivation of the expression for k and E in polar
coordinates 169
EVOLUTES AND INVOLUTES
110. Definition of an evolute . 170
111. Properties of the evolute 172
CHAPTER XV
Singular Points
112. Definition of a singular point 179
113. Determination of singular points of algebraic curves o . 179
XIV CONTENTS
ARTICLE PAGB
114. Multiple points 181
115. Cusps 182
116. Conjugate points 184
CHAPTER XVI
Envelopes
117. Family of curves
118. Envelope of a family of curves ....
119. The envelope touches every curve of the family
120. Envelope of normals of a given curve
121. Two parameters, one equation of condition
187
188
189
190
191
INTEGRAL CALCULUS
CHAPTER I
General Principles of Integration
122. The fundamental problem 195
123. Integration by inspection 196
124. The fundamental formulas of integration .... 198
125. Certain general principles 199
126. Integration by parts 203
127. Integration by substitution 205
128. Additional standard forms 209
129. Integrals of the form f M^ + -g)^£ 210
*" y/ax^ -{-bx+ c
130. Integrals of the form f- '^^ ... 212
JiAx + B) Vax2 -i-bx-^c
CHAPTER n
131. Reduction Formulas 215
CHAPTER m
Integration op Rational Fractions
132. Decomposition of rational fractions 223
183. Case L Factors of the first degree, none repeated . . 225
CONTENTS XV
ARTICLE - PAGE
134. Case II. Factors of the first degree, some repeated . . 226
135. Case III. Occurrence of quadratic factors, none repeated . 228
136. Case IV. Occurrence of quadratic factors, some repeated . 229
137. General theorem on the integration of rational fractions . 230
CHAPTER IV
Integration by Rationalization
138. Integration of functions containing the irrationality Vax -\- b 231
139. Integration of expressions containing Vax^ -\-hx + c . . 232
140. General theorem on the integration of irrational functions . 236
CHAPTER V
Integration of Trigonometric and Other Tran-
scendental Functions
141. Integration by substitution 238
142. Integration of i sec^" x dx, j cosec^** xdx . . . . 238
143. Integration of J sec* x tan^" +'^xdx, \ coseC" x cot^™ ^'^xdx . 239
144. Integration of \ tan« x dx, \ cot" xdx 240
145. Integration of j sin^a: cos"a:c?a: 242
244
246
146. Integration of ( ; , i t—. — . . . ,
J a-\-o cos X J a -{• 0 sm x
147. Integration of J e«=* sin nx dx, \ e"^ cos nxdx . ' .
CHAPTER VI
Integration as a Summation
148. The definite integral 248
149. Geometrical interpretation of the definite integral as an area 253
150. Generalization of the area formula. Positive and negative
area 255
151. Certain properties of definite integrals 256
152. Definition of the definite integral when f(x) becomes infinite.
Infinite limits 257
XVi CONTENTS
CHAPTER VII
Geometrical Applications
ARTICLE PAGB
153. Areas. Rectangular coordinates 260
154. Areas. Second method 260
155. Precautions to be observed in evaluating definite integrals . 263
156. Areas. Polar coordinates 267
157. Length of curves. Rectangular coordinates .... 269
158. Length of curves. Polar coordinates 271
159. Measurement of arcs by the aid of parametric representation . 273
160. Area of surface of revolution . 274
161. Volume of solid of revolution * . .277
162. Miscellaneous applications . . 279
CHAPTER VIII
Successive Integration
163. Successive integration of functions of a single variable . . 286
164. Integration of functions of several variables .... 288
165. Integration of a total differential 289
166. Multiple integrals .292
167. Definite multiple integrals 293
168. Plane areas by double integration 294
169. Volumes 295
DIFFERENTIAL CALCULUS
3j»iC
CHAPTER I
FUNDAMENTAL PRINCIPLES
1. Elementary definitions. A constant number is one that
retains the same value throughout an investigati6n in which
it occurs. A variable number is one that changes from one
value to another during an investigation. When the varia-
tion of a number can be assigned at will, the variable is called
independent; when the value of one number is determined
by that of another, the former is called a dependent variable.
The dependent variable is called a function of the indepen-
dent variable.
E.g., 3 x% 4 Va; — 1, cos x, are all functions of x.
Functions of one variable x will be denoted by the sym-
bols /(a;), <i>(x)y •••; similarly, if 2 be a function of two
variables a?, ^, it will be denoted by such expressions as
z =f(P^^ y')^ 2J = F(x, y) '".
When a variable approaches a constant in such a way that
the difference between the variable and the constant may
become and remain smaller than any fixed number, pre-
viously assigned, the constant is called the limit of the
variable.
There is nothing in this definition which requires a vari-
able to attain the value of its limit, or not to attain it. The
1
2 DIFFERENTIAL CALCULUS [Ch. I.
examples of limits met with in elementary geometry are
usually of the second kind ; i.e. the variable does not reach
the limit. The limiting values of algebraic expressions are
more frequently of the first kind.
E.q., the function has the limit 1 when x becomes zero : it has
the limit 0 when x becomes infinite. The function sin x has the limit 0
when x becomes zero ; tan x has the limit 1 when x becomes y.
4
EXERCISES
1. Let xp (x, y) = Ax + By -\- C ; show that ij; (x, y) =0,\f/ (y, —x)=0
are the equations of two perpendicular lines.
2. If f(x) = 2 xVl- x\ show that ffain-^ = sin x =ffcoa-\
3. If 4> (x) = ^^, show that <f>(^)-<t>(y) = E^LIL.
4. K f{x) = log f^, show that f{x) +f{y) =fl^±JL\.
1 -\- X \1 + xy/
5. Given /(x) = Vl^^, find /( VF^^).
6. If f(xy) =/(x) +f(y), prove that /(I) = 0.
7. Given f(x + y)=f(x)+f(y), show that /(O) = 0, and that
pf(^x) =f(px), p being any positive integer.
8. Using the same notation as in the last example, prove that
/(mx) = mf(x), m being any rational fraction.
2. Infinitesimals and infinites. A variable that approaches
zero as a limit is an infinitesimal . In other words, an infini-
tesimal is a variable that becomes smaller than any number
that can be assigned.
The reciprocal of an infinitesimal is then a variable that
becomes larger than any number that can be assigned, and
is called an infinite variable.
E.g., the number (J)" is an infinitesimal when n is taken larger and
larger ; and its reciprocal 2'» is an infinite variable.
1-2.] FUNDAMENTAL PRINCIPLES 3
From the definitions of the words " limit " and " infinitesi-
mal" the following useful corollaries are immediate inferences.
Cor. 1. The difference between a variable and its limit
is an infinitesimal variable.
Cor. 2. Conversely, if the difference between a constant
and a variable be an infinitesimal, then the constant is the
limit of the variable.
For convenience, the symbol = will be used to indicate
that a variable approaches a constant as a limit ; thus the
symbolic form x = a is to he read " the variable x approaches
the constant a as a limit."
The special form a; = oo is read "a? becomes infinite."
The corollaries just mentioned may accordingly be sym-
bolically stated thus :
1. li X = a, then x = a + a, wherein a = 0 ;
2. li X = a -\- a, and a = 0, then x = a.
It will appear that the chief use of Cor. 1 is to convert
given limit relations into the form of ordinary equations,
so that they may be combined or transformed by the laws
governing the equality of numbers ; and then Cor. 2 will serve
to express the result in the original form of a limit relation.
In all cases, whether a variable actually becomes equal to
its limit or not, the important property is that their differ-
ence is an infinitesimal. An infinitesimal is not necessarily
in all stages of its history a small number. Its essence lies
in its power of decreasing numerically, having zero for its
limit, and not in the smallness of any of the constant val-
ues it may pass through. It is frequently defined as an
"infinitely small quantity," but this expression should be
interpreted in the above sense. Thus a constant number,
however small it may be, is not an infinitesimal.
4 DIFFERENTIAL CALCULUS [Ch. I.
3. Fundamental theorems concerning infinitesimals and
limits in general. The following theorems are useful in
the processes of the calculus ; the first three relate to in-
finitesimals, the last four to limits in general.
Theorem 1. The product of an infinitesimal a by any
finite constant k is an infinitesimal ;
i.e.^ if a = 0,
then ka = 0.
For, let c be any assigned number. Then, by hypothesis, a
can become less than - ; hence ka can become less than c, the
k
arbitrary, assigned number, and is, therefore, infinitesimal.
Theorem 2. The algebraic sum of any finite number n
of infinitesimals is an infinitesimal ;
I.e., if a = 0, yS= 0, •••,
then a-}-/3-f---- = 0.
For the sum of the n variables does not at any stage
numerically exceed n times the largest of them, but this
product is an infinitesimal by theorem 1 ; hence the sum
of the n variables is either an infinitesimal or zero.
Note. The sum of an infinite number of infinitesimals may be
infinitesimal, finite, or infinite, according to circumstances.
E.ff.f if rt be a finite constant, and if n be a variable that becomes
infinite; then — , -, — , are all infinitesimal variables; but
n2 n „i
— + ^ + .•• to n terms = -, which is infinitesimal,
n* n^ n
while - + - + ••• to n terms = a, which is finite,
n n
and — + — + ••• to n terms = an^, which is infinite.
3-4.] FUNDAMENTAL PRINCIPLES 5
Theorem 3. The product of two or more infinitesimals
is an infinitesimal.
Theorem 4. If two variables x, y be always equal, and
if one of them, x^ approach a limit a, then the other ap-
proaches the same limit.
Theorem 5. If the sum of a finite number of variables
be variable, then the limit of their sum is equal to tlie sum
of their limits ;
I.e., lim (a; + ?/+•••)= lim a: + liin^ + •••.
For, let x = a^ y = ^-> **••
Then a; = « + «, y = h-\-^, •-, [Art. 2, Cor. 1.
wherein « = 0, yS = 0, •••;
hence x + y^ .- =(a + J + •••) + (« + ^+ -•);
but «4.^4.... = 0, [Th. 2.
hence, by Art. 2, Cor. 2,
lim(x-|- ?/+ •••)= a + h -\ =lim a: + lim y + •••.
Theorem 6. If the product of a finite number of varia-
bles be variable, then the limit of their product is equal to
the product of their limits.
Theorem 7. If the quotient of two variables rr, y be
variable, then the limit of their quotient is equal to the
quotient of their limits, provided these limits are not both
infinite, or not both zero.
4. Comparison of variables. Some of the principles just
established will now be used in comparing variables with
each other. The relative importance of two variables that
are approaching limits is measured by the limit of their
ratio.
6 DIFFERENTIAL CALCULUS [Ch. I.
Definition. One variable a is said to be infinitesimal,
infinite, or finite, in comparison with another variable x when
the limit of their ratio a : a; is zero, infinite, or finite.
In the first two cases, the phrase " infinitesimal or infinite
in comparison with " is sometimes replaced by the less pre-
cise phrase " infinitely smaller or infinitely larger than.'*
In the third case, the variables will be said to be of the same
order of magnitude.
The following theorem and corollary are useful in com-
paring two variables :
Theorem 8. The limit of the quotient of any two varia-
bles a;, y is not altered by adding to them any two numbers
a, y8, which are respectively infinitesimal in comparison with
these variables;
i.e.,
provided
For, since
it follows, by theorems 4, 6, that
1+-
,. x-\-a ,. X ,. X
lim 7^ = lim — • lim r; ;
y^^ y 1+^
y
but, by theorems 7, 5, and hypothesis,
lim 5=1;
y
therefore, lim ^ = lim -•
y+/3 y
,. x-\- a
y-\-^
lim-,
y
X y
= 0.
X ■\- a X
1 + ^
X
y-\-^~y
1 ,^'
4-5.] FUNDAMENTAL PRINCIPLES 7
Cor. If the difference between two variables rr, y be
infinitesimal as to either, the limit of their ratio is 1, and
conversely ;
i.e., if
^"^-0, then ^ = 1.
y y
For, since
x-y^x
y y
hence
- - 1 = 0, and - = 1. [Art. 2, Cor. 2.
y y •■
Conversely,
if
""^1, then^~^=0.
y y
For, by Art.
2,
Cor. 1,
? 1^0;z...,^-^-0.
y y
5. Comparison of infinitesimals, and of infinites. Orders of
magnitude. It has already been stated that any two variables
are said to be of the same order of magnitude when the limit
of their ratio is a finite number ; that is to say, is neither
infinite nor zero. In less precise language, two variables
are of the same order of magnitude when one variable is
neither infinitely larger nor infinitely smaller than the other.
For instance, Tc^ is of the same order as y8 when h is any
finite number; thus a finite multiplier or divisor does not
affect the order of magnitude of any variable, whether
infinitesimal, finite, or infinite.
In a problem involving infinitesimals, any one of them, a,
may be chosen as a standard of comparison as to magni-
tude ; then a is called the principal infinitesimal of the first
order, and a~^ is called the principal infinite of the first
order.
S DIFFEBENTIAL CALCULUS [Ch. I.
To test for the order n of any given infinitesimal yS with
reference to the principal infinitesimal a on which it depends,
it is necessary to select an exponent n such that
lim ^ _ r.
a = 0 ^ "" '*^'
wherein ^ is a finite constant, not zero.
When n is negative, ^ is infinite of order — n. An
infinitesimal, or infinite of order zero, is a finite number.
E.g., to find the order of the variable 3 x* — 4 x^, with reference to x
as the principal infinitesimal.
Comparing with x^, x\ x^, in succession :
lim 3x^-4x3 ^ lim (3 ^2 _ 4 ^) = q, not finite ;
x = 0 X
lim 3 x^ — 4 xS _ lim
a; = 0 r4 a;
hence 3 x* — 4 x^ is an infinitesimal of the same order of smallness as x*;
that is, of the third order.
The order of largeness of an infinite variable can be tested
in a similar way. For instance, if x be taken as the principal
infinite, let it be required to find the order of the variable
3 2^ _ 4 2^. Comparing with a^ and a^ :
lim Sa^-4a^^ lim (32;_4)=qo;
Hm 3 2:* -4 2:3 lij^
X = 00 ^ X
r«(3-g=3;
hence 3 a:* — 4 a;^ jg ^n infinite of the same order of largeness
!is 3^, that is, of tlie fourth order.
The process of finding the limit of the ratio of two in-
finitesimals is facilitated by the following principle, based
5-6]
FUNDAMENTAL PRINCIPLED
9
on theorem 8 of Art. 4 : The limit of the quotient of two
infinitesimals is not altered by adding to them (or subtract-
ing from them) any two infinitesimals of higher order,
respectively.
lim 3 a;2 + a:* _ lim 3 a:2 _ 3
a; = 0
E.g.,
4:X^—2X^
x = 04^-2 4
From these definitions the following theorems are at once
established :
Theorem 1. The product of two infinitesimals is another
infinitesimal whose order is the sum of the orders of the
factors.
Theorem 2. The quotient of an infinitesimal of order m
by an infinitesimal of order n is an infinitesimal of order m—n.
Theorem 3. The order of an infinitesimal is not altered
by adding or subtracting another infinitesimal of higher order.
6. Useful illustrations of infinitesimals of different orders.
lim sin 0
Theorem 1.
-1 . lim tan 6
6
With 0 as a center and OA = r
as radius, describe the circular
arc AB. Let the tangent at A
meet OB produced in D ; draw
BO perpendicular to OA, cutting
OA in O. Let the angle AOB = 6
in radian measure,
then arc AB = rO,
CB<^rcAB<AD,
i. e, , r sin 6 <r6 <r tan 6,
ain6<0< tan 9,
1.
B D
OA
Fia.l.
by geometry,
i.S^'
10 DIFFERENTIAL CALCULUlS [Ch. I.
By dividing each member of these inequalities by sin 6^
n
sin u
but sec ^ = 1, when ^ = 0,
v^/^««« lina ^ 1 „„ J lim sin 6 ^
hence ^^^__=1, and ^^^-^=1.
Similarly, by dividing the inequalities by tan 0^
tan 0
hence li?l_^ = l, and /Pli^=l.
^ = ^tan^ ^ = ^ ^
Cor. 1. The numbers 6, sin^, tan ^ are infinitesimals of
the same order.
Cor. 2. The expressions sin 6 — 0, tan 0 — 0 are infinitesi-
mal as to 0.
Theorem 2. If one angle 0, of a right triangle, be an
infinitesimal of the first order, then the hypotenuse r and
the adjacent side x are either both
finite, or they are infinitesimals of
the same order ; and the opposite
side 1/ is an infinitesimal of order
one higher than that of r and x.
For - = cos 0, which approaches the value 1 as ^ = 0 ;
T
hence x, r are infinitesimals of the same order ; which may
be the order zero.
Also y = r sin 0^
and sin 0 is of order 1 ; therefore y is of order one higher
than r, by theorem 1, Art. 5.
6.] FUNDAMENTAL PBINCIPLES 11
Cor. In the same case, if 6 be of the first order, and
if r and x be of the order n, then the difference between
r and x is an infinitesimal of order n + 2.
For 7^ — x^ = y'^ = 7^ sin^ ^, r — x = ;
r -\-x
but the orders of /^, sin^ ^, r + a:, are respectively 2 ^, 2, w ;
therefore by theorem 2, Art. 5, r — a; is of order
2n+2—n=n+±
Theorem 3. The difference between the length of an
infinitesimal arc of a circle and its chord is of at least the
third order when the arc is the first order.
For, let QD be the arc, and CB^ DB, tangents at its
extremities. Then by elementary geometry
chord CD < arc CD < BB + BC.
Let the angle B OD = ^ be taken as the principal infini-
tesimal. Then, since arc p
CD = 2 rd^ and r is finite,
shence;arc CD is of order 1.
Again, since AD is of o
order 1 (Th. 2), and
angle ADB = ^ is of or-
der 1, hence DB is of Fig. 3.
order 1, and DB — DA is of order 3 (Th. 2, Cor.); therefore
{DB -{-BC)- chord CD is of order 3.
Hence arc CD — chord CD is of order, at least, three.
Theorem 4. The difference between the length of any
infinitesimal arc (of finite curvature) and its chord, is an
infinitesimal of, at least, the third order.
Note. The curvature is said to be finite when the limiting ratio of
the length of a small chord to the acute angle between the tangents at
its extremities is finite, and not zero.
12
DIFFERENTIAL CALCULUS
[Cii. I.
li FQ be such an arc, the chord PQ and the angle TSF
are, by hypothesis, infinitesimals of the same order.*
Let the angle TSP be the
principal infinitesimal. Then,
since
TSF = aS'^^ + RFS,
it follows that the greater of
the latter two angles, say SQR,
is of the first order, while the
other may be of th§ first or
a higher order. Also, the
greater of the two segments
HQ, FM, say the latter, is of
the first order, while MQ may
be of the first or higher order.
Again, by theorem 2, QR, QS are of the same order, and
FR^ FS are of the same order.
Fig. 4.
Now arc QF - chord QF<QS+SF- QF,
i.e., < iQS- QE) + (SF-RF);
but since QS-QR= QS(1 - cos /3) = 2 QS sin^^.
feeo
m.
and, similarly.
AS'P-^P=2AS'Psin2^,
and, since each of these products is, at least, of the third
order, hence arc QF — chord QF is of, at least, the third
order.
* If TSiT were of higher order than P^, tho curvature would be zero ;
If of lower order, the curvature would be infinite ; the former is the case at
an inflexion, the latter at a cusp.
6-7.] FUNDAMENTAL PRINCIPLES 13
EXERCISES
1. Let ABC be a triangle having a right angle at C; draw CD per-
pendicular to AB, DE perpendicular to CB, EF perpendicular to DB,
FG perpendicular to EB ; let the angle BA C be an infinitesimal of the
first order, AB remaining finite. Prove that :
CD, CB are of order 1 ;
DB, DE are of order 2 ;
EB, EF, (CB - CD) are of order 3 ;
FB, FG, (DB - DE) are of order 4.
2. Of what order is the area of the triangle ^5C? BCD'} CDE'i
3. A straight line, of constant length, slides between two rectangular
straight lines, CAA', CB'B. Let AB, A'B' be tw^o positions of the line.
Show that, in the limit, when the two positions coincide,
AAf ^ CB
BB' CA
7. Continuity of functions. When an independent variable
x^ in passing from a to 5, passes through every intermediate
value, it is called continuous.
A function f(x) of an independent variable x is said to
be continuous at any value x^ when f(x^ is finite, real, and
determinate, and such that in whatever way x approach rcp
in which f(x-^ is independent of the law of approach.
From the definition of a limit it follows that corresponding
to a small increment of the variable the increment of the
function is also small, and that corresponding to any number
€, previously assigned, another number h can be determined,
such that when h remains numerically less than 3 the
difference ^, ,^ ^, -
is numerically less than e.
14 DIFFERENTIAL CALCULUS [Ch. I.
E.g., the function f(x) = x^ + 3 x -\- 2
is continuous at the value a: = 1.
/(1)=6, /(l + A)=6 + 5A + *«
/(I + h)-f(l)= bh + h^=h(_5 + h).
If the difference /(I + h) — f(l) is to be less than, say, unr^rnnf* i* Js
only necessary that
I ^ 1
(5 + ^)1000000*
If 8 = Ttnihinrsi ^^^^ ^^^ every value of h such that
it is evident that /(I + ^) — /(I) is less than nnrixnn)'
When a function is continuous at every value of x within
the interval from a to 5, it is said to be continuous within
that interval.
When a value x^ exists at which any one of the preceding
conditions is not fulfilled for a given function (f>(x)^ the
function is said to be discontinuous at a;= x^
E.g., the function may become infinite, as ^^ , when x = 2;
the function may be imaginary, as Vd — x^, when x^ > 9 ;
the function may be indeterminate, as sin -, when a: = 0 ;
X
finally, the value of the function may depend upon the manner in
which the variable approaches the value x^, as in the function
A^)=
1
2-3*.
V
1-3^
when X = + A, /(x)= 1 ; when x = - A, /(- A) = 2 as A = 0.
A continuous function actually attains its limit for any
value of the variable within the region of continuity, and
the variable may be substituted directly.
7-8.] FUNDAMENTAL PRINCIPLES 15
It may be shown as on p. 14 that any polynomial
aaf^ + ^2:""^ H [w a positive integer.
is continuous for every finite value of x.
The ordinary functions involving radicals and ratios are
continuous only for certain intervals.
The trigonometric functions sin x and cos x are continuous
for all real finite values of x ; the other trigonometric func-
tions are rationally expressible in terms of sine and cosine.
Show that tan x is discontinuous when x= ^ir.
The exponential function a^ and the logarithmic function
logo; are each continuous, the former for all finite values
of a;, the latter for all finite positive values of x [D. C, p. 31].
8. Comparison of simultaneous infinitesimal increments of
two related variables. The last few articles were concerned
with the principles to be used in comparing any two infini-
tesimals. In the illustrations given, the law by which each
variable approached zero was assigned, or else the two vari-
ables were connected by a fixed relation ; and the object was
to find the limit of their ratio. The value of this limit gave
the relative importance of the infinitesimals.
In the present article the particular infinitesimals com-
pared are not the principal variables x^ y themselves, but
simultaneous increments A, k of these variables, as they start
out from given values rr^, y-^ and vary in an assigned manner,
as in the familiar instance of the abscissa and ordinate of a
given curve.
The variables x^ y are then to be replaced by their equiva-
lents x^ -i-h, y^-\- Jc^ in which the increments h, k are them-
selves variables, and can, if desired, be both made to approach
zero as a limit ; for since y is supposed to be a continuous
16 DIFFERENTIAL CALCULUS [Ch. I.
function of x^ its increment can be made as small as desired
by taking the increment of x sufficiently small.
The determination of tlie limit of the ratio of k to A, as h
approaches zero, subject to an assigned relation between x
and ?/, is the fundamental problem of the Differential
Calculus.
E.g.^ let the relation be
let a:^, y-^ be simultaneous values of the variables rr, y ; and
when X changes to the value x^ + A, let y change to the
value y^ + h. Then
y^j^k = Qx^+ A)2 = x^ + 2x^h^ A2.
hence k=2 x-Ji + h^.
This is a relation connecting the increments h, k.
Here it is to be observed that the relation between the
infinitesimals A, k is not directly given, but has first to be
derived from the known relation between x and y.
Let it next be required to compare these simultaneous
increments by finding the limit of their ratio when they
approach the limit zero.
By division,
-=2x^-\-h\
hence, A^O^=^^i-
This result may be expressed in familiar language by
saying that when x increases through the value x^^ then y
increases 2 x^ times as much as x ; and thus when x continues
8.]
FUNDAMENTAL PRINCIPLES
17
to increase uniformly, 1/ increases more and more rapidly.
For instance, when x passes through the value 4, and y
through the value 16, the limit of the ratio of their incre-
ments is 8, and hence y is changing 8 times as fast as x ; but
when X is passing through 5, and ^ through 25, the limit of
the ratio of their increments is 10, and ^ is changing 10
times as fast as x.
The following table will numerically illustrate the fact
that the ratio of the infinitesimal increments A, Jc approaches
nearer and nearer to some definite limit when h and Jc both
approach the limit zero.
Let x^, the initial value of x.he 4. Then 1/^, the initial
value of t/, is 16. Let A, the increment of a;, be 1. Then k,
the corresponding increment of ^, is found from^
16 + A: = (4 + A)2;
thus Ar=9, and y = ^' Next let h be successively diminished
h
to the values .8, .6, .4, •••. Then the corresponding values of
Jc
Jc and of - are as shown in the table :
Ji
x = 4 + A
y = 16-\-k
k
k
h
4+ 1
25
9
9
4 +.8
23.04
7.04
8.8
4 + .6
21.16
5.16
8.6
4 +.4
19.36
3.36
8.4
4 +.2
17.64
1.64
8.2
4+.1
16.81
.81
8.1
4+ .01
16.0801
.0801
8.01
4 + A
16-f 8A + A2
8^ + A2
8 + A
18 DIFFERENTIAL CALCULUS [Ch. I.
Thus the ratio of corresponding increments takes the
successive values 8.8, 8.6, 8.4, 8.2, 8.1, 8.01, •••, and can
be brought as near to 8 as desired by taking h small enough.
As another example, let the relation between x and y be
Then y,^ = x,^
hence, by expansion and subtraction,
2 y^^ + ^2 _ 3 x2Ji + 3 ^.^^2 + ^8^
*(2y, + k)=h (dx^^+SxJi + A2),
k^dx,^-{- dx,h + h^
h 2yi+k
Therefore lim f = lim ^^i'+ ^^'^ / ^', as A = 0, * = 0,
and, by Art 4, theorem 8,
lim^ = 5^^
h 2y,
The " initial values " of x, y^ have been written with
subscripts to show that only the increments A, k vary
during the algebraic process, and also to emphasize the
fact that the limit of the ratio of the simultaneous incre-
ments depends on the particular values through which the
variables are passing, when they are supposed to take
these increments. With this understanding the subscripts
will hereafter be omitted. Moreover, the increments A, k
will, for greater distinctness, be denoted by the symbols
Aa:, Ay, read "increment of x," "increment of y."
Ex. 1. If x2 + 2/2= a\ find Hm^. I^t the initial values of the
variables be denoted by x, y, and let the variables take the res}>ective
increments Ax, Ay, so that their new values x + Ax, y •¥ ^y shall still
satisfy the given relation. Then
(x + Ax)2 + (y + Ay)2 = a«.
8.] FUNDAMENTAL PRINCIPLES
By expansion, and subtraction,
2 a: . Aa: + (Aa:)^ + 2 y • Ay + (Ay)2 =
hence Ax (2 or + Aa:) = - Ay (2 y + Ay),
and
Therefore
19
0,
Ay_ 2a; + Aa:
Aa: 2y + Ay
lim Ay _ lim 2 a: + Aa:
Aa; = OAx~ Ax = 02^+ Ay~
a:
2^
The negative sign indicates that when
Aa: and the ratio x:y are positive, Ay is
negative ; that is, an increase in x produces
a decrease in y. This may be illustrated
geometrically by drawing the circle whose
equation is a;^ -f y2 == a^ (Fig. 5).
Ex.2. If a;2 + y = y2_2a:,
prove lim Ay^2a: + 2,
^ Aa:=OAar 2y-l
Fig. 5.
Similarly, when the relation between x and y is given in
the explicit functional form
then
and
hence
y -{- Ay = cl>(x -{- Ax),
Ay = (i>{x + Ax) — <i>(x) = A<^(a;),
lim ^ = lim '^(^ + A^)-</'('^).
Ax Ax
When the form of <f> is given, the limit of this ratio can be
evaluated, and expressed as a function of x. This function
is then called the derivative of the function <t>(x) with
regard to the independent variable x.
The formal definition of the derivative of a function with
regard to its variable is given in the next article.
20 DIFFERENTIAL CALCULUS [Ch. I.
9. Definition of a derivative. If to a variable a small
increment be given, and if the corresponding increment of
a continuous function of the variable be determined, then
the limit of the ratio of the increment of the function to
the increment of the variable, when the latter increment
approaches the limit zero, is called the derivative of the
function as to the variable.
If (i>{x) be a finite and continuous function of a:, and Aa;
a small increment given to x^ then the derivative of <f>(x) as
to X is
lim
Ax = 0
(i>(x + Aa:) — <f>(x') 1 ^ lim Aj>(a;)
Aa; /-Aa: = 0 ^^
It is important to distinguish between lim ^^ ^ and
— - — ^^ ^ ; that is, between the limit of the ratio of two
lim Aa;
infinitesimals and the ratio of their limits. The latter is
indeterminate of the form - and may have any value ; but
the former has usually a determinate value, as illustrated in
the examples of the last article.
EXERCISES
1. Find the derivative of a;^ — 2 a: as to x.
2. Find the derivative of 3 x* — 4 a: + 3 as to x.
3. Find the derivative of — as to x.
4x
3
4. Find the derivative of x* — 2 H — as to x.
/[^ 10. Geometrical illustrations of a derivative. Some con-
ception of the meaning and use of a derivative will be
afforded by one or two geometrical illustrations.
Let 1/ = <^(a;) be a function of x that remains finite and
continuous for all values of x between certain assigned con-
9-10.]
FUNDAMENTAL PRINCIPLES
21
stants a and b ; and let the variables x, y be taken as the
rectangular coordinates of a moving point. Then the rela-
tion between x and y is represented graphically, within the
assigned bounds of continuity, by the curve whose equation is
y=^^{x).
Let (a;^, y-^^ (x^^ y^ be the coordinates of two points Py,
Then it is evident that the ratio
Pg? oil this curve
-y\
Xc — x^
is equal to tan a, wherein a is the inclination angle of the
secant line P^^'^ to the a;-axis. Let P^ be moved nearer and
nearer to coincidence with P^ so that x^ = x^ y<^ = y-^. Then
the secant line PiP^ approaches nearer and nearer to coinci-
dence with the tangent line drawn at the point
the inclination-angle a of
the secant approaches as
a limit the inclination
angle </> of the tangent
line.
Hence,
tan a = tan <f>.
Py and
Pi carg.y*)
Thus
X,
when X,
— -^ = tan <^,
2 — ^v Vi — yy
—X
Fig. 6.
It may be observed that if x^ be put directly equal to x^
and y2 to y^ the ratio on the left would, in general, assume
the indeterminate form -, as in other cases of finding the
limit of the ratio of two infinitesimals ; but it has just been
shown that the ratio of the infinitesimals y^ — y^, x^ — x^ has,
nevertheless, a determinate limit, viz., tan <^.
22 DIFFERENTIAL CALCULUS LCh. 1.
They are thus infinitesimals of the same order except
when <^ is 0 or — •
If the differences x^ — ajj, y^ — y^ be denoted by Aa;, Ay,
then x^ = x^-\- Aa:, 3^2 = ^i + ^^ 5
but, since y = <^(a:),
it follows that ^j = ^^(^i)^ ^2 ~ *^(^2)»
hence the ratio of the simultaneous increments may be
written in the various forms
Aa; x^ — x-^ x^ — x^ Ax
In the last form x is regarded as the independent variable
and Aa; as its independent increment ; the numerator is the
increment of the function <^(a;), caused by the change of x
from the value a;^ to the value x^ + Aa;. The limit of this
ratio, as Aa; = 0, is the value of the derivative of the function
</>(a;) when x has the value x^. Here x^ stands for any
assigned value of x. Thus the derivative of any continuous
function <l>(x) is another function of x which measures the
slope of the tangent to the curve y = </)(a;), drawn at the
point whose abscissa is x.
0
Ex. Find the slope of the tangent line to the curve y = — at the
point (1, 2).
- lim - 2(2 X + Ax) ^ _ 4.
Ax = 0 x\x + Ax)a x«
10.]
FUNDAMENTAL PRINCIPLES
23
Hence tan ^ = — 4, when a; = 1 ; and the equation of the tangent line at
the point (1, 2) is y - 2 = - 4(a; - 1).
As another illustration, if the coordinates of P be (x, y)^
and those of §, (a: + Aa;, «/+ A^), then
MN=PR = ^x, and P>S=EQ=Ai/.
It the area OAPM be denoted by
«, then z is evidently some function
of the abscissas;; also if area OAQN
be denoted by z + Az^ then the
area MNQP is Az ; it is the incre-
ment taken by the function z^ when
X takes the increment Ax, But MNQP lies between the
rectangles MR^ MQ ; hence
Y
S
/
R
"
A
X
0
M N
Fig. 7.
and
1/Ax <Az<(jj + Ay')Ax^
Therefore, when Aa?, A^, Az all approach zero,
v Az
lim — = v»
Ax ^
Thus, if the ordinate and the area be each expressed as a
function of the abscissa, the derivative of the area function
with regard to the abscissa is equal to the ordinate function.
Ex. If the area included between a curve, the axis of x^ and the
ordinate whose abscissa is a:, be given by the equation
z — a:*,
find the equation of the curve
Here y
lim ^ - li™ (a; + Aar)8 - ofi
Aar Aic = 0 Aa:
lim
= ^^^ Q [3 a;2 + 3 arAa: + (Aa:)2] = 3 a;=
24 DIFFERENTIAL CALCULUS [Ch. I.
11. The Operation of differentiation. It has been seen in a
number of examples that when the operation indicated by
lim j>(2; + Aa;) - <t)(x)
Ax = 0 ^x
is performed on a given function <t>(x)^ the result of the
operation is another function of x. The latter function may-
have properties similar to those of (^(a;), or it may be of an
entirely different class.
The operation above indicated is for brevity denoted by
the symbol — ^-^? and the resulting derivative function by
(^'(x)', thus,
#(^)_ lim A0(a:)_ lim <^(a:4- Aa;)- <^(a;)
C?a; ~ Aa; = 0 Aa: Aa: = 0 ^^
<\>\x^.
The process of performing this indicated operation is
called the differentiation of <t>(x) with regard to x. The
symbol * — , when spoken of separately, is called the differ-
dx
entiating operator, and expresses that any function written
after it is to be differentiated with regard to rr, just as the
symbol cos prefixed to <f>Qx^ indicates that the latter is to
have a certain operation performed upon it, namely, that
of finding its cosine.
The process of differentiating <^(a;) consists of the follow-
ing steps :
1. Give a small increment to the variable.
2. Compute the resulting increment of the function.
3. Divide the increment of the function by the increment
of the variable.
4. Obtain the limit of this quotient as the increment of
the variable approaches zero.
* This symbol is sometimes replaced by the single letter D.
11-12.] FUNDAMENTAL PRINCIPLES 25
EXERCISES
Find the derivatives of the following functions :
1. 5 3/8 - 2 2/ + 6 as to 3^. 3. 8 w^ - 4 w + 10 as to 2 m.
2. 7 /2 _ 4 ^ - 11 ^8 as to «. sX 4. 2 a;2 _ 5 a: + 6 as to x - 3.
This process will be applied in the next chapter to all the
classes of functions whose continuity within certain inter-
vals has been pointed out in Art. 7. It will be found that
for each of them a derivative function exists ; that is, that
lim —^ — - has a determinate and unique value, and that the
curve 1/ = (t>{x) has a definite tangent within the range of
continuity of the function.
A few curious functions have been devised, which are continuous and
yet possess no definite derivative ; but they do not present themselves in
any of the ordinary applications of the Calculus. Again, there are a few
functions for which lim \} ^ has a certain value when Ax = 0 from
Aa;
the positive side, and a different value when Aa: = 0 from the negative
side ; the derivative is then said to be non-unique.
Functions that possess a unique derivative within an as-
signed interval are said to be differentiahle in that interval.
Ex. Show that the four steps of p. 24 do not apply at a discontinuity.
12. Increasing and decreasing functions. A good example
of the use of the derivative is its application to finding the
intervals of increasing or decreasing for a given function.
A function is called an increasing function if it increases
as the variable increases and decreases as the variable de-
creases. A function is called a decreasing function if it
decreases as the variable increases, and increases as the
variable decreases.
E.g., the function a;^ + 4 decreases as x increases from — oo to 0, but
it increases as x increases from 0 to + oo. Thus a:'^ + 4 is a decreasing
26
DIFFERENTIAL CALCULUS
[Ch. I.
function while x is negative, and an increasing function while x is posi-
tive. This is well shown by the locus of the equation y=x^-\-^ (Fig* 8).
Fig. 8.
Again, the form of the curve y = - shows that - is a decreasing func-
tion, as X passes from — co to 0, and also a decreasing function, as x
passes from 0 to + oo. When x passes through 0, the function changes
discontinuously from the value — co to the value + oo (Fig. 9).
Most functions are
increasing functions
for some values of the
variable, and decreas-
ing functions for
others.
Fia. 10. ^'9"> v/2 rx- x^ is an
increasing function from
a; = 0 to a; = r, and a decreasing function from x = rio x=2r (Fig. 10).
A function is, said to be an increasing or decreasing func-
tion in the vicinity of a given value of x according as it
increases or decreases as x increases through a small interval
including this value.
12-13.]
FUNDAMENTAL PRINCIPLES
27
13. Algebraic test of the intervals of increasing and de-
creasing. Let 1/ = </>(a;) be a function of x^ and let it be real,
continuous, and differentiable for all values of x from a to h.
Then by definition i/ is increasing or decreasing at a point
X = x^, according as
is positive or negative, where Ax is a small positive number.
The sign of this expression is not changed if it be divided
by Ax, no matter how small Ax may be ; hence 0(:?^) is an
increasing or a decreasing function at the value x^, accord-
ing as
Ax
^= . lim f <^(^i + Ax) - cl>(x{) I ^ ^,^^^^
dx
is positive or negative.
Thus the intervals in which (i>(x) is an increasing function
are the same as the intervals in which <t>'(x) is positive.
Ex. Find the intervals in which the function
, f^{x) = 2 a:^ - 9 a:2 + 12 a; - 6
is increasing or decreasing. The derivative is
<^'(a:)= 6a:2 - 18a; + 12 = 6(a: - l)(a: - 2);
hence, as x passes from -co to 1, the derived function ^'(x), is positive
and ^{x) increases from ^( — go) to f^(l),
i.e., from <f>= — ao to <^=— 1; as x passes
from 1 to 2, <f>'(x) is negative, and <l>(x)
decreases from ^(1) to <^(2), i.e., from
— 1 to - 2 ; and as x passes from 2 to +qo,
<j!)'(x) is positive, and <f>(x) increases from
<f}(2) to <^(od), i.e., from — 2 to + oo.
The locus of the equation y = cf>(x) is
shown in Fig. 11. At points where
<l>'(x)=0, the function <^(x) is neither
increasing nor decreasing. At such points
the tangent is parallel to the axis of x.
Thus in this illustration, at a: = 1, x = 2,
the tangent is parallel to the x-axis. Fig. 11.
28 DIFFERENTIAL CALCULUS [Ch. I.
EXERCISES
1. Find the intervals of increasing and decreasing for the function
^{x) = a;3 + 2 a;2 + a: - 4.
Here <^'(a;) = 3 a:2 + 4 x + 1 = (3 a: + 1) (a: + 1).
The function increases from x = —co to x = — 1; decreases from x = — 1
to a; = — I ; increases from a; = — ^toa; = oo.
2. Find the intervals of increasing and decreasing for the function
y = a;8 - 2 a;2 + a: - 4,
and show where the curve is parallel to the a;-axis.
3. At how many points can the slope of the tangent to the curve
3/ = 2a;8-3a:2+l
be 1 ? - 1 ? Find the points.
4. Compute the angle at which the following curves intersect :
3^ = 3 a:2 - 1, y = 2 x^ + 3.
14. Differentiation of a function of a function. Suppose
that y, instead of being given directly as a function of a;,
is expressed as a function of another variable w, which is
itself expressed as a function of x. Let it be required to
find the derivative of y with regard to the independent
variable x.
Let y=f(u)'> in which tt is a function of x. When x
changes to the value x + Aa;, let u and y^ under the given
relations, change to the values w-f-Aw, y + Ay. Then
A^ _ A^ Aw _ f(u-\-Au) — f(u') ^ Aw .
Ax A?/ Ax An Ax
hence, equating limits,
^y _ ^U ^^^ _ df(u) ^ du
dx du dx du dx
13-14.] FUNDAMENTAL PRINCIPLES 29
This result may be stated as follows;
The derivative of a function of u with regard to x is equal to
the product of the derivative of the function with regard to u,
and the derivative of u with regard to x.
EXERCISES
1. Given y-'^u^-l, w = 3a;2+4:; find ^
dx
dy ^ du ^
du dx
2. Given y = 8 m2 _ 4 m + 5, m = 2 a:^ - 5 ; find $^
dx
3. Given ^ = i, m = 5 x2 - 2 a: + 4; find $•
u dx
4. Given, = 3„^ + ^,„=^+|; find g.
CHAPTER II
DIFFERENTIATION OF THE ELEMENTARY FORMS
In recent articles, the meaning of the symbol -^ was ex-
plained and illustrated; and a method of expressing its
value, as a function x^ was exemplified, in cases in wliich y
was a simple algebraic function of a;, by direct use of the
definition. This method is not always the most convenient
one in the differentiation of more complicated functions.
The present chapter will be devoted to the establishment
of some general rules of differentiation which will, in many
cases, save the trouble of going back to the definition.
The next five articles treat of the differentiation of alge-
braic functions and of algebraic combinations of other differ-
entiable functions.
15. Differentiation of the product of a constant and a
variable.
Let
y=^cx.
Then
^ + A?/ = (?(a; + Aa;),
Ay = cQc -f Aa;) — ca: = cAr,
therefore
t-
SO
Ch. II. 15-16.] DIFFERENTIATION OF ELEMENTARY FORMS 31
Cor. If y = cu^ where w is a function of x^ then, by
Art. 14, ^, ^
dx doc
The derivative of the product of a constant and a variable is
equal to the constant multiplied hy the derivative of the variable.
16. Differentiation of a sum.
Let y=f(x) + <i>(x) + 'f{x).
Then y + ^y =f(x + i^x) + <i>(x + A:r) + -^(x + Lx),
Ay ^fCx+Ax}-f(x} ^ (f>(x-hAx}-(l)(x}
Ax Ax Ax
^'\lr(x + Ax}-ylr(x')
Ax
Therefore, by equating the limits of both members,
g =/'(^) + 4,'Cx^ + ^'(x-). [Art. 3, Th. 5.
Cor. 1. \i y = u -^ V + w^ in which u^ v, w are functions
of X, then ^ ^ \,
The derivative of the sum of a finite number of functions is
equal to the sum of their derivatives.
Cor. 2. \i y = u ■\- c^ c being a constant, then
y -\- Ay = w + Au + c ;
hence, Ay = Aw,
and ■ dy^du
dx dx
The last equation asserts that all functions which differ
from each other only by an additive constant have the same
derivative.
32 DIFFERENTIAL CALCULUS [Ch. II.
Geometrically, the addition of a constant has the effect of
moving the curve y = u(x) parallel to the y axis ; this opera-
tion will obviously not change the slope at points that have
the same x,
^ ,-. dy du , dc
trom(2), 3^ = T- + :r»
dx dx dx
but from the fourth equation above,
dy _ du^
dx dx*
dc
hence, it follows that — - = 0.
dx
The derivative of a constant is zero.
If the number of functions be infinite, theorem 5 of Art. 3 may not
apply ; that is, the limit of the sum may not be equal to the sum of the
limits, and hence the derivative of the sum may not be equal to the sum
of the derivatives. Thus the derivative of an infinite series cannot always
be found by differentiating it term by term.
17. Differentiation of a product.
Let y=/(a^)</>(^)-
Then ^ = /(y + Aa;)</)(a: + Aa:) -/(a;) j>(a;)
Ax Ax
By subtracting and adding f(x)^Qx + Ax) in the numer-
ator, this result may be rearranged thus :
Now let Ax approach zero, using Art. 3, theorems 5, 6,
and noting that the first factor (\){x -h Ax) approaches <j)(x)
since by hypothesis (f>(x) is continuous (Art. 7). Then
16-18.] DIFFERENTIATION OF ELEMENTARY FORMS 33
Cor. 1. By writing u = (^{x)^ v=f(x)^ this result can
be more concisely written,
d{uv) _ dv du ^gv
doc ~ doc doc ^
The derivative of the 'product of two functions is equal to the
sum of the products of the first factor by the derivative of the
second^ and the second factor hy the derivative of the first.
This rule for differentiating a product of two functions
may be stated thus : Differentiate the product, treating the
first factor as constant, then treating the second factor as
constant, and add the two results.
Cor. 2. To find the derivative of the product of three
functions uvw.
Let y = uvw.
/ dv
= w[ u—-
\ dx
du\ , dw
dxj dx
The result may be written in the form
dCuvw) dw , du , dv ...
By application of the same process to the product of
4, 5, '", n functions, the following rule is at once deduced:
The derivative of the product of any finite number of factors
is equal to the sum of the products obtained by multiplying the
derivative of each factor by all the other factors.
18. Differentiation of a quotient.
Let y
*(^)
Then y + Ay=lp^,
84 DIFFERENTIAL CALCULUS [Ch. 11.
fjx^^x-) fix-)
Ay _ <l>(x-{-Ax) <t>(x)
Ax" Ax
^ j^jx^fCx + Ax) ^f(x)(t>(x H- Ax-)
Ax(t)(x)(t)(x-\-Ax)
By subtracting and adding (\)(x)f{x) in the numerator,
this expression may be written
... [f(x-\-Ax)-f(ix)\ l^(x+Ax')-<t>{x)\
Ax <t){x)<j)(x -{- Ax)
Hence, by equating limits,
dy <l>(x)f{x) -f(ix)<i>'(x) p. q Tha 6 7
di [f(x)Y "-Art. d, 1 hs. b, 7.
This result may be written in the briefer form
du dv
d / '* \ _ dx dx i-gN
dQc\v)~ v^ '
The derivative of a fraction^ the quotient of two functions, is
equal to the denominator multiplied hy the derivative of the
numerator minus the numerator multiplied hy the derivative
of the denominator, divided hy the square of the denominator,
19. Differentiation of a commensurable power of a function.
Let y = w^ in which w is a function of x. Then there are
three cases to consider.
1. w a positive integer.
2. n a negative integer.
8. n a commensurable fraction.
1. w a positive integer.
This is a particular case of (4), the factors w, v, w, ••. all
being equal. Thus
dx dx
18-19.] DIFFERENTIATION OF ELEMENTARY FORMS 35
2. wa
negative
integer.
Let w =
= -m,
in
which m\8 a. positive integer.
Then
y=„.=,*-»=i,
and
dx
hence
dx dx
3. w a commensurable fraction.
P
Let w=— , where jt?, q are both integers, which may be
either positive or negative.
p
Then y = u'' = 'uP\
hence y^ = u^^
(ia; ^ dx
Solving for the required derivative,
dx q dx^
hence ^ = nw" -^ ~ (6)
doc dx
The derivative of any commensurable power of a function is
equal to the exponent of the power multiplied hy the power with
itB exponent diminished by unity, multiplied by the derivative
of the function.
•'■
* If two functions be identical, their derivatives are identical.
36 DIFFERENTIAL CALCULUS [Ch. II.
These theorems will be found sufficient for the differentia-
tion of any function that involves only the operations of
addition, subtraction, multiplication, division, and involu-
tion in which the exponent is an integer or commensurable
fraction.
The following examples will serve to illustrate the theo-
rems, and will show the combined application of the general
forms (1) to (6).
1. y =
ILLUSTRATIVE EXAMPLES
3^'- 2. ^^^ dy
X + 1 ' dx
dy (x+l)£(B.-2)-(3x»-2)£(x
+ 1)
by (6)
dx (x+l)-^
|(^^^-2)=£(^^^>-|(2>
by (2)
= 6x.
by (1). (0)
i.(x+l) = ^ = l.
dx dx
by (2)
Substitute these results in the expression for -^' Then
dy _ (x + 1)6 x - (3 x^^ - 2) ^ 3 a:2 + 6 37 + 2
dx {x + 1)2 {x + 1)2
2. m = (3«2h-2)V1 + 552; find—.
ds
^= (3«2 + 2)4 Vl + 5.2 + v/TfSTS. 4(3*2+2). by (3)
ds ds ds
4vi + 5«2= 4(1 + 5*2)*
€U ds
= |(l + 5«2)-il(l + 5«2) by (6)
5»
Vl + 5«2
4(8««4-2) = 6«. by (6)
ds
19.] DIFFEBENriATlON OF ELEMENT AUY FORMS 37
Substitute these values in the expression for — • Then
ds
3. y^VTJ^^+Vljz:^^^ fi^^ rf^.
VI + x^ _ Vl - x^ ^a;
First, as a quotient,
dy dx
dx~ (\/I+^^-V'n^)2
</
( vTT^ + VT^=^) — ( vr+^ - VI - a;2)
^^" by (6)
(VH-a:2- Vl-x2)2
— (VTT^ + VI - a;2) = -^L VTT^^ + iL VI - a;2. by (2)
c?a; dx dx
-f VrT^= -f (1 + :r2)^ = i(l + x'^y-^^il + a;2). by (6)
dx dx ^ dx
— (1 + x2) = 2 X. by (2) and (6>
dx •
Similarly for the other terms. Combining the results,
dx x^ \ Vl — x^f
Ex. 3 may also be worked by first rationalizing the denominator.
EXERCISES
Find the a:-derivatives of the following functions:
1. 2/ = xio. ^ y X
2. y = x~^.
3. y = cV^. 9. y
8/—
10. y=(x+ 1) Vx + 2.
4.
V^ 3
5.
y=^t^.
6.
?/ = (x + a)«.
7.
y = a;« + a«.
11. y= ^«+^
Va + Vx
--=Vr^-
38 DIFFERENTIAL CALCULUS [Ch. II.
X TQ 3x8+2
13. y = 19. y= r
x+Vl- x^ x{x^-\-\y
14. y = (2 J + x^) VJ77. ^20. y = ^x^ + 1)*(4 x^ - d).
.„ 21. y=du^-7.
15. y=\ [ '
^ I 1- x^
16
17. y^
y/l _ a;2 > 22. ?^ = 4 m8 - 6 m2 + 12 M - 3.
23. y=(l-3w2+6M*)(l + u2)3.
24. y = wx.
X«_4_l 25. 3^ = m2 ^ 3 a;ti2 ^ a;4.
x«- 1
y =
18. y = -^ 1— . '"^"^'^
(a + xl"* (& + x)« 27. y = M%8w.
28. Given (a + x)^ = a^ + 5 a^x + 10 a^x^ + 10 a^x^ + 5 ax* + x^ ; find
(a + x)* by differentiation.
29. Show that the slope of the tangent to the curve y = x* is never
negative. Show where the slope increases or decreases.
V 30. Given b^x^ + aY = a^^^ find -^ : (1) by differentiating as to x ;
(2) by differentiating as to y; (3) by solving for y and differentiating
as to X. Compare the results of the three methods.
31. Show that form (1), p. 31, is a special case of (3).
^ 32. At what point of the curve y^ = ax* is the slope 0? — 1? +1?
r 33. Trace the curve iy = x^ + 3 x^ + x — 1.
34. V = '^ "' -^ "^ and M = 5 x2 - 1 ; find ^.
V7 m2 + 5 rfa:
y 35. At what angle do the curves y^ =: 12 x and y^ -\- x^ + Q x - QS = 0
intersect ?
20. Elementary transcendental functions.
The following functions are called transcendental func-
tions :
Simple exponential functions, consisting of a constant
number raised to a power whose exponent is variable,
as 4', a** ;
19-21.] DIFFERENTIATION OF ELEMENTARY FORMS 39
general exponential functions, involving a variable raised
to a power whose exponent is variable, as x^^^ ;
the logarithmic * functions, as log« x^ log^ u ;
the incommensurable powers of a variable, as x^^, u^ ;
the trigonometric functions, as sin w, cos u ;
the inverse trigonometric functions, as sin~^ w, tan~^ x.
There are still other transcendental functions, but they
will not be considered in this book.
The next four articles treat of the logarithmic, the two
exponential functions, and the incommensurable power.
21. Differentiation of loga oc and loga u.
Let
g = log„x.
Then
g + Ag = log^(^x -{- Ax},
^y _ ^^^a (^ + Ax} - l0g« X
Ax Ax
1 , (x^Ax\
-AxM X }
For convenience writing h for Ax, and rearranging.
Ax
* The more general logarithmic function log„ w is not classified separately,
as it can be reduced to the quotient -2E^.
loga V
' /
/
40 DIFFERENTIAL CALCULUS [Ch. II.
X
To evaluate the expression ( 1 + - 1 when A = 0, expand it
by the binomial theorem, supposing - to be a large positive
integer m.
The expansion may be written
l^W ~ »» 1-2 ™2+ 1.2.3 «t3+ '
which can be put in the form
V^mJ ^ ^1 2 ^1 2 8 ^
1 2
Now as m becomes very lar^e, the terms — , — , ••• become
-^ ^ mm
very small, and when w« = oo the series approaches the limit
1 + 1+ — + — + — +-.
2! 3! 4!
The numerical value of this limit can be readily calculated
to any desired approximation. This number is an important
constant, which is denoted by the letter e, and is equal to
2.7182818...; thus
lim
m
^^ (l + -X = e = 2.7182818
= «>\ mj
The number e is known as the natural or Naperian base ;
and logarithms to this base are called natural or Naperian
logarithms. Natural logarithms will be written without a
• This method of obtaining e is rather too brief to be rigorous ; it assumes
that — is a positive integer, but that is equivalent to restricting Aa; to
Ax
approach zero in a particular way. It also applies the theorems of limits to
the sum and product of an infinite number of terms. The proof is completed
on p. 316 of McMahon and Snyder's '* Differential Calculus."
ex-
21-22.] DIFFERENTIATION OF ELEMENTARY FORMS 41
subscript, as log x ; in other bases a subscript, as in log^ x,
will generally be used to designate the base. The logarithm
of e to any base a is called the modulus of the system whose
base is a.
X
If the value, ;^^o(l+-l = e, be substituted in the
pression for -~, the result is
di/ 1 ,
More generally, by Art. 14,
£l„„« = l.log„e.f. (7)
In the particular case in which a = e.
The derivative of the. logarithm of a function is the product
of the derivative of the function and the modulus of the system
of logarithms, divided hy the function.
22. Differentiation of the simple exponential function.
Let y =1 a^.
Then log y = u log a.
Differentiating both members of this identity as to a;,
, l^=log«.f^, byforin(8),
y ax ax
dy 1 du
■dx^'^^'^y'Tx'
therefore ^a'* = log a • a** • ~ (9)
In the particular case in which a = «,
eu = e** • -j— (10)
doc dx
42 DIFFERENTIAL CALCULUS - [Ch. II.
The derivative of an exponential function with a constant base
is equal to the product of the function, the natural logarithm of
the base^ and the derivative of the exponent.
23. Differentiation of the general exponential function.
Let y = u^^
in which -m, v are both functions of x.
Take the logarithm of both sides, and differentiate. Then
logi/ = v log u,
ydx dx udx
dx \__ dx u dx] '
therefore dx^^ ~ ^^ ^^^ ^ d^^ ^^^^dx ^^^^
The derivative of an exponential function in which the base
is also a variable is obtained by first differentiating^ regarding
the base as constant^ and again, regarding the exponent as
constant, and adding the residts.
In the differentiation of any given function of this form it
is usually better not to substitute in the formula directly
but to apply the method just used in deriving (11), i.e., to
differentiate the logarithm of the function by the preceding
rules.
Ex. y = (4 0:2 - 'jy+^^^a^ fi^^ %
ax
logy = (2 + Vz2-r5) log (4 x^ - 7).
^ = (4x^- 7)«+^'^« .^rlog (4x^-7) 8(2 + VJ^^Ts)-!
*i^ I V^a^^ 4x»-7 J"
\ \
22-24.] DIFFERENTIATION OF ELEMENTARY FORMS 43
24. Differentiation of an incommensurable power.
Let 1/ = w",
in which n is an incommensurable constant. Then
log 1/ = n log u^
\d]£_ n ^ du
y dx u dx^
du y du
dx u ax
d n n-\ du
dx dx
This has the same form as (6), so that the qualifying word
'' commensurable " of Art. 19 can now be omitted.
EXERCISES
Find the x derivatives of the following functions :
1. y = log(a:+a). i
16. y = e^+*.
2. y=zlog(ax + b).
3. y = log(4a;2-7a: + 2). ^^- ^ = i^PgT
^. y= log ii-^. 18. y = e''(l- x^).
1 — X
1 l + 3;2 19. y = - —
6. 2,= xlog^. 20. ./ = log(6^-.-»).
7. 2, = x-log:r. 21. 2, = log (^ + .^).
8. y = 3-^ logx- 22. y = x"a»=. ^
9. y = log\/]r^^. 23. y = \og^±^'
10. y = y/x - log (Va: + 1). -i
11. y = log«(3x2-V2T^). 24. 3/ = j^-
12. 3/ = log,„ {x^ + 7 x). 25. 2/ = (log xy.
13. ?/ = log^ a. 26. 2/ = log (log x).
14. 2/ = e^^"- 27. y = xf'.
15. y = e^x+s^ 28. .y = a^og==.
44
DIFFERENTIAL CALCULUS
[Ch. II.
The followiug functions can be easily differentiated by first taking
the logarithms of both members of the equations.
29. v =
(^-1)
(2:-2)^(a:-3)*
30. y = a;Vl-a:(l + x).
31. , = £lldL£!l.
32. y = x\a + 3 xy{a - 2 x)^.
33. y:
ovy-''
Articles 25-31 will treat of the differentiation of the
Trigonometric Functions.
25. Differentiation of sin u.
Let y = sill u.
Ay _ sin (u + Au) — sin 2^ A?*
Then
Ax Au Ax
_ 2 cos |( 2 1^ + A?^) sill I- Au ^ Ai*
A?^
Ax
r . 1 A \ Sill J Au Au
. s= COS (^ + i At*) r-^ •
^ ^ lAu Ax
But, when Au = 0, cos (?* H- | Au) = cos w, and —^. = 1
by Art. 6 ; hence, passing to the limit.
iAu
d . du
(12)
The derivative of the sine of a function is equal to the prod-
uct of the cosine of the function and the derivative of the
function.
26. Differentiation of cos u.
Let ^ = cosw = sinf ^ — wj-
'^'>-|=f/Kl-")=-<l-")rif-4
d , du
(18)
24-28.] DIFFERENTIATION OF ELEMENTARY FORMS 45
The derivative of the cosine of a function is equal to minus
the product of the sine of the function and the derivative of the
function.
27. Differentiation of tan u.
sin u
Let ?/ = tan u
GOSU
d ■ . d
cos u • — - sm u — sin u • — cos u
Then -/-= 2 ^1 (5)
ax coa^u ^ V ^
» du , . o du du
aa^ aa: dx
caa^u
(12), (13)
that is, ^ tan M = sec^ m ^- • (14)
The derivative of the tangent of a function is equal to the
product of the square of the secant of the function and the
derivative of the function.
28. Differentiation of cot u.
Let y = cot u = •
tan u
c^a; tan^tfc dx U\n^u dx''
-5- cot t* = - csc^ u -T— (15)
The derivative of the cotangent of a function is equal to minus
the product of the square of the cosecant of the function and
the derivative of the function.
46 DIFFERENTIAL CALCULUS [Ch. II.
29. Differentiation of stcu.
Let y = sec u =
cosu
rjy. dy —1 d , smu du
Then -f- = — k- • -j- cos w = H k- 3-,
dx cos^u dx cos^ udx
^8ecu = tsLnusecu~ (16)
dx dx
The derivative of the secant of a function is equal to the
product of the secant of the function^ the tangent of the func-
tion^ and the derivative of the function,
30. Differentiation of cscu,
1
Let y — CSC u
sin w
rp, ^„ dy — 1 d . COS u du
Then -f- = ^--- • — sin w = - -^— - -—•
ax sm^w ax mn^udx
-P-CSCM =-C8Ct«C0tW^- (17)
dx dx
The derivative of the cosecant of a function is equal to mimis
the product of the cosecant of the function^ the cotangent of the
function^ and the derivative of the function.
31. Differentiation of verst*.
Let y = vers w = 1 — cos u.
Then -^ = — — cos t«.
dx dx
^Ter8W = 8inM^. (18)
dx dx
The derivative of the versed-sine of a function is equal to
the produ>ct of the sine of the function and the derivative of
the function.
29-32.] DIFFERENTIATION OF ELEMENTARY FORMS 47
EXERCISES
Find the x derivatives of the following functions :
1. y = sin 7 x, 18. y = sin (m + li) cos (m — 6) /
2. y = cos 5x. 19 — s^^"* ^^, ^
3. y = sin x^. * cos« wa:
4. 3^ = sin 2 ar cos a:, 20. «/ = x + log cosf a; - | j-
5. y — sin^ x. i:j ^
7. y=3m2 7i. 22. j, = sin (sin «).
a 2/ = itan»x-tanx. 23. i, = sin^ e"".
9. y = sin'acosi. 24, y = sin «- • logx.
10. 2, = tanrt+8ec:t. 25. t, = ^^ii^. '
mx
12. , = tan(3-5.T. 27. , = csc» 4 x. -
13. ?/ = tan^x — logrsec^a;). _o ,. o\?! ^
•^ ^ ^ ^ 28. ?/ = sec (4 x — 3)2.
^14. y = log tan(ia: + iTr). or* « •>
^ ° vz ' 5 y 29. y = vers a;^.
15. y — log sin Vx. «^ .l 9 . /- -*>
^ ° J 30. y = cot x^ + sec vx. ^
^ 16. y = tan a*. 31. ?/ = sin xy.
\ « 4 17. y = sinna: sin«a:. 32. y = tan (x + y).
•^ 32. Differentiation of Sin- li^. oM/l^>'^>^
Let ^ = sin~^w.
Then sin y = u^
and, by differentiating both members of this identity,
hence,
dy du
dx dx
dy _ 1 du 1 du ^
dx cos y dx j. VI — sin^^ dx
6? . _i \ du
i.e,j —-sin ^u=± — -— '
dx Vl — u^ ^^
The ambiguity of sign accords with the fact that sin""^ u
is a many- valued function of u, since, for any value of u be-
48 DIFFERENTIAL CALCULUS [Ch. II.
tween — 1 and 1, there is a series of angles whose sine is u :
and, when u receives an increase, some of these angles in
crease and some decrease ; hence, for some of them,
du
is positive, and for some negative. It will be seen that,
when sin~^w lies in the first or fourth quarter, it increases
with tfc, and, when in the second or third, it decreases as w
increases. Hence, for the angles of the first and fourth
quarters,
-^sin-^^ = 4- ^ . ^sin-ii.^4-— 1— ^. (19)
In the other quarters the minus sign is to be used before
the radical.
33. Differentiation of the remaining inverse trigonoikfitric
forms.
The derivatives of the other inverse trigonometric- func-
tions can be easily obtained by the method employed in the
last article. The results are as follows :
^ co8-iw - -^ f" (in 1st and 2d quarters).
(20)
doc Vl _u'i^iio ^ ^ ^
/.**-"'" =, + «^^: (in all quarters).
(21)
^'''^-x'^'u^Z 0" all quarters).
(22)
i^ s€c-»M - * "J** (in 1st and 3d quarters).
(23)
dx uVu^^ ^dx ^ ^ ^
** CSC »t« - ~* ^^ (in 1st and 3d quarters).
(24)
ax uVu^ -idx ^
^ vers-i w%: * ^** rin 1st and 2d quarters^
dx V2u-U'^^^
(26)
The radicals in forms (20), (23), (24), (25) receive the
opposite signs respectively when the angles are taken in the
quarters other than those stated.
32-34.] DIFFERENTIATION OF ELEMENTARY FORMS 49
EXERCISES
Find the a:-derivative of each of the following functions:
1. y = sin~^ 2 x^. 16. y = tan x • tan~^ x.
I. y = cos- 1 VI - x^. *H\1. y = x sin-ix.
;6. y =
Vl7. v =
/ 3. 2^ = sin- 1(3 a: - 1).
18. v = e
tau-'x
4. ?/= sin-i(3x — 4x8). -.
y< 19. y = csc-^
/ 5. w=sm-i^ —• '^^ ^
/ . ., ^20. 2/ = sec-i^-±-!^.
6. 3/ = vsin-ix. "^ x^—1
- 7. j, = tan-'«-. ^21. y = taD-'^ + l".
S. y = cos-i log ar. 1 - Vaa:
</9. w = sin-i(tana;). ^^ ,ex_e-x
^ ^ ' 22. 2/ = cos-i^ —
10. y=sec-i ^ c*-fe-^
VI - a:2 V 23. ?/ = tan-i(n tana:).
•^11. V = vers-i 2 x2. ' -iz o \
^ 24. y = cos '(cos 2 a:).
12. y = tan-if ^ j» y 25. ?/ = cos-i(2cosa:).
/ 0^:2 26. w = tan-i(Vr+^- a:).
V 14. w = sin-1 Vsin x.
. y = sin-'vsin a:. '1 +
i/ 27. y = 2tan-i /izi^.
^1 + a:
15. y = tan-i Jln:^^. 28. ^ = tan-i?f^^+ tan-i?lzJ
^ ^1 + cosa: . ^Va ^y/3
34. Table of fundamental forms.
d(cu) =c^' (1)
doe dx »
#(« + t.4t«)=^+^+^- (2)
dx dx dx dx
d(uv) =u^+v^. (3)
dx dx dx ^
^{uvw) =uv^+uw^-\^vw^' (4)
dx dx dx dx
^du_^dv
d u _ dx dx
dx V ~ «2 '
(6)
#M" =nw" 1^. (6)
dx dx
60
DIFFERENTIAL CALCULUS
l""^""
_ loga e du
u dx
L''^'^
^Idu^
u dx
dx
dx
dx
dx
dx
dx
dx
da?
dx
da?
f taut*
dx
da?
dx
da?
f secw
dx
= sec 11 tan t^ ^•
da?
dx
= -CSCMCOtl*^.
dx
-^versi*
dx
= 8int.f*.
da?
^sin-ii*
1 du
da?
Vi-u^dx
dx
-1 dw
^tan-i'ti
da?
^ 1 du
l^u^dx
/-cot-it*
dx
-1 du
1 + 1*2 da?
^ sec^u
1 dw.
dx
i« Vt*2 - 1 ^^^
:^C8C »w
-1 du
dx
uy/u^ _ 1 <*a5
^yern-^u
1 du
dx
V2n -«€«<*«
■VVU^'
.\du
dx
[Ch. n.
(7)
(8)
(»)
. (10)
(11)
(12)
(13)
(U)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(28)
(24)
(26)
34.] DIFFERENTIATION OF ELEMENTARY FORMS 51
EXERCISES ON CHAPTER II
Find the x derivatives of the following functions :
1. 3, = 3 :.^ + 5 x' - 7. 4ig_ 3, = (^ + „) tan-> Ji_ V5S.
2.^ = 1 + 1-1. ^^
^' ^" ' 16. , = eot-l±^^I±^.
3. y=(x-{- 5)V^^^. ^ X
4. ^ = xVcfi-x^. ^ 17. 2/ = tan'*a:-2tan2a;+log(sec4a;).
5. y = x log sin x. ^ ^g. y = ^M^ + log(l - x\
1 — X
6. y = ^y/a'^- a:2.
-^ 7. 2/ = -eV ' ""' "^ 5 + 3cosa:
-^8. 3, = tan2z, 2 = tan-i(2x-l).i^20- 2^ = ^^^ (l^) -|tan-i^.
V 9. y = .V«^ n = log sinz. . ; ■g^ •'- ^ j^g (^ + V^2i:^2)+ sec
19. v = cos-i§^Ll£2^.
5 + 3 cos X
l + x\^ 1
10. y = log-. '22. y = e% t* = logo:. ^ .. ^;^
1 _ a;2 23. y = log s^ + e«, s = sec a;.
11
Vl + a;2 24. x^ + yS _ 3 ^^-^ _ q.
12. y = e* cos X. 25. a:2y2 + ar^ + ?/8 _ q.
13. ?/ = vers-M-). 26. a:^ + a: = y + 3/8.
4 sin 3. 27. xy^-\-x^y = x + y.
14. v = tan-i-^^i5_£_.
3 + 5 cos a; 28. y = sm(2u- 7), m = log x^.
tion?
29. For what values of x is the function —^ — - an increasing f unc- (l/jl
n? a + a; ^^_^
/ Vl + a:2 — 1\
30. Prove that tan-^ ( j always increases with x.
. Show that the a:-derivative of tan-^ A/ ~ ^^^ ^ is not a function
' 1 + cos a:
31
oix. '1 +
32. Find at what points of the ellipse — + ^ = 1 the tangent cuts off
equal intercepts on the axes. ^
33. Find the points at which the slope of the curve y = tan x is twice
that of the line y = x.
34. Find the angle which the curves y = sin x and y = cos x make
with each other at their point of intersection.
jj^
CHAPTER III
• SUCCESSIVE DIFFERENTIATION
35. Definition of nth derivative. When a given function
1/ = <l>(^x) is differentiated witli regard to x by tlie rules of
Cliapter I, tlien the result
is a new function of x which may itself be differentiated by
the same rules. Thus,
dx\dxj dx
Cpy
The left-hand member is usually abbreviated to -y^, and
the right-hand member to <f>"(x)\ that is,
Differentiating again and using a similar notation,
and so on for any number of differentiations. Thus the
d^y
symbol -t4 expresses that y is to be differentiated with
regard to x, and that the resulting derivative is then to be
d^y
differentiated. Similarly, ^ indicates the performance of
52
Ch. III. 35.] SUCCESSIVE DIFFERENTIATION 53
the operation — three times, T-fyf-^))- ^^ general, the
^"v/ *^ ctx\dx\dx//
symbol -j— means that y is to be differentiated n times in
succession with regard to x.
Ex. 1. If y = a:^ + sin 2 x,
3^ = 4a:8+2cos2a:,
dx
g=12x2-48in2ar,
^ = 24 ar - 8 cos 2 ar,
'-^ = 244-16sin2x.
dx*
If an implicit equation between x and «^ be given and the
derivatives of i/ with regard to x are required, it is not
necessary to solve the equation for eithef variable before
performing the differentiation.
Ex.2. Given x^-{-y^-\-4:a^xy=0', find ^.
±^(x^ + y* + ^a^xy) = 0,
4a:«+4/^ + 4a2a:^ + 4a2^ = 0.
dx dx ^
The last equation is now to be solved for -^,
dy x^ + a^y
dx y^ + a^x
Differentiating again,
(1)
G?a;2~ dx\y^ + a^xl
(y^ + a^x) J- (x^ + a^y) - (x^ + a^y) -^ (?/« + a^x)
(z/« + «2x) ^3 z2 + a2 ^) - (a:8 + a^y) (3 .^^ ^^ + a^^
54 DIFFERENTIAL CALCULUS [Ch. III.
The value of -^ from (1) is now to be substituted in the last equa-
tion, and the resulting expression simplified. The final form may be
written :
dhf _2 a^xy - 10 a^:i^y^ - a\x* + j/*)- 3 a;V(x* + yQ
In like manner higher derivatives may be found.
36. Expression for the nth derivative in certain cases. For
certain functions, a general expression for the nth. derivative
can be readily obtained in terms of w.
Ex. 1. If y = ^, then -^ = e*, -fi^ = e^ ..., -^- e',
^ dx dx^ dx"
where n is any positive integer. If ^ = e'*", -r-^ = a"e'^.
Ex. 2. If y= sin x,
-i^ = cos a; = sin ( X + - )>
dx \ 2/
g=cos(..|) = sin(..V1,
dx» \ 2 1
U 3/=sinax, — ^ = a» sin ( ax + n - V
c/x" \ 2/
EXERCISES ON CHAPTER III
1. y=3a:*+5x2+3a:-9; find^. 6. y = e'\ogx', find ^.
2. y = 2x2+ 3a: + 5; find 0 7. y = xMogx; find ^
3. y = 1 ; find g. / 8. y = sec^x ; find ^3.
/ 4. y = x«-i; find 0. 9. y = logsinx; find ^3.
5. y = c* ; find -t-|. 10. y = sin x cos x ; find ^.
35-36.] SUCCESSIVE DIFFEUENTIATION 55
. / 11. y = ,J1^ ; find '% ^19. y = cos mx ; find ^.
12. y = xMoga;2; find^,- . / 20. ?/=— J— -; find ^.
13. y = sin a; ; find ^,. "/2I. «/ = log (a + x)-; find ^.
14. y = log («- + e--) ; find ^. 22. y^=2px; find ^.
15. ^, = (x^-3x + 3).2.;findg. 23. ^'+ |-I= 1 ; find g.
16. y = xMoga;; find ^. 24. x^ + z/3 = 3 ax^/ ; find ^.
17. 2/ = e«* ; find |^. 25. e^+» = a:y ; find ^.
18. V = -ir; find ^' 26. «/ = 1 + a:e»; find ^.
^ a: - 1 ' rfx** ^ ' 6?a;2
rf^y dy
27. y = e* sin a: ; prove y| — 2-p + 2y = 0.
28. y=:aa:sina;; prove a:2 ^ - 2 a: ^ + (a:^ + 2) 3/ = 0.
rf2?/
29. y = aa;"+i + Ja;-*^ ; prove x^j~ = n(n+ 1) z/.
30. y= (sin-i a:)^; prove (1 " ^^) ^2 " ^ ^ = 2.
31. y = ^;±^; prove ^=l-y'- 33. y = a:-^ log x ; find ^.
. 32. 2, = -r4-T; find ^. - 34. y = 1^; find ^f-
^ ix^-l dx'^ ^ 1 + a;' c?x«
35. y = xV; prove ^ = 2|^-'^^ + 2...
36. w = cos^ X ; find -r-^'
rfv 1 d"^!! dy^
37. From the relation -^ = -— -, prove that -^ = y-r-rg'
CHAPTER IV
EXPANSION OF FUNCTIONS
It is sometimes necessary to expand a given function in a
series of powers of the independent variable. For instance,
in order to compute and tabulate the successive numerical
values of sin x for different values of x^ it is convenient to
have sin x developed in a series of powers of x with coeffi-
cients independent of x.
Simple cases of such development have been met with in
algebra. For example, by the binomial theorem,
(« + xy = a" + na^-^x + ^^!^ ~ "^^^""'^ + '" » (^)
J. * ^
and again, by ordinary division,
1
1-x
=^\+x-\'x'^ + a^-\- .... (2)
It is to be observed, however, that the series is a proper
representative of the function only for values of x within a
certain interval. For instance, the identity in (1) holds
only for values of x between — a and + a when n is not a
positive integer ; and the identity in (2) holds only for
iralues of X between — 1 and + 1. In each of these ex-
amples, if a finite value outside of the stated limits be given
to a:, the sura of an infinite number of terms of the series will
be infinite, while the function in the first member will be
finite.
66
Ch. IV. 37.] EXPANSION OF FUNCTIONS 57
37. Convergence and divergence of series.* An infinite
series is said to be convergent or divergent according as the
sum of the first n terms of the series does or does not
approach a finite limit when n is increased without limit.
Those values of x for which a series of powers of x is con-
vergent constitute the interval of convergence of the series.
For example, the sum of the first n terms of the geometric
series
* a + ax -^ aoc^ -\- ao^ -\- •••
IS s„ = -^- ^•
1 — X
First let x be numerically less than unity. Then when n
is taken sufficiently large, the term x^ approaches zero ;
hence li"V^,^=^.
Next let X be numerically greater than unity. Then rr" be-
comes infinite when n is infinite ; hence, in this case
Thus the given series is convergent or divergent according
as X is numerically less or greater than unity. The condition
for convergence may then be written
-l<a;<l,
and the interval of convergence is between — 1 and + 1.
Similarly the geonietric series
1 _ 3^ + 93,2 _ 27 2:3^...^
* For an elementary, yet comprehensive and rigorous, treatment of this
subject, see Professor Osgood's " Introduction to Infinite Series" (Harvard
University Press, 1897).
58 DIFFERENTIAL CALCULUS [Ch. IV.
whose common ratio is — 3 a;, is convergent or divergent
according as 3 a; is numerically less or greater than unity.
The condition for convergence is — 1 < 3a;< 1, and hence
the interval of convergence is between — J and + J.
38. General test for interval of convergence. The follow-
ing summary of algebraic principles leads up to a test that
is sufficient to find the interval of convergence for a series of
the most usual kind, that is, a series consisting of positive
integral powers of a;, in which the coefficient of a;" is a known
function of n,
1. If 8„ is a variable that continually increases with w, but
for all values of n remains less than some fixed number ^,
then «„ approaches some definite limit not greater than h
[This follows from the definition of a limit.]
2. If one series of positive terms is known to be conver-
gent, and if the terms of another series be positive and less
than the corresponding terms of the first series, then the
latter series is convergent. [Use 1.]
3. If, after a given term, the terms of a series form a
decreasing geometric progression, then :
(a) The successive terms approach nearer and nearer to
zero as a limit ;
(5) The sum of all the terms approaches some fixed con-
stant as a limit. [Use method of last article.]
4. If the terms of a series be positive, and if after a given
term the ratio of each term to the preceding be less than a
fixed proper fraction, the series is convergent. [Use 2 and 3.]
5. If there be a series A consisting of an infinite number
of both positive and negative terms, and if another series B,
obtained therefrom by making all the terms positive, is
known to be convergent, then the series A is convergent.
37-38.] EXPANSION OF FUNCTIONS 59
For the positive terms of A must form a convergent series,
otherwise the series B could not be convergent; similarly
the negative terms of A must form a convergent series.
Let the sums of these convergent series be u and — v. Let
the first n terms of series A contain m positive terms and p
negative terms. Let 2^, — T^, S„ denote the sum of the
positive terms, the sum of the negative terms, and the sum
of all n terms respectively. Then aS^^ = 2^ — Tp. Now
when n approaches infinity, m and p also approach infinity
and hence
lim cr __ lim y _ lim m { ^ S-v - v
Therefore the series A is convergent.
Definitions. The absolute value of a real number x is
its numerical value taken positively, and is written \x\. The
equation | a: | = \a\ indicates that the absolute value of x is
equal to the absolute value of a. When, however, x and a
are replaced by longer expressions, it is convenient to write
the relation in the form a; | = | a, in which the symbol | = | is
read "equals in absolute value." In like manner, the sym-
bols I < I, I > I will be used to indicate that the expression on
the left has respectively a smaller, or larger, numerical value
than the one on the right.
Any series of terms is said to be absolutely or uncon-
ditionally convergent when the series formed by their abso-
lute values is convergent. When a series is convergent, but
the series formed by making each term positive is not
convergent, the first series is said to be conditionally
convergent.*
*The appropriateness of this terminology is due to the fact that the terms of
an absolutely convergent series can be rearranged in any way, without altering
the limit of the sum of the series ; and that this is not true of a conditionally
60 DIFFERENTIAL CALCULUS [Ch. IV.
E.g., the series ——— + — —••• is absolutely convergent; but the
series \ — \ + \— ••• is conditionally convergent.
6. If there be any series of terms in which after some fixed
term the ratio of each term to the preceding is numerically
less than a fixed proper fraction ; then,
(a) the successive terms of the series approach nearer
and nearer to zero as a limit ;
(J) the sum of all the terms approaches some fixed con-
stant as a limit ; and the series is absolutely convergent.
[Use 3, 4, 5.]
Ex. 1. Find the interval of convergence of the series
l+2.2a:+3.4z2+4.8a:3 4-5.16a:4+....
Here the nth term m„ is n2^-^x'^-'^, and the (n-f l)st term u„+i is
(n + l)2"a;", hence
Mn+i ^ (n + 1) 2»a:" ^ (n + 1)2 a:
M„ - n2~-ix« 1 ~ n '
therefore when n = co, -^^ = 2x.
It follows by (6) that the series is absolutely convergent when
— 1 < 2 X < 1, and that the interval of convergence is between — | and
+ J. The series is evidently not convergent when x has either of the
extreme values.
Ex. 2. Find the interval of convergence of the series
X x^ x^ £l_ 4. . (— 1)"
1.3 3-38 5.36 7-3^ (2n-l)32'-i
® w, '"^2n + 1 * 32«+i ' a:2«-i~ 2n + 1 ' 3«*
hence --^-^ = :77,» when n = co
w«+i . ar-*
convergent series. Thus the numerical value of the series - + •••
12 2* .3'^
is independent of the order or grouping ; but the value of the series
\ — \-\- \ — \-\- •'• can be made equal to any nunjber whatever by suitable
rearrangement. [For a simple proof, see Osgood, pp. 43, 44.]
38-39.] EXPANSION OF FUNCTIONS 61
thus the series is absolutely convergent when — < 1, i.e., when — 3 < a: < 3,
32
and the interval of convergence is from — 3 to + 3. The extreme values
of X, in the present case, render the series conditionally convergent.
E.. 3. Show that the seHes i(|)- |^(|)% J.4|)^-^(|)V ...
has the same interval of convergence as the last ; but that the extreme
values of x render the series absolutely convergent.
39. Remainder after n terms. The last article treated
of the interval of convergence of a given series without
reference to the question whether or not it was the develop-
ment of any known function. On the other hand, the series
that present themselves in this chapter are the developments
of given functions, and the first question that arises is
concerning those values of x for which the function is
equivalent to its development.
When a series has such a generating function, the differ-
ence between the value of the function and the sum of the
first n terms of its development is called the remainder
after n terms. Thus if fQx') be the function, S„(^x} the
sum of the first n terms of the series, and i2„(a;) the
remainder obtained by subtracting S^C^") from /(a;), then
in which S„(x'), Jl„(^x') are functions of n as well as of x,
" „'L"'co^»C^) = 0, then J™^^,(:,)=/(^);
thus the limit of the series S„(^x^ is the generating function
when the limit of the remainder is zero. Frequently this
is a sufficient test for the convergence of a series V" ^nC^^)-
If a series proceed in integral powers of x—a^ the pre-
ceding conditions are to be modified by substituting x — a
for X ; otherwise each criterion is to be applied as before.
62 DIFFERENTIAL CALCULUS [Ch. IV.
40. Maclaurin's expansion of a function in a power-series.*
It will now be shown that all the developments of functions
in power-series given in algebra and trigonometry are but
special cases of one general formula of expansion.
It is proposed to find a formula for the expansion, in
ascending positive integral powers of x — a^ of any assigned
function which, with its successive derivatives, is continuous
in the vicinity of the value x = a.
The preliminary investigation will proceed on the hypothe-
sis that the assigned function f(x) has such a development,
and that the latter can be treated as identical* with the
former for all values of x within a certain interval of equiva-
lence that includes the value x = a. From this hypothesis
the coefficients of the different powers of a: — a will be de-
termined. It will then remain to test the validity of the
result by finding the conditions that must be fulfilled, in
order that the series so obtained may be a proper representa-
tion of the generating function.
Let the assumed identity be
f(x)^ A 4- B(^x -d)+ C(x - ay -f- I)(x - ay
+ J57(a;-a)4H-..., (1)
in which A^ B^ (7, ••• are undetermined coefficients indepen-
dent of X,
Successive differentiation with regard to x supplies the
following additional identities, on the hypothesis that the
derivative of each series can be obtained by differentiating
it term by term, and that it has some interval of equivalence
with its corresponding function :
• Named after Colin Maclaurin (1698-1746), who published it in his
♦'Treatise on Fluxions" (1742) ; but he distinctly says it was known by
Stirling (1690-1772), who also published it in his " Methodua Differcntialis "
(1730), and by Taylor (see Art. 41).
40.] EXPANSION OF FUNCTIONS 63
f{x)=B-{-2C(^x-a')+ SB{x-ay+ 4UCx-ay + '"
f"(x)=^ 2(7 +2^'2D{x-a) + 4:-^ E(x-aY+-:
f"'(x)= 3.2i) +^'2>'2E(x-a) +-
If, now, the special value a be given to x^ the following
equations will be obtained :
/(a)= A, /'(«)= B, f\a)=^2C, f\a)= 3 • 2 D, ....
Hence,
^ =/(«), 5 =/'(«), 0 = ^^, I> = i^, -.
Thus the coefficients in (1) are determined, and the re-
quired development is
/(ic) = /(a) + /'(a)(a5 - a) + ^^ (05 - a)2 + '^^^(a5 - «)»
+ -+^-^^(^ -«)" + -. (2)
This series is known as Maclaurin's series, and the theo-
rem expressed in the formula is called Maclaurin's theorem.
Ex. 1. Expand log x in powers oi x - a.
Here f{x) = \ogx, f(x) = I r(x)= - 1, f"(x) = ^ ••.,
Hence, /(a)=loga, /(«)=^, /"(«)=-i:2' /'"(«) = ^-'
^..)^,)^(-l)-Hn-l)!
and, by (2), the required development is
log X = log a + 1 (;r - a) - ^^(2: - «)2 + ^3(0: - a)8 - ...
+ i — i-i — (a: - a)« + .-..
64 DIFFERENTIAL CALCULUS [Ch. IV.
The condition for the convergence of this series is
lim r {x-aY^^ . (^-«)"]|^|i.
n = ooL(n + l)a"+' * na'' J' ' '
I.e., • -|<|1,
a
x-a\<,a,
0<x<2a.
This series may be called the development of log x in the vicinity of
X = a. Its development in the vicinity of a: = 1 has the simpler form
\ogx = x-l-lix-lY + \{x-iy-...,
which holds for values of x between 0 and 2.
Ex. 2. Show that the development of - in powers of a: — a is
- = - - ^ (^ - «) + -3 (^ - ay- \ {X - ay + ...,
X a a- a^ a^
and that the series is convergent from x = 0 to x = 2a.
Ex. 3. Develop e^ in powers of x — 2.
Ex. 4. Develop x^ — 2x'^ -{■ ox — 7 in powers of x — 1.
Ex, 5, Develop 3 ?/2 _ 14 ?^ _|_ 7 in powers of y — 3.
The expansion of a function /(a:) in a series of ascending
powers of x can be obtained at once from formula (2) by
giving a the particular value zero. The series then becomes
fix) = /(O) + r (0) X + ^^x^ + - + /^"'^<|)^" + ♦•■ (3)
Ex. 6. Expand sin x in powers of x, and find the interval of conver-
gence of the series.
Here /(x) = sin x, /(O) = 0,
/'(.r)=cosx, /'(0)=1,
/"(.r)=-sinx, /'(0)=0,
/"(x) = -cosx, /"(0) = -l,
/'v(x)=8inx, /'^•(0) = 0,
/v(2-) = cosx, /^(0)=1,
40.] EXPANSION OF FUNCTIONS 65
Hence, by (3),
sin X = 0 + 1 . ar + 0 . a;2 - i-ar3 + 0 . x* + ^x^ -" ;
'~^r^ 3! j,i 5!
thus the required development is ' ,
,,., = , _|.^+l.,._i., + ... + ^^.^-x + ....
To find the interval of convergence of the series, use the method of
Art. 38. The ratio of Un+i to m„ becomes
Un '~'(2n+ 1)! ■ (2n- 1)! (2n+ l)2n*
This ratio approaches the limit zero, when n becomes infinite, however
large be the constant value assigned to x. This limit being less than
unity, the series is convergent for any finite value of x, and hence the
interval of convergence is from — co to +00-
The preceding series may be used to compute the numerical value of
sinx for any given value of x. Take, for example, x = .5 radians. Then
'^^ 2.3 2.3.4.5 2.3.4.5.6.7 '
= .5000000
- .0208333
+ .0002604
- .0000015
+ .0000000
sin (.5) = .4794256...
Show that the ratio of u^ to 1/4 is i^l^ ; and hence that the error in stopping
at W4 is numerically less than u^l^h + (^i^)^ + "•]» ^^at is, <^\7U^'
Ex. 7. Show that the development of cos x is
and that the interval of convergence is from — co to + 00.
Ex. 8. Develop the exponential functions a*, e*.
Here
f(x) = a^, f(x) = a-loga, f"(x)=a-(\oga)% -, /('')(x)=a*(loga)«;
hence /(0)= 1, /'(0)= loga, /"(0) = (loga)2, -, /f«)(0) = (loga)»,
and a— 1 + (log a) x + Q^x^ + ...+ (M^a;» + ....
At Til
66 DIFFERENTIAL CALCULUS £Ch. IV.
As a special case, put a = e.
Then log a = log e = 1,
and «.= i + .+|! + |!+...^£2+....
These series are convergent for every finite value of a?.
41. Taylor's series. If a function of the sum of two num-
bers a and x be given, f(a + :r), it is frequently desirable to
expand the function in powers of one of them, say x.
In the function f(a + a;), a is to be regarded as constant,
so that, considered as a function of a;, it may be expanded by
formula (3) of the preceding article. In that formula, the
constant term in the expansion is the value which the func-
tion has when x is made equal to zero, hence the first term
in the expansion of f(a -F x') may be written f(a). In the
same manner the coefficients of the successive powers of x
are the corresponding derivatives of f(a + x) as to rr, in
which X is put equal to zero after the differentiation has
been performed. The expansion may therefore be written
/(a + a^)=/(a)+/'(«)aJ + ^^ajn...+^^a5" + -.
This series, from the name of its discoverer, is known as
Taylor's series, and the theorem expressed by the formula is
known as Taylor's theorem.
Ex. Expand sin (a + x) in powers of x.
Here /(o + a:) = sin (a + x),
hence /(a)=sina,
and f (n) = cos a,
T¥ • X . \ • sin a n cos n o ,
Heuce sin (a + a:) = sina + cos a • x — KT~^ TT"
40-42.] / EXPANSION OF FUNCTIONS 67
EXERCISES
1. Expand tan x in powers of x.
/ 2. Compare the expansion of tan x with the quotient derived by
dividing- the series for sin x by that for cos x. ^ -t/"
See Exs. 6 and 7, Art. 40. a^.u*J) ^"^^^ %
i. 3. Prove log^ ^ ^^^
^ X 6 180
A^rt. 40. M..^^^^ ^ Ai
3 x^
4. Prove log(a; + Vl + a;2)^a:--:^+-^^-....
^ 5. Prove log cosa: = --- — --gj g^-....
6. Expand by division, making use of the exponential series.
7. Find the expansion of e* log (1 + x) to the term involving a:^ by
multiplying together a sufficient number of terms of the series for e* and
for log (1 + x).
8. Expand Vl — a;^ in powers of x.
9. Expand ^ "^ in powers of x.
\ — x
10. Arrange (3 + xY - 5(3 + xy + 2(3 + a;)2 _ (3 + a:) - 2 in powers
42. K necessary restriction imposed upon the series so
that it may be a correct representative of the generating
function, is that the remainder after n terms may be made
smaller than any given number by taking n large enough.
Before deriving the general form for this remainder it is
necessary to prove the following theorem.
Rolle's theorem. If f(x) and its first derivative are con-
tinuous for all values of x between a and 6, and if /(a),/(5)
both vanish, then f'(x) will vanish for some value of x be-
tween a and h.
By supposition f(x) cannot become infinite for any value
of x^ such that a<x<h. If f (x) does not vanish, it must
always be positive or always be negative ; hence, f(x) must
68 DIFFERENTIAL CALCULUS [Ch. IV.
continually increase or continually decrease as x increases
from a to 5 (Art. 13).
This is impossible, since by hypotheses /(a) = 0 and
/(5) = 0 ; hence, at some point x between a and 6, f(x) must
cease to increase and begin to decrease, or cease to decrease
and begin to increase.
This point x is defined by the equation f'(x^= 0.
To prove the same thing geometrically, let t/ =f(x) be
the equation of a continuous
curve, which crosses the ic-axis
at distances a:=a, a;=6> from the
origin. Then at some point P
between a and h the tangent to
Pj^ j2 ^^® curve is parallel to the
ic-axis, since by supposition
there is no discontinuity in the slope of the tangent. Hence
at the point P
g =/'(..) =0.
43. Form of remainder in Maclaurin's series. Let the
remainder after n terms be denoted by Rni^Xt «)? which is
a function of x and a as well as of n. Since each of the
succeeding terms is divisible by (x — a)", R^ may be con-
veniently written in the form
i2„ (x, a) = y^^ V ^ (x, a),
n I
The problem is now to determine ^(a;, a) so that the
relation
fix) =/(a) +/'(«)(:. - «) + £^ Cx - a)« + ]
(n — 1)! n!
42-43.] EXPANSION OF FUNCTIONS 69
may be an algebraic identity, in which the right-hand mem-
ber contains only the first n terms of the series, with the
remainder after n terms. Thus, by transposing,
fix-) -fia-) -fiaXx -a}- £^ (x - af - ...
-fr^i^-ay-^-i^^ix-ay^O. (2)
(» — 1) ! n:
Let a new function, FQi), be defined as follows :
J-CZ) =f{x-) -fiz) -fiz)(x - 2) - -f^ (X - Zy - ...
(n—l)l nl
This function F(^z) vanishes when^ 5ir as is seen by
inspection, and it also vanishes when'^psa, since it then
becomes identical with the left-hand member of (2) ; hence,
by Rolle's theorem, its derivative F'^z') vanishes for some
value of z between x and a, say Zy But
-f"(2)=-/'(^)+/'(2)-/"(^)(a'-2)+/"(^)(^-z)--^J
in-iy.^'' ^■> +(„_!)! ^"^ '■> ■
These terms cancel each other in pairs except the last
two; hence
^'^'^ = %"-% ^'^^''' «)-/'"'-(^)]-
Since F'(^z} vanishes when z = z^ it follows that
<^(^,«)=/'">(z,). (4)
In this expression z^ lies between x and a, and may thus
be represented by
z^^a + eCx-a-), AMrj\Ji\j
~i±-^
70 DIFFERENTIAL CALCULUS [Ch. IV.
where ^ is a positive proper fraction. Hence from (4)
(^(a:, «)=/(«>[« + <? (2: -a)],
A T> r \ f''^\a + e(x-a)'\ , .„ *
and R^ (x, a) = ^ — > — -i—-^ ^ (x — a)". *
The complete form of the expansion of f(x) is then
finc>)=fia)+f'{a) (a? - a) + f^ (05 - a)^ + ...
in which w is any positive integer. The series may be car-
ried to any desired number of terms by increasing /i, and the
last term in (5) gives the remainder (or error) after the first
n terms of the series. The symbol /^"^ (a + ^(2; — a)) indi-
cates that f(x) is to be differentiated n times with regard to
a;, and that x is then to be replaced hj a ■\-6(x — a),
44. Another expression for the remainder. Instead of put-
ting Rf^ (x, a) in the form
(x — a)" , , N
^^ r-^(f>(x, a),
n\
it is sometimes convenient to write it
i2„ (x, a) = (a; - a) i/r (x, a).
Proceeding as before, the expression for F'(z) will be
F'(z) = - /"^^^^^ , (X - g)»-' + ir{x, a),
(w - 1) !
In order for this to vanish when z = Zj, it is necessary that
in which z^=a + 6(x — d), x — z^ = (x — a) (\ — 0).
* This form of the remainder was found by Lagrange (1736-1813), who
published it in the M^moires de TAcad^mie des Scieuces k Berlin, 1772.
43-44.] EXPANSION OF FUNCTIONS 71
Hence i/r (x, a) = /"^^ + 6>(a:- a)) ^^ _ Qy-w^ __ ^y-i^
(?^ — 1) !
"and i2„(^, a) = (l-^)«-i£!l£±%=^(^_a)«.*
(n-1)!
An example of the use of this form of remainder is fur-
nished by the series for log x in powers of a; — a, when x — a
is negative, and also in the expansion of (a + a;)"*.
1. Find the interval of equivalence for the development of log a; in
powers of a: — a, vv^hen a is a positive number.
Here, from Art. 40, Ex. 1,
hence /X. („ + ,(._„)),= |_£|r_IlL_,
and,.,Art.43, ^•^(^,»^l = \;:^^^f^^J = ll[^^I^J-
First let a: — a be positive. Then when it lies between 0 and a it is
numerically less than a + 0{x — a), since ^ is a positive proper fraction ;
hence when n = oc
r ?Lp^ — T^ 0, and i?„ (x, a) = 0.
Again, when a; — a is negative and numerically less than a, the second
form of the remainder must be employed. As before,
hence i^.(., a)|H(l - ^)-- ■ ^^^^/^l^j.
1= 1(1 _ ^)n-i (a-x)-
._. r(a - x) - 6(a — x)!"*-^ a — x
'~'L a-d(a-x) J 'a-e(a-x)
* This form of the remainder was found by Cauchy (1789-1857), and first
published in his "Legons sur le calcul infinitesimal," 1826.
72 DIFFERENTIAL CALCULUS [Ch. IV.
The factor within the brackets is numerically less than 1, hence the
(n — l)st power can be made less than any given number, by taking
n large enough. This is true for all values of x between 0 and a.
Therefore, log x and its development in powers oi x — a are equiva-
lent within the interval of convergence of the series, that is, for all
values of x between 0 and 2 a.
Ex. 2. Show that the development of arz in positive powers oi x — a
holds for all values of x that make the series convergent; that is, when
X lies between 0 and 2 a.
If the function is expanded in powers of a;, the complete
form will be
/(^) =/C0) +/'(0> + =^ ;^ + - + ^^ :^->
^^-^^ (1)
for the first form of remainder, and
/W =/(0) +/'(0> + ^^'a? + - + -^^af-^
for the second form of remainder.
Similarly, the complete form of Taylor's series (Art. 41)
becomes
/(a + ^)=/(a)+/'(a> + ^^r'+ - +^^^-'
for the first form of remainder, and
/(a + rr) =/(«)+/'(«> + =^^ a? + - +^--^^'
(»-l)!
^^tS:^^^^-^)"-'-^ W
for the second form of remainder.
44.] EXPANSION OF FUNCTIONS 73
Ex. Expand (a + x)"" in ascending powers of x, and determine the
interval within which R^ has the limit zero.
Here /(a + x) = (a + a:)"»,
hence f(x) = x»",
and f'(x)= mx^-i, f"(x) = m(m - l)a;«-2, ...,
/(«)(x)= m(m - 1) ••• (m-n + l)x'»-»;
Aa)=a-^, f'{a)=ma-^-\ f"{a)= m{m - \)a-^-\ ...,
f^^\a) = 7?i(m — 1) ••• (m — n + 1)0"^-".
Therefore (a + x)*" = a"* + ma'^-^x + ^^^^~ ^^ a'"-^^;^ + ...
^ r/?(m-l)...(m-n + 2) ^«-«+i^n-i ^ ^^(^^ ^^^
(w - 1) !
in which, from the first form of remainder,
R,(x, a) = Mm-\)'"(m-n + l) ^^ ^ Qx^-nx-
n !
^ m{m - 1) - (m - n + 1) ^^ ^ Sxyl-^Y^
n\ \a + dxf
Consider the ratio
7t„ m — n + 1 a;
When m is greater than - 1, the factor ^~ ^ is less than unity
for every value of n greater than I (m+1), and when x lies between 0
and a the second factor is also less than unity. Their product is there-
fore less than some fixed proper fraction k for every such value of n.
Hence Bn<f<'Rn~i< F/^n-s - < ^'^t- lt>l(m + l).
Since /?, is finite and ^'»-« can be made as small as desired by taking
n sufficiently large, it follows that
H"^ Rn(x,a) = 0
n= CO "^ ' ^
when m>— 1, and 0<a:<a.
When m is not greater than - 1, let it be replaced by — p, where p
is equal to or greater than 1. The ratio R^ to Rn-i may now be written
- 1
(-i)(^+^)-«-fr/
74 DIFFERENTIAL CALCULUS [Ch. IV.
The factor ( — f- 1 j is equal to or greater than unity, but in
the latter case becomes more and more nearly unity as n increases.
The factor r— is numerically less than 1 for any value of n when
a-\-ex ^ ^
z is given any value between zero and a. The product is therefore
greater than unity for values of n less than some finite number iV, but
less than unity for values of n greater than N. Then, in the same way
as for the preceding case,
Ry+2<kR!f+l<kmff,
in which Rjf is a finite number and k a proper fraction. Hence
for every value of m, provided x lies between 0 and a.
Since the interval of convergence is, by Art. 38, from x = — a to a: = a,
it remains to examine the value of Rn{x, a) when x lies between 0
and — a. For this purpose it is necessary to use the second form of
remainder, which may be written,
a
But when - is negative and less than 1, the expression Z-Z. is a
proper fraction, hence its (n — l)st power approaches zero as a limit.
The expression (^ - 1) • • -(^ - ^ + 1) (A'"'^ is made up of n - 1 factors
(n-l)l \a/
of the form ^ ~ . -, each of which is finite and, when k is sufficiently
k a
large, is numerically less than 1 ; hence the limit of the entire product
is zero. Therefore
R„(Xf a) = 0, when n = oo,
if X lies between — a and + a.
Therefore, for values of x within this interval the function (a + x)*^ and
its expansion are equivalent.
Y
/-^
^
H^,.-^:^^^''^
^
M
X
0
N R
44-45.] EXPANSION OF FUNCTIONS 75
45. Theorem of mean value. Let /(a;) be a continuous
function of x which has a derivative. It can then be repre-
sented by the ordinates of a curve
whose equation is i/ =f(x).
In Fig. 13, let
x = ON,x-\-h = OR,
f(x) = NR, fix + h)= RK,
Then f(x-\-K) -f(x) = MK, and
h HM
But at some point S between H and K the tangent to the
curve is parallel to the secant HK. Since the abscissa of S
is greater than x and less than rr + A it may be represented
by a; + Oh^ in which ^ is a positive number less than unity.
The slope of the tangent at 8 is then expressed by /'(a;+^A),
hence
from which
f(x + }i)=f(x) + hf(x + eJi),
The theorem expressed by this formula is known as the
theorem of mean value.
The theorem of mean value can also be established from Taylor's
theorem by putting w = 1 in equation (3) of Art. 44.
JiA^^
EXERCISES ON CHAPTER IV
1. Expand cos(x + It) in powers of h.
2. Expand tan(x + h) in powers of x.
JL- 3. By expanding cos(a: + h) in powers of one of its variables, prove
the theorem cos(a: + A ) = cos x cos k — sin x sin h.
4. Expand log (a; + ^) by Taylor's theorem in powers of h.
5. Expand {x + y)^ in powers of y.
76
DIFFERENTIAL CALCULUS
[Ch. IV. 45.
6. Prove log(l + e*) = log 2 + | + ^
v^ 7. Prove log
2 28 28.41
1 1
+ i2.
x-\ x-l 2{x-iy 3(a;-l)'
+ i2.
'V. 8. Prove log(l + sin ar) = a: - ^ + — - ^ + i2.
9. If f(x)=f(— x), prove that the expansion of f(x) in powers of x
will contain only even powers of x, as cosx, y/l - x'^; if f(x)= _y(_x),
the expansion oi f(x) will involve only odd powers of Xy as sin a:, tanar,
X
1 + a:2*
10. Show that in the expansion for sin x in powerr? of x,
Ijm
R^(x)=0.
M
1-
' *■- jf/ >i-t
/.. /
1?
TL.
"', -/
.3. -2'3
^V 1^,
2l,
CHAPTER V
INDETERMINATE FORMS
46. Hitherto the values of a given function f{x)^ corre-
sponding to assigned values of the variable x^ have been
obtained by direct substitution. The function may, how-
ever, involve the variable in such a way that for certain
values of the latter the corresponding values of the function
cannot be found by mere substitution.
For example, the function
sma;
for the value a;=0, assumes the form -, and the correspond-
ing value of the function is "thus not directly determined.
In such a case the expression for the function is said to
assume an indeterminate form for the assigned value of the
variable.
The example just given illustrates the indeterminateness
of most frequent occurrence ; namely, that in which the
given function is the quotient of two other functions that
vanish for the same value of the variable.
Thus if /(:,)=^,
and if, when x takes the special value a, the functions 4>{x)
and ^(x) both vanish, then
is indeterminate in form, and cannot be rendered determinate
without further transformation.
• 77
78 DIFFERENTIAL CALCULUS [Ch. V.
47. Indeterminate forms may have determinate values.
A case has already been noticed (Art. 9) in which an ex-
pression that assumes the form - for a certain value of its
variable takes a definite value, dependent upon the law of
variation of the function in the vicinity of the assigned
value of the variable.
As another example, consider the function
_x^ — a^
^ ~~ X— a'
If this relation between x and y be written in the.forms
y{x — a)= x^ — a\ (x — a)(^y — x — a) = 0^
it will be seen that it can be represented graphically, as in
the figure (Fig. 14), by the pair of lines
/ X— a = 0,
* y — X — a = 0.
Hence when x has the value of a there
— X is an indefinite number of corresponding
points on the locus, all situated on the
^°- ^*' line x = a; and accordingly for this
value of X the function y may have any value whatever, and
is therefore indeterminate.
When X has any value different from a, the corresponding
value of y is determined from the equation y = x + a. Now,
of the infinite number of different values of y corresponding
to x = a, there is one particular value AP which is con-
tinuous with the series of values taken by y when x takes
successive values in the vicinity of x= a. This may be
called the determinate value of y when x = a. It is ob-
tained by putting x= a in the equation y = a: -{- a, and is
therefore y = 2 a.
47.] INDETERMINATE FORMS 79
This result may be stated without reference to a locus as
follows : When a; = a, the function
00^ — a^
is indeterminate, and has an infinite number of different
values; but among these values there is one determinate
value which is continuous with the series of values taken by
the function as x increases through the value a ; this deter-
minate or singular value may then be defined by
lim ^- CL^
X = a X — a
In evaluating this limit the infinitesimal factor x — a may be
removed from numerator and denominator, since this factor
is not zero, while x is different from a ; hence the determi-
nate value of the function is
lim X + a _ i) ^
Ex. 1. Find the determinate value, when x = 1, of the function
x^+2x^-nx
Sx^-dx^- X + 1'
which, at the limit, takes the form —
0
This expression may be written in the form
(x^+^x)(x- 1)
(3x2- l)(ar-l)'
x^ 4- 3 x
which reduces to jr— 5 r^. When x = 1, this becomes # = 2.
3 x2 - 1 ' ^
Ex. 2. Evaluate the expression
x^ + ax^ + a^x + a»
x^ + xb^ + ax^ + ab^
when X = — a.
80 DIFFERENTIAL CALCULUS [Ch. V.
Ex. 3. Determine the value of
a;3 _ 7 3:2 + 3 a: + 14
a;3 + 3 x-^ - 17 a; + 14
when X = 2.
Ex. 4. Evaluate - — ^ — - when x= 0.
x^
(Multiply both numerator and denominator by a + Va^ — r^.)
48. Evaluation by development. In some cases the com-
mon vanishing factor can be best removed after expansion
in series.
Ex. 1. Consider the function mentioned in Art. 46,
When numerator and denominator are developed in powers of x, the
expression becomes
21 3! V 2\ S\ I
X-
3,3
3!
+ •
••
2X + |;X3
+
...
2
,.-,...
3.8
^-31 +
...
-
1
-i^-'
which has the determinate value 2, when x takes the value zero.
Ex. 2. As another example, evaluate, when a: = 0, the function
X — sin-'a:
sin^a:
By development it becomes
Removing the common factor, and then putting x = 0, the result is J.
47-49.] INDETERMINATE FORMS 81
In these two examples the assigned value of x^ for which
* the indeterminateness occurs, is zero, and the developments
are made in powers of x. If the assigned value of x be
some other number, as a, then the development should be
made in powers of x — a.
Ex. 3. Evaluate, when x = -, the function
cos a:
1 — sin X
By putting a: — - = A, x = --\- h, the expression becomes
eo,(|+*) . ^ -A + ^-... -1 + f
\ 2 / — sm A 6 6
l-sin(|+^) 1
cos A '^_A!4- ^_A!
2 24 2 24
which becomes infinite when A = 0, that is, when a; = ^.
TT hm cos X , ^
Hence „ . ^ : = ± co,
•*^ — z 1 — sin a:
according as h approaches zero from the negative or positive side.
49. Evaluation by differentiation. Let the given function
be of the form ,; i^ and suppose that /(a) =0, </>(a) = 0.
It is required to find j^l^^^i^.
As before, let/(2j), <^(a;) be developed in the vicinity of
x = a^ by expanding them in powers oi x — a. Then
fix) /(«)+/'(«)(^-«) + ^^(*-«)' + -
^^'"^ ~ <^(«) + f (a)(^ - a) + *^ (a; - «)2 + ...
/'(a)(r.-a)+•^(a.-a)''+••.
82 DIFFERENTIAL CALCULUS [Ch. V.
By dividing hj x—a and then letting a: = a, it follows
that
lim Ax) ^f{a)
^ = ^<t)(x) (l>\a)'
The functions /\'a), <^'(a) will in general both be finite.
If /(a) = 0, <^'(a) ^ 0, then ^ = 0.
If /(a) ^ 0, <^'(a) = 0, then ^^ = oo.
If /'(a) and <^'(a) are both zero, the limiting^ value of
f(x)
\\ ^ is to be obtained by carrying Taylor's development
one term farther, removing the common factor (^x — a)^, and
then letting x approach a. The result is f,^ ^-
Similarly, if /(a), /(a), /'(«); (^(a), </>'(«), </>''(«) all
vanish, it is proved in the same manner that
lim f(x}_f"(a-)
and so on, until a result is obtained that is not indeterminate
in form.
Hence the rule :
To evaluate an expression of the form -, differentiate numer-
ator and denominator separately/ ; substitute the critical value
of x in their derivatives^ and equate the qw)tient of tJie deriva-
tives to the indeterminate form,
Ex. 1. Evaluate ^"^P^^ when ^ = 0.
Pat /(^) = l-cosd, <!>($) = e^.
Then /'($) = sin 0, <f>'(e) = 2 0,
and /(O) = 0, <^'(0) = 0.
49.] INBETERMINATE FORMS 83
Again, f>(e)=oosO, <^"(^)=2,
/"(0)=1, <^"(0)=2,
hence /!"„ kl_^ = 1.
Ex. 2. Find lim f!±iIl±2^2^^zJ.
X - u 3,4
lim e" + e-'' + 2 cos a: — 4 _ lim e=' — e-» — 2 sin a?
a;=:0 ^4 - x = 0 4 ^8
_ Mm 6» + e-^ — 2cosa:
- X = 0 22 x2
_ lim e» _ e-x _^ 2 sin a;
-x = 0 24x
_ lim e» + e-*+ 2 cos a?
"" a; i 0 24
Ex. 3. Find lip ^^-sinxcosar,
a; = u ™8
Ex. 4. Find "P. a;^-2xa-4^^+9a:-4.
+
In this example, show that x—1 is a factor of both numerator and
denominator.
1?^ R T?;^^ li°i Stana:- Sar-aH'
^ Ex. 5. Find ^ ^ 0 -^
In applying this process to particular problems, the work
can often be shortened by evaluating a non-vanishing factor
in either numerator or denominator before performing the
differentiation.
Ex.6. Find "ro^^-^>'^^^^.
The given expression may be written
^i°^ rr_4^2tana:_ lim , ..3 ^i^ tanar
a: = 0 ^^ *>' —^ -x = 0C^-*; a; = 0 "^
= 16 . 1 = 16.
84 DIFFERENTIAL CALCULUS [Ch. V.
In general, if fCx')= 'f(x)x(p^^^ ^^^ ^^ '^(^)= ^» xC^)"?^^'
<^(a) = 0, then
For
lim ^(a^)x(^) _ lim ^. . . lim ^(^) _ ^,^^. . ^'(<^)
Ex.7. Find
lim sin x cos^ a;
Ex.8. Find,^!P^/"-^)^^^^(^-^).
* — ^ sin (x — 1)
^ p-i / EXERCISES
^ / Evaluate the following expressions :
9. Lz.£2i£ when a: = 0. 15. ^ "r e-^ -2 ^^^^^ ^^^
sin X x^
when a; = 0. , ^ tan a:- sin x cos a:
sma:
16. "^"•^-^^"•^^"'^-^ when a: = 0.
11. £i=li when a: = 1.
X —\ __ sin~^a:
17. -ii^i— ± when ar = 0.
12. ?l::ilwhen a: = 0. *^°"'^
6- -1
, « sin aa: , «« a 18. ^' ^^° ^ ~ ^ ~ ^^ when a: = 0.
s"i^ll a:2 + X log (1 - X)
14. (l±£)i:ii when X = 0. 19- ^"^l^^"^ when a: = 0.
X x*
There are other indeterminate forms than -• They are
g,oo-oo,0«, r, ao».
00
50. Evaluation of the indeterminate form ^.
oo
Let the function ^rr^ become — when x^a. It is re-
quired to find Jf'^lg.
49-50.] INDETERMINATE FORMS 85
This function can be written
<l>(x) 1 '
which takes the form - when x = a^ and can therefore be
evaluated by the preceding rule.
When a; = a,
1 <i,<ix')
lira fjx) ^ lim i>(j() _ lira \.4>(^)J
f(x)
If both members be divided by , , {^ the equation becomes
-. ^ lim /(a:) <^'(a:)
therefore l-JZ^l^^^^. (2)
This is exactly the same result as was obtained for the
form -; hence the procedure for evaluating the indetermi-
nate forms -, — , is the same in both cases.
When the true value of ~r-^ is 0 or oo, equation (1) is
satisfied, independent of the value of T77-T; but (2) still
I' iCxi
gives the correct value. For, suppose ^i" ,) ( = 0. Con-
X — a ^ ^^^
sider the function
86 DIFFERENTIAL CALCULUS [Cb. V.
which has the form — when x = a» and has the determinate
00
value <?, which is not zero. Hence by (2)
lim /(^) + gj)(^) _ /(«) + c<^^(«) _ /^(g) ■ ^
Therefore, by subtracting c,
lim/C^^ZW
If r '^«x^ = <»i ^en jiP ^ = 0, which can be treated
as the previous case.
51. Evaluation of the form oo • 0.
Let the function be (\>{x) "^(x)^ such that <^(a) = oo,
Vr(a) = 0.
This may be written ^^ ^, which takes the form - when
a is substituted for a?, and therefore comes under the above
rule. (Art. 49.)
52. Evaluation of the form oo — oo.
The form oo — oo may be finite, zero, or infinite.
For instance, consider y/a? + ax— x for the value x = cc.
It is of the form oo — oo, but by multiplying and dividing by
^x^ 4- ax -f a; it becomes ^ , which has the form
00 1 ^x^ -{- ax -\- X
5o when a; = oo.
Again, by dividing both terms by x, it takes the form
., which becomes ? when rr = oo.
4U
Jl+' + l
> X
There is here no general rule of procedure as in the
previous cases, but by means of transformations and proper
50-52.] INDETERMINATE FORMS 87
grouping of terms it is often possible to bring it into one
of the forms -, — . Frequently a function which becomes
oo — 00 for a critical value of x can be put in the form
u t
V V)
in which v, w become zero. This can be reduced to
uw — vt
which is then of the form -•
Ex. 1. Find 3.^5 (sec x — tan x).
This expression assumes the form 00 — 00, but can be written
1 _ sin a- _ 1 — sin a:
cos X cos X ~ cos X
which is of the form -, and gives zero when evaluated.
Hence ^ ^^ (sec x — tan a:) = 0.
Ex. 2. Prove ^ ^^n (sec^a: — tan»a:) = 00, 1, 0, according as
\^
EXERCISES ^
Evaluate the following expressions :
2. !^S^ when x = 0. 6. ^5£^ when a; = 5-
cot x sec 5 a; 2
3. — when a: = 00. 7. (a^ — x^) tan — when x = a.
- tan X „i V 0/1 +„^ ^\ o«« o «. T^kz^r. «._'''
when ar = -• 8. (1 — tan ar) sec 2 a; when a: = -•
tan5x 2 4
88 DIFFERENTIAL CALCULUS [Ch. V.
y g 1-loga; ^YiQn x = 0. '^12. — ^ i— when x = l.
e' — e X — 1
yr 10. ^-^ when :r = 00. N 13- csc^x - 1 when x = 0.
J 11. -i ^ when a; = 1. /!*• r^ " r^ ^^^^ ^ = 1*
^ logx a:-l *^ logx loga;
53. Evaluation of the form 1*.
Let the function u — [</>(a;)]'''^'^^ assume the form 1* when
x — a.
In order to evaluate this expression, take the logarithm of
both sides. Then
log u = ^(x) ■ log ^(x) = l2E|M.
This expression assumes the form - when a; = a, and can
be evaluated by the method of Art. 49.
If the reduced value of this fraction be denoted by tw,
then log w = m and u = g"*.
Note. The form 1° is not indeterminate, but is equal to 1.
For, let [^(x)] '/'(*) assume the form 1° when x — a.
Put u = [</>(x)]<^(*).
Then logu = ^(a:)log[<^(a;)],
which equals zero when x ■=. a\
hence log u = 0, u — e^—\,
54. Evaluation of the forms oo^ 0^.
Let [^^{xy^'''^ become oo^ when x — a.
Put w = [</)(a:)]'^<->.
Then log u = y^(x) log 4>(x) = ]2K^^,
52-54.] INDETERMINATE FORMS 89
This is of the form — , and can be evaluated by the method
of Art. 50. Similarly for the form 0^.
Note. The form 0* is not indeterminate, but is equal to 0.
For let u = [<^(x)]'l'(^> become 0"° when x = a.
Then log u = i//(a:) log <f}(x)= — oo, and u =e-* = 0.
Similarly, the form 0-* is equal to co.
This completes the list of ordinary indeterminate forms.
The evaluation of all of them depends upon the same
principle, namely, that each form (or its logarithm) may be
brought to the form -, and then evaluated by differentiating
numerator and denominator separately. In finally letting
x= a, the two directions of approach should be compared,
so as to reveal any discontinuity in the function.
EXERCISES
Evaluate the following indeterminate forms :
1. a:'""' when x = 0.
2. (cosaa:)«8«'''=* when a: = 0.
6
3. (1 4- axy when a: = 0.
6
4. (1 4- ax)" when a: = co.
/1\8in« / d \x
5. I - 1 when x = 0. 9, ( _ 4. 1 J when x =
6.
^2--^"'- when x = a.
7.
X" when x = 0.
8.
1
o(f when x = co.
EXERCISES ON CHAPTER V
Evaluate the following indeterminate forms :
^ 1. ^^^ whena: = 0. ^ ^, ^^^"^ whena: = 0. '
X V(e* - 1)8
^ 2. -^ when x = 0. ^5. 1^S£ when x = 0.
sin X CSC X
gmx _ gma / 6. 6"* log X whcU a: = 00.
^ 3. — when a; = a. ^„ y-. .
(x —ay y 1. X — vx^ — ax when a: = oo.
90
[Ch. V. 64.
DIFFERENTIAL CALCULUS
8. (cot x)"" * when a; = 0.
10. 1 - coto; when a: = 0. ^^ ^t^^^ ^j^^^ ^ ^ ^^
11. -^ 1_ ^hen a; = 1. 15. V2-sinx-cosa: ^^^^ ^^tt
x2 - 1 a; — 1 log sin 2 X 4
T 16. a; tan x — J sec a: when a; = J*
2 2
^ 12. 2* sin ^ when a: = oo
13.
17. fl2^Ywhena: = oo.
/- /- J \ X I
^ whena:=6. is. x
X ) ""r
• x^ log [ 1 + - ) when a:= oo.
CHAPTER VI
MODE OF VARIATION OF FUNCTIONS OF ONE VARIABLE
55. In this chapter methods of exhibiting the march or
mode of variation of functions, as the variable takes all
values in succession from — oo to + oo, will be discussed.
Simple examples have been given in Art. 13 of the use
that can be made of the derivative function <^' (a;) for this
purpose.
The fundamental principle employed is that when x in-
creases through the value a, ^{x) increases through the
value </>(«) if </>'(«) is positive, and that ^{x) decreases
through the value <^(«) if <^' («) is negative. Thus the
question of finding whether ^{x) increases or decreases
through an assigned value ^(a), is reduced to determining
the sign of </>' (a).
1. Find whether the function
increases or decreases through the values ^(3) = 2, ^(0) = 5, ^(2)=|,
^(— 1)= 10, and state at what value of x the function ceases to increase
and begins to decrease, or conversely.
56. Turning values of a function. It follows that the
values of x at which ^{x) ceases to increase and begins to
decrease are those at which <^' (x) changes sign from positive
to negative ; and that the values of x at which </> {x) ceases
to decrease and begins to increase are those at which <^' (x)
changes its sign from negative to positive. In the former
case, (^ {x) is said to pass through a Tyvaximum, in the latter,
a nbinimum value.
91
92
DIFFERENTIAL CALCULUS
[Ch. VI.
Fig. 15.
Ex. 1. Find the turning values of the function
<^ (x) = 2 a;8 - 3 a;2 - 12 x + 4,
and exhibit the mode of variation of the function by sketch-
X ing the curve y = <f>{x).
Here <f>' (x)= Qx^ - Qx - 12 = Q(x -^ 1) (x - 2),
hence <f>' (x) is negative when x lies between — 1 and + 2,
and positive for all other values of x. Thus <fi(x) increases
from X = — x toa: = — 1, decreases from ar = — ltoa: = 2,
and increases from x = 2 to x = cc. Hence <^(— 1) is a
maximum value of <^ (x), and <^ (2) a minimum.
The general form of the curve y = <f>(x) (Fig. 15) may-
be inferred from the last statement, and from the following simultaneous
values of x and y :
x = -oo, -2, -1,0, 1, 2, 3, 4, 00.
y = - 00, 0, 11, 4, - 9, - 16, - 5, 36, oo.
Ex. 2. Exhibit the variation of the
function
<f>(x) = ix-l)'-^2,
especially its turning values.
Since <f>'(x) = ? ,
3(x-l)^
hence <f>'(x) changes sign at a:= 1, being
negative when a: < 1, infinite when x = 1,
and positive when x>l. Thus <^(1) = 2
is a minimum turning value of <f>(x).
The graph of the function is as shown in Fig. 16, with a vertical tangent
at the point (1, 2).
Ex. 3. Examine for maxima and minima th6 function
<t>(x) = (x- 1)^ + 1.
Here </)'(x) = i ? ,
3(x-l)S
hence <t>'(x) never changes sign, but is
always positive. There is accordingly no
turning value. The curve y = <f>(x) has a
vertical tangent at the point (1, 1), since
'-^ = 4*'(x) is infinit<* when x = l. (Fig. 17.)
Fig. 16.
Fio. 17.
1
56-58.] VARIATION OF FUNCTIONS 93
57. Critical values of the variable. It has been shown that
the necessary and sufficient condition for a turning value of
<l>(x) is that <t>'(^x) shall change its sign. Now a function
can change its sign only when it passes through zero, as in
Ex. 1 (Art. 56} , or when its reciprocal passes through zero,
as in Exs. 2, 3. In the latter case it is usual to say that the
function passes through infinity. It is not true, conversely,
that a function always changes its sign in passing through
zero or infinity, e.g., y = x^.
Nevertheless all the values of re, at which (f>\x) passes
through zero or infinity, are called critical values of a;, be-
cause they are to be further examined to determine whether
<t>'(x) actually changes sign as x passes through each such
value ; and whether, in consequence, 4>(x) passes through a
turning value.
For instance, in Ex. 1, the derivative (i>'(x) vanishes when
a; = — 1, and when a: = 2, and it does not become infinite for
any finite value of x. Thus the critical values are — 1, 2,
both of which give turning values to (f>(x). Again, in
Exs. 2, 3, the critical value is x = 1, since it makes <l>'(x')
infinite ; it gives a turning value to <t>(^x) in Ex. 2, but not
in Ex. 3.
58. Method of determining whether <{>'(») changes its sign
^ » in passing through zero or infinity. Let a be a critical value
i^ of a:, in other words let (^'(a) be either zero or infinite, and
let ^ be a very small positive number, so that a — h and a-\-h
^re two numbers very close to a, and on opposite sides of it.
^in order to determine whether <^'(a;) changes sign as x in-
Acreases through the value a, it is only necessary to compare
^the signs of <^'(a + h) and <^'(a — h). If it is possible to
^ take h so small that <^'(a — h} is positive and <^'(a + h)
V negative, then (j)'(ix) changes sign as x passes through the
94
DIFFERENTIAL CALCULUS
[Ch. VI.
value «, and <t)(x) passes through a maximum value <^(a).
Similarly, if ^'(a — A) is negative and <^'(a + h) positive,
then <t>(jc) passes through a minimum value <^(a).
if <^'(a — h) and <^'(a + h) have the same sign, however
small h may be, then <^(a) is not a turning value of <^(a;).
Ex. Find the taming values of the function
Here <l>'(x)= 2{x - l)(a: + 1)8 + 3(x - V)\x + 1)«
= (z-l)(a; + l)2(5a:-l).
Hence 4*'(x) becomes zero at a: = — 1, |, and 1 ; it does not becdme
infinite for any finite value of x.
Thus, the critical values are — 1, |, 1.
F
Fio. 18.
When X = — 1 — h^ the three factors of <^'(^) ^^^® ^-l^® signs —
and when a: = — 1 + A, they become —
thus ^'{x) does not change sign as x increases through — 1 ;
<^(— 1)= 0 is not a turning value of <^(x).
When r = ^ — A, the three factors of ^'(•^) h*^® signs -
and when x = \ + h, they become —
thus <l>'{x) changes sign from + to — as x increases through \y and
^(i) = 1 • 11052 is a maximum value of </)(x).
+ -,
+ -;
hence
+ -,
+ +;
68-59.]
VARIATION OF FUNCTIONS
95
Finally, when x=l — h, the three factors of <f>' (x) have the signs — + + ,
and when x = 1 + h they become + + + ;
thus <f>'(x) changes sign from — to + as a; increases through 1, and
<^(1)= 0 is a minimum value of <f>{x).
The deportment of the function and its first derivative in the vicinity
of the critical values may be tabulated as follows, in which inc., dec. stand
for increasing, decreasing, respectively :
1 +h
+
inc.
The general march of the function may be exhibited graphically by
tracing the curve y = <f>(x) (Fig. 18), using the foregoing result and
observing the following simultaneous values of x and y :
X
-l-Jl
-1
-l + h
\-^
\
l + h
l-h
1
<i>'{x)
+
0
+
4-
0
-
-
0
<f>(x)
inc.
inc.
iiic.
max.
dec.
dec.
min.
y =
1, 0, i, 1, 2, 00.
0, 1, 1.1..., 0, 27, 00.
59 Second method of determining whether <t>'(i») changes
sign in passing through zero. The following method may be
employed when the function and its derivatives are continu-
ous in the vicinity of the critical value x = a.
Suppose, when x increases through the value a, that <f)'(^x)
changes sign from positive through zero to negative. Its
change from positive to zero is a decrease, and so is the
change from zero to negative ; thus (^'(a:) is a decreasing
function at x = a^ and hence its derivative <t>"(x) is nega-
tive 2it X— a.
On the other hand, if <\>\x^ changes sign from negative
through zero to positive, it is an increasing function and
(^"(a?) is positive at a: = a; hence :
The function <^(a;) has a maximum value (f>(a'), when <^'(a) = 0
and <j>"(a^ is negative ; <^(a:) has a minimum value 4>(cl)-, when
<^'(a)= 0 and <^"(a) is positive.
96 DIFFERENTIAL CALCULUS [Ch. VI.
It may happen, however, that <i>" (ci) is also zero.
In this case, to determine whether <\>(x) has a turning
value, it is necessary to proceed to the higher derivatives.
If (i>(x) is a maximum, <t>"(x) ^^ negative just before vanish-
ing, and negative just after, for the reason given above ; but
the change from negative to zero is an increase, and the
•change from zero to negative is a decrease; thus (t>"(x)
I changes from increasing to decreasing as x passes through a.
Hence <f>"'(x') changes sign from positive through zero to
negative, and it follows, as before, that its derivative <^'^(a;)
is negative.
Thus <^(a) is a maximum value of <t>(x) if <^'(a)=0,
<^'^(a)=0, </>'"(«)= 0, <^'''(a) negative. Similarly, <^(a) is
a minimum value of <i>Qc) if </>'(a) = 0, <^''(a) = 0, <^'^'(a) = 0,
and </>'^(a) positive.
If it happen that (^'^(a) = 0, it is necessary to proceed to
still higher derivatives to test for turning values. The
result may then be generalized as follows:
The function <\>(x) has a maximum {or minimum') value at
x= a if one or more of the derivatives <^'(a), <f>"{a}, (i>"'{cL)
vanish and if the first one that does not vanish is of even order ^
and negative {or positive),
Ex. Find the critical values in the example of Art. 58 by the second
method.
<^"(:r) = (x+l)2(5a:-l) + 2(x-l)(x+l)(5x-l)+5(a:-l)(z+l)a,
= 4(5x» + 3a:a-3x- 1),
<^"(1)= 16, hence ^(1) is a minimum value of <^(a:),
^"(— 1) = 0, hence it is necessary to find <!>'"(— 1) ;
<^"'(x)=12(6x« + 2x-l),
^'"(— 1)=24, hence <^(- 1) is neither a maximum nor a minimum
value of <f>(j:)-
Again, <^"(i) = H^ - l)(i + 1)* ^ negative, hence <f>(^) is a maximum
value of <ft(x).
69-60.] VARIATION OF FUNCTIONS 97
60. Conditions for maxima and minima derived from Tay-
lor's theorem. In this article, as in the preceding, the func-
tion and its derivatives are supposed to be continuous in the
vicinity of x = a; otherwise the method of Art. 58 must be
used.
If <l>Qa) be a maximum value of <^(a;), it follows from the
definition that </>(«) is greater than either of the neighboring
values, </>(« + A), or <j>(^a — A), when h is taken small enough.
Hence <^(a + A)— </>(«) and <^(a — A)— </>(«) are both
negative.
Similarly, these expressions are both positive if <^(«) is a
minimum value of (f>(^x^.
Let <l>(x + K) and (f>Qc — K) be expanded in powers of h by
Taylor's theorem.
Then <^(:i:+A) = </>(2:)+<^'(a:)A+^^A2 + *^^A3+...,
A . o I
If X be replaced by a, and ^(a) transposed, the result is
The increment h can now be taken so small that h(i>'(a')
will be numerically larger than the sum of the remaining
terms in the second member of either of the last two equa-
tions, and its sign will therefore determine the sign of the
entire member. Since these signs are opposite in the two
equations, <^(a + A) — <^(a) and <^(a — A) — <^(a) cannot have
the same sign unless <^'(a) is zero, hence the first condition
for a turning value is <^' (a) = 0.
98 DIFFERENTIAL CALCULUS [Ch.VI.
In case <f)'(a)= 0 the preceding equations become
and h can be taken so small that the first term on the right
is numerically larger than either of the second terms, hence
(j>(a + K) — <^(«) and (f>(a — K)— (f>(cL) are both negative when
</)"(«) is negative, and both positive when <j>"(a} is positive.
Thus <^(«) is a maximum (or minimum) value' of <t>{x)
when <^'(a) is zero and <i>" (a) is negative (or positive).
If it should happen that <i>" (a) is also zero, then
and by the same reasoning as before, it follows that for a
maximum (or minimum) there are the further conditions
that <f>"'(cL) equals zero, and that <^*^(«) is negative (or
positive) .
Proceeding in this way, the general conclusion stated in
the last article is evident.
Ex. 1. Which of the preceding examples can be solved by the general
rule here referred to ?
Ex. 2. Why was the restriction imposed upon <i>'{x) that it should
change sign by passing through zero, rather than by passing through
infinity?
61. The maxima and minima of any continuous function
occur alternately. It has been seen that the maximum and
60-62.] VABIATION OF FUNCTIONS 99
minimum values of a rational polynomial occur alternately
when the variable is continually increased, or diminished.
This principle is also true in the case of every continuous
function of a single variable. For, let c/>(<x), 4>(h^ be two
maximum values of <^(a;), in which a is supposed less than
h. Then, when x = a-\- h, the function is decreasing ; when
x=h — h, the function is increasing, h being taken suffi-
ciently small and positive. But in passing from a decreas-
ing to an increasing state, a continuous function must, at
some intermediate value of x, change from decreasing to
increasing,, that is, must pass through a minimum. Hence,
between two maxima there must be at least one minimum.
It can be similarly proved that between two minima there
must be at least one maximum.
62. Simplifications that do not alter critical values. The
work of finding the critical values of the variable, in the
case of any given function, may often be simplified by means
of the following self-evident principles.
1. When c is independent of x, any value of x that gives
a turning value to c(\>(x) gives also a turning value to
(t>(x); and conversely. These two turning values are of
the same or opposite kind according as c is positive or
negative.
2. Any value of x that gives a turning value to c-f- <l)(x}
gives also a turning value of the same kind to (/>(a;); and
conversely.
3. When n is independent of x^ any value of x that gives
a turning value to [</>(2;)]" gives also a turning value to
(/>(a;); and conversely. Whether these turning values are
of the same or opposite kind depends on the sign of n^ and
also on the sign of [^(a^)]""^-
y\A
iajL^
100 DIFFERENTIAL CALCULUS [Ch. VI.
EXERCISES
Find the critical values of x in the following functions, determine the
nature of the function at each, and obtain the graph of the function.
/
1. M = ar(a;2-1).
6.
M = ar(ar+1)2-
^ 2. M = 2 a:8 - 15 a;2 + 36 a: -
-4.
1 7.
M = 5 + 12 a: - J
3. M= (x- 1)8 (a; -2)2.
8.
u = !2££.
^ 4. w — sin X + cos x.
5. «=(^^^.
a-2x
9.
X
u = sin^ X cos8 X.
^'
10. Show that a quadratic integral function always has one maxi-
mum, or one minimum, but never both.
11. Show that a cubic integral function has in general both a maxi-
mum and a minimum value, but may have neither.
12. Show that the function (x — by has neither a maximum nor a
minimum value.
^ ' 63. Geometric problems in maxima and minima. The
theory of the turning values of a function has important
applications in solving problems concerning geometric
maxima or minima, i.e., the determination of the largest
or the smallest value a magnitude may have while satisfying
certain stated geometric conditions.
The first step is to express the magnitude in question
algebraically. If the' resulting expression contains more
than one variable, the stated conditions will furnish enough
relations between these variables, so that all the others
may be expressed in terms of one. The expression to
be maximized or minimized, being thus made a func-
tion of a single variable, can be treated by the preced-
ing rules.
62-63.] VARIATION OF FUNCTIONS 101
Ex. 1. Find the largest rectangle whose perimeter is 100. Let x, y
denote the dimensions of any of the rectangles whose perimeter is 100.
The expression to be maximized is the area
u = xy, (1)
in which the variables x, y are subject to the stated condition
2a;+2?/=100,
le,, y = ^0-x; (2)
hence the function to be maximized, expressed in terms of the single
variable x, is
M = <^ (a;) = a; (50 - x) = 50 a; - x"^. (3)
The critical value of x is found from the equation
<^'(a:) = 50-2ar=0
to be a; = 25. When x increases through this value, <^'(x) changes sign
from positive to negative, and hence ^ {x) is a maximum when x = 25.
Equation (2) shows that the corresponding value of y is 25. Hence the
maximum rectangle whose perimeter is 100 is the square whose side is 25.
Ex. 2. If, from a square piece of tin whose side is a, a square be cut
out at each corner, find the side of the latter square in order that the
remainder may form a box of maximum capacity, with open top.
Let a: be a side of each square cut out. Then
the bottom of the box will be a square whose side
is a - 2 a:, and the depth of the box will be x.
Hence the volume is
v = x{a-2xy,
which is to be made a maximum by varying «.
Here ^= (a - 2ar)2 - 4a:(a - 2a:)
d^ Fio. 19.
= (a-2a:)(a-6a:).
This derivative vanishes when x = -, and when x = -. It will be found,
2 0
by applying the usual test, that a: = ^ gives v the minimum value zero, and
O 3
that x = - gives it the maximum value — ^. Hence the side of the
6 27
square to be cut out is one sixth the side of the given square.
102
DIFFERENTIAL CALCULUS
[Ch. VI.
Ex. 3. Find the area of the greatest rectangle that can be inscribed
in a given ellipse.
An inscribed rectangle
will evidently be sym-
metric with regard to
the principal axes of the
ellipse.
Let a, b denote the
lengths of the semi-axes
OA, 05 (Fig. 20); let2.T,
2y he the dimensions of
an inscribed rectangle.
Then the area is
Fig. 20.
u = 4:xy,
(1)
in which the variables x, y may be regarded as the coordinates of the
vertex P, and are therefore subject to the equation of the ellipse
t+t = l
ft2
(2)
It is geometrically evident that there is some position of P for which
the inscribed rectangle is a maximum.
The elimination of y from (1), by means of (2), gives the function of
X to be maximized,
(3)
«=i*a:V^^^
By Art. 62, the critical values of x are not altered if this function be
4 A
divided by the constant — , and then squared. Hence, the values of a?
a
which render u a maximum, give also a maximum value to the function
Here
<f> {x) = x\a^ - x2) = a2x2 - x^,
<f>f(x) =2a^x -ix» = 2x(a^- 2x^,
<l>"(x) = 2a^- 12x2;
hence, by the usual tests, the critical values x = ± — - render ^(x), and
\/2
therefore the area u, a maximum. The corresponding values of y are
given by (2), and the vertex P may be at any of the four points
denoted by
V2 y/f
63.]
VABIATION OF FUNCTIONS
103
giving in each case the same maximum inscribed rectangle, whose
dimensions are aV2, by/2, and whose area is 2 aft, or half that of the
circumscribed rectangle.
Ex. 4. Find the greatest cylinder that can be cut from a given right
cone, whose height is h, and the radius of whose base is a.
Let the cone be generated by the
revolution of the triangle GAB ^^B
(Fig. 21), and the inscribed cylinder
be generated by the revolution of the
rectangle AP.
Let OA =hj AB = ttj and let the
coordinates of P be (x, y). Then the
function to be maximized is Try^Qi — x)
subject to the relation - — t' - ^^' ^'
This expression becomes
V =
h^
x\h - x).
The critical value of a; is f A, and F =
27 '
EXERCISES ON CHAPTER VI
1. Through a given point within an angle draw a straight line which
shall cut off a minimum triangle. Solve this problem by the method of
the calculus, and also by geometry.
[Take given lines as coordinate axes.]
2. The volume of a cylinder being constant, find its form when the
entire surface is a minimum.
3. A rectangular court is to be built so as to contain a given area c^,
and a wall already constructed is available for one of its sides. Find its
dimensions so that the expense incurred may be the least possible.
4. The sum of the surfaces of a sphere and a cube is given. How do
their volumes compare when the sum of their volumes is a minimum ?
5. What is the length of the axis of the maximum parabola which
can be cut from a given right circular cone, given that the area of a
parabola is equal to two thirds of the product of its base and altitude ?
6. Determine the greatest rectangle which can be inscribed in a given,
triangle whose base is 2 6 and whose altitude is a.
.ir .
104 DIFFERENTIAL CALCULUS [Ch. VI. 63.
7. The flame of a candle is directly over the center of a circle whose
radius is 5 inches. What ought to be the height of the flame above the
plane of the circle so as to illuminate the circumference as much as pos-
sible, supposing the intensity of the light to vary directly as the sine of
the angle under which it strikes the illuminated surface, and inversely
as the square of its distance from the illuminated point ?
y^ 8. A rectangular piece of pasteboard 30 inches long and 14 inches
wide has a square cut out at each corner. Find the side of this square
so that the remainder may form a box of maximum contents.
9. Find the altitude of the right cylinder of greatest volume in-
scribed in a sphere whose radius is r.
10. Through the point (a, b) a line is drawn such that the part inter-
cepted between the rectangular coordinate axes is a minimum. Find its
length.
^ 11. Given the slant height a of a right cone ; find its altitude when
the volume is a maximum.
\ 12. The radius of a circular piece of paper is r. Find the arc of the
sector which must be cut from it so that the remaining sector may form
the convex surface of a cone of maximum volume.
13. Find relation between length of circular arc and radius in order
that the area of a circular sector of given perimeter should be a maximum .
14. On the line joining the centers of two spheres of radii r, R, find
the distance of the point from the center of the first sphere from which
the maximum of spherical surface is visible.
15. Describe a circle with its center on a given circle so that the
length of the arc intercepted within the given circle shall be a maximum.
CHAPTER VII
RATES AND DIFFERENTIALS
64. Rates. Time as independent variable. Suppose a par-
ticle P is moving in any path, straight or curved, and let s
be the number of space units passed over in t seconds. Then
s may be taken as the dependent variable, and t as the inde-
pendent variable.
If As be the number of space units described in the addi-
tional time A^ seconds, then the average velocity of P during
As
the time A^ is — ; that is, the average number of space units
described per second during the interval.
The velocity of P is said to be uniform if its average
As
velocity — is the same for all intervals A^. The actual
velocity of P at any instant of time t is the limit which the
average velocity approaches as A^ is made to approach zero
as a limit.
Thus t;= 1/^^^=^
^t = ^M dt
is the actual velocity of P at the time denoted by t. It is
evidently the number of space units that would be passed
over in the next second if the velocity remained uniform
from the time t to the time ^ -}- 1.
It may be observed that if the more general term, " rate
of change," be substituted for the word "velocity," the
above statements will apply to any quantity that varies with
the time, whether it be length, volume, strength of current,
105
106 DIFFERENTIAL CALCULUS [Ch. VII.
or any other function of the time. For instance, let the
quantity of an electric current be C at the time ^, and C-{-AO
at the time t + A^. Then the average rate of change of cur-
AC
rent in the interval A^ is ; this is the average increase
in current-units per second. And the actual rate of change at
the instant denoted by t is
lim AO^dO
At = 0 /^t dt'
This is the number of current-units that would be gained
in the next second if the rate of gain were uniform. from the
time t to the time t-\-l. Since, by Art. 14,
dy _dy ^ dx
dx dt dt
hence -^ measures the ratio of the rates of change of y and
ax
of X,
It follows that the result of differentiating
y=f(?d (1)
may be written in either of the forms
!=/'(-), (2)
The latter form is often convenient, and may also be
obtained directly from (1) by differentiating both sides
with regard to t. It may be read : the rate of change of
y is f'(x) times the rate of change of x.
Returning to the illustration of a moving point P, let its
coordinates at time t hQ x and y. Then — measures the
rate of change of the a;-coordinate.
Since velocity has been defined as the rate at which a point
64.]
RATES AND DIFFERENTIALS
107
is moving, the rate — may be called the velocity which the
Cit
point P has in the direction of the a;-axis, or, more briefly,
the rc-component of the velocity of P.
It was shown on p. 105 that the actual velocity at any
instant t is equal to the space that would be passed over in
a unit of time, provided the velocity were uniform during
that unit. Accordingly, the a;-component of velocity — -
at
may be represented by the distance FA (Fig. 22) which P
would pass over in the direction of the a;-axis during a unit
of time if the velocity remained uniform.
Similarly -^ is the y-component of the velocity of P, and
may be represented by the distance PB.
ds
The velocity — of P along the curve can be represented
civ
by the distance P(7, measured on the tangent line to the
curve at P. It is evident that
PC is the diagonal of the rec-
tangle PA, PB.
Since PC^ = PA^ + P&,
it follows that
m-m-m- <•>
rdS.
dt
dx
dt
Fig. 22.
Ex. 1. If a point describe the straight line 3 x + 4 ?/ = 5, and if x
increase h units per second, find the rates of increase of y and of s.
2^ = 1 -far,
dy _ S dx
dt ~ 4:dt'
dx
dt
it follows that ^ = -ih, ^
dt ^ dt
Since
hence
When
108 DIFFERENTIAL CALCULUS [Ch. VII.
Ex. 2. A point describes the parabola y"^ = 12 x in such a way that
when a; = 3 the abscissa is increasing at the rate of 2 feet per second ; at
what rate is y then increasing? Find also the rate of increase of s.
Since
y^=12x,
then
2y^=12^,
dt dt
dy_^dx_ 6 dx,
dt y dt Vl2 X dt '
hence, when x =
= 3,
and
— =2, it follows that ^y
dt dt
±2.
^^^'- {%'- {llT^ (I) ' ^-- I = ^^ '-' rer second.
Ex. 3. A person is walking toward the foot of a tower on a horizontal
plane at the rate of 5 miles per hour ; at what rate is he approaching the
top, which is 60 feet high, when he is 80 feet from the bottom?
Let X be the distance from the foot of the tower at time t, and y the.
distance from the top at the same time. Then
x^ + 60-2 = y\
and x^ = y^-
dt ^ dt
When a: is 80 feet, y is 100 feet ; hence if — is 5 miles per hour, -^
• . .. . ^^ dt ^ dt
IS 4 miles per hour.
65. Abbreviated notation for rates. When, as in the above
examples, a time derivative is a factor of each member of an
equation, it is usually convenient to write, instead of the
symbols --^, -^, the abbreviations dx and dy, for the rates
dt dt
of change of the variables x and y. Thus the result of
differentiating v ^^ n , /1^
/=/(a^) (1)
may be written in either of the forms
dy __
dx
f'(x), (2)
|=/'(.)|, (3)
dy=f'ix-)dx. (4)
64-65.] BATES AND DIFFERENTIALS 109
It is to be observed that the last form is not to be re-
garded as derived from equation (2) by separation of the
symbols dv, dx; for the derivative -^ has been defined as
dx
the result of performing upon «/ an indicated operation rep-
resented by the symbol — ; and thus the di/ and dx of the
7 (XX
symbol -^ have been given no separate meaning. The di/
dx
and dx of equation (4) stand for the rates, or time deriva-
tives, -^ and — occurring in (3), while the latter equation
at ctt
is itself obtained from (1) by differentiation with regard
to t, by Art. 14.
In case the dependence of ^ upon x be not indicated by a
functional operation /, equations (3), (4) take the form
dy _dy dx
dt dx dt
dy = -^ dx.
dx
In the abbreviated notation, equation (4) of the last
article is written
ds^ = dx^ + dy'^.
Ex. 1. A point describing the parabola y^ — lpx is moving at the
time t with a velocity of v feet per second. Find the rate of increase
of the coordinates x and y at the same instant.
Differentiating the given equation with regard to f,
ydy = pdx.
But dx, dy also satisfy the relation
rfa;2 + dy'^ = u^ ;
hence, by solving these simultaneous equations,
dx = — ^ V, dy — ^ y, in feet per second.
110 DIFFERENTIAL CALCULUS [Ch. VII.
Ex. 2. A vertical wheel of radius 10 ft. is making 5 revolutions per
second about a fixed axis. Find the horizontal and vertical velocities of
a point oil the circumference situated 30° from the horizontaL
Since a: = 10 cos 6, y = lO sin ^,
then dx = -10 sin Odd, dy = 10 cos OdO.
But f?^ = 10 TT = 31.416 radians per second,
hence dx = — 314.16 sin ^ = — 157.08 feet per second,
and dy = 314.16 cos $ = 272.06 feet per second.
Ex. 3. Trace the changes in the horizontal and vertical velocity in a
complete revolution.
66. Differentials often substituted for rates. The symbols
dx, dy have been defined above as the rates of change of x
and y per second.
Sometimes, however, they may conveniently be allowed
to stand for any two numbers, large or small, that are pro-
portional to these rates; the equations, being homogeneous
in them, will not be affected. It is usual in such cases to
speak of the numbers dx and dy by the more general name
of differentials; they may then be either the rates them-
selves, or any two numbers in the same ratio.
This will be especially convenient in problems in which
the time variable is not explicitly mentioned.
Ex. 1. When x increases from 45° to 45° 15', find the increase of
logjo sin x, assuming that the ratio of the rates of change of the function
and the variable remains sensibly constant throughout the short interval.
Here dy = logjo^ . cot xdx = .4343 cot xdx = .4343 dx.
Let dx = 15' = .004363 radians.
Then dy = .001895,
which is the approximate increment of log,o sin x.
But log,o sin 45° = - J log 2 = - .150516,
therefore log^o sin 46° 15' = - .148620.
65-66.] BATES AND DIFFERENTIALS 111
Ex. 2. Expanding logj^ sin {x + h) as far as A^ by Taylor's theorem,
and then putting x = .785398, h = .004363, show what is the error made
by neglecting the thii'd term, as was done in Ex. 1.
Ex. 3. When x varies from 60° to 60° 10', find the increase in sin x.
Ex. 4. Show that log^QX increases more slowly than ar, when x > logj^e,
that is, X > .4343.
Ex. 5. Two sides a, 6 of a triangle are measured, and also the in-
cluded angle C; find the error in the computed length of the third side
c due to a small error in the observed angle C.
[Differentiate the equation c^ = a^ + b^ ~2ab cos C, regarding a, b as
constant-!
/
Ex. 6. A vessel is sailing northwest at the rate of 10 miles per hour.
At what rate is she making north latitude ?
^ Ex. 7. In the parabola y^ = 12 x, find the point at which the ordinate
and abscissa are increasing equally.
Ex. 8. At what part of the first quadrant does the angle increase
twice as fast as its sine?
/^ Ex. 9. Find the rate of change in the area of a square when the side
b is increasing at a ft. per second.
r Ex. 10. In the function y = 2 x^ -\- Q, what is the value of x at the
point where y increases 24 times as fast as x ?
l/ Ex. 11. A circular plate of metal expands by heat so that its diameter
increases uniformly at the rate of 2 inches per second ; at what rate is
the surface increasing when the diameter is 5 inches?
/ Ex. 12. What is the value of x at the point at which x^ — 5 x^ -{■ 17 x
and x^ — S X change at the same rate?
/
Ex. 13. Find the points at which the rate of change of the ordinate
7/ = x^ — 6 a:2 + 3 ar + 5 is equal to the rate of change of the slope of the
tangent to the curve.
jI Ex. 14. The relation between s, the space through which a body falls,
and t, the time of falling, is s-16t^; show that the velocity is equal
to 32 t.
The rate of change of velocity is called acceleration ; show that the
acceleration of the falling body is a constant.
112 DIFFERENTIAL CALCULUS [Ch. VII. 66.
. Ex. 15. A body moves according to the law s = cos (nt + e). Show
that its acceleration is proportional to the space through which it has
moved.
Ex. 16. If a body be projected upwards in a vacuum with an initial
velocity Vq, to what height will it rise, and what will be the time of
ascent ?
^ Ex. 17. A body is projected upwards with a velocity of a feet per
second. After what time will it return ?
v/ Ex. 18. If A be the area of a circle of radius x, show that the circum-
dA
ference is Interpret this fact geometrically.
dx
^ Ex. 19. A point describing the circle x^ + y^ = 25 passes through
(3, 4) with a velocity of 20 feet per second. Find its compon^it veloci-
ties parallel to the axes.
CHAPTER VIII
DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES
Thus far only functions of a single variable have been
considered. The present chapter will be devoted to the
study of functions of two independent variables x^ y. They
will be represented by the symbol
If the simultaneous values of the three variables a:, y, z be
represented as the rectangular coordinates of a point in space,
the locus of all such points is a surface having the equation
67. Definition of continuity. A function z oi x and ^,
2 = /(ic, ?/), is said to be continuous in the vicinity of any
point (a, 5) when /(a, 5) is real, finite, and determinate, and
such that
however A and h approach zero.
When a pair of values a, h exists at which any one of these
properties does not hold, the function is said to be discon-
tinuous at the point (a, 6).
£.a., let 2 = ^^tl.
x-y
When a: = 0, then z = — 1 for every value of y ; when y = 0 then
a = + 1 for every value of x. In general, if y = wa:,
1 — m
and z may be made to have any value whatever at (0, 0) by giving an
appropriate value to m.
113
114 DIFFERENTIAL CALCULUS [Ch. VIII.
68. Partial differentiation. If in the function
a fixed value y-^ be given to ?/, then
is a function of x only, and the rate of change in z caused
by a change in x is expressed by
dz = ^dx, (1)
dx
in which — is obtained on the supposition that y is constant.
dx
To indicate this fact without the qualifying verbal state-
ment, equation (1) will be written in the form
d^ = ^Ux. (2)
dx
The symbol — represents the result obtained by differ-
entiating z with regard to x^ the variable y being treated as
a constant; it is called the partial derivative of z with
regard to x.
From the definition of differentiation. Art. 11, the partial
derivative is the result of the indicated operation
50^ lim f(x + Aa:, .y) —f(x, y).
dx ^^ = ^ Aa:
Similarly, the symbol — represents the result obtained
by differentiating z with regard to y, the variable x being
treated as a constant ; it is called the partial derivative of z
with regard to y.
The partial derivative of z with regard to y is accordingly
the result of the indicated operation
68.] FUNCTIONS OF TWO VARIABLES 115
Bz_^ lim fix,y-\-Ai/)-f(x,^}
dgZ = — dx is called the partial x-differential of 2, and
ox
dz
dyZ = —-dy\^ called the partial y-differential of z.
if
Geometrically, the two equations
define the curve of section of the surface z=f(x^y) made
by the plane y — y^ The derivative — defines the slope of
the tangent line to this curve.
Similarly, when rr has a given constant value, x = x^^ the
partial derivative — is the expression for the slope of the
dy
tangent to the curve cut from the surface z^fQc^y') by
the plane x = x-^.
The equations of these two tangent lines at the point
(p^v Vv ^1) are
y = y^, z-z^ = ^(x-x{),
dz
x = x^, z-z^=-A.(y-y{),
and hence the equation of the plane containing these two
intersecting lines is
The plane is called the tangent plane to the surface
^ =f(p^^ y) at the point {xy, y^ z^.
116 DIFFERENTIAL CALCULUS [Ch. VIII.
EXERCISES
1. Given u = x^ + Z ^V - 7 xy\ prove that a:^ + y ^ = 4 1*.
dx dy
2. Given u = tan-i ^, show that a: ^ + j/ ^ = 0.
a; dx ^ dy
3. M = log (e- + eO ; find ^ + ^.
^ 4. M = sinx3/; find ^ + ^.
aa; dy
' 5. M = log (x + Va:2 + ?/2) ; find a: ^ + y ^.
aa: ay
6. u = log (tan x + tan y + tan z) ; show that
sin 2a:^ + sin 2^/^ + sin 23 ^ = 2.
dx dy dz
7. M = log(a: + w); show that ^ + ^ = -•
^ ^^ ^^ dx dy e-
69. Total differential. If both x and y be allowed to vary
in the function z — f{x^ y), the first question that naturally
arises is to determine the meaning of the differential of 2.
Let 2i =/(a^r Vx)-'
and Zj + A2 = /(ajj + Aa:, i/j + A!/)
be two values of the function corresponding to the two pairs
of values of the variables Xy, y^ and x-^ + Aa:, y^ + Az/.
The difference
Az =/(a:i 4- Aa;, y^ + ^y)-f(x^, y{)
may be regarded as composed of two parts, the first part
beUig the increment which z takes when x changes from x^
to ajj + Aa:, while y remains constant (y = yj), and the sec-
ond part being the additional increment which z takes when
68-69.] FUNCTIONS OF TWO VARIABLES 117
t/ changes from i/j to ^^ + Ay, while x remains constant
(^x = x^ + Ax~). The increment Az may then be written
Az =f(x^ + Ax, y^ + A^^) -f(x^ + Ax, y{)
+/(^i + Aa:, yi)-fCx^, yO
^/(^i + Ax, y^ + Ay) -/(a^i + Ax, y^
Ay ^
/(^l + Ax, y^-f(^x^, y^-) ^^
Ax
From the theorem of mean value, Art. 45, the last equation
may be written
Az = — /(a^i + OAx, y{)Ax + —fC^i + Aa;, y^ + O^Ay^Ay.
dx ay
It represents the actual increment Az which the dependent
variable z takes when the independent variables x and y take
the increments Ax and Ay.
In the preceding equation let Ax, Ay, Az be replaced by
€ • dx, € • dy, € ' dz respectively, in which dx, dy are entirely
arbitrary. After removing the common factor €, let €
approach zero. The result is
d^ = ^f(^dx + ^^^^d2,. (1)
The differential dz defined by this equation is called the
total differential of z. It is not an actual increment of z,
but the increment which z would take if the change con-
tinued uniform while x changed from x-^^ to x-^ + dx and y
changed from y^ to y^ + dy. Geometrically speaking this is
the increment which z would take if the point (x, y, 2) should
move from the position (x^, y^ ^i) in the tangent plane of
the surface z = fix, y) instead of on the surface itself.
118 DIFFERENTIAL CALCULUS [Ch. VIII.
Equation (1) may be written in the form
dz = — dx -\ dy,
dx dy ^
from which the following theorem can be stated : the total
differential of a function of two variables is equal to the sum
of the partial differentials.
The same method can be directly applied to functions of
three or more variables. Thus, if 2 be a function of the
variables a;, ^, w,
z = <l)(x, y, u),
then dz = ^-idx + ^dy-^^du.
dx dy du
Ex. 1. Given z = axy^ + bx^y + cx^ + ey,
then dz = (ay"^ -\-2bxy + 3 cx^)dx + (2 axy + bx^ + e)dy.
Ex. 2. Given z = x*, then d^ = yx^-^dx, and dyZ = x^ log x dy.
Hence dz = yx^~^dx + x" log x dy.
Ex. 3. Given u = tan-i?^, show that du = """^f ~ ^/^.
x x^ -\- y^
Ex. 4. Assuming the characteristic equation of a perfect gas, vp = Kt,
in which v is volume, p pressure, t absolute temperature, and R a constant,
express each of the differentials dv, dp, dt, in terms of the other two.
Ex. 5. A particle moves on the spherical surface x^ -{■ y^ -\- z^ = a^ in
a vertical meridian plane inclined at an angle of 60° to the zx plane.
If the x-component of its velocity be ^ a per second when x = \a, find
the y-component and the ^-component velocity.
Since
then dz = —
xdx ydy
y/a^ - x^- y^ Va^ - x^ - y^
But since dx = ^a, and the equation of the given meridian plane is
y = X tan 60^ hence dy = dxy/S = ^ V3, and y = ^. Therefore
(fa = --^-^ = -^ in feet per second.
69-70.] FUNCTIONS OF TWO VARIABLES 119
70. Language of differentials. The results of the preced-
ing articles may be stated thus :
The partial ^-differential due to a change in x is equal to
the ri:-differential multiplied by the partial a;-derivative.
The partial ;3-differential due to a change in ^ is equal to
the ^/-differential multiplied by the partial ^-derivative.
The total ^-differential is equal to the sum of the partial
^-differentials.
One advantage of writing the equation in the differential
form is that it may be divided when necessary by the dif-
ferential of any other variable s, to which x and y may be
related, and then, remembering that the ratio of two differ-
entials (or rates) may be expressed as a derivative, the
equation would become
dz_dzdx^ d^^
ds dx ds dy ds
In particular, if y be not independent, but is a function of
x, then s may be chosen as x itself, and the preceding equa-
tion becomes , ^
dz_oz^.dz dj£
dx dx dy dx
If the functional relation between x and y be given,
y = ^(^)^
dz
then the same result would be obtained, whether -— be
dx
determined by the present method, or y be first eliminated
from the relation
2'=/(^, «^),
and the resulting equation be differentiated as to x. The
method of this article frequently shortens the process.
It is here well to note the difference between — and -— .
dx dx
The former is the partial derivative of the functional ex-
120 DIFFERENTIAL CALCULUS [Ch. VIII.
pression for z with regard to a:, on the supposition that y
is constant. The latter is the total derivative of z with
regard to a;, when account is taken of the fact that y is
itself a function of x,
Ex. 1. Given z= Va;^ + y\ y = \ogx', find — •
dx
(Iz
^,
dx y/x^ + V* y/x^ 4- y^ ^"^
rfar a;'
hence ^Il^_^±]L..
fi^ xy/x^ + y^
Ex. 2. If 2 = tan-i JL and ^x'^-{-y^=l, show that — = — •
71. Differentiation of implicit functions. If in the relation
z—f{x^y)^ z be assumed to be constant, then
hence ^dx-\-^dy=0, (1)
Bx By ^ ^ ^
from which ^ = _|£. (2)
dx ^
dy
In all such cases either variable is an implicit function of
the other, and thus the last equation furnishes a rule for
finding the derivative of an implicit function.
Ex. 1. Given x* + y« + 3 axy = c, find -p
Since (3a:^ + 3a,) + (3,« + 3a.)^^ = 0, | = -^.
Ex.2. /(ax+%)=c; |^= a/(a:r + 6y) ; ^= 6/'(«a: + 6y); ^=- ^^
Ex. 3. If aa:2 + 2 ^ary + fty" + 2 ^rar + 2/y + c = 0, find ^
Ex. 4. Given x* - w« = c, find ^.
ax
70-73.] FUNCTIONS OF TWO VARIABLES 121
It is to be noticed that the result of differentiating any
implicit function of x, y by the method of the present article
agrees with the result of differentiation according to the
rules of Chapter II. Compare Ex. 2, p. 53.
72. Successive partial differentiation. The expressions
— , — which were defined in Art. 68 are functions of both
bx dy
X and y.
If — be differentiated partially as to x^ the result is written
This expression is called the second partial derivative of
z as to X,
Similarly, the results of the operations indicated by
dy\dx/ dx\dy/ dy\dy)
S^z d^z b^z
are written , , — r, respectively.
by dx dx dy dy^
Beginning with the left, these expressions are called the
second partial derivative of 2 as to a; and y^ the second par-
tial derivative of z SiS to y and x, and the second partial
derivative of z as to y.
73. Order of differentiation indifferent.
Theorem. The successive partial derivatives
d^z d^z
,
dy dx dx dy
are equal for any values of x and y in the vicinity of which
z and its first and second partial x- and ^-derivatives are
continuous.
For, in z = f{x, y), first change x into a; + A, keeping y
122 DIFFERENTIAL CALCULUS [Ch. VIII.
constant. Then by the theorem of mean value (Art. 45),
the increment of the function is equal to the increment of
the variable multiplied by the derivative taken for some
value intermediate between x and x-{-h\ that is,
fQc + h, y) -f(x, y^=h ±f(x + ^A, ?/) . [0 < ^ < 1.
Next let y change to y + k^ x remaining constant, and
take the increment of the function on the left. Then by
the theorem of mean value applied to — f(x + dh^ y) as a
function of y with the increment ^,
lf(x + li,y + h) -fix, y + m- if(x + h y) -fix, y)]
= kh4-^f(x + eh,y + e^k-).
dy ax
Now let these increments be given in reversed order. Then
[/(^ + h,y + 1c)-f(ix + h, y)] - IfCx, y + k^ -fix, y)]
^^k^-^f(x-\-eji,y{.ejcy,
dx dy
hence
i. ±fQc + eh, y + e,k-)= i- -^fCx + e,h, y + ejc).
dy ox dx dy
This relation is true for any values of h and k for which
all the functions mentioned are continuous.
When h, k approach zero,
x-\- Oh, y + B^k, and x + 6Ji^ y + 6jc
approach a;, y, and
fix + eh,y + e^k), f(x + e^h, y + ejc^
approach /(a;, y)\ and similarly for the derivatives; hence
Tyl-J^-yH.Ty^^^^y^^
or, since /(a;, y) = 2,
d^z ^ ^z
dy dx dx dy
73.] FUNCTIONS OF TWO VARIABLES 123
Cor. It follows directly that under corresponding con-
ditions the order of differentiation in the higher partial
derivatives is indifferent.
dh dh a%
Kg.,
dx by dx dx^ by dy dx^
EXERCISES
^1. Verify that -^ = -^, when u = xY-
dx dy oy ox
y 2. Verify that ^^ = ^^, when w = a:2z/ + a:j^8
^ 3. Verify that -^- -^, when w = y log (1 + :^^).
dx dy dy dx
^ 4. In Ex. 3 are there any exceptional values of x, y for which the
relation is not true ?
V- 5. Given m = (x^ + y^p, verify the formula
dx"^ dx dy dy^
/ 6. Given m = (x^ + 3/^)^, show that the expression in the left member
of the differential equation in Ex. 5 is equal to — —
/ 7. Given u =(x'^-\- y'^ + s^)-^ . prove that |!^ + ^ + |!^ = 0.
dx^ dy^ dz^
8. Given m = sec (v + ax) + tan Cy —-ax) ; prove that -— = a^—--
dx^ dy^
9. Given m = sm a: cos y ; verify that = :,:,:,:, = TTK~i'
dy^ dx'^ dx dy dx dy dx^ dy^
^ 10. Given u = (4 a6 - c^)"^ ; prove that |^' = ^^.
CHAPTER IX
CHANGE OF VARIABLE
74. Interchange of dependent and independent variables.
If ^ be a continuous function of a;, defined by the equation
/(a?, y) = 0, the symbol -^ represents the derivative of y
dx
with regard to x^ when one exists. If a; be regarded as a
function of y, defined by the same equation, the symbol —
dy
represents the derivative of x with regard to y, when one
exists. It is required to find the relation between -^
and ^. •""
dy
Let x^ y change from the initial values x^^ y^ to the values
x^ H- Aa;, y^ + Ay, subject to the relation /(a;, y) = 0.
Then, since
Aa: Aa;'
Ay
it follows, by taking the limit, that
dx dx ^^
dy
Hence, if y and x he connected hy a functional relation the
derivative of y with regard to x is the reciprocal of the derivative
of X tvith regard to y.
This process is known as changing the independent varia-
ble from X to y. The corresponding relations for the higher
124
Ch. IX. 74-75.]
CHANGE OF VARIABLE
125
derivatives are less simple. They are obtained in the follow-
ing manner :
To express —4 in terms of — -, — -r, differentiate (1) as
dx^ ay dy^
to a;; ^ if
^_d^
dx^ dx
fl 1
d
fi-i
dx
dy
dx
idy\
[dy}
dy d { 1 ^
dx dy
dx
Vdy}
dx
But
hence
dJh
d^ rj_1 _ dy'^
dx\^'
dy
dx
<dy\
In a similar manner,
d^
d7?
dxV
dy)
dy\
fdxS^
\dy)
d^x dx _ o fd^x^
dy^ dy Wy V ^
dx\^
dy.
(2)
(3)
75. Change of the dependent variable, li y is sl function
of 25, let it be required to express -^, — ^, ••• in terms of
^, ^, ....
dx dx^
Suppose y = (f>(z). Then
dy_^dydz^^Mt(^^^dz^
dx dz dx dx
dx^ dx
(^'^43
126 DIFFERENTIAL CALCULUS [Ch. IX.
g = r(.)(|)V^'(.)S. (4)
The higher a;-derivatives of y can be similarly expressed
in terms of a:-derivatives of z.
76. Change of the independent variable. Let t^ be a
function of x^ and let both x and y be functions of a new
variable t. It is required to express -^ in terms of -^,
TO 1 -n
and -T~ in terms of -^ and -=4*
do!^ dt dt^
By Art. 14,
dy__ dt
dx ~ dx^
dt
d^y dx
cPy dt^ dt
dh;d]i
dt^ dt
dx^ ~ fd
x\^
(1)
hence u^ u^^_u^ — u^i_u^^ ^2)
In practical examples it is usually better to work by the
methods here illustrated than to use the resulting formulas.
EXERCISES
1. Change the independent variable from a: to 2 in the equation
A.
^»+^l + 2' = ''' ''^"" ' = *••
dy
dz
dy
dx~
dz
dH.
Hence x2^, + z^ + y = 0 becomes ^i + y = 0.
7r>-76.] CHANGE OF VARIABLE 127
* 2. Interchange the function and the variable in the equation
/ 3. Interchange x and y in the equation
R
' 4. Change the independent variable from a: to ^ in the equation
Jd^y _dl fu _d^fdyy^ ^^
\dx^/ dx dx^ dx'^XdxJ
*^ 5. Change the dependent variable from y to 2 in the equation
d^y , 2n+y)fdyy ^
/ 6. Change the independent variable from x to y in the equation
x^ \- X 1- M = 0, when y = log x.
dx^ dx
7. If y is a function of x, and x a function of the time t, express the
^-acceleration in terms of the a:-acceleration, and the x-velocity.
Since dy^dy^Ix^
dt dx dt
hence d^ ^d_yd^^lx , ^dJJ\
dt^ dx dt^ dt dAdxJ
But ^(^Ie\ = jL(^]^I^ = ^^j
dt\dxJ dxKdxl dt dx'^ dt
hence d^^dj,d^ d^nix\\
dt^ dx dt^ dxAdt)
In the abbreviated notation for ^derivatives,
«4. 8. Change the independent variable from x to u in the equation
d'^y 2 X dy y ^1
-t4 + ^ 5 -r- + ,, 0x0 = 0, when x = tan u.
dx^ 1 -\- x2 t/a; (1 + x^y
128 DIFFERENTIAL CALCULUS [Ch. IX. 76.
^ 9. Change the independent variable from x to f in the equation
^^-'^'^S-'^i^^' ""^^^ :r = cos^
10. Show that the equation
remains unchanged in form by the substitution 2: = -•
r 11. Interchange the variable and the function in the equation
dx^ \dxi y\dxi "
^ 12. Change the dependent variable from y to 2 in the equation
APPLICATIONS TO GEOMETRY
CHAPTER X
TANGENTS AND NORMALS
77. It was shown in Art. 10 that if -/(a;, «/) = 0 be the
dy
equation of a plane curve, then -~ measures the slope of the
tangent to the curve at the point x, y. The slope at a partic-
ular point (a^j, ^j) will be denoted by -~=^ meaning that x^ is
to be substituted for x^ and y^ for y in the expression for -^.
(XX
78. Equation of tangent and normal at a given point.
Since the tangent line passes through the given point (a^j, y-[)
and has the slope -^, its equation is
The normal to the curve at the point (x^^ y{) is the straight
line through this point, perpendicular to the tangent.
Since the slope of the normal is
:zl=_^, [Art. 74,
dy dy
dx
(2)
its equation is
dx-, r ^
y y,= ^(^ x,y
i.e..
<^^-^i)+ii^2'-2'>>=
129
130
DIFFERENTIAL CALCULUS
[Ch. X.
79. Length of tangent, normal, subtangent, subnormal.
The segments of the tangent and normal intercepted be-
tween the point of tangency and the axis OX are called,
respectively, the tangent length and the normal length,
and their projections on OX are called the subtangent and
the subnormal.
Fig. 23 a.
Fig. 23 h.
Thus, in Fig. 23, let the tangent and normal to the curve
P(7 at P meet the axis OX in T and N, and let MP be the
ordinate of P. Then
TP is the tangent length,
PN the normal length,
TM the subtangent,
JOT the subnormal.
These will be denoted, respectively, by t, w, t, v.
Let the angle XTP be denoted by <^, and write tan<^=m.
1
Then
cos<^ =
vr+
; sin<^ =
m
m"
VI +
m'^
t = -^ = ^ — ; w = -^=yiVl H-TTi^;
sin 9 m cos 9
T = j,,cot^ = y,^^J=^; r = y, tan <^=«/,^= my,.
79.] TANGENTS AND NORMALS 131
The subtangent is measured from the intersection of the
tangent to the foot of the ordinate ; it is therefore positive
when the foot of the ordinate is to the right of the intersec-
tion of tangent. The subnormal is measured from the foot
of the ordinate to the intersection of normal, and is positive
when the normal cuts OX to the right of the foot of the
ordinate. Both are therefore positive or negative, according
as (f) is acute or obtuse.
The expressions for t, v may also be obtained by finding
from equations (1), (2), Art. 78, the intercepts made by the
tangent and normal on the axis OX. The intercept of the
tangent subtracted from x-^^ gives r, and x-^^ subtracted from
the intercept of the normal gives v.
Ex. Find the intercepts made upon the axes by the tangent at the
point (x^, ?/j) on the curve y/x + Vy = -\/a, and show that their sum is
constant.
Differentiating the equation of the curve,
y/x y/y ^^
Hence the equation of the tangent is
The X intercept is x^ + V^^p and the y intercept is y^ + Vx^~y[, hence
their sum is
If a series of lines be drawn such that the snm of the intercepts of
each is the same constant, account being taken of the signs, the form
of the parabola to which they are all tangent can be readily seen.
EXERCISES
1. Find the equations of the tangent and the normal to the ellipse
^ + ^ = 1 at the point (xp y^) . Compare the process with that employ*^.d
in analytic geonietry to obtain the same results.
132 DIFFERENTIAL CALCULUS [Ch. X.
2. Find the equation of the tangent to the curve x^{x-\-y)=a\x—y)
at the origin.
/ 3. Find the equations of the tangent and normal at the point (1, 3)
on the curve y^ = 9 x^.
y 4. Find the equations of the tangent and normal to each of the
following curves at the point indicated:
(a) V = ) at the point for which a: = 2 a.
(^) y2 _ 2 a:2 — z*, at the points for which x = 1.
(y) ?/2 _ ^py.^ at the point (/>, 2p).
5. Find the value of the subtangent of y'^ = ^x'^—12 at a; = 4.
Compare the process with that already given in analytic geometry.
SI 6. Find the length of the tangent to the curve ^y^ — 2 a: at a; = 8.
~^ 7. Find the points at which the tangent is parallel to the axis of a:,
and at which it is perpendicular to the x axis for each of the following
curves :
(a) ax^-\-2hxy+hy'^ = l.
08) y =
ax
(y) y^ = x\2a-x).
-w 8. Find the condition that the conies ax'^ + hy^=l, a'x^ + l/y^= 1
shall cut at right angles.
T 9. Find the angle at which x^ = y^ + 5 intersects Sx^-\-lSy^= 144.
Compare with Ex. 8.
10. Show that in the equilateral hyperbola 2 xy = a^ the area of the
triangle formed by a variable tangent and the coordinate axes is constant
and equal to a^.
11. At what angle does y^ = Sx intersect 4 a:* + 2 y^ = 43 ?
12. Determine the subnormal to the curve y» = a*-^ x.
13. Find the values of x for which the tangent to the curve
?/8 = (x-a)2(x-c)
is parallel to the axis of x.
14. Show that the subtangent of the hyperbola xy = a^ is equal to
the abscissa of the point of tangency, but opposite in sign.
/ 15. Prove that the parabola y* = 4 aa: has a constant subnormal.
79-80.]
TANGENTS AND NORMALS
133
16. Show analytically that in the curve x^ + y^ = a^ the length of the
normal is constant.
17. Show that in the tractrix, the length of the tangent is constant,
the equation of the tractrix being
2 ^e + V^^37
18. Show that the exponential curve y = ae" has a constant sub-
tangent.
19. Find the point on the parabola y^ = 4:px at \v^hich the angle
between the tangent and the line joining the point to the vertex shall be
a maximum.
POLAR COORDINATES
80. When the equation of a curve is expressed in polar
coordinates, the vectorial angle 6 is usually regarded as the
independent variable. To determine the direction of the
curve at any point, it is most convenient to make use of
the angle between the tangent and the radius vector to the
point of tangency.
Let P, Q be two points on the
curve (Fig. 24). Join P, Q with
the pole 0, and drop a perpendic-
ular PM from P on OQ. Let /?,
0 be the coordinates of P; p+Ap^
6+AO those of Q. Then the angle
P0$ = A(9; Pil[/ = /3sinA6>; and
MQ=OQ-OM=p + Ap-p cos Ad.
Fig. 24.
Hence
tan MQP =
p sin A^
p + Ap — p cos A^
When Q moves to coincidence with P, the angle MQP
approaches as a limit the angle between the radius vector
and the tangent line at the point P. This angle will be
designated by yjr.
DIFFERENTIAL CALCULUS
lira
134
Thus
But p(l - cos Ae)=2p sin2 1 A^,
[Ch. X.
t^^r-A^-Op + Ap_pcosA^
hence
^^^i^ = Jlo
p sin A^
A(9
. 1 ./J siniA^ , Ap
Since . l^^ ^ ^^" ^ = i^ the preceding equation reduces to
^ , p dO
tant = ^=P^-.
dd
Ex. 1. A point describes a circle of radius p.
Prove that at any instant the arc velocity is p times
the angle velocity,
(It P dt
dt
Fia.2t).
Fio. 27.
Ex. 2. When a point describes a given
curve, prove that at any instant the velocity
^ — has a radius component -^ and a com-
''* dt ^ dt
ponent perpendicular to the radius vector
p — , and hence that
dt
C08i^ = ^, 8mtf/ = p^, tan.Zr = p^.
ds ds dp
81. Relation between ^ and ^•
dx dp
If the initial line be taken as the axis
of X, the tangent line at P makes an
angle (f) with this line.
Hence 6 + yjr = ^;
80-82.]
TANGENTS AND NOBMALS
135
82. Length of tangent, normal, polar subtangent, and polar
subnormal. The portions of the tangent and normal inter-
cepted between the point of tangency P and the line through
the pole perpendicular to the radius vector OP, are called
the polar tangent length and the polar normal length;
their projections on this perpendicular are called the polar
subtangent 'dudi polar subnormal.
Fig. 28 a. Fig. 28 6.
Thus, let the tangent and normal at P meet the perpen •
dicular to OP in the points JV and M. Then
PN is the polar tangent length,
PM is the polar normal length,
ON is the polar subtangent,
OMi^ the polar subnormal.
They are all seen to be independent of the direction of
the initial line. The lengths of these lines will now be
determined.
Since PN= OP . sec OPN= psec^jr^ pyjp^f^J + 1
-'f.M%i
dp
hence polar tangent length = p -^\P^ + ( -^
dO
136 DIFFERENTIAL CALCULUS [Ch. X.
Again, 0N= OP tan OPN= /o tan -f = p2 ^,
dp
hence polar subtangent = /o^ -—•
dp
PM= OP ' CSC (9PiV^= pGSGylr= \p^ + f^\
hence polar normal length = \p^ "*■ V;^) *
0M= OP cot OP]Sr= ^,
hence polar subnormal = -^*
The signs of the polar tangent length and polar normal
length are ambiguous on account of the radical. The direc-
tion of the subtangent is determined by the sign of p^ — .
dS ^P
When -- is positive, the distance ON^ should be measured
dp
to the right, and when negative, to the left of an observer
placed at 0 and looking along OP; for when 0 increases
with p, —- is positive (Art. 13), and ylr is an acute angle (as
dp ^Q
in Fig. 28 h) ; when 0 decreases as p increases, — is negative,
and i/r is obtuse (Fig. 28 a). ^
EXERCISES
1. In the curve p = a sin ^, find \p.
2. In the spiral of Archimedes p = a$f show that tan \^ = 0 and find
the polar subtangent, polar normal, and polar subnormal. Trace the
curve.
3. Find for the curve p^ = a*co8 2d the values of all the expressions
treated in this article.
4. Show that in the curve pO = a the polar subtangent is of constant
length. Trace the curve.
82.] TANGENTS AND NORMALS 187
5. In the curve p=a(l — cosO), find i/r and the polar subtangent.
6. Show that in the curve p = b • e^cota the tangent makes a constant
angle a with the radius vector. For this reason, this curve is called the
equiangular spiral.
yC 7. Find the angle of intersection of the curves
p = a(l + cos 6), p = 6(1 — cos^.
8. In the parabola p = a sec^ -, show that ^ + ^ = w.
CHAPTER XI
DERIVATIVE OF AN ARC, AREA, VOLUME, AND SURFACE
OF REVOLUTION
83. Derivative of an arc. The length s of the arc AP of
a given curve 1/ =/(a:), measured from a fixed point A to any
point P, is a function of the abscissa x of the latter point,
and may be expressed by a relation of the form s = (K^^)-
The determination of the function (f> when the form of
/ is known, is an important and sometimes difficult problem
in the Integral Calculus. The first step in its solution is
ds
to determine the form of the derivative function -— = <t>'(x)^
ax
which is easily done by the methods of the Differential
Calculus.
Let PQ \)Q two points on the curve (Fig. 29); let x^ y
be the coordinates of P ; a: + Aa:,
y -\- ^y those oi Q\ s the length
of the arc AP ; s + A« that of
the arc AQ. Draw the ordinates
MP, NQ ; and draw PR parallel
— 2C_ to MN. Then PR= Ax, RQ= Ay;
M N
J.IQ29. arcP^=As. Hence
Chord PQ = V(Aa:)2 + (Ay)2,
Ax
T..„,.„=.^.^.|iV..(g)'
As _ As
Ax PQ Ax PQ
138
Ch. XI. 83-84.] DERIVATIVES OF ARC, AREA, ETC.
139
Taking the limit of both members as Ax approaches zero
and putting ^^"Iq-^^^ 1, by Art. 6, Th. 4, and Art. 4,
Th. 8, Cor., it follows that
%'Mf)'-
Similarly
Moreover, from Art. 65
or in the differential notation
dx^ fdy^
dt \dt
)'■
(1)
(2)
(3)
(4)
84. Trigonometric meaning of
ds ds
dx dy
Since
PQ
As PQ As
it follows by taking the limit that
dx ,
-— = cos 9,
wherein <j), being the limit of the angle RPQ, is the angle
which the tangent at the point (x, y) makes with the a;-axis.
Similarly, -^ = sin <^ ; whence — - = sec <^, — - = esc </>.
ds - - r ^ - ^^
Using the idea of a rate or dif-
ferential, all these relations may
be conveniently exhibited by Fig.
30.
These results may also be de-
rived from equations (1), (2) of
Art. 83, by putting ^ = tan </>.
dy
Y
dy
y.
I-
^1
0
X
Fia. 30.
140
DIFFERENTIA L CALCUL US
[Ch. XI.
85. Derivative of the volume of a solid of revolution. Let
the curve APQ revolve about the a:-axis, and thus generate
a surface of revolution ; let V
be the volume included between
this surface, the plane generated
by the fixed ordinate at A, and
the plane generated by any ordi-
nate MP.
Let A I^ be the volume gener-
ated by the area PMNQ. Then
A V lies between the volumes of the cylinders generated
by the rectangles PMNR and SMNQ; that is,
iry'^a^x < A r< 7r(y -h Ay)2Aa;.
Dividing by Aa; and taking limits.
F
•
^
1
-JL^
F
r
R
X
0
A
{ I
V
Fia. 30 a.
dV
— m-ll^
86. Derivative of a surface of revolution. Let S be the
area of the surface generated by the arc AP (Fig. 31), and
AS that generated by the arc PQ whose length is As.
Draw PQ\ QP' parallel to OX
and equal in length to the arc PQ.
Then it may be assumed as an
axiom that the area generated by
PQ lies between the areas gen-
erated by PQ' and P'Q\ i.e.,
2 iryAs < AaS' < 2 irQy + Ay) A«.
Dividing by A« and passing to the limit,
dS
M N
Fio. 31.
!;-'-'>•
f-f-l— V^^
(1)
(2)
85-87.] DERIVATIVES OF ARC, AREA, ETC. 141
87. Derivative of arc in polar coordinates.
Let /3, 6 be the coordinates of P ; p -\- A/a, 6 + A^ those
oi Q ; s the length of the arc KP ;
As that of arc PQ ; draw Pilif per-
pendicular to OQ. Then
PM=p sin AO,
MQ=OQ-OM==p^Ap-p cos Ae
=p(l — GosA6) + Ap o^
= 2/)sin2iA<9-hA/).
Hence PQ^=Cp sin Al9)2 -f (2 /o sin2 J A(9 + A/o)2,
Replacing the first member by ( — -^ * t4 ) ' passing to the
\ As Ad J
limit when A^ = 0, and putting lim — ^ = 1, lim ^^^ - = 1,
1 A/? ^^ ^^
lim^i^4^= 1, it follows that
^Ad
Fig. 32.
©■='■- gj
*•'•' d0
Mtl
In the rate or differential notation this formula may be
conveniently written
d»^ = dp^ + p^dff^.
This relation may be readily deduced also from Fig. 26,
Art. 80.
142
DIFFERENTIAL CALCULUS
[Ch. XI.
88. Derivative of area in polar coordinates. Let A be
the area of OKP measured
from a fixed radius vector OK
to any other radius vector OP ;
^x let A A be the area of OPQ.
Draw arcs PM, QN, with 0 as
a center. Then the area POQ .
lies between the areas of the
sectors OPiltf and ONQ\ i.e..
Fig. 33.
1 /32A(9 < A^ < lip + Ap)2 A^.
Dividing by A^ and passing to the limit, when A^ = 0, it
follows that
dA
dd
= \p'-
For the derivative of the area of a curve in rectangular
dA
coordinates, see Art. 10. The result is —— = y.
dx
EXERCISES ON CHAPTER XI
1. In the parabola w^ = 4 ax, find — , - — , -— , — —
dx dx dx dx
2. Find — and — for the circle x^-{-y^ = a\
dx dy ^
ds
3. Find — for the curve e^ cos x = \.
dx
4. Find the x-derivative of the volume of the cone generated by
revolving the line y — ax about the axis of x.
5. Find the ar-derivative of the volume of the ellipsoid of revolution,
X^ 2/2
formed by revolving -^+ rj = 1 about its maior axis. ^
6. In the curve p = a^ find -^- ^(ffi " ^
Im^^
de
ds
7. Given p = a(H-co8^); find ^.
d$
8. In p« = a3 cos 2d, find
d$
c^
CHAPTER XII
ASYMPTOTES
89. Hyperbolic and parabolic branches. When a curve
has a branch extending to infinity, the tangents drawn at
successive points of this branch may tend to coincide with
a definite fixed line as in the familiar case of the hyperbola.
On the other hand, the successive tangents may move farther
and farther out of the field as in the case of the parabola.
These two kinds of infinite branches may be called hyperbolic
and parabolic.
The character of each of the infinite branches of a curve can
always be determined when the equation of the curve is known.
90. Definition of a rectilinear asymptote. If the tangents
at successive points of a curve approach a fixed straight line
as a limiting position when the point of contact moves farther
and farther along any infinite branch of the given curve,
then the fixed line is called an asymptote of the curve.
This definition may be stated more briefly but less pre-
cisely as follows: An asymptote to a curve is a tangent
whose point of contact is at infinity, but which is not itself
entirely at infinity.
DETERMINATION OF ASYMPTOTES
91. Method of limiting intercepts. The equation of the
tangent at any point (x^, y{) being
143
144 DIFFERENTIAL CALCULUS [Ch. XII.
the intercepts made by this line on the coordinate axes are
0)
Suppose the curve has a branch on which x==oo and
y = Qo. Then from (1) the limits can be found to which
the intercepts rr^, i/q approach as the coordinates x^^ y-^ of the
point of contact tend to become infinite. If these limits be
denoted by a, 5, the equation of the corresponding asymptote is
a 0
Except in special cases this method is usually too compli-
cated to be of practical use in determining the equations of
the asymptotes of a given curve. There are three other
principal methods, which will always suffice to determine the
asymptotes of curves whose equations involve only algebraic
functions. These may be called the methods of inspection,
of substitution, and of expansion.
92. Method of inspection. Infinite ordinates, asymptotes
parallel to axes. When an algebraic equation in two co-
ordinates X and y is rationalized, cleared of fractions, and
arranged according to powers of one of the coordinates, say
y, it takes the form
ayn + (hx 4- c)5^"-^-f (c?r2 -f ex +f^y^-^+ ... + u„_,y 4- w„ = 0,
in which w„ is a polynomial of the degree n in terms of the
other coordinate a;, and w„_i is of degree n — 1.
When any value is given to a:, the equation determines n
values for y.
Let it be required to find for what value of x the corre-
sponding ordinate y has an infinite value.
91-92.] ASYMPTOTES 145
For this purpose the following theorem from algebra will
be recalled :
Given an algebraic equation of degree n
a^» + /3?/"-^ + 7«/""' + - = 0.
If a = 0, one root i/ becomes infinite ; if a = 0 and /3 = 0,
two roots 1/ become infinite ; and in general if the coefficients
of each of the k highest powers of t/ vanish, the equation will
have k infinite roots.
Suppose at first that the term in y" is present; in other
words, that the coefficient a is not zero. Then, when any
finite value is given to x, all of the n values of y are finite,
and there are accordingly no infinite ordinates for finite
values of the abscissa.
Next suppose that a is zero, and 6, c, not zero. In this
case one value of «/ is infinite for every finite value of x^
and hence the curve passes through the point at infinity
on the ^ axis.
There is one particular value of x, namely, x = ——, for
which an additional root of the equation in t/ becomes
infinite. For, when x has this value, the coefficient hx + o oi
the highest power of ^ remaining in the equation vanishes.
Geometrically, every line parallel to the i/ axis has one
point of intersection with the curve at infinity, but the
line bx + c = 0 has two points of intersection with the i
curve at infinity. A line having two coincident points of /
intersection with a curve is a tangent to the curve, and ^
when the coincident points are at infinity, but the line Ij ^
itself not altogether at infinity, the tangent is an asymp- a
tote. Hence an ordinate that becomes infinite for a defi- \ J"
nite value of x is an asymptote. (
Again, if not only a, but also h and c are zero, there are
146
DIFFERENTIAL CALCULUS
[Ch. XII.
two values of x that make «/ infinite ; namely, those values of
X that make dx^-{-ex-\-f = 0^ and the equations of the infinite
ordinates are found by factoring this last equation ; and so on.
Similarly, b}^ arranging the equation of the curve accord-
ing to powers of a;, it is easy to find what values of 1/ give
an infinite value to x.
Ex. 1. In the curve
2 x^ + x^y + xy^ z= x^ - y^ - 5,
md the equation of the infinite ordinate, and determine the finite point
in which this line meets the curve.
This is a cubic equation in which the coefficient of y^ is zero.
Arranged in powers of y it is
f (x+1) + yx^ + (2 a:8 - a:2 + 5) = 0.
I the equation for y becomes
0-1/2 + 2, +2 = 0,
the two roots of which are y = 00, y = — 2 ; hence the equation of the
infinite ordinate is x + JL=-0. — Xhe infinite ordinate meets the curve
again in the finite poin^( — 1^ — 2J
~^n5e~the~T«rm m^r'Tf^^TBSBtl^ there are no infinite values of x for
finite values of y.
Ex. 2. Show that the lines x = a, and y = 0 are asymptotes to the
curve a^x = y(x- ay (Fig. 34).
¥iQ. 34.
Ex. 3. Find the asymptotes of the curve x"^ (y - a) + xy^ = o*
92-93.] ASYMPTOTES 147
93 Method of substitution. Oblique asymptotes. The
asymptotes that are not parallel to either axis can be found
by the method of substitution, which is applicable to all
algebraic' curves, and is of especial value when the equation
is given in the implicit form
/(^,^) = 0. (1)
Consider the straight line
y =mz + b, (2)
and let it be required to determine m and b so that this line
shall be an asymptote to the curve f{x, ?/) = 0.
Since an asymptote is the limiting position of a line that
meets the curve in two points that tend to coincide at
infinity, then, by making (1) and (2) simultaneous, the
resulting equation in x,
fix, ma: + 5) = 0,
is to have two of its roots infinite. This requires that the
coefificients of the two highest powers of x shall vanish.
These coefficients, equated to zero, furnish two equations,
from which the required values of m and h can be deter-
mined. These values, substituted in (2), will give the
equation of an asymptote.
Ex. 4. Find the asymptotes to the curve y^ = x^{2a ~ x).
In the first place, there are evidently no asymptotes parallel to either
of the coordinate axes. To determine the oblique asymptotes, make the
equation of the curve simultaneous with y = mx -\- b, and eliminate y.
Then
{mx-{- by = x^(2a - x),
or, arranged in powers of x,
(1 + w8) x^ + (3 m% - 2 a) x2 + 3 b^x + &» = 0.
Let m8 + 1 = 0 and dm^b- 2a = 0.
148
DIFFERENTIAL CALCULUS
[Ch. XII.
Then
hence
-a: +
3 '
2a
3
is the equation of an asymptote.
The third intersection of this line with the given cubic is found from
the equation 3 mb^x + &^ = 0, whence x = — •
Y
This is the only oblique asymptote, as the other roots of the equation for
m are imaginary.
Ex. 5. Find the asymptotes to the curve y (a* + x^) = a^(a — x).
Fio. 3«.
Here the line y = 0 is a horizontal asymptote by Art. 92. To find
the oblique asymptotes, put y = mx ■\- h.
93-94.] ASYMPTOTES 149
Then (mx + b) (a^ + x^) = a^ (a - x),
i.e., mx^ + bx^ + (ma^ +a^)x + {aPh - a^) = 0 ;
hence m = 0, & = 0, for an asymptote.
Thus the only asymptote is the line y = 0 already found.
94. Number of asymptotes. The illustrations of the last
article show that if all the terms be present in the general
equation of an nth. degree curve, then the equation for
determining m is of the nth degree and there are accord-
ingly n values of m^ real or imaginary. The equation for
finding h is usually of the first degree, but for certain
curves one or more values of m may cause the coefficient
of a^ and a;""^ both to vanish, irrespective of h. In such
cases any line whose equation is of the form y = m^x + c
will have two points at infinity on the curve independent
of c; .but by equating the coefficient of a;""^ to zero, two
values of h can be found such that the resulting lines
have three points at infinity in common with the curve.
These two lines are parallel; and it will be seen that in
each case in which this happens the equation defining m has
a double root, so that the total number of asymptotes is
not increased. Hence the total number of asymptotes, real
and imaginary, is in general equal to the degree of the
equation of the curve.
This number must be reduced whenever a curve has a
parabolic branch.
Since the imaginary values of m occur in pairs, it is evi-
dent that a curve of odd degree has an odd number of real
asymptotes ; and that a curve of even degree has either no
real asymptotes or an even number. Thus, a cubic curve
has either one real asymptote or three ; a conic has either
two real asymptotes or none.
150 DIFFERENTIAL CALCULUS [Ch. XII.
95. Method of expansion. Explicit functions. Although
the two foregoing methods are in all cases sufficient to find
the asymptotes of algebraic curves, yet in certain special
cases the oblique asymptotes are most conveniently found
by the method of expansion in descending powers. It is
based on the principle that a straight line will be an asymp-
tote to a curve when the difference between the ordinates
of the curve and of the line, corresponding to a common
abscissa, approaches zero as the abscissa becomes infinite.
It will appear from the process of applying this principle
that a line answering the condition just stated will also
satisfy the original definition of an asymptote.
The principal value of the method of expansion is that
it exhibits the manner in which each infinite branch ap-
proaches its asymptote.
Ex. Find the asymptotes of the curve
a:— 3
TT n \ X/\ Xl
Here y^ = :r^^ ,
Hence the oblique asymptotes are y = ±{x — 1) (Fig. 37).
The sign of the next term shows that when z = + x), tlie curve is above
the first asymptote and below the second; and vice versa when a; == — oo.
95.]
ASYMPTOTES
151
The same method may be applied to cases in which x is an explicit
function of y.
This method can also be extended so as to apply to curves defined by
an implicit equation, f(x, y) = 0. [See McMahon and Snyder's " Differ-
ential Calculus," p. 234.]
Fig. 37.
EXERCISES ON CHAPTER XII
Find the asymptotes of each of the following curves :
1. y(a^ - a;2) = b(2 x -\- c). ■' 7. (x + a)f = (y + h)x\
8. x'
x^+ X + y.
\f
2 2^q^(a:-a)(a:-3a).
■ ^ x'^-2ax
3. a:Y ^ a\x'^ - y^).
^ 4. y = a+ .
(X - C)2
/ 5. / = x%a - x).
^ 6. y\x-l) = x^.
15. x^ + 2x^y - xy^-2y^ + 4:y^ + 2xy + y=l.
9. xy^ + x^y = a^.
10. 2/(^2 + 3 a2) = a;8.
A 11. a:3 - 3 aar?/ + ^8 _ q.
12. x3 + ?/8 = aS.
13. x4 - a; V + a'^^^ + 6* = 0.
CHAPTER XIII
DIRECTION OF BENDING. POINTS OF INFLEXION
96. Concavity upward and downward. A curve is said to
be concave downward in the vicinity of a point P when,
for a finite distance on each side of P, the curve is situated
below the tangent drawn at that point, as in the arcs -42),
FH. It is concave upward when the curve lies above the
tangent, as in the arcs DF^ HK.
By drawing successive tangents to the curve, as in the
figure, it is easily seen that if the point of contact advances
to the right, the tangent swings in the positive direction of
rotation when the concavity is upward, and in the negative
direction when the concavity is downward. Hence upward
concavity may be called a positive bending of the curve, and
downward concavity, negative bending.
A point at which the direction of bending changes con-
tinuously from positive to negative, or vice versa, as at F oi
102
Ch. XIII. 96-97.] DIRECTION OF BENDING 163
at D, is called a point of inflexion^ and the tangent at such
a point is called a stationary tangent.
The points of the curve that are situated just before and
just after the point of inflexion are thus on opposite sides of
the stationary tangent, and hence the tangent crosses the
curve, as at i>, #, H.
97. Algebraic test for positive and negative bending. Let
the inclination of the tangent line, measured from the right-
hand end of the a;-axis toward the forward (right-hand) end
of the tangent, be denoted by </>. Then <f> is an increasing
or decreasing function of the abscissa according as the bend-
ing is positive or negative ; for instance, in the arc AD^ the
angle <f> diminishes from + — through zero to — — ; in the
arc i>jP, <f) increases from — -j through zero to — ; in the arc
FH^ (f) decreases from -f — through zero to — — J ^^^ i^ *^®
arc HK^ </> increases from — — through zero to + — •
2i 4
At a point of inflexion </> has evidently a turning value
which is a maximum or minimum, according as the concavity
changes from upward to downward, or conversely.
Thus in Fig. 38, <^ is a maximum at JP, and a minimum at
D and at S,
Instead of recording the variation of the angle </>, it is
generally convenient to consider the variation of the slope
tan<^, w^hich is easily expressed as a function of x by the
equation
tan <f) = -^•
^ ax
Since tan <^ is always an increasing function of <^, it follows
that the slope function -=^ is an increasing or a decreasing
ax
154 DIFFERENTIAL CALCULUS [Ch. XIII.
function of a;, according as the concavity is upward or down-
ward, and hence that its a;-derivative is positive or negative.
Thus the bending of the curve is in l^e positive or nega-
tive direction of rotation, according as the function -t4 is
positive or negative.
dii
At a point of inflexion the slope -^ is a maximum or
minimum, and therefore its derivative -t4 changes sign from
positive to negative or from negative to positive. This
latter condition is evidently both necessary and sufficient in
order that the point (x^ y) may be a point of inflexion on
the given curve.
Hence, the coordinates of the points of inflexion on the
curve y^fix)
may be found by solving the equations
and then testing whether f'Qx) changes its sign as x passes
through the critical values thus obtained. To any critical
value a that satisfies the test, corresponds the point of
inflexion (a^f(a)').
Ex. 1. For the curve
find the points of inflexion, and show the mode of variation of the slope
and of the ordinate.
Here i = *^(^"-l)'
g = 4(3.-l).
hence the critical values for inflexions are ar = ± — . It will be seen
1 ^
that as a; increases through — =-, the second derivative changes sign from
\/3
positive to negative, hence there is an inflexion at which the concavity
changes from upward to downward. Similarly, at x = + — ^ the con-
V3
97.]
DIRECTION OF BENDING
155
X
y
dy
dx
— CO
+ 00
— c»
+
_ 2
+ 25
-24
+
-1
0
0
+
1
V3
-1
8
3V3
0
0
1
0
—
+ 1
V3
-t
8
3\/3
0
1
0
0
+
+ CO
+ 00
+ QO
+
cavity changes from downward to upward. The following numerical
table will help to show the mode of variation of the ordinate and of the
slope, and the direction of bending.
As X increases from — oo to
V3
the bending is positive, and the slope
continually increases from -co through
zero to a maximum value — — , which
3V3
is the slope of the stationary tangent
drawn at the point ( , - ) •
^ V V3 9/
As X continues to increase from
to H 3 , the bending is nega-
V3 V3
tive, and the slope decreases from
Q
H r: through zero to a minimum
3V3
value ^^, which is the slope of the stationary tangent at f + -— , - J-
Finally, as x increases from + — to + oo, the bending is positive
V3
and the slope increases from the value
Q
through zero to + co.
3V3
The values a: = - 1, 0, +1, at
which the slope passes through zero,
correspond to turning values of the
ordinate.
Ex. 2. Examine for inflexions the fiq. 39.
curve ar + 4 = (y - 2)\
In this case
2^ = 2+(x + 4)^,
Fig. 40.
^ is positive, and when a:>- 4, ^ is negative.
dy
Hence, at the point (— 4, 2), ~
d^v
and -r4 are infinite. When j;<— 4,
dx^
dh,
156
DIFFERENTIAL CALCULUS
[Ch. XIII.
Thus there is a point of inflexion at ( — 4, 2), at which the slope is
infinite, and the bending changes from the positive to the negative
direction.
Ex. 3. Consider the curve y=x*.
dx dx^
Fig. 41.
At (0, 0), ^ is zero, but the
curve has no inflexion, for — ^ never
dx^
changes sign (Fig. 41).
98. Analytical derivation of the test for the direction of
bending. Let the equation of a curve be i/ =/(2;), and let
P, ^j, ^j), be a point upon it. Then the equation of the
tangent at P is
Suppose that when x changes from x^ to x^ + ^, the ordi-
nate of the tangent change from
^j to y', and that of the curve
from 2/i to y'' ; then it is pro-
posed to determine the sign of
the difference of ordinates y — «/'
corresponding to the same ab-
scissa x^ + h.
By Taylor's theorem,
and from the above equation of the tangent,
Hence / = yi + ¥'(^i) = /(^i)+ ¥'(^i)'
and it follows that
y"-y = f/"(^i)+--
Fig. 42.
P
97-99.]
DIRECTION OF BENDING
167
When h is made sufficiently small, /''(a:j)+ ••• will have
the same sign as /'^(^i); but the factor h^ is always positive,
hence when f(x-^ is positive, y" — y' is positive, and thus
the curve is above the tangent at both sides of the point of
contact, that is, the concavity is upward. Similarly, when
f"{x{) is negative, the concavity is downward.
This agrees with the former result.
99. Concavity and convexity towards the axis. A curve
is said to be convex or concave toward a line, in the vicinity
of a given point on the curve, according as the tangent at
the point does or does not lie between the curve and the
line, for a finite distance on each side of the point of contact.
Fig. 43 a. Fig. 43 6.
First, let the curve be convex toward the a^axis, as in the
left-hand figure. Then if y is positive, the bending is positive
and —^■ is positive ; but if y is negative, the bending is neg-
ative and — - is negative. Hence in either case the product
y-zTT, is positive.
Next, let the curve be concave toward the a:-axis, as in
the right-hand figure. Then if y is positive, the bending is
negative and -^ is negative ; but if y is negative, the bend-
ing is positive and -j^ is positive. Thus in either case the
product y-^ is negative. Hence:
158 DIFFERENTIAL CALCULUS [Ch. XIII. 99.
In the vicinity of a given point (x^ y) the curve is convex or
concave to the x-axis, according as the product y — ^ is positive
. . dor
or negative.
EXERCISES ON CHAPTER XIII
1. Examine the curve ?/ = 2 — 3(a; — 2)5 for points of inflexion.
2. Show that the curve a^y = x(cfi — a:^) has a point of inflexion at
the origin.
3. Find the points of inflexion on the curve y = ; — •
TO
4. In the curve ay = i!^, prove that the origin is a point of inflexion
if m and n are positive odd integers.
5. Show that the curve y =csin - has an infinite number of points of
inflexion lying on a straight line.
6. Show that the curve y{x^ ■\- 0."^) = x has three points of inflexion
lying on a straight line ; find the equation of the line.
7. If 3^2 =f(x) be the equation of a curve, prove that the abscissas of
its points of inflexion satisfy the equation
[f'(x)y = 2f(x)^f"(xy
8. Draw the part of the curve a^y = ^- ax^ + 2 a* near its point of
3
inflexion, and find the equation of the stationary tangent.
CHAPTER XIV
CONTACT AND CURVATURE
100. Order of contact. The points of intersection of the
two curves
are found by making the two equations simultaneous ; that
is, by finding those values of x for which
Suppose ic = « is one value that satisfies this equation.
Then the point x — a^ q^ = (j) (^a} = yjr (^a) is common to the
curves.
If, moreover, the two curves have the same tangent a^
this point, they are said to touch each other, or to have
contact of the first order with each other. The values of y
and of -^ are thus the same for both curves at the point in
question, which requires that
<^ (a) = i/r (a), •
If, in addition, the value of -t4 be the same for each
curve at the point, then
<^"(«) = f"(a),
and the curves are said to have a contact of the second
order with each other.
If <^(a) = i/r(a), and all the derivatives up to the nth.
order inclusive be equal to each other, the curves are said
159
160 DIFFERENTIAL CALCULUS [Ch. XIV.
to have contact of the nth order. This is seen to require
n-^l conditions. Hence if the equation of the curve
y = <i>(x) be given, and if the equation of a second curve
be written in the form y = '>^(x)', in which "^^Qc) proceeds
in powers of x with undetermined coefficients, then n-\-l
of these coefficients could be determined by requiring the
second curve to have contact of the nth. order with the
given curve at a given point.
101. Number of conditions implied by contact. A straight
line has two arbitrary constants, which can be determined
by two conditions ; accordingly a straight line can be drawn
which touches a given curve at any specified point. For if
the equation of a line be written i/=mx-\-b, then
hence, through any arbitrary point x = a on a given curve
«/ = (^(ic), a line can be drawn which has contact of the first
order with the curve, but which has not in general contact
of the second order ; for the two conditions for first-order
contact are
ma -\-b = <l> (a),
m = <^'(a),
which are just sufficient to determine m and b.
In general no line can be drawn having contact of an
order higher than the first with a given curve ; but there
are certain points at which this can be done. For example,
the additional condition for second-order contact is 0 = <f>"(a)^
which is satisfied when the point a: = a is a point of inflexion
on the given curve y = <t>(x)' Thus the tangent at a point
of inflexion on a curve has contact of the second order
with the curve.
100-102.] CONTACT AND CURVATURE 161
The equation of a circle has three independent constants.
It is therefore possible to determine a circle having contact
of the second order with a given curve at any assigned
point.
The equation of a parabola has four constants, hence a
parabola can be found which has contact of the third order
with the given curve at any point.
The general equation of a central conic has five inde-
pendent constants, hence a conic can be found which has
contact of the fourth order with a given curve at any
specified point.
As in the case of the tangent line, special points may be
found for which these curves have contact of higher order.
102. Contact of odd and of even order.
Theorem. At a point where two curves have contact of
an odd order they do not cross each other ; but they do
cross where they have contact of an even order.
For, let the curves y = (f>(x)^ yz='\^(x) have contact of
the nth order at the point whose abscissa is a ; and let ^j,
^2 be the ordinates of these curves at the point whose
abscissa is a + A. Then
and by Taylor's theorem
2,1 = .^(a) + f (a) . A +^!^ . ^2 +...
*^-^"+7;Sm^-«-^--
-^•^"-(^•^-'(«>--
162 DIFFERENTIAL CALCULUS [Ch. XIV.
Since by hypothesis the two curves have contact of the
wth order at the point whose abscissa is a, hence
and y^-y^= (^ + 1)!^'^°^'^"^ + - " ^'^''^"'^ — '^ \
but this expression, when h is sufficiently diminished, has
the same sign as
Hence, if n be odd, y^ — y^ does not change sign when h is
changed into — A, and thus the two curves do not cross each
other at the common point. On the other hand, if ti be
even, y-^ — y^ changes sign with h ; and therefore when the
contact is of even order the curves cross each other at their
common point.
For example, the tangent line usually lies entirely on one
side of the curve, but at a point of inflexion the tangent
crosses the curve.
Again, the circle of second-order contact crosses the
curve except at the special points noted later, in which
the circle has contact of the third order.
EXERCISES
1. Find the order of contact of the curves r/A'
4iy = x^ and y = x — 1. f jy*^
2. Find the order of contact of the curves /» 'vtVr v'l^^t^^
x^f and x + y + 1 = ^,fifX^ fi^^^
3. Find the order of contact of the curves
43^ = ^:2-4 and x^-2y = S-i/^
4. Determine the parabola having its axis parallel to the y-axis,
which has the closest possible contact with the curve ah/ = x^ at the
point (a, a).
102-104.] CONTACT AND CURVATUEE 163
5. Determine a straight line which has contact of the second order
with the curve
y = a:8-3a;2- 9a; + 9.
j6. Find the order of contact of
y = log (x - 1) and x^ - Qx -{■ 2y -{■ S = 0
at the point (2, 0).
7. What must be the value of a in order that the curves
y =: X + 1 + a(^x — ly and xy = 'dx — 1
may h^pvB contact of the second order?
103. Circle of curvature. The circle that has contact of
the closest order with a given curve at a specified point is
called the osculating circle or circle of curvature of the
curve at the given point. The radius of this circle is called
the radius of curvature, and its center is called the center
of curvature at the assigned point.
104. Length of radius of curvature; coordinates of center
of curvature. Let the equation of a circle be
(X-«)2+(r-;S)2 = i2^ (1)
in which R is the radius, and ct, /3 are the coordinates of the
center, the current coordinates being denoted by X, T to
distinguish them from the coordinates of a point on the
given curve. —
It is required to determine i2, «, yS, so that this circle
may have contact of the second order with the given curve
at the point (a;, y).
From (1), by successive differentiation, it follows that
(2)
164 DIFFERENTIAL CALCULUS [Ch. XIV.
If the circle (1) has contact of the second order at the
point (a;, ^) with the given curve, then when X=x it is
necessary that
Y=y. 1
dX dx dX^ d^ J
Substituting these expressions in (2),
(a:-«) + (2/-^)g = 0,
(4)
whence
^ \dx) d:^ \dx) J ..
and finally, by substitution in (1),
v<m
t^a;2
105. Direction of radius of curvature. Since, at any point
P on the given curve, tlie value of -^ is the same for the
dx
curve and the osculating circle for that point, it follows that
they have the same tangent and normal at P, and hence
that the radius of curvature coincides with the normal.
Again, since the value of -^ is the same for both curves at
dar
P, it follows from Art. 97, that they liave tlie same direction
104-105.]
CONTACT AND CURVATURE
165
of bending at that point, and hence that the center of
curvature lies on the concave side of the given curve
(Fig. 44).
It follows from this fact and Art. 102 that the osculating
circle is the limiting position of a circle passing through
three points on the curve when these points move into
coincidence.
The radius of curvature is usually regarded as positive or
negative according as the bending of the curve is positive
Fig. 44.
Fig. 45.
or negative (Art* 97), that is, according as the value of
dx^
is positive or negative ; hence, in the expression for H, the
radical in the numerator is always to be given the positive
sign. The sign of It changes as the point P passes through
a point of inflexion on the given curve (Fig. 45). It is
evident from the figure that in this case M passes through
an infinite value ; for the circle through the points iV, P, Q
approaches coincidence with the inflexional tangent when N
and Q approach coincidence with P, and the center of this
circle at the same time passes to infinity.
166 DIFFERENTIAL CALCULUS [Ch. XIV.
106. Total curvature of a given arc; average curvature.
The total curvature of an arc PQ (Fig. 46) in which the
bending is continuous and in one direc-
tion, is the angle through which the
tangent swings as the point of contact
moves from the initial point P to the
terminal point Q\ or, in other words,
it is the angle between the tangents at
P and §, measured from the former to
the latter. Thus the total curvature of a given arc is posi-
tive or negative according as the bending is in the positive
or negative direction of rotation.
The total curvature of an arc divided by the length of the
arc is called the average curvature of the arc. Thus, if the
length of the arc PQ be As centimeters, and if its total
curvature be A<^ radians, then its average curvature is — ^
radians per centimeter.
107. Measure of curvature at a given point. The mea^sure
of the curvature of a given curve at a given point P is the
limit which the average curvature of the arc PQ approaches
when the point Q approaches coincidence with P,
Since the average curvature of the arc P^ is — ^, the
measure of the curvature at the point P is
lim A^ d^
'^ = A. = o;^-^,'
and may be regarded as the rate of deflection of the arc from
the tangent estimated per unit of length; or again, as the
ratio of the angular velocity of the tangent to the linear
velocity of the point of contact.
106-108.]
CONTACT AND CURVATURE 167
To expri
ess K in terms of x^ y^ and their derivatives, observe
that
ax
Whence
dx
and
as d8\ dx)
. -£('•»-'£) -f
da^ 1
\dxj dx
therefore « = ^ = — -. [Art. 83
108. Curvature of osculating circle. A curve and its oscu-
lating circle at P have the same measure of curvature at
that paint.
For, let K^ k' be their respective measures of curvature at
the point of contact (a:, y). Then from Art. 107,
dx^
K =
l'-(l)T
But this is the reciprocal of the expression for the radius
of curvature (Eq. (6), p. 164) ; hence
1
168 DIFFERENTIAL CALCULUS [Ch. XIV.
That is: the measure of curvature k at a point P is the
reciprocal of the radius of curvature R for that point. Since
a curve and its osculating circle have the same radius at
their point of contact, it follows from this result that the
measure of curvature is also the same for both.
It is on account of this property that the osculating circle
is called the circle of curvature. This is sometimes used as
the defining property of the circle of curvature. The radius
of curvature at P would then be defined as the radius of the
circle whose measure of curvature is the same as that of the
given curve at the point P. Its value, as found from Art.
106 and Art. 107, accords with that given in Art. 104.
EXERCISES
1. Find the radius of curvature of the curve y^ = 4cax at the origin.
2. Find the radius of curvature of the curve y^+3fi-^a(x'^-\-y^) = a^y
at the origin.
3. Find the radius of curvature of the curve a^y = hx^ + car^y at the
origin.
Find the radius of curvature for each of the following curveer:
4. xy = m^. Rectangular hyperbola.
X^ 7/2
^' ^-|2=1- Hyperbola.
6 a^-^y = x**. General parabola.
7. y/x ■\-y/y =\/a. Parabola.
V 8. x^ + y^ = al Hypocycloid.
9. y2 = ^ ^ . Cisaoid.
2a — X
10. y = ^ (€• + c"«). Catenary.
K^^^-^
r
108-109.
CONTACT AND CURVATURE
169
109. Direct derivation of the expressions for k and M in
polar coordinates. Using the notation of Art. 81,
</> = 6> + ^/r,
hence
"" c^s ds_
dd
'* O-S)
dO
(•-^)
[^m
But tan -^ = p — , y^r — tan-^
dp
je^
tlierefore, by differentiating as to 6 and reducing,
(dp\^ cPp
d^ \deJ Pd0^
which, substituted in (1), gives
p'-Pw + \Te)
(1)
[Art. 87.
^-©7
Since «: = — , it follows that
R
E =
['■-s:
de»^\d0)
K =
170 DIFFERENTIAL CALCULUS [Ch. XIV.
When M == - is taken as dependent variable, the expres-
sion for K assumes the simpler form
Since at a point of inflexion k vanishes and changes sign,
hence the condition for a point of inflexion, expressed in
in
polar coordinates, is that u 4- -j^ shall vanish and change
sign.
EXERCISES
Find the radius of curvature for each of the following curves :
1. p = a^. 3. /o = 2acos^-a. 5. p^cos2e=o^,
2. p? = aaco82^. 4. pcos2^^ = a. 6. p = 2a(l-co8^.
7. p6 = a.
'\\J\^^^ E VOLUTES AND INVOLUTES
oi 110. Definition of an evolute. When the point P moves
along the given curve, the center of curvature Q describes
another curve which is called the evolute of the first.
Let f(x^ y)= 0 be the equation of the given curve. Then
the equation of the locus described by the point C is found
by eliminating x and y from the three equations
dx
X — a =
d^
1 +
y-P =
<i]L
109-110.]
CONTACT AND CURVATURE
171
and thus obtaining a relation between a, yS, the coordinates
of the center of curvature.
No general process of elimination can be given; the
method to be adopted depends upon the form of the given
equation f(x^ y) = 0.
Ex. 1. Find the evolute of the parabola y^ = 4px.
Since y = 2p^x^, -^=p^x~^, T^ = - t^P^^"^*
ax dx^ 2
hence x— a = —p^x~^ (1 +px~^) 2p~ix^ = — 2{x + p),
and y - )8 = (1 + px-^) 2p~^x^ = 2 (p~^x^ + /> W) ;
therefore a = 2p-\-3Xf P = - 2p~^xk
Fig. 47.
The result of eliminating x between the last two equations is
M^-^py=Kp^m
t.e., 4(a-2py = 2^pl3^y
172 DIFFEEENTTAL CALCULUS [Ch. XIV.
which is the equation of the evolute of the parabola, a, /3 being the
current coordinates.
Ex. 2. Find the evolute of the ellipse
Here £+iL.^ = 0, f^^_^,
a^ b^ dx dx a'^y
y-x
dy
d^y fe2 ^ -dx -by b^xH -6^2 2. ;2 2N -^*
dx^~ a^ " ~ -o o\./ I o. I— .s.9\"-a /— •> a>
-by , b^xH -h\ .... 2, -6*
a^y^V a^y j ahj^^ ^ ' a^y^
whence
^ ^ a%* \ b^ ^ a'r\ M ^ 2)-'r
Therefore ^ B = ^—r^y^ (2)
Similarly, a = ^—J^x^- (3)
Eliminating x, y between (1), (2), (3), the equation of the locus
described by («, p) is
(aa)t + (bjS)^ = (a2 - J^)!. (Fig. 52)
111. Properties of the evolute. The evolute has two im-
portant properties that will now be established.
I. The normal to the curve is tangent to the evolute. The
relations connecting the coordinates (a, yS) of the center of
curvature with the coordinates (a:, ?/) of the corresponding
point on the curve are, by Art. 104,
rr-« + (y-/3)g = 0, (1)
By differentiating (1) as to x, consideriucf «, /Q, y as
functions of rr.
110-111.] CONTACT AND CURVATURE 173
Subtracting (3) from (2),
*! + ^^ = 0, (4)
ax ax ax
whence di^_dx^
da dy
But -y- is the slope of the tangent to the e volute at (a, y8),
dir
and — ;t- is the slope of the normal to the given curve at
(a;, y). Hence these lines have the
same slope; but they pass through
the same point (a, /8), therefore they
are coincident.
II. The difference "between two
radii of curvature of the given curve^ \ p ^
which touch the evolute at the points
Cy, C^ (^Fig. 4^), is equal to the arc O^Q^ of the evolute.
Since B is the distance between points (x^ «/), (a, yS),
hence
(x-ay^+<j,-py^=ii?. (6)
When the point (x^ y) moves along the given curve, the
point (a, /3) moves along the evolute, and thus a, y8, i2, y
are all functions of x.
Differentiation of (5) as to x gives
(.-«)(i-|)-^c.-.)(|-f)=«f; (6)
hence, subtracting (6) from (1),
174 DIFFERENTIAL CALCULUS [Ch. XIV.
(8)
Again, from (1) and (4),
da
dx
• =
d^
dx
= ='
Hence, each of these fractions is equal to
w^
dg\^ da
^"^ =±^, (9)
V(a;-«)2+(2/-/3)2 B
in which a is the arc of the evolute. (Compare Aft. 64.)
Next, multiplying numerator and denominator of the first
member of (8) by a; — a, and those of the second member by
«/ — yS, and combining new numerators and denominators, it
follows that each of the fractions in (8) is equal to
(^_a)2+(^_^)2 '
which equals —
j.dR
dx
-^ by (7) and (5).
By combining
: with (9),
,
da_^dR^
dx ■*" dx
that is,
ax
Therefore
a ±R = constant,
(10)
wherein <r is measured from a fixed point A on the evolute.
Now, let (7^, C2 be the centers of curvature for the points
111.]
CONTACT AND CURVATURE
175
^1' -^2 ^2 "~ ^2 '
Pj, Pg o^ t^^ given curve ; let Pj (7^
let the arcs -4 6\, ^ 63 be denoted by o-^, o-g. Then
0-1 ± i^i = 0-2 ± E^, by (10);
that is, o-j — o-g = ± (i?2 — -^1)5
hence, arc C-^ C^= R^ — H^.
Thus, in Fig. 49,
PjCj + CjCg = P2^2'
P9^'3!+ ^9^a = Pa^ai ^tc.
and
(11)
2^2
2^3
Fig. 49.
Hence, if a thread be wrapped
around the evolute, and then be
unwound, the free end of it can be ^^
made to trace out the original curve.
From this property the locus of the
centers of curvature of a given curve
is called the evolute of that curve, and the latter is called
the involute of the former.
When the string is unwound, each point of it describes a
different involute ; hence, any curve has an infinite number
of involutes, but only one evolute.
Any two of these involutes intercept a constant distance
on their common normal, and are called parallel curves on
account of this property.
Ex. Find the length of that part of the evolute of the parabola which
lies inside the curve.
From Fig. 47 the required length is twice the difference between the
tangents C3P3 and PqC^, both of which are normals to the parabola.
To find the coordinates of the point P^, write the equation of the tan-
gent to the evolute at C3, and find the other point at which it intersects
the parabola.
The coordinates of Cg, the point of intersection of the two curves, are
(8 JO ; 4:j9 V2), and the equation of the tangent at Cg is
176 DIFFERENTIAL CALCULUS [Ch XIV.
This tangent intersects the parabola at the point (2p, — 2V2jo),
which is Pg.
The value of the radius of curvature is -v^"^_^) , hence PqCq = 2p,
PgCg = 6\/3jt?, hence the arc C^C^ is 2jt)(3V3-l), and the required
length of the evolute is therefore 4:p (3 V3 — 1).
EXERCISES
Find the coordinates of the center of curvature for each of the follow-
ing curves :
'^1. x^-\-y^=a^. 3. y^=a^x.
2. x=alog^±:^^iHj!_V52T:p. 4. y = |(J+r-).
Find the equations of the evolutes of the following curves :
/" 5. xy=a^. / 6. a^y'^ -1^"^ = - a%\ /7. a:t + ?/l = ai
8. Show that the curvature of an ellipse is a minimum at the end
of the minor axis, and that the osculating circle at this point has con-
tact of the third order with the curve.
Fio. fiO.
This circle of curvature must be entirely outside the ellipse (Fig. 50).
For, consider two points Pj, P,, one on each side of -C, the end of the
111.]
CONTACT AND CURVATURE
177
minor axis. At these points the curvature is greater than at J5, hence
these points must be farther from the tangent at B than the circle of
curvature, which has everywhere the same curvature as at J5.
9. Similarly, show that the curvature at Ay the end of the major
axis, is a maximum, and that the circle of curvature at A lies entirely
within the ellipse (Fig. 50).
10. Show how to sketch the circle of curvature for points between A
and B. The circle of curvature for points between A and B has three
coincident points in common with the ellipse (Art. 104), hence the circle
crosses the curve (Art. 102). Let K, P, L be three points on the arc,
such that K is nearest A and L nearest B. The center of curvature for
Fig. 51.
P lies on the normal to P, and on the concave side of the curve. The
circle crosses at P, lying outside of the ellipse at K (on the side towards
A^, and inside the ellipse at L ; for the bending of the ellipse increases
from 5 to P and from P to K, while the bending (curvature) of the
osculating circle remains constant (Fig. 61). *
11. Two centers of curvature lie on every normal. Prove geometfi- *^
cally that the normals to the curve are tangents to the evolute.^ , ' "^
12. Show that the entire length of the evolute of the ellipse is
4(~ ). [From equation (11) above, take i^j, P, as the radii of
curvature at the extremities of the major and minor axes.]
178
DIFFERENTIAL CALCULUS
[Ch. XIV. 111.
13. If E be the center of curvature at the vertex A (Fig. 52), prove
that CE = ae% in which e
is the eccentricity of the
ellipse ; and hence that CD,
CA, CF, CE form a geo-
metric series whose com-
mon ratio is e. Show also
that DA, AF, FE form a
similar series.
14. If H be the center of
curvature at B, show that
the point // is without or
within the ellipse, according
as a > or < bV2, or accord-
ing as e^ > or < ^.
15. Show by inspection
of the figure that four real
normals can be drawn to
the ellipse from any point
within the evolute.
CHAPTER XV
SINGULAR POINTS
112. Definition of a singular point. If the equation
f(x^ ^) = 0 be represented by a curve, the derivative -5^,
CLX
when it has a determinate value, measures the slope of the
tangent at the point (a;, ?/). There may be certain points
on the curve, however, at which the expression for the
derivative assumes an illusory or indeterminate form ; and,
in consequence, the slope of the tangent at such a point can-
not be directly determined by the method of Art. 10. Such
values of x^ y are called singular values^ and the corre-
sponding points on the curve are called singular points.
113. Determination of singular points of algebraic curves.
When the equation of the curve is rationalized and cleared
of fractions, let it take the form f(x^ ^) = 0.
This gives, by differentiation with regard to a;, as in
Art. 71, , , ,
• ^+^^ = 0
dx dy dx '
dl
whence ^=-^. (1)
dy
In order that -^ may become illusory, it is therefore
necessary that §^= ^' F=^- (^)
179
180 DIFFERENTIAL CALCULUS [Ch. XV.
Thus to determine whether a given curve f(x^ ^) = ^
df df
has singular points, put -^ and -^ each equal to zero and
solve these equations for x and i/.
If any pair of values of x and ^, so found, satisfy the
equation /(a;, ^) = 0, the point determined by them is a
singular point on the curve.
To determine the appearance of the curve in the vicinity
of a singular point (x^, y^), evaluate the indeterminate form
di/ _ dx _0
^"""^■"0'
by finding the limit approached continuously by the slope
of the tangent when x^x^^ y = yv
Hence dy^_dx\dxj
dx i_(^
dx\dy)
0 ^ ^ dy
dx^ dxdy dx r . .„ „
"- ay ^fdy [Arts. 49, 72.
dx dy dy^ dx
at the point (a^j, y{).
This equation cleared of fractions gives, to determine the
slope at (xy, ^i), the quadratic
This quadratic equation has in general two roots. The
only exceptions occur when simultaneously, at the point in
question,
Bt? ^' dxdy ' dy^ ' ^^
113-114.] SINGULAR POINTS 181
in which case -:r- is still indeterminate in form, and must be
ax
evaluated as before. The result of the next evaluation is a
cubic in -^, which gives three values to the slope, unless all
the third partial derivatives vanish simultaneously at the
singular point.
The geometric interpretation of the two roots of equation
(3) will now be given, and similar principles will apply
when the quadratic is replaced by an equation of higher
degree.
The two roots of (3) are real and distinct, real and coin-
cident, or imaginary, according as
(:
dx By) dx^ 5^2
is positive, zero, or negative. These three cases will be
considered separately.
114. Multiple points. First let H be positive. Then at
df df
the point (x, y) for which ^ = 0, ^ = 0, there are two values
ijx oy
of the slope, and hence two distinct singular tangents. It
follows from this that the curve goes through the point in
two directions, or, in other words, two branches of the curve
cross at this point. Such a point is called a real double
point of the curve, or simply a node. The conditions, then,
to be satisfied at a node (a^j, y-^ are
and H(x-^, y{) > 0.
Ex. Examine for singular points the curve
3 x^ - xy - 2 y^ + x^ - 8y^ = 0,
182
DIFFERENTIAL CALCULUS
[Ch. XV.
Here |f = 6a: - v + 3x2, ^= - a: - 4 v - 24 v^.
dx oy ^ ^
The values x = 0, ?/ = 0 will satisfy these three equations, hence
(0, 0) is a singular point.
Since
1^=6 + 6a: = 6 at (0,0),
bxby *
^ = _4-48y=-4at (0,0),
FiQ. 53.
hence the equation determining the slope is, from (3),
-(ir-(i)-«-.
of which the roots are 1 and — f . It follows that (0, 0) is a double
point at which the tangents have the slopes 1, — |.
115. Cusps. Next let 5"= 0. The two tangents are then
coincident, and there are two cases to consider. If the
curve recedes from the tangent in both directions from the
point of tangency, the singular point is called a tacnode.
Two distinct branches of the curve touch each other at
this point. (See Fig. 54.)
If both branches of the curve recede from the tangent in
only one direction from the point of tangency, the point is
called a cusp.
114-115.]
SINGULAR POINTS
183
Here again there are two cases to be distinguished. If
the branches recede from the point on opposite sides of the
double tangent, the cusp is said to be of the first kind ; if
they recede on the same side, it is called a cusp of the second
kind.
The method of investigation will be illustrated by a few
examples.
Ex. 1. f(x, y) = aY - «^^* + a:« = 0.
dx dy
The point (0, 0) will satisfy /(x, y)= 0, ^ = 0, ^ = 0 ; hence it is a
singular point. Proceeding to the second derivatives,
- 12 a%2 + 30 a:* = 0 at (0, 0),
^"f =0
dxdy *
dy''
The two values of -r- are therefore coincident, and each equal to zero.
dx
From the form of the equation, the curve is evidently symmetrical with
regard to both axes; hence the point (0, 0) is a tacnode.
No part of the curve can be at a greater distance from the y-axis than
± a, at which points -^ is infinite. The maximum value of y corre-
dx
sponds to x = ±aV\, Between a; = 0, ar = aV| there is a point of
inflexion (Fig. 54).
Ex.2. /(a:,y)=3^2-.x8=0;
|f=-3:r^, f =2^.
dx dy
ay
dx^
Hence the point (0, 0) is a singu-
lar point.
Further, If,:
ay
dxby
6a:=0at(0,0);
0- ^-2
Fig. 54.
184
DIFFERENTIAL CALCULUS
[Ch. XV.
Therefore the two roots of the quadratic equation defining -^ are both
dx
equal to zero. So far, this case is exactly like the last one, but here no
part of the curve lies to the left of the axis y. On the right side, the
curve is symmetric with regard to the a:-axis. As x increases, y increases;
there are no maxima nor minima, and no inflexions (Fig. 55).
Ex.3.
/(x, y)z=x^- 2ax'^y - axy^ + aV ^ q.
The point (0, 0) is a singular point, and the roots of the quadratic defining
dx
are both equal to zero.
Let a be positive. Solving the equation for y,
When X is negative, y is imaginary ; when a: = 0, y = 0 ; when x is
positive, but less than a, y has two positive values, therefore two branches
Pig. 66.
Pro. 66.
are above the a:-axis. When ar = a, one branch becomes infinite, having
the asymptote x = a] the other branch has the ordinate \ a. The origin
is therefore a cusp of the second kind (Fig. 56).
116. Conjugate points. Lastly, let H be negative. In
this case there are no real tangents ; hence no points in the
immediate vicinity of the given point satisfy the equation of
the curve.
Such an isolated point is called a conjugate point.
115-116.]
SINGULAR POINTS
185
Ex. f(x, y) = ay^ — a;^ + hx^ = 0.
a singular
Here (0, 0) is a singular point of the
locus, and
dx
both roots being imaginary if a and b
have the same sign.
To show the form of the curve, solve
the given equation for y.
Then
=±x4
Fig. 57.
and hence, if a and b are positive, there
are no real points on the curve between x = 0 and x = b. Thus 0 is an
isolated point (Fig. 57).
These are the only singularities that algebraic curves can
have, although complicated combinations of them may ap-
pear. In each of the foregoing examples, the singular point
was (0, 0) ; but for any other point, the same reasoning will
apply.
Ex. f(x, 7/)= x^ -{- S y^ - U y^ - 4:x + 17 y - S = 0,
^=2x-4 ^:
dx ' dy
y2_2Qy + l\
At the point (2, 1), /(2, 1)= 0, %. = 0, ¥ = 0; hence (2, 1) is a
singular point. ^
dV _
^'^^ S = ^' ^y = '-^ W='''-''^ =-8 at (2,1).
Hence --p- — ±\\ and thus the equations of the two tangents at the
node (2, 1) are y - 1 = i(a: - 2), y - 1 = - K^ - 2).
When H is negative, the singular point is necessarily a
conjugate point, but the converse is not always true. A
singular point may be a conjugate point when 11=0.
[Compare Ex. 4 below.]
186 DIFFERENTIAL CALCULUS [Ch. XV. 116.
EXERCISES ON CHAPTER XV
Examine each of the following curves for multiple points and find the
equations of the tangents at each such point :
1. a2x2 = ftV + ^ V-
2.
^ 2a-x
3.
xt + yl =ai.
4.
^2(x2 _ a-2) = x\
5. y z=za + X -\- hx^ ± cx'i.
When a curve has two parallel asymptotes it is said to have a node at
infinity in the direction of the parallel asymptotes. Apply to'Ex. 6.
6. (x^-y^)2-^y^+y = 0. ■
7. a;4 - 2 a?/8 _ 3 a2^2 _ 2 a2a.2 ^ (j4 = 0.
8. y^ = x(x-\-ay.
9. a;8 - 3 axy -\-y^ = 0.
10. y^ = x*~{-x^.
11. Show that the curve y = x\ogx has a terminating point at the
origin.
CHAPTER XVI
ENVELOPES
117. Family of curves. The equation of a curve,
usually involves, besides the variables x and ^, certain coeffi-
cients that serve to fix the size, shape, and position of the
curve. The coefficients are called constants with reference
to the variables x and «/, but it has been seen in previous
chapters that they may take different values in different
problems, while the form of the equation is preserved. Let
a be one of these "constants." Then if a be given a series
of numerical values, and if the locus of the equation, corre-
sponding to each special value of a be traced, a series of
curves is obtained, all having the same general character,
but differing somewhat from each other in size, shape, or
position. A system of curves so obtained is called a family
of curves.
For example, if A, h be fixed, and p be arbitrary, the equa-
tion Qy — k')'^ = 2p(x— K) represents a family of parabolas,
each curve of which has the same vertex (7a, A;), and the
same axis y=h^ but a different latus rectum. Again, if k
be the arbitrary constant, this equation represents a family
of parabolas having parallel axes, the same latus rectum, and
having their vertices on the same line x = h.
The presence of an arbitrary constant a in the equation of
a curve is indicated in functional notation by writing the
187
188 DIFFERENTIAL CALCULUS [Ch. XVI.
equation in the form /(a;, y^ «) = 0. The quantity «, which
is constant for the same curve but different for different
curves, is called the parameter of the family. The equa-
tions of two neighboring curves are then written
f(x, y, a) = 0, f{x, y, a + h')= 0,
in which A is a small increment of a. These curves can be
brought as near to coincidence as desired by diminishing h.
118. Envelope of a family of curves. A point of inter-
section of two neighboring curves of the family tends toward
a limiting position as the curves approach coincidence. The
locus of such limiting points of intersection is called the
envelope of the family.
Let f(x,y,a)=0, /(x, y, a+ h)==0 (1)
be two curves of the family. By the theorem of mean value
(Art. 45)
f(x, y,a-hh^ = fCx, y, a)-\-h^Cx, y, a-^-OK), (2)
da
which, on account of equation (1), reduces to
Hence, it follows that in the limit, when A = 0,
is the equation of a curve passing through the limiting
points of intersection of the curve /(a:, ?/, a) = 0 with its
consecutive curve. This determines for any assigned value
of a a definite limiting point of intersection on the corre-
sponding member of the family. The locus of all such
117-119.] -EN VEL OPES 189
points is then to be obtained by eliminating the parameter
a between the equations
/(a;, y, «)= 0, Z(a;, «/, «)= 0.
da
The resulting equation is of the form F(x^ y) = 0, and
represents the fixed envelope of the family.
119. The envelope touches every curve of the family.
I. Geometrical 'proof. Let A, B^ Q be three consecutive
curves of the family ; let A^ B intersect in P ; B^ C inter-
sect in Q. When P, Q approach coincidence, PQ will be
the direction of the tangent to the envelope at P ; but since
P, Q are two points on B that approach coincidence, hence
P(> is also the direction of the tangent to B at P, and
accordingly B and the envelope have a common tangent at
P. Similarly for every curve of the family.
II. More rigorous analytical proof. Let — f(x^ y, a) = 0
da
be solved for a, in the form a= <t>(x, «/). Then the equation
of the envelope is
Equating the total rr-derivative to zero,
dx dy dx d<p\Sx by dx)
190 DIFFERENTIAL CALCULUS [Ch. XVI.
but -=^ = — = 0, hence the slope of the tangent to the
d(p da
envelope at the point (a;, y) is given by
dx by dx
But this equation defines the direction of the tangent to
the curve f(x^ y^ a) = 0 at the same point, and therefore a
limiting point of intersection on any member of the family
is a point of contact of this curve with the envelope.
Ex. Find the envelope of the family of lines
obtained by varying m.
Differentiate (1) as to w,
y = mx+|' . (1)
0 = --^- (2)
To eliminate m, multiply (2) by m and square; square (1) and sub-
tract the first from the second. The envelope is found to be the parabola
y^ = 4c px.
120. Envelope of normals of a given curve. The evolute
(Art. 110) was defined as the locus of the centers of curva-
ture. The center of curvature was shown to be the point of
intersection of consecutive normals (Art. Ill), whence by
Art. 118 the envelope of the normals is the evolute.
Ex. Find the envelope of the normals to the parabola y"^ = 4jt)a;.
The equation of the normal at (Xj, yy) is
or, eliminating x^ by means of the equation y^ = 4/>Xj,
i/-v, = ^-^ (1)
119-121.]
ENVELOPES
191
The envelope of this line, when y^ takes all values, is required
Differentiating as to y^
~ Sp^ 2p
Substituting this value for y^ in (1), the result^
27 py^ = i(x - 2 py,
is the equation of the required evolute. *•
121. Two parameters, one equation of condition. In many
cases a family of curves may have two parameters which are
connected by an equation. For instance, the equation of
the normal to a given curve contains two parameters x^, y-^
which are connected by the equation of the curve. In such
cases one parameter may be eliminated by means of the given
relation, and the^ other treated as before.
When the elimination is difficult to perform, both equa-
tions may be differentiated as to one of the parameters a,
regarding the other parameter yS as a function of a. This
dB
gives four equations from which a, y8 and -^ may be elim-
da
inated, the resulting equation being that of the desired
envelope.
Ex. 1. Find the envelope of the line
a b
the sum of its intercepts remaining constant.
The two equations are
X y ^
- + I = 1,
a 0
a+b = c.
192
DIFFERENTIAL CALCULUS
[Ch. XVI.
Differentiate both equations as to a ;
1 + ^ = 0.
da
Eliminate
da
Then — = ^^ which reduces to
^ I X y_
a b a b \ . , — , , —
- = J- = 7 = -; whence a — y/cx, b = Vcy.
a b a+b c'_ ^
Therefore Va; + Vy = Vc
is the equation of the desired envelope. [Compare Ex. p. 131.]
Ex. 2. Find the envelope of the family of coaxial ellipses having a
constant
Here
121.] ENVELOPES 193
For symmetry, regard a and b as functions of a single parameter t.
Then ^da+^db = 0,
bda •+ ac?6 = 0 ;
hence —- = ^ = --,
a=±a:V2, b=±yV2,
and the envelope is the pair of rectangular hyperbolas xy =±^ k^.
Note. A family of curves may have no envelope ; i.e., consecutive
curves may not intersect; e.g., the family of concentric circles x^ + y^=r^,
obtained by giving r all possible values.
If every curve of a family has a node, and the node has
different positions for different curves of the family, the
envelope will be composed of two (or more) curves, one of
which is the locus of the node.
Ex. Find the envelope of the system
/= (y-\y + x^-x^ = 0,
in which A is a varying parameter.
Here -^ = — 2(y — A,) = 0 ; by combining with /= 0 to eliminate X,
we obtain
a;2 = 0, X - 1 = 0, x + 1 = 0.
From Art. 114 it is seen that
x = 0, y = X
is a node on /; moreover, the various curves of the family are obtained
by moving any one of them parallel to the y-axis. The lines a: — 1 = 0,
a: + 1 = 0 form the proper envelope, and a: = 0 is the locus of the node.
EXERCISES ON CHAPTER XVI
1. Find the envelope of the line x cos a + y sin « = jo, when ce is a
parameter.
2. A straight line of fixed length a moves with its extremities in two
rectangular axes. Find its envelope.
194
DIFFERENTIAL CALCULUS [Ch. XVI. 121.
^ 3. Ellipses are described with common centers and axes, and having
the sum of the serai-axes equal to c. Find their envelope.
\/ 4. Find the envelope of the straight lines having the product of their
intercepts on the coordinate axes equal to k\
/ 5. Find the envelope of the lines y — ^ = m(x — a) + rVl + m^, m
being a variable parameter.
6. A circle moves with its center on a parabola whose equation is
y2 = 4 axj and passes through the vertex of the parabola. Find its
envelope.
7. Find the envelope of a perpendicular to the normal to the parabola
y2 = 4 ax, drawn through the intersection of the normal with the x-axis.
8. Show that the curves defined by the equations
^ + ^=1, a + p = c,
X y "^
in which a and j8 are parameters, all pass through four fixed points ; find
them.
9. In the « nodal family '' {y - 2ay={x - aY + Sx^ - y\ show that
the usual process gives for envelope a composite locus, made up of the
"node-locus " (a line) and the envelope proper (an ellipse).
INTEGRAL CALCULUS
CHAPTER I
GENERAL PRINCIPLES OF INTEGRATION
122. The fundamental problem. The fundamental prob-
lem of the Differential Calculus, as explained in the preced-
ing pages, is this :
Given a function f (^x), of an independent variable x, to
determine its derivative f'(x).
It is now proposed to consider the inverse problem, viz. :
Given any function f'(x), to determine the function fix)
having f {x) for its derivative.
The study of this inverse problem is one of the objects
of the Integral Calculus.
The given function f'(x) is called the integrand^ the
function f(x) which is to be found is called the integral^ and
the process gone through in order to obtain the unknown
function f(x) is called integration.
The operation and result of differentiation are symbolized
by the formula ,
£/w=/'(^), (1)
or, written in the notation of differentials,
dfix-)=f'ix)dx. (2)
195
196 INTEGRAL CALCULUS [Ch. I.
The operation of integration is indicated by prefixing the
symbol j to the function, or differential, whose integral it
is required to find. Accordingly, the formula of integration
is written thus :
Following long established usage, the differential, rather
than the derivative, of the unknown function f(x) is written
under the sign of integration. One of the advantages of so
doing is that the variable, with respect to which the inte-
gration is performed, is explicitly mentioned. This is, of
course, not necessary when only one variable is involved,
but is essential when several variables enter into the inte-
grand, or a change of variable is made during the process
of integration.
123. Integration by inspection. The most obvious aid to
the problem of integration is a knowledge of the rules and
results of differentiation. It frequently happens that the
required function f{x) can be determined at once by recol-
lecting the result obtained in some previous differentiation.
For example, suppose it to be required to find
/
cos X dx.
It will be recalled that cos x dx is the differential of sin x^
and thus the answer to the proposed integration is directly
obtained. That is,
cos xdx ss sin x.
f'
Again, suppose it is required to integrate
j x"dx^
122-123.] GENERAL PRINCIPLES OF INTEGRATION 197
where n is any constant (except — 1). This problem imme-
diately suggests the formula for differentiating a variable
affected by a constant exponent [(6), p. 49]. When this
formula is written
or, what is the same thing,
it becomes obvious that
/=
x^dx =
n + 1
An exception to this result occurs when n has the value
— 1. For in that case it is apparent from (8), p. 50, that
'dx
ix'^dx— \ — = log X.
The method indicated in the above illustration may be
designated as the method of integration hy inspection. This
is in fact the only method of practical service available.
The object of the various devices suggested in the subse-
quent pages is to transform the given integrand, or to
separate it into simpler elements in such a way that the
method of inspection can be applied.*
* When all has been done that can be accomplished in this direction, it
will be found that a large portion of the field is yet unexplored and unknown,
and that many functions exist whose integrals cannot be found. By .this we
mean that such integrals cannot be expressed in terms of functions already
known. To illustrate, let it be imagined that the integral calculus had been
discovered before the logarithm function was known. It would then have
I* (It
been impossible to express the integral \ — in terms of known functions.
This integral might in consequence have led to the discovery of the function
log X. An exactly analogous thing, in fact, has happened in the attempt to
integrate other expressions, and many important and hitherto unknown func-
tions have been discovered in this way which have greatly enriched the entire
field of mathematics.
198 INTEGRAL CALCULUS [Ch. I.
124. The fundamental formulas of integration. When the
formulas of differentiation (l)-(26), pp. 49-50, are borne in
mind, the method of inspection referred to in the preceding
article leads at once to the following fundamental integrals.
Upon these sooner or later every integration must be made
to depend.
1^
I. \u^du = ^^^^*
n + 1
II. f** = log«.
III. frt''efw = -^^.
J log a
lY. (e^du = e^.
T, f cos u du = sin u,
VI. \^inudu = -eosu*
Til. f sec^ udu = tan u,
VIII. f cosec'-^ udu = - cot u,
IX. f sec u tan w <fw = sec u,
X. ( cosec u cot u du = — cosec u,
XI. r_^^_ = sin-i w, or -cos-^i*.
XII. f-^^^ = tan-it*, or -cot-^w.
XIII. f — ^^ = sec-' u, or -cosec-^M,
•^ u^u^ i- 1
XIV. J
^^ = rers-i u.
V 2 u - u«
l:f
: /.
L
124-125.] GENERAL PRINCIPLES OF INTEGRATION 199
125. Certain general principles. In applying the above
formulas of integration certain principles which follow from
the rules of differentiation should be borne in mind.
(a) The integral of the sum of a finite number of functions
is equal to the sum of the integrals of each function taken
separately.
This follows from Art. 16.
For example,
j__dx=^^xdx-}- = --logx.
(5) A constant factor may he removed from one side of the
sign of integration to the other.
For, since
d(G * u)= c ' du,
it follows that
j cdu = c \du = cu.
To illustrate, let it be required to integrate
f'
Sx^dx.
The numerical factor 5 is first placed outside the sign of
integration, after which formula I is applied. Accordingly,
f5x^dx=5fx^dx = 5'^'
a
Again, suppose the integral
J x^4-l
_ dx
is to be found. It is readily noticed that except for the
constant factor 2 the numerator of the integrand is the exact
derivative of the denominator, and formula II would be
200 INTEGRAL CALCULUS [Ch. I.
applicable. All that is required, then, in order to reduce
the given integral to a known form, is to multiply inside the
sign of integration by 2 and outside by J. This gives
r xdx . r2xdx 1 rdOt^ + r) ., /^ , in
In this connection it must not be forgotten that an expres-
sion containing the variable of integration cannot be removed
from one side of the sign of integration to the other.
(c) An arbitrary constant may be added to the result of
integration.
For, the derivative of a constant is zero, and hence
du = d(u + c),
from which follows
I du=: i d(u -{-c^=u-\-o.
This constant is called the constant of integration.
It will be seen from this that the result of integration is
not unique, but that any number of functions (differing from
each other, however, only by an additive constant) can be
found which have the same given expression for derivative.
[Compare Art. 16, Cor. 2.]
Thus, any one of the functions rr^ — 1, rc^ + 1, a^ + a^,
(a; — a)(a; + a), etc., will serve as a solution of the problem of
integrating j 2 a; dx.
It often happens that different methods of integration lead
to different results. All such differences, however, can occur
only in the constant terms.
For example,
f^Cx + ^ydx = 8 f(x + 1)^^(2^ + 1)=(3' f i)'
=2^^.nx^-^iix-\-i.
125.] GENEBAL PRINCIPLES OF INTEGRATION 201
Integration of the terms separately gives
a result which agrees with the preceding except in the con-
stant term.
Again, from formula XII,
/— — - = tan~^a;, or — cot~^a:.
It does not follow from this that tan~^a; is equal to — cot~^a;.
But they can differ at most by an additive constant. In
fact, it is known from trigonometry that
— cot~^a? = tan~^a; + ^tt -f — ,
where h is any integer.
In a similar manner the different results in formulas XI
and XIII can be explained.
EXERCISES
Integrate the following :
. \Vxdxi= \x^ dx\. 8. J {ax + by dx,
J J x-\- a
3 r_^. 10. fl^LzJEl^
J ^ * J lax — X
rmx-^-\ 11 fsec^arc^a:
*• \ — -^dx. • J tanx
^ Vx
J 12 r siti X dx
(a^ — x^Y dx. '^ 1 + cos X
W', 6. p^'-3^+ldx. . 13. f_^^('=fJI.)
/t" —-'''/ J x^ -^ X log x\ J log x/
7. ^x^x'^ + a^ydx. 14- J^m*
dx
202 INTEGRAL CALCULUS [Ch. L
,15. Jtanarrfyf^-J'^^^^^A 19. ^ {a -^ h)^+^ dx,
'^,16. ^cotxdx. 20. JcosHarrfxf^J^^t^^^c^xY
17. Xe^dx. 21. J sin (m + n)a:c?a:.
"^ 18. j e==^ X dx. 22. J sin a;^ . a; dx.
23. j cos^arc^xf = \ cosa:(l — sin^a;)^^].
24. j sin*a;<fa:. 25. |sin2a:6fa;.
V"'
27
^6. Ttan^ a: c?a: r = ("(sec^ a; - 1) <^a;l .
I tan^ X sec^ x dx. 28. | cosec^ (aa; + 6) rfa.
'^ 29. i Vcot a? cosec^ a: rfa:.
30 r <^3: /_ C^ec'^xdxY
J sin a; cos x\ ^ tan a; /
^
. J sec*MtanM<iM.
C . J ( _ fcosec u cot udu\
'J \ ^ cosecM /
_ rtan M r?M
J sec M
f '^ = C
35.
36
37
f^^^-. 38. f ^^ (=f-
1^
dx
C du 39. f
f ^^ 40. f
^ a* + />%2 -^ xV^«
(ar-l)Va:2-2x
125-126.] GENERAL PRINCIPLES OF INTEGRATION 203
126. Integration by parts. If u and v are functions of a?,
the rule for differentiating a product gives the formula
d (uv) = V du + u dv,
whence, by integrating and transposing terms,
\udv = uv — \vdu.
This formula affords a most valuable method of integra-
tion, known as integration by parts. By its use a given
integral is made to depend on another integral, which in
many important cases is of simpler form and more readily
integrable than the original one.
Ex.1.
Jlog
xdx.
Assume
'
u -
= log a:,
dv = dx.
Then
du:
dx
V = X.
By substituting
in the formula for in
itegration by parts,
Jloga
:dx =
xlogx-
-j-..
=
a: log a: ■
-X = x (log X -
-1)
X
= X (log X — log e) = a: log -.
Ex. 2. ixe^'dx.
Assume u = x, dv — eFdx.
Then du = dx, v = e*,
and
j xe'^dx = xc* — i e'^dx = e*(a; — 1).
Suppose that a different choice had been made for u and dv in the
present problem, say
u = e*, dv = X dx.
204 INTEGRAL CALCULUS [Ch. I.
From this would follow
du = e'dx, y = — ,
2
and J xe'dx = ^ arV — \ ^e*dx.
— e*dx is less simple in
fomi than the original one, and hence the present choice of u and do
is not a fortunate one.
No general rule can be laid down for the selection of u and dv.
Several trials may be necessary before a suitable one can be found.
It is to be remarked, however, that as far as possible dv should be
chosen in such a way that its integral may be as simple as possible,
while u should be so chosen that in differentiating it a material sim-
plification is brought about. Thus in Ex. 1, by taking u = log a:, the
transcendental function is made to disappear by differentiation. In
Ex. 2, the presence of either x or e* prevents direct integration. The
first factor x can be removed by differentiation, and thus the choice
u = X is naturally suggested.
Ex. 3. Kx^a'dx.
From the preceding remark it is evident that the only choice which
will simplify the integral is
u = x^, dv = a'^dx,
qX
Hence du = 2x dx, v = ,
log a
and (x^a'dx = ^^ - -^ (xa'dx.
J log a log a J
Apply the same method to the new integral, assuming
M = x, dv — a'dXf
whence du = rfx, v = ,
log a
and (xa' dx = -^^ - -^ f o-rfar
J log a log a J
logrt {\o%ay
By substituting in the preceding formula,
J logaU logo (log a)'' J
126-127. J GENERAL PRINCIPLES OF INTEGRATION 205
EXERCISES
1. (Birr'^xdx. 7. Kxcoi-'^xdx,
J ^ ^ 8. J a: SI
2
3. \ x"^ cos X dx.
4. j x^ log X dx.
5. (x^t&n-^xdx. ^^' J
'^
6.
sin 3 X dx,
\ e* cos X dx,
e" sin X dx.
j sec a; tan a: log cos a: rfa:. 11. i cos a: cos 2 a; rfa:.
127. Integration by substitution. It is often necessary to
simplify a given differential f'(x)dx by the introduction of
a new variable before integration can be effected. Except
for certain special classes of differentials (see, for example,
Arts. 138, 139) no general rule can be laid down for the
guidance of the student in the use of this method, but some
aid may be derived from the hints contained in the problems
which follow.
Ex.1.
J
xdx
Va2 - a;2
Introduce a new variable z by means of the substitution a'^ — x^ = z.
Differentiate and divide by — 2, whence xdx = Accordingly
The details required in carrying out this substitution are so simple
that they can be omitted and the solution of the problem will then take
the following form :
r_^d^ = ((a^-x^r^xdx = - ^ ((a^-x^rH-2xdx) = - (a^-x^^.
In this series of steps the last integral is obtained by multiplying inside
the sign of integration by - 2 and outside by - J, the object being to
206 INTEGRAL CALCULUS [Ch. I.
make the second factor the differential of a^ — x\ Thinking of the
latter as a new variable, the integrand contains this variable affected by
an exponent {— \) and multiplied by the differential of the variable, in
which case formula I can be applied.
Ex. 2. (^^^dx.
J X
Assume log a; = 2.
Then — = dz,
and |l2^rf. = pI = | = (!o|£l^
Here again it is not necessary to write out the details of the substitu-
tion, as it is easy to think of log a; as a new independent variable and to
perform the integration with respect to that. It is then readily seen
that the expression to be integrated consists of the variable logx mul-
tiplied by its differential — , and that the integration is accordingly
X
reduced to an immediate application of the first formula of integration.
Thus
Ex.3. fgtan-^x dx
J 1 + X2
GoK^y
logx ' d(logx)z= ^
Think of tan-^ a: as a new variable and apply formula IV. Thus
Ex.4. r^iBli:
+
'sin"^ X dx
dx
Think of sin-^ a: as a new variable and — — ^^^ as the differential of
that variable. Apply formula I. ^^ ~ ^*
Ex.5. j'(a;2+2a; + 3)(x + l)«?x.
Multiply and divide by 2. The integral then takes the form
i JCar^ + 2 ar + 3) • (2 x + 2)dx.
Observing that (2 a: + 2)^a: is the differential of ar^* + 2 a: -|- 3, and think-
ing of the latter expression as a new variable, it is seen that formula I is
directly applicable, leading to the result
127.] GENERAL PRINCIPLES OF INTEGRATION 207
Ex. 6. flog cos (x^ + 1) sin (x^ +1)'X dx.
Make the substitution
ar2+l=2;.
The given integral takes the form
J j log cos 2; siw zdz.
Make a second change of variable,
cos z = y.
Then sin zdz=: — dy.
The transformed integral is
-lyf^gydy,
to which the result of Ex. 1, Art. 126, can be at once applied.
It will be observed that two substitutions which naturally suggest
themselves from the form of the integrand are made in succession. The
two together are obviously equivalent to the one transformation,
cos (x^ + 1) = 3/.
Ex.7, f /^ .
Either put x = az, or else divide numerator and denominator by a, and
write in the form
/
<^
v-(iy
Regarding - as a new variable, this comes under XI and gives the
result
C dx • ^ X , ^
•^ Va2 - x^ a
= - cos-i- + C^.
In a similar manner treat Exs. 8-10.
Ex.8, r ^^ .
J x'^ + a^
Ex.9, f— ^=.
•^ xy/x^ - a2
Try also the substitution x = —
z
208 INTEGRAL CALCULUS fCn. I.
Ex.10, f- ^^
V2 ax - a;2
Try also the substitution 2: - a = x.
Ex. 11. • ^^
/:
Vx2 ± a2
Make the transformation
From this follows, by differentiation,
(1+ ^ -\dx = dz-r
dx
that is, (v^2±a2 + a;) ""^ = c?2,
or, • = = — •
Ex.12, r^^.
Assume x — a _ ^ . ^^^ is, ar = a — i-?»
a: + a 1 — «
The reasons for the choice of substitution made in this and the pre-
ceding example will be made clear in Arts. 133 and 139.
Ex.13. jcosecx£?x.
Multiply and divide by cosec x — cot x. It will be readily seen that
rdz
the integral then takes the form \ —
Another method would be to use the trigonometric formula
sin a: = 2 sin ^ cos |,
aec^ -dl-\
whence I cosec xdx = I = I -
• ' 2 sin ^ cos r *^ tan ?
:. 14. J
Ex. 14. i sec a: dx.
Put « = « - J, and use Ex. 18.
127-128.] GENERAL PRINCIPLES OF INTEGRATION 209
Solve the problem also by means of substitutions similar to those
used in the preceding example.
ii^"^ Ex. 15. f ^£ = f i°^^
= 2f-5 P^ (s=2ax + 6)
2
tan-i ^«^ + A-, if 4 ac - 52 >0.
V4 ac - 52 V4 ac - ¥•
= — ^::::r=r log ' > if 4 ac — 6^ < 0.
Vft-^ - 4 ac 2 aa; + 6 + V62 _ 4 ac
Ex.16, f ^?£ l=f 2rf^ =f ^(2^) .
^2a;2+2a;+3 ^ 4^2+ 4a: + 6 ^(2a:+l)2+5
In this form it is ea^ to integrate by taking 2 a: + 1 as a new variable.
Ex.17, f ^ .
^3a:2-2a:+5
Ex.18, f i^^ .
Ex. 19. ^—^ ^ .
V-9a;2+ 30ar-24
Ex. 20. f (3 a: - 2) cos (3 a: - 2) dx.
Ex.21, f
adx
xVcfi + hx
Substitute y/a^ + hx — z, and use Ex. 12.
\\)J 128. Additional standard forms. The integrals in Exs.
7-14 of the preceding article, and in Exs. 15-16 of Art. 125,
are of such frequent occurrence that it is desirable to collect
the results of integration into an additional list of standard
forms.
210 INTEGRAL CALCULUS [Ch. I.
du
Va2 _ ^2 a - a
_„ r du ., 1 u
XT. J— =:=r = 6iii-i-, or -cos
XVI. f— z^^=log(i* + V^i2X^).
XYII. f_^ = ltan-i^, or -icofi
XIX. f — ,^^ ,=^sec-*^, or -£:Cosec
du 1 it* 1 1 ««
,.= = — sec-*— , or — cosec~* —
t*Vw2_«2 a a a a
XX. i ^ ==Yer8-^--
XXI. j tan udu = - log cos «* = log sec u,
XXII. j cot u du = log sin u.
XXI 11. f sec udu = log (sec i* + tan i*) = log tan (^ + t) •
XXIY. f cosec udu = log (cosec u - cot u) = log tan ^ •
129. Integrals of the form
r (Ax. + B)dx
^ y/axi^ -\-bx + c
Integrals of this form are of such frequent occurrence as
to deserve special mention. The integration is readily effected
by the substitution of a new variable which reduces the
radical to a simpler form. Two cases are to be considered
according as a is positive or negative.
Case I. a positive. In this case by dividing out the
coefficient of a^ the radical may be written
^ a a ^\ 'la J 4a^
128-129.] GENERAL PRINCIPLES OF INTEGRATION 211
The given integral then takes the form
( Az 4-B— \dz
1 r* (Ax-\-B)dx - ^ A ^^J ( _ ^
_ A r zdz
V
2 , -nac
z^-\-
P
+
2aB-bA
4:0,'
C-
dz
^ac — W
= —Wax^ + bx-[- c-\ — log X + - — \'\x^ -\--x-\--
^ 2aVa ""K 2a ^ a a
Case II. a negative. When a is negative, by dividing
out the positive number — a the radical becomes
^ a a 4a2 \ 2 a/
and in consequence the integral takes the form
1 y- (Ax + B)dx
^'^-hi-J
V
(Az+B-^)dz
!_ A 2aJ
— a*^
— -tao
JblszA
^ ■ia'
V 2a
_ A r» zdz
^ -ia
+
2 a^ - 6^
— 4:ac 2 2 aV
<3
- 0^^ n dz
\ A „1
V
— a^ 4 a^
ac 2.2 a5 — bA ' _i
4^2
2a2
ac 2
z^
2 a V — « V^^ — 4 <xc
-■^ / 2 , I . . 2aB-bA . _i 2a2; + 6
= — Vax^ + bx-{-c-\ sin ^ ■•
^ 2«V-a V62-4ac
212 INTEGRAL CALCULUS [Ch. 1.
Ex.1. f-(^^ + ^)^^ .
-^ V3 a;2 + 3 a; + 2
On dividing numerator and denominator by VS the integral reduces to
— X H 1 dx
a/8 a/8/
f
which by means of the substitution a: + J = s can be written
•' Vz2 + ^ V3-' -^ Vz2 ^ ^j
= J_ i5i±^il5 + V3 log (2 + Vi2T5)
= _?_Va:2+a; + f +\/31og(a: + i + Va:2 + x + |).
V3
Ex.2, r (2x + l)dx _,
•^V-2a;2-3a:-l
Divide numerator and denominator by v^. The integral becomes
^^±ldx
= - V2 V^^Tia _ _i_ sin-i 42
2V2
= - V-2x^-Sx^ 1 - -i- sin-i (4 z + 3).
2\/2
130. Imtegrals of the form j*
da?
(^05 + B) y/ax^ + 6aj + c
Integrals of this type can be reduced to the form given in
the preceding article by means of the reciprocal substitution
z
129-130.] GENERAL PRINCIPLES OF INTEGRATION 213
From this follow the relations
and Vaa^ '\-hx-\- c — — Vas^ ^ yg^ + 7,
in which a = aW' - hAB + cA^,
j3 = -'2aB + hA,
ry — a.
When these expressions are substituted in the given inte-
gral it reduces to
-f '^^ ,
which has the form discussed in Art. 129.
EXERCISES ON CHAPTER I
^^+0^ + 1 ^^_
1. f ^^^ . a f-^
•^ Va;2 + 2 a: + 2 V (^ + 1) Va:^ + 2 a: + 3
2 r ^^ , [Divide the numerator by a; + l.J
•^ xVo a:2 — 4 a: + 1
^1 + ^ + ^?^^.
3 r C2a:-3)rfar, 9- J ^
•^ V3 a:2 + a; - 2
. C (4:x + 5)dx ^0- 1—7==^
4. I .^ ^ "^ a;Va;4 + a;2 +
V8 + 4 ar - 4 a;2
'■J
1 -.
a: dx [Assume x + i- = s. ]
V-ar2 + 2ar + l
6. (^jmdx.
[Rationalize the numerator.]
X
11. fe«*e*rfx
'5 x^ dx
12. f5^1£E. / h
J4 + a;8 <t^f|AC-^
W^. fV^rfi. 13. fja+l^)^
T
214
"J
dx
INTEGRAL CALCULUS
25
[Ch. I.,1
VI- e^
[Put e' = z.]
dx
15
r dx
"I
8x2
■<Za:
27
(ix
Vl -a:*
17. f-^^.
.If
^•'xVl - log a:
C e'^dx
* J gX _f. g-X
18. 1 X* tan-i a: dx,
Tx^ - 2 X + 3
J a:2+l
20. f^^-
21. r f r-p-tM ^'' -^
Jsin^+lL J cos2^ J
22- ll
dO
29 r COS X dx ^
•^ Vl + cos^ X — sin X
Cf secx y^i.
J\a — 6tanx/ -^JV*
v5t
{x— a) dx'
Va* -a^(x- ay- {x - a)*
23. r_*5
J a +
cost;
tan X dx
33. I vers^ ::•
J a y/x
24.
i"rT
6 tan^ x
dx
-]•;
^x
cot X
[Put 1 + cotx = -.J
a:2 V3 a:2 + 2 a: + 1
I Substitute a: = — J
/ 35. Jlog {X + \/^^^r^2) ^. '^
r
-h
CHAPTER II
REDUCTION FORMULAS
131. In Arts. 129, 130 the integration of certain simple ex-
pressions containing an irrationality of the form Vaa^ -{-bx-i-c
has been explained. As was shown in Art. 129, the radical
can be reduced to the form -V ± a^ ± a^ by a change of vari-
able. It remains to show how the integration can be per-
formed in such cases as, for example.
CxW± a^ ± a^ dx, C-
x^dx
V± x^ ± a2
n being any integer.
For this purpose it is convenient to consider a more general
type of integral of which the preceding are special cases, viz.,
^x'^ia + hx'^ydx, (1;
in which m, w, 'p are any numbers whatever, integral or frac-
tional, positive or negative.
It is to be remarked in the first place that n can, without
loss of generality, be regarded as positive. For, if n were
negative, say n =— n'^ the integrand could be written
^""(^ + 4^)" = ^""f^"^" ^ ^T = x'^-J'^'Xh -h ax»y.
This expression, which is of the same type as x^(a + hx^'y^ is
such that the exponent of x inside the parenthesis is positive.
215
216 INTEGRAL CALCULUS [Ch. II.
It will now be proved that an integral of the type (1) can
in general he reduced to one of the four integrals
(a) A^x^'-'^i^a -f hx'^ydx, (b) Afx^'-^Xa + hx'^ydx,
(c) ^ J^r'^Ca + hsf'y-^dx, (d) A Cx%a + hx'^y^Hx,
plus an algebraic term of the form
Bx\a + haf^y.
Here J., j5, \, /x are certain constants which will be deter-
mined presently.
Observe that in each of the four cases the integral to
which (1) is reduced is of the same type as (1), but that
certain changes have taken place in the exponents, viz., ""
the exponent m of the monomial factor is increased or
diminished by w,
or, the exponent p of the binomial is increased or dimin-
ished by unity.
The values of \ and /* are determined by the following
rule :
Compare the exponents of the monomial factors in the given
integral and in the integral to which it is to he reduced. Select
the less of the, two numbers and increase it by unity. The
result is the value of X. In like manner^ compare the exponents
of the binomial factors in the two integrals^ select the less^ and
increase by unity. This gives fx.
Thus, if it is desired to reduce the given integral to
A Cx"'-^(a + ba^ydx, '
first write down the formula
Cx^'Ca -h bx"ydx = A Cx'^-^i^a + bx^ydx -f Bx^(a -f- b7f*y.
131.] BEDUCTION FORMULAS 217
The exponents of the monomial factors in the two integrals
are m and m— n respectively, of which m — yi is the less.
This, increased by unity, gives the value of \; that is,
\ = m — n -{-1,
Again, the exponent of the binomial factor in each integral
is the same, namely jo, so that there is no choice as to which
of the two is the less. Increase this number p by unity to
obtain the value of /i. Hence {jl = p -\-\,
The above formula may now be written
= A^x'^-^ia + haf'ydx + Bx'''-^''\a + haf'y^^. (2)
In order to determine the values of the unknown constants
A and B^ simplify the equation by differentiating both
members. After dividing by x^~^(^a + hx^y the resulting
equation reduces to
x"" = A-\- Ba(m — n -\- 1) + Bh(m + np -\- l)a;".
By equating coefficients of like powers of x in both members,
the values of A and B are found to be
j^ — ^(^ — 9^ + 1) ^_ 1
h(m + np + \y h(m + np + 1^
When these values are substituted in formula (2), it
becomes
fx'^^a + bsfydx
= ~ Tr~. ^^ ) ^ (^ + hx'^ydx + w . , A ' [a]
Notice that the existence of formula (2) has been proved
by showing that values can be found for A and B which
make the two members of this equation identical.
218 INTEGBAL CALCULUS [Ch. II.
There is one case, however, in which this reduction is
impossible, viz., when
w -f wjt? + 1 = 0,
for in that case A and B become infinite. [See Ex. 4, p. 221.]
In a similar manner the three following formulae may be
derived :
/'
x"'(a + bx^ydx ^
^ ^ fx 1 x"'+''(a + hx''ydx-\ \ ,/ . fB]
x%a + hsfydx
w + wp + 1^ ^ m + np + 1
= — , ^7^ I x'^Ca + 52;")^+^(^2; V-^ — ^^ [D]
a/i(j9 + l) J ^ ^ «n(jt? + l) ■" -'
/■
The cases in which the above reductions are impossible
are,
For formulae [A] and [C], when m + np -^1 = 0;
for formula [B] , when m + 1 = 0 ;
for formula [d] , when j9 + 1 = 0.
Ex.1. (xWa"^- x^dx.
K the monomial factor were x instead of x^, the integration could
easily be effected by using formula I. Since in the present case rn = 3,
n = 2, forjnula [A], which diminishes m by n, will reduce the above
integral to one that can be directly integrated.
Instead of substituting in [A], as might readily be done, it is best to
apply to particular problems the same mode of procedure that was used
in deriving the general formula. There are two advantages in this.
First, it makes the student independent of the formulas, and second,
when several reductions have to be made in the same problem, the work
ia generally shorter. [See Ex. 4.]
131.] BEDUCTION FORMULAS 219
Accordingly assume
(x\a'^ - x^^dx = A(x(a^ - x^)^dx + Bx%a^ - x^)^,
the values of A and fj. having been determined by the previously given
rule.
Differentiate, and divide the resulting equation by x (cfi — x^p. This
^^^®^ x^=A-{-B(2a^-6 x^),
from which, by equating coefficients of like powers of x,
and hence,
(x^VaT^x^dx = —^(a^ - x'^^xdx - \ x\a^ - ar^)^
= - ^3(2 a2 + 3a;2)(a2 - x'^)^,
Ex.2. fVa;2-2a;-3(/ar.
By following the suggestions of Art. 129, this integral can be reduced
to the form
in which z = ar — 1.
Assume
J(22 _ ^)\dz = A^(z^- ^)-^dz + Bz(f - 4)i
In determining X notice that m = 0 in both integrals, so that
X = 0+1 = 1. Also, /A = -i+l = i.
Ex.3. (V2ax-x^dx.
The mode of procedure of Ex. 2 may be followed. Another method
can also be used, as follows :
On writing in the form
(xi(2a-x)^dxy
and observing that
vers
-1
X C dx
f ^^ = f ^-^(2a - x)-i dx.
« ^ V2ax-x^
it will be seen that the integration may be effected in the present case
by reducing each of the exponents m and p by unity. This is possible
since n = 1 and m can accordingly be diminished by 1. Hence assume
(x^(2a- x)^ dx = A'(x-i(2a~ x)idx + B'xi(2 a - x)i.
220 INTEGRAL CALCULUS [Ch. II.
The exponent of the binomial in the new integral may be reduced in
turn by assuming
(x-^(2 a-x)^dx = A "(x-^(2 a - xy^dx + B"x^i2 a - x)^.
When this expression is substituted for the integral in the second
member of the preceding equation, the result takes the form
f y/2ax-x^dx = A f ^^ + Bx^ {2 a - x)^ + Cx^{2 a-x)\
J -^ y/2ax — x^
in which A, B, C are written for brevity in the place of A' A", A'B", B'
respectively. The values oi A, B, C are calculated in the usual manner
by differentiating, simplifying, and equating coefficients of like powers
of X.
The method just given requires two reductions, and heilce is less
suitable than that employed in Ex. 2, which requires but one reduction.
The rule for determining the values of X and fi may now
be advantageously abbreviated. Let m, p be the exponents
of the two factors in the given integral, and m', p' the corre-
sponding exponents in the new integral. Of these two
pairs, m, p and m'^ p\ one of the numbers in the one pair is
less than the corresponding number in the other pair. This
fact will be expressed briefly by saying that the one pair is
less than the other pair. With this understanding the
preceding rule may be expressed as follows :
Select the less of the two pairs of exponents m, p and
m', p'. Increase each number in the pair selected by unity.
This gives the pair of exponents X, fi,
Ex.4. ^ ^'^^
^■^7^.
(x2 + a2)^
Assume successively
f a:<(x2 + a«)~* dx = A' (x*(x* + a^)"* dx + B'x*(x^ + a«)~*,
^x*(x^ + a^)'^dx = Af jx^x^ + a^yhx + B"x^(x^ + a^)*,
(x^(x^ + a2)-i dx = A '" f (z* + a2)~i dx + B'"x(x^ + a^)K
131.] REDUCTION FORMULAS 221
These equations may be combined into the single formula
(x\x^ + d^)~^ dxz=A({x'^+ a2)-i dx + Bx (x2 + a2)i
+ CxXx"^ + a2)i + Dx^x"^ + a2)~*.
The values of the coefficients are found to be
Hence
•^ 2Va;2-j- a2
In this example three reductions were necessary ; first, a reduction of
type [1>J, second, and third, a reduction of type \A'\ . Can these reduc-
tions be taken in any order ?
The different possible arrangements of the order in which these three
reductions might succeed each other are
(1) [-4], [^], [D] ; (2) [^], [1>], [^]; (3) [1>], [^], [^],
of which number (3) was chosen in the solution of the problem. Of
the other two arrangements, (2) can be used, but (1) cannot. For,
after first applying [^] (which would be done in either case), the new
integral is
( x\a^ + x'^y^ dx.
If [^] were now applied, it would be necessary to assume
Ja:2(a2 + x"^)"^ dx = ^ f (a2 + x'^)~^-\- Bx{cfi + x'^)'^.
This equation, when difEerentiated and simplified, becomes
a:2 = ^ + 5a2,
a relation which it is clearly impossible to reduce to an identity by
equating coefficients of like powers of x, since there is no x^ term in
the right member to correspond with the one in the left member. It
will be observed that this is the exceptional case mentioned on page 218,
in which m + np + 1 = 0.
Ex. 5. Show thaib the integral fC^ -^ ) ^x can be integrated by
four reductions. Prove that these can be arranged in six different
orders, and determine those which can be used.
222 INTEGRAL CALCULUS [Ch. II. 131.
Ex. 6. f — ^-— . (T-Ex. 13. (V^^Tadx.
J (a:'-' + iy J
Ex. 8. (•_^!*L_. ■^L^ .^..^ J ^ V^
y^' rt^x. 9. iV^^^^dx, Ov
J^^ , .. Ex.16. f_^=. D>V
•^ xWa' - x^
Ex.11, f-i^^ / Ex.17. r,^f ^o^«' K^
Ex. 12. f (a2+ x2)^rfar. Ex. 18. f Vl - 2 x - x^ rfar. ^
^'
Ex. 19. Show that
r ^^ = 1 r ^ +(2n-3)f ^-^ 1
''^^'
l>
CHAPTER III
INTEGRATION OF RATIONAL FRACTIONS
132. Decomposition of rational fractions. The object of the
present chapter is to show how to integrate fractions of the
form , ^
wherein (f)(^x) and '>jr(^x) are polynomials in x.
The desired result is accomplished by the method of sepa-
rating the given fraction into a sum of terms of a simpler
kind, and integrating term by term.
If the degree of the numerator is equal to or greater than
the degree of the denominator, the indicated division can be
carried out until a remainder is obtained which is of lower
degree than the denominator. Hence the fraction can be
reduced to the form
^ = a." + 5.«- + ... + ^,
in which the degree oif(x') is less than that of '^(x)'
As to the integration of the remainder fraction ^^ ^ , it is
to be remarked in the first place that the methods of the
preceding articles are sufficient to effect the integration of
such simple fractions as
x—a (x — ay^'' ' x^±d?'' (x^±a^y^^ ' x^-\-'mx-{-n
Now the sum of several such fractions is a fraction of the
kind under consideration, viz., one whose numerator is of
223
224 INTEGRAL CALCULUS [Ch. III.
lower degree than its denominator. The question naturally
arises as to whether the converse is possible, that is : can every
fraction ^^^£2. he separated into a sum of fractions of as simple
types as those given in (1) f
The answer is, yes.
Since the sum of several fractions has for its denomina
tor the least common multiple of the several denominators,
it follows that if •^^ ^ can be separated into a sum of
simpler fractions, the denominators of these fractions must
be divisors of 'yjr(x^. Now it is known from Algebra that
every polynomial '^^(x) having real coefficients (and only those
having real coefficients are to be considered in what follows)
can he separated into factors of either the first or the second
degree^ the coefficients of each factor heing real.
This fact naturally leads to the discussion of four different
cases.
I. When '>^(x') can be separated into real factors of the
first degree, no two alike.
E.g. , '^^(2?) = (x — a)(x — 6) (x — c).
II. When the real factors are all of the first degree, some
of which are repeated.
III. When some of the factors are necessarily of the
second degree, but no two such are alike.
U.g., ylr^x) =^Cx^ + a^Ca^-hx + l)(rr -hXx- c^.
IV. When second degree factors occur, some of which
are repeated.
Kg., f(x) = (:t-2 + a2)V-^ + 0(^-^)-
132-133.] INTEGRATION OF RATIONAL FRACTIONS 225
133. Case I. Factors of the first degree, none repeated.
When ylr(x) is of the form
^fr(x') = (x — d)(x —h}(x— 6) ••• (x — 7l),
assume
i|r(a;) X — a x — h x — c x — n
in which A^ B^ C^ '"^ N are constants whose values are to be
determined on condition that the sum of the terms in the
right-hand member shall be identical with the left-hand
member.
Ex.l. (tpl^d^.
Dividing numerator by denominator, -i— - = x ,
^ ^ x2-3a: + 2 x^~^x+2
Assume -^ ^ = ^d_+ ^
(a:-l)(a;-2) x-1 x-2
By clearing of fractions,
(1) x = A(x-2) + B(x-iy
In order for the two members of this equation to be identical it is
necessary that the coefficients of like powers of x be the same in each.
Hence 1 = A-^B, 0 = -2A-B,
from which ^ = — 1, B=2.
Accordingly the given integral becomes
Kx + ^^-^)dx=:aL+\og(x-l)~2log(x--2)'rC
A shorter method of calculating the coefficients can be used. Since
equation (1) is an identity, it is true for all values of x. By giving x
the value x=l the equation reduces to 1 = ^(— 1), or A=: — l.
Again, assume x = 2. Whence 2 = B.
226 JtcVf ^ -^INTEGRAL CALCULUS ^^K '"^ [Ch. IH
Ex. 2M ^^ Ex.10 ri?^±il^. V
^a;2-a2 J 2x2 + 3 3:- 2
Ex.3, flu^dor. Ex.11. f-M+^&l^^. lAL
J x^-x X ^ a; (a-- -a) (a: + 6) ^
Ex.4
Cix^-\2)dx ^ Ex. 12. r_i£±ii^. \i
Ja;2+4a:+3 J2x-a:2-a;8 , l^
Ex.6.
f a:</a; Ex.14, f ^ ^^
)x^-4.x + i Ja^x^-b^
Ex 7 r_i£!^iM£_ Ex.15, f (^^-^-1)^.
J(a:2-4)(4i2-l)" J(x2-3x+2)V23r«
P^ « ra:2-2ca: + qc-a&+6c^^ [Separate f~^~\ into par-
^^- «: J ^,-aXx-b)(x-c)'^'' tialfractions.]^'- 3 ^+ 2
(a;2-4)(4a;2-l)
Ex.8, f
Ex.9.
fx2(.+a)-i(:.+6)-i^.. Ex.16, f (^^+^^-^^
Vx2+4a:+7
134. Case II. Factors of the first degree, some repeated.
Ex 1 rC5a:2-3ar+ l)</x.
■ J a: (x - 1)8
Assume
rn 5a:2-3a:4 1^^ BCD
^^ a:(x-l)8 X a:- 1 (a; -1)2"^ (x- 1)8*
To justify this assumption, observe that :
(a) In adding the fractions in the right-hand member, the least com-
mon multiple of the denominators will be x (x — ly, which is identical
with the denominator in the left-hand member.
(6) Further, the expressions a:, a: — 1, (a: — 1)^, (x — 1)8 are the only-
ones which can be assumed as denominators of the partial fractions,
since these are the only divisors of x(x — 1)8.
(c) When equation (1) is cleared of fractions, and the coefficients of
like powers of x in both members are equated, four equations are ol)-
tained, which is exactly the right number from which to determine the
four unknown constants A, B, C, D.
Instead of the method just indicated in (c) for calculating the coeffi-
cients, a more rapid process would be as follows :
By clearing of fractions, the identity (1) may be written
6 a;« - 3 X + 1 = ^ (x - 1)8 -h Bx (x - 1)2 + Cx (x - 1) + 7)a;.
133-134.] INTEGRATION OF RATIONAL FRACTIONS 227
Putting a; = 1 gives at once
3 = i>.
Substitute for D the value just found, and transpose the correspondmg
term. This gives
b x'^ - Q X -t I = A {x - ly -{- Bx {x - \y + Cx {x - 1).
It can be seen by inspection that the right-hand member of the result
is divisible by a? — 1. As this relation is an identity, it follows that the
left-hand member is also divisible by x—\. When this factor is re-
moved from both members, the equation reduces to
bx- \ = A(x-\y-\-Bx(x-l)+ Cx.
Now put a; = 1. Then
C = 4.
Substitute the value found for C, transpose, and divide by a: — 1.
The result is \ = A{x-l) + Bx.
By giving x the values 0 and 1 in succession, it is found that
A = -l, 5 = 1.
Accordingly,
r(5x^-3x+l)dx^ rf_l + _j_ + __^_ + _3_U
J x{x-iy J\ X x-i (x-iy (x-iy)
= log.^-l ^^-^
X 2(x - 1)2
Ex. 2. (—J^ J Ex. 9. r^£!±^^!±r«±l}^±«rf^.
J(x^iyix+1) J x\a+x)
Ex. 3. rC^^- 11^+26)^0:. ^^^^^ nx^-l)dx,
^'^' *• ^{x^-a^y Ex.ll.'|(ax2+6;r8)-irfx.'\
Ex. 5. r(^2a:+l)rfa;, ^^ ^^ f ix^-x'^-r)dx .
•^ x\x -f V2)a ' J (a; _ 1)2 Va:^ - 2 a: + 2
^^- ^- f-^i^fl^^- [Separate ^Izz^iz-l into partial
/Ex 7 r^{^i±_«Mi!^
J a:4 - 2 a%2 + a*
yEx. B. I ^2rf.
N^
fractions.]
Ex.13. { ^' rfa?.
-^ (a; — a)8
(2 -I- v^ - V^ a:)8 [Substitute a: - a = «.]
'm-
228 INTEGRAL CALCULUS [Ch. III.
135. Case III. Occurrence of quadratic factors, none
repeated.
' • • J(x2+l)(a;2+2a;+2)*
Assume
(Vi 4a;2 + 5a: + 4 _ Ax ■{■ B Cx-{- D
^^ (a;2+l)(a:2 + 2x+2) x^^-1 x^-\-2x^-2
Then
(2) 4a;« + 5a:4-4 = (^a: + ^)(a;2 + 2a: + 2) + (Ca: + 7))(a:2+l).
By equating coefficients of like powers of x
0 = J+C, 5 = 2^ + 2£ + C,
4 = 2^+5+ A 4 = 25 + A
from which
^ = 1, 5 = 2, C = -l, i) = 0.
Hence the given integral becomes
C{x + 2)dx_C xdx ^2tan-ia;+tan-i(a:+l)+ilog /'+^ .
To make clear the reasons for the assumption which was made con-
cerning the form of equation (1), observe that since the factors of the
denominator in the left member are a-^ + 1 and a;^ + 2 a: + 2, these must
necessarily be the denominators in the right. member. Also, since the
numerator of the given fraction is of lower degree than its denominator,
the numerator of each partial fraction must be of lower degree than its
denominator. As the latter is of the second degree in each case, the
most general form for a numerator fulfilling this requirement (i.e., to be
of lower degree than its denondnator) is an expression of the first degree
such as Ax + J5, or Cx + B.
Notice, besides, that in equating the coefficients of like powers of x in
opposite members of equation (2), four equations are obtained which
exactly suffice to determine the four unknown coefficients A, B,Cf D.
1 (Adx_, E^.6 C{^x-^)dx^
' J x^ + ^x J x^ + 2x^
Ex.2.
Ex 3 f ^^^ Ex 7 f ^^^
" ^(a;+l)(a;2+l) ' ' J x^ -^ x'^ ■\- 1
Ex.5, f {a'-^')dx . E,.9. r {^ + ^^x + 2)dx .
i\\x}^
135-136.] INTEGRATION OF RATIONAL FRACTIONS 229
136. Case IV. Occurrence of quadratic factors, some
repeated. This case bears the same relation to Case III
that Case II bears to Case I, and an exactly analogous mode
of procedure is to be followed.
J (a;2 + 2)8
Ex
Assume
,j. 2x^- x^ i- 8x^ + 4: ^ Ax -{- B Cx + D Ex + F
^ ^ (:r2 + 2)3 a:2 + 2 (a;2 + 2)^ ^ {x^ + 2)3*
Whence, by clearing of fractions,
2 x^ - x^ + 8 x^ + 4: ={Ax + B){x'^ + 2)2 + (Ca:+ D){x^ + 2)+Ex-\- F.
Instead of equating coefficients of like powers of x, as might be done,
the following method of calculating the values of J., ^, C, ••• is briefer.
Substitute for x^ the value — 2, or, what is the same thing, let
x = V— 2. This causes all the terms of the right member to drop out
except the last two, and equation (1) reduces to
_ S\/^^ = EV^ + F.
By equating real and imaginary terms in both members,
- 8 = ^, ^ = F.
Substitute the values found for E and F in (1), and transpose the
corresponding terms. Both members will then contain the factor x^-^2.
On striking this out the equation reduces to
2a:8-a:2+4a: + 2 = (^Ax + B^{xP- + 2)+ Ca: + 2).
Proceed as before by putting x^ = — 2. Whence
4=CV^r2 + D,
and therefore 0 = C, 4 = Z).
Substitute these values, transpose, and divide by a:^ + 2. This gives
2x-\ = Ax-\- B,
whence ^ = 2, 5 = — 1.
The given integral accordingly reduces to
"^xdx
J a:? 4- 2 J (x^ + 2)2 J {x
(a;2 + 2)2 J {x^ + 2)8
230 INTEGRAL CALCULUS [Ch. III. 136-137.
The first term becomes
- The second, integrated by the method of reduction (Chap. II), gives
^ +_l.taii-i^
x^ + 2 V2 y/2
Finally, by applying formula I the last term integrates immediately
Hence
J {x^ +2)8 - ^ '^ ^ ^ ^ a:2 + 2 ^ (x2 + 2)2
Ex. 2. a^Y dx. Ex. 5. f(-^/ + V^.
J\x2 + 1/ J a:2(x2+l>2
>v E^. 3. C{x + ay+a^ ^^^ ^x. 6. Cl^±l^^^^sU^ dx.
V Ex.4, f ^^^ Ex.7, f ^'^^ .
\ J(l + a:)(l + ar2)2 J (1 + a:2)8
The principles used in the preceding cases in the assump-
tion of the partial fractions may be summed up as follows ;
^ach of the denominators of the partial fractions contains
one and only one prime factor of the given denominator. When
a- repeated prime factor occurs., all of its different powers must
he used as denominators of the partial fractions.
The numerator of each of the assumed fractions is of degree
one lower than the degree of the prime factor occurring in the
corresponding denominator.
137. General theorem. — Since every rational fraction can
be integrated by first separating, if necessary, into simpler
fractions in accordance with some one of the cases considered
above, the important conclusion is at once deducible :
The integral of every rational fraction can he found., and is
expressihle in terms of algehraic, logarithmic, and inverse-trigo-
nometric functions.
CHAPTER IV
INTEGRATION BY RATIONALIZATION
At the end of the preceding chapter it was remarked that
every rational algebraic function can be integrated. The
question as to the possibility of integrating irrational func-
tions has next to be considered. This has already been
/touched upon in Chapter II, where a certain type of irra-
tional functions was treated by the method of reduction.
In the present chapter it is proposed to consider the
simplest cases of irrational functions, viz., those containing
^ax + h and -\/ ax^ -\-hx -\- c^ and to show how, by a process
of rationalization, every such function can be integrated.
138. Integration of functions containing the irrationality
\/aa? + 6. When the integrand contains ^ ax -f- 5, that is,
the wth root of an expression of the first degree in x^ but no
other irrationality, it can be reduced to a rational form by
means of the substitution
Ex.1.
le
dx
p + 3-1
ASSUDC
that is,
Then
and
V2a; + 3 = 2,
2a;+3 = z2.
dx = z dz,
C dx _Czdz -^^^^^^ 1^
•^V2a: + 3-l *^^-l
= V2 a; + 3 + log ( V2 a; -f 8 - 1).
231
232 INTEGRAL CALCULUS [Ch. IV.
Ex.2. Jl-f^"-^"- v^^^^
^. t:»_ « r 1 + a;6_— a;s — v^
a:* + a;
It would appear at first sight that this integrand contains several
irrationalities, viz., Vx, Vx, Vx, It is readily seen, however, that they
are all powers of Vx, and hence the substitution Vx = z will rationalize
the expression to be integrated. iryJ
tK Ex.3, f ^^ .^y^^^^l ()^Ex.6. f— ^^ tvL^^^'' ^
Ex 4 f ^^ - Ex.7, r ^^ ^ ♦> A'^'
Ex. 5. f ^? Ex. 8. f 4^^^-
/tH
When two irrationalities of the form Wax + 5, Vc^+^
occur in the integrand, the first radical can be made to dis-
appear by the substitution
} "Vax + 5 = 2.
The second radical then reduces to
^ a
r\ and the method of the next article can be applied.
y^-^
Integration of expressions containing Vax^ -\-bx + c.
Every expression containing ^ax^ + bx + c^ but no other
irrationality, can be rationalized by a proper substitution.
In order to make the necessary steps clearer, a geometrical
interpretation of the problem will be very useful.
To this end let the given radical be represented by y;
that is, let
i/^ = aa^-^bx + o, (1)
138-139.] INTEGRATION BY RATIONALIZATION
233
If now (a;, ^) be regarded as the rectangular coordinates of
a point in a plane, equation (1) represents a conic (Fig. 60).
Let (A, k'), or Q, be a given
point on this curve. The equa-
tion of any line through this
point is
i/-k = z(x-h), (2)
X
Fig. 60.
in which z is the slope of the
line. The line (2) will inter-
sect the conic in a second point
P. It is geometrically evident
that the coordinates (rr, «/) of P depend on the value of 2,
and in such a way that to each value of z corresponds only
one pair of values x^ y.
Consequently the variables x and y can be rationally
expressed in terms of the variable z. This is done by treat-
ing equations (1) and (2) as simultaneous, and solving for
X and y in terms of z.
For example, suppose it were desired to rationalize an
expression containing Vic^ _ 5 ^j ^ g.
Let
^2=a;2_5a.^.8,
and select (1, 2) for the point Q,
Then y-1 = z{x-V)
represents any line passing through Q. In solving these
two equations simultaneously for x and y^ the elimination
of y gives
z\x-Vf^-^z{x-r) = ^-bx^\,
This quadratic equation in x has two roots, one of which
should be a; = 1, since this is the value of x at C one of the
234 INTEGRAL CALCULUS [Ch. IV.
points of intersection. The other root, corresponding to the
variable point P, is
22—42 — 4
X = •
22-1
From this follows
or y = V;»^-5^ + 8=-'^^'-^^-^-
Two particular cases of the method given above deserve
to be noticed.
(a) When the conic intersects the x-axis.
In this case the quadratic expression aoc^ + hx-\-c has real
factors, say,
ax^ -\-hx-\- e = a(x — a) (a: — /3).
The conic (1) intersects the a;-axis in the two points
(a, 0) and (y8, 0), either one of which may be conveniently
selected for the point Q.
The equation of any line QP through the first point is
y = z(x-a^, (^)
and the equation of any line through the second point,
y = z(x-^y (A')
Either one of these equations, combined with (1), will
effect the desired rationalization.
(h) When the conic is an hyperbola.
This case occurs when the coefficient of a^ is positive.
The curve extends to infinity in two different directions,
namely, the directions of the asymptotes. If one of the
points at infinity on the curve be taken for the point Q, the
lines QP passing through this point are parallel to that
139. J INTEGRATION BY RATIONALIZATION 235
asymptote which touches the curve at Q. The equations of
the asymptotes are
Accordingly the lines parallel to the one asymptote are
and those parallel to the other
Either of these equations used in place of (2) will serve
equally well in expressing x and y{=-^a^ -\-hx-\- c) ration-
ally in terms of a new variable z,
Ex.l. ^ ^^
(x + V ar2 + 2 ar - 1)^
The conic y = y/x^ + 2 a: — 1 is an hyperbola and formula (B) can be
applied. This gives
Vx^ + 2 a: - 1 = x + Zy
whence by squaring and solving for x,
z^+1
^-2(l-z)'
and accordingly
Vx +2x 1- 2^^_^^
When these expressions are substituted in the given integral, it becomes
= i[-^ + 41og(l + .) + ^]
= l(x-y/x^+2x-l) + +21og[H-Vrr2+2a;-l-a:].
l_a:+Va;2+2ar-l
Since the conic y = Vx^ + 2 a: — 1 cuts the ar-axis, formula (A) [or
(^')] could be used for the purpose of rationalization.
236 INTEGRAL CALCULUS [Ch. IV.
Ex
-•/:
\/\ + xdx
The denominator being rationalized, the integrand takes the form
vT^^^
(1-xy
The conic
y = VI - a:2
intersects the a:-axis in two points (i 1, 0).
If the point (1, 0) be chosen for Q, the equation of any line passing
through this point is
y = z{x- 1).
The simultaneous solution of these two equations gives
22 _ 1 _2z
whence fVl^^, ^ f- 2.^.
= 2(— 2 + tan-^2)
\ a: — 1 X — \ I
Ex. 3.
J
<fx
(l_a:)(l-Vl-a:2)
Ex.4, r ^^
V2 a:2 _ .3 a; + 1 [V2 x2 - 8 a: + 1 + v^(a: - 1)]
140. From what precedes, combined with the theorem of
Art. 137, it follows that every rational function depending
only on x and the square root of a polynomial of not higher
than the second degree in x can be integrated, and the result
expressed in terms of known functions.
EXERCISES ON CHAPTER IV
^ r (-x^ + ^x)dx 2. r \ix-a)^-\'\dx
' J (x2 + 2)2Vx^Zri ' J 2(x - a)t - (x - a)i
[Substitute >/i^^n: = z.i 3. r.l£i+^Lzil^.
•'(x2+l)2(xa+2)i
139-140.]^
4. C ^
•^ X + y/x — 1
EC^MATION BY RATIONALIZATION 237
dx
dx
x + Vx^-1
-J
dx
8.
(a + a:)^
(2 - 3 a:-^) rfa:
J
a; — 3 376 + 5 a;i!
v^
'•I
Vx2 - lVv^TT + Va:-l
10
[Assume Vx + 1 + Va; — 1 = 2.]
of a:
J(a;2+a2)
Va;2 - a2
[A ssume x = a sec ^.]
11.
1+V^
Vx
ri + v
Jl + v
^a:.
12.
(/x
+ x (1 + a;)2
I Substitute ^-^ = ^8.1
L 1 + a; J
CHAPTER V
INTEGRATION OF TRIGONOMETRIC AND OTHER TRAN-
SCENDENTAL FUNCTIONS
141. In regard to the integration of trigonometric func-
tions, it is to be remarked in the first place that every
rational trigonometric function can be rationally expressed
in terms of sine and cosine.
It is accordingly evident that such functions can be inte-
grated by means of the substitution
sin x=- 2.
After the substitution has been effected, the integrand
may involve the irrationality
Vl — 252(= cos a;).
This can be removed by rationalization, as explained in the
preceding chapter, or the method of reduction may be
employed.
The substitution cos a; = 2 will serve equally well.
It is usually easier, however, to integrate the trigonometric
forms without any such previous transformation to algebraic
functions. The following articles treat of the cases of most
frequent occurrence.
142. jsec^**xdx, j cosec^^xdx.
In this case n is supposed to be a positive integer.
If sec^^a? dx be written in the form
Bec^'^x . sec'a; dx = (l-\- tan'a;)"-^ (tan a;),
(/
Ch. V. 141-143.] TRIGONOMETBIC FUNCTIONS 239
the first integral becomes
C(i2iYi^x + l)«-i<:7(tan x).
If (tanV + 1)""^ be expanded by the binomial formula
and integrated term by term, the required result is readily
obtained.
In like manner,
J cosec^"2J dx= \ cosec^"~^a; • cosec^a; dx
= -C^cot^x + l)"-^^(cotrc).
This last form can be integrated, as in the preceding case, '
by expanding the binomial in the integrand. (^ <
The same method will evidently apply to integrals of thei ^ j
foim \ f-
rtan"*a;sec^«a;cZrr, Jcot"*a; cosec^^o: 6^ic, V^/V
in which m is any number. \^ X "^^
dx K rn-cosxW^
cof*a:
^•1
2. J cosec^a; dx, "^ } ^^^^^ ^^^^^ (cos4^ _ sin^a:)^'"/^^
3. (sec^xdx, ' ^' f . ^"^ (=(tan-^xsec^xdx). f
J J sin^a; cos x -^ <\
Jdx g rcos% dx XJ^^^^
143. fsec'**ictan2»* + iicc?ic, J cosec"*a?cot2'» + ia5^iC.
In these integrands n is a positive integer, or zero, so that ^
2 w + 1 is any positive odd integer, while m is unrestricted./-- >w
240 INTEGRAL CALCULUS [Ch. V.
The first integral may be written in the form
J sec"*"^a; tan^"a; • sec x tan x dx
= j sec"*"^a; (sec^a; — V)'*d (sec a:),
which can be integrated after expanding (sec^ic — l)** by the
binomial formula.
Similarly,
J cosec"*a; cot^"^^a; dx = j cosec^""^a; cot^"a; • cosec x cot x dx
= — I cosec"*"^a; (cosec^a; — l)"c?(cosec2;).
EXERCISES
f^
1. j sec% tan^a; dx. 5. \ tan^x dx.
2. fcosecSarcotSarda;. 6. (^^^^^^^=(sec''-^xta.n^xdx\
J J cos'*x •'
o fsec ax -.^ ^ c ,
4. J sin X Go\?x dx. 8. J cot x dx.
y 144. (tun^^ocdx, KcoV^xdx,
The first integral can be treated thus :
j tan** a? dx= j tan""^ • tan^a; dx
= J tan"-2 a; (sec^ a; — 1 ) ia;
= ^.^^-Ct^n^-^xdx.
When w is a positive integer, the exponent of tan x may
be diminished by successive applications of this formula
until it becomes zero (when n is even), or one (when n
is odd).
143-144] TRIGONOMETRIC FUNCTIONS 241
In like manner,
J cot" a? dx=\ GoV^-^x Qoi^x dx
s= j cot'*~^a;(cosec2a;— V)dx
= — — — I GoV'^x dx.
n — 1 *^
Since tana; and cot a; are reciprocals of each other, the
above method is sufficient to integrate any integer power
of tan a;, or cot x.
Another method of procedure would be to make the
substitution tan x = z, whence
2" dz
ft3in''xdx=f^
P
If the exponent w is a fraction, say n = —^ the last integral
can be rationalized -by the substitution z = u^.
It is evident from this that any rational power of tangent
or cotangent can be integrated.
EXERCISES
1. J cot*xdx.
2. \ta.ii^(ixdx.
3. J (tan X — cot xy dx,
4. j'(taii«ar + tan«-2a;)da;.
5. I tan^ X dx.
When w is a positive integer show that^
6. f tan^n X dx = ^^^^^^ - *^B!!i!5 + •.•+(- l)«-i (tan x - x).
J 2n-l 2n-3 v / v /
7. (t^n^n+i^dx = *^^^ - ^-^^+ ... + (- l)~-Kitan2a:+logcos:r).
242 INTEGRAL CALCULUS [Ch. V.
145. Tsinw* X COS" x dx,
(a) Either m or n a. positive odd integer.
If one of the exponents, for example m, is a positive odd
integer, the given integral may be written
gijjw-i ^ (3Qgn ^ gjjj xdx= — j (1 — cos^ x) ^ cos" a:c?(cos x).
Since m is odd, w — 1 is even, and therefore ^ ~ is a
positive integer. Hence the binomial can be expanded into
a finite number of terms, and thus the integration can be
easily completed.
Ex. 1. j sin^a:Vcosarc?a:.
According to the method just indicated this integral can be reduced to
— J sin* a; Vcos x d(cos x) = — j (1 — cos2a;)2'(cosa;)^<f(cosa:)
= — I cos* a: + ^ cos2 X - ^cos i* a;.
Ex. 2. (sin^xdx. Ex. 5. f sin^xrfx
"Vcos a:
Ex.3, i sin^ a: cos* a; dx. ^ • a ^
•^ Ex.6. |_^1L£££_.
r^^cB^ ^^ — cos a;
Ex.4. (^2t£dx.
J sin a:
\ Y^ Q>) w + ^ an even negative integer.
In this case the integral may be put in the form
/sin"* X C
CQgm+» xdx= \ tan™ x sec'^™"^"^ x dx^
cos"* X ^
which can be integrated by Art. 142, since the exponent
— (w + w) of sec a; is an even positive integer.
145.] TRIGONOMETRIC FUNCTIONS 243
Ex.7, f^^rfa:.
COS^ X
The integration is effected in the following steps :
^ Vcosa; cos^ x
= I tan^ x (tan2 x + l)d (tan a;)
= 2tanta:(^+ ^tan2a;).
Ex. 8. i^-^^dx. Ex.11, f . /^ .
-^ sin* a; •^ sin* a; cos^ x
Ex. 9. f-;^. Ex.12, r
sin^a; -^ Vsin^ a; cos^ x
Ex.10. f5!:!?l^dx. Ex.13. fHH^dx.
-^ sni^a: -^ cos^+^a;
(c) Multiple angles.
When m and n are both even positive integers, integration
may be effected by the use of multiple angles. The trigo-
nometric formulas used for this purpose are
.2^ _
sma; cos a; =
Ex. 14. j sin^a; cos* a: c?a:.
1 — cos 2 X
sin^ X = )
1 -h COS 2 X
2 '
sin 2 a;
fsin^ X cos* xdx= \ (sin a: cos xY cos^ x dx
_ rsin2 2a: 1 + cos 2 a; .
~~J 4 2 ^
= i f sin2 2xdx+ t^ f sin2 2 a; cos 2 a; c? (2 a:)
1 ri — cos 4 a: ,^ , i sin^ 2 x
= -jJ^ a; — ^ sin 4 a: + j^ sin^ 2 a:.
244 INTEGRAL CALCULUS [Ch. V.
Ex.15. J cos* ar sin 2 a; rfar. Ex.17, j sin* a; cos* arrfx.
Ex.16, i silica: cos® a: rfa;. Ex.18. J (sin* a; — cos* x)*c?a;.
Integrate the two following by the aid of multiple angles.
Ex
.19- f^
J sin*
dx
<l
sin* X cos* X *
Ex. 20.
J sma:
dx
cos"x— sin" a: cos x
Integrate the following by any of the preceding methods.
Ex. 21. r^inlf dx = r(l - cos^a:)* rf^r = f (sec^ x-2 + cos^ a:) dx,
^cos^a; ^ cos^a; ^^ ^
Ex.24. (xW^^^dx.
Ex. 25. r
cos^
Ex.22. r22£!£j^.
•^ sin* x
J Ex. 23. f^'''-^%/x.
^ x^
[Substitute a: = a sin OJ]
doc
dx
xWx^ - a*
[Substitute x = a sec 0."]
dx
146. f f' , f- , ,
Write a H- 6 cos a; in the form
a^cos2 1 + sin2 1^ + j^cos^ | - sin^ |)
Then f ^^ ^ _1_ f
•/ a 4- ^ cos a; a — bJ a -\-
sec^- c?( -
Utan'l
a — 0 2
VS^TTp
tan"
tan
^ a —
146-146.] TBIGONOMETRIC FUNCTIONS 245
This result has a real form provided that a is numerically
greater than 5, since and a^ — b^ are then positive.
a — 0
When a is numerically less than b the integral may be
written
/-
sec^ r-dl -
dx ^ 2 /- 2 V2
b cos a; a — bj 2^__ ^ + <^
2 5 — a
a;
-1 , 2 ^5-a
log
tan- + \T-^--
2 ^b — a
The integration of
ci?a;
f—
6 sin a?
is effected by making the substitution
2^ = 3^ + 2'
which reduces the integral to
_^
b 0,0^ y
The preceding results may then be applied.
EXERCISES
2 C dx 2 r ^^
J a^sin^a: + ft^cos^ar * J a sin a: + 6 cos*
Suggestion. Write the denominator of Ex. 2 in the form
2a sin| cos?+ 6(cos2|- sin2^),
divide numerator and denominator by cos'^ ^, and replace tan r by a new
variable.
246.
INTEGRAL
CALCULUS [Ch.V.
3.
r dx
J5 + 3cos2a;
6. f ^^
J (asinx + ficosar)^
4.
J 5 - Ssina;
7. f ''^
J 1 + COS^I
5
(/a;
Jl-2sin2a:
147. \ e^ sin nx doc, \ ef*^ cos nx dx.
Integrate J e'*^ sin nx dx by parts, assuming
u = sin wa;, and dv = e°^ c?a;.
This gives
J e'*'^ sin nxdx = - e^ sin wa; j g"* cos nx dx, (1)
Integrate the same expression again, assuming this time
u = e"^, dv = sin nxdx.
Then
j e*** sin nxdx — e*^ cos nx-\-- Ae"^ cos 7ia:6?a;. (2)
Multiply (1) by - and (2) by - and add. The integrals
n CL
in the right members are eliminated, and the result is
r«« sin nx dx = «°"(«si»>»a;-ncns«^).
•^ • d?' + n^
By subtracting (1) from (2), the formula
Ce-' cos rmdx^ """ <^ "^" ^^ + ^ ^^^ ^^^^
is obtained.
EXERCISES ON CHAPTER V
1. Show that
(1 + n) J 8ec'»+2 xdx = tan x sec" a: + n J sec a: dx.
Integrate by parts, taking u = sec" x, dv = sec'* x dx.
146-147.] TRIGONOMETRIC FUNCTIONS 247
2. Show that
(1 + n) I cosec^+2 x dx = — cot x cosec« a; + «n J cosec" x dx,
J sin a; COS a; * J cos^ar
l 4 C ^^ 8. fc^cos-rfa;. .
[Put a; = cos ^.] 9. p^^^^dx, ^i
■ 5- J"^SiJF" "^ 10. j^'^sin"
•sin a: c?a;
xdx.
dx
6. i :~;r~* ' 11 r^'' sin 2 a: sin a: dx.
Jcoszsin^a; J
[Suggestion. 2 sin 2 a; sin a; = cos x — cos 3 a:.]
12. Show that
f sin aa: sin fta: ^a: = HLl^-^IL^ _ ££i^±*^
J 2(a-b) 2 (a + 6)
Use the trigonometric formula
sin a sin ^ = ^ [cos (a — P) — cos (a + ^)].
13. Show that
f sin ax cos 5a: ^a: = - "^,^ (^ " ^> - "",%(^ + ,^X
^ 2 (a - 6) 2 (a + 6)
14. Show that
f cos ax cos 6ar dx = ^^" ^^ " ^>/ + ^^^ f ^ "^ \)^.
J 2(a-b^ 2(a + b^
2(a-b) 2(a + b)
. j sin« a: cos* a; <?a;. 17. j (tan a; + cota:)®rfa:.
C # ■■■. /• i^a:
•^ sin2a;cos2a; * J(
(1 + cos a;) 8
,^
v
CHAPTER VI
INTEGRATION AS A SUMMATION
148. In the preceding five chapters various methods of
integration have been explained. The final object in every
case has been to determine a function F(x) such that its
derivative should be identical with a given function f(x).
It is now proposed to analyze this idea a little more fully,
and to show that it readily leads to a view of integration
which is of the highest interest and importance.
In order to obtain the derivative of a function F(x) it is
necessary in the first place to determine the increment
F{x + ^x)-F{x) (1)
which the function F{x) takes when the independent vari-
able X takes the increment Aa;.
The expression (1) can be put in a form more convenient
for present purposes, if it be assumed that for all values of x
under consideration F(x + Aa;) can be expanded by means of
Taylor's theorem [Art. 41, p. 66] . This expansion is
F(x + Aa;)
, = Fix) + F>(x)^x -f ?^ (i^xy + ^^^^(Ar)3 + ....
From this, by transposing FQx')^ the increment (1) is ob-
tained in the form of a series, viz.,
F{x-\-/:^')-F(x)
= F(.)A. + A.[^A.4-^(A.y-H ...J
n=/(a:)Aa;4-<^(a:)Aa;. (2)
248
Ch. VI. 148.] INTEGRATION AS A SUMMATION 249
In the last expression (/>(a;) has been written for brevity in
place of the series in brackets, and f^x) is the equivalent of
F'(x)^ since by supposition /(a;) is the derivative of F(x).
Suppose now that the variable x starts with a given value
a and increases until it reaches another given value h. The
function F(x) will change accordingly, beginning with the
value F(^a) and ending with F(h), The difference between
these two, viz.,
F(h) - F{a)
can be determined by the aid of (2) in the following manner.
Let the variation of x from a to 5 be imagined to occur in
successive steps, first from a to a-\- Ax^ then from a + Ax to
a-\-2 Axy and so on. The increment which the function F(^x}
takes at the first step of the change is F(a-\- Ax^ — F(a).
Its value is found by giving x the value a in formula (2).
That is,
F(ia + Arr) - F^a} =f(a)Ax + 0(a)Aa;.
The increment that F(x^ takes at the second step is
F(a + 2 Ax} - F(a + Ax) =f(a + A2:)A2; + <^(« + Aa;)A2;,
the right member of which is found by substituting x=a-\-Ax
in (2). In like manner, by giving x the values a-\-2Ax,
a + ^Ax, ..., a-{-(n — V)Ax, the additional equations are
found:
F{a + 3 A^^) -1^(^ + 2 Ax) =f(a + 2 A^^) Aa; + (/>(« + 2 A:r) Aa^,
F(a -{-4: Ax)- FQa + 3 Ax) ^f{a + 3 A2;)Aa; + ^(a + 3 Aa;) Aa;,
F{a-\-n Ax)-'F{a->rn-\ Ax)^ fia -\- n-\ Ax)Ax
•f <^(a + 71—1 Aa;)Aa;.
Assume a-\-nAx = 'b^ (3)
250 INTEGRAL CALCULUS [Ch. VI.
and substitute in the first term of the preceding equation.
The addition of the above n equations then gives
= Aa:[/(a)+/(a + A:r)+/(a + 2Aa;)+ .•• +/(« + ^T^l Arr)]
H-Aa;[(/)(a) + <^(a + A2;) + <^(a + 2A2:)H f-</)(« + w-l Aa:)].
This expression for F(h') — F(^a)^ while depending on the
given function /(a:), contains also a series of successive
values of <j)(x)^ viz.,
A2:[(^(a) + </)(«+ Aa^)+ .•• +(/>(« + /i— lAa:)].
This latter can be gotten rid of by taking its limit as Lx
approaches zero.
For, since
., . F"(x). , F'"{x^,. .2 ,
<^(^) = — ^Aa:+^^^(A2:)2+ ...,
it follows that
and hence, if ^ denote the numerically greatest term of the
series
<^(a)+(^(a + Arc)+ •-,
then
Ax [</)(a) + ^(a + Aa;) + •••] | < | Aa:[^ + O ••• (n terms)]
|<|Aa;-w<l>
\<\(h-a)^.
But since, on account of (4),
a"2o*=o.
it follows that
and hence
lim
= Ax'i 0 [/(«)+/(« + ^^) +•••+/(<» + »- 1 Ax)]A2:. (5)
148.] INTEGRATION AS A SUMMATION 251
The second member of (5) is denoted for brevity by the
symbol
and is called the definite integral of fQc) between the limits
a and h.
Suppose one of the limits, say the upper limit 5, is
regarded as variable, while the other has a fixed value.
To emphasize this assumption concerning the variability of
5, let it be replaced by the letter x. Then equation (5)
may be written
lim
•^(*) = Ail 0 [/(«) +/(«> + Aa;) + - +/(a + n-16.x)-\^x
+ Fia). (6)
Here the term FQa) has a fixed, although arbitrary, value
depending on the particular choice that is made for the con-
stant a. It may be regarded as a constant of integration.
Formula (6) expresses in two steps the solution of the
problem of determining the function F(x') :
(1) Find the sum of the series of n terms
fia), f{a + A:r), f(a + 2 Ao;), .-., /(a -V{n- l)A:i:),
these being the values of the given function f{x) corresponding
to the n equidistant values of x^
a, a + Arr, a + 2 Ax, •••, a -{- (n — 1) Aa;.
(2) Find the limit of the product of this sum hy Ax, as Ax
approaches zero while n increases to infinity, subject to the con-
dition nAx = X — a.
The addition of an arbitrary constant of integration makes
the solution the most general possible.
The method just formulated for determining the integral
F(x) of a given function f(x) is not suitable for the actual
252 INTEGRAL CALCULUS [Ch. VI.
work of integration, since, with few exceptions (cf. Exs. 1,
2 below), the summation of the series in the right-hand
member of (6) presents insuperable difficulties.
On the other hand, formula (5) admits of a very simple
geometrical or physical interpretation in most of the applica-
tions of the calculus, and herein lies one of its chief merits.
It places before one a very convenient and useful formulation
of many of the problems of geometry, mechanics, physics,
etc., the final solution of which is most readily effected by
the evaluation of the definite integral
'f(x)dx
in the following manner. First obtain the function F(x)
by integrating f{x)dx according to the methods already
explained in the preceding chapters.
Determine T'Q)) and JP(a) by substituting the limits h and
a in the result. Finally subtract ^(a) from FQ)), This
gives
£f(x)dx = F(h^ - Fii^oL)
as the value of the definite integral.
Ex. 1. Given /(x) = e*, find F{x) by the method of summation.
For the sake of brevity write Ax = A. Then formula (6) gives
^(^) = ^^^Q [e" + c*+* + e*-^" + - + e«+(«-i>*]A + F(a).
The sum in the right member may be written
gan ^. g» 4. g8» 4. ... 4. c(H-i)»]A = e«l^=^ • h
1 — e*
(by the formula for summing a geometric series)
1 — c*
= <-(^--i).-jf^-
148-149.]
INTEGRATION A 8 A SUMMATION
253
As h is made to approach zero the factor becomes indetermi-
e*— 1
nate. Its limit is found by the method of Chapter V (p. 77) to be
lim ^
4
1.
A = 0 e» _ 1
Hence ^^^^ e« [1 + e* + . . . + e(«-i)»]A = c« _ e«,
and accordingly F(x) = \ e'^dx z= e'^ — e** -\- F(a) = e* -j- C,
in which C{= F(a) — e«) may be regarded as an arbitrary constant of
integration.
Ex. 2. Given /(a:) = ax, find \ axdx by the method of summation.
149. Geometrical interpretation of the definite integral as an
area. Let the values of the function f(x) be represented by
the ordinates to a curve. Its equation would then be
It is proposed to find an expression for the area bounded by
this curve, the a;-axis,
and two ordinates AP
and BQ^ correspond-
ing to two given
values oi x^ x = a and
a; = 5, respectively.
Let the interval
from J. to 5 be
divided into n equal
intervals AA^^ ^i-^g,
• • •, An-\B each of
magnitude Aa;, so that
interval AB = h — a = nAx.
At each of the points of division A, A^, - - -, B erect ordi-
nates, and suppose that these meet the curve in the points P,
Pj, • • •, Q, Through the latter points draw lines PB^^
PiB^, • • •, P„_i-B^ parallel to the a;-axis.
A, A, A^^B
FiQ. 61.
254 INTEGRAL CALCULUS [Ch. VI.
A series of rectangles PA-^^ ^1^2.^ • • • is thus formed, each
of which lies entirely within the given area. These will be
referred to as the interior rectangles. By producing the
lines already drawn, a series of rectangles SA-^<, ^1^2' • • • is
formed which will be called the exterior rectangles. It is
clear that the value of the given area will always lie between
the sum of the interior, and the sum of the exterior rec-
tangles, or, expressed in a formula,
PA^ + P^A^ + ... + Pn-iB < area APQB < SA^ + S^A^
+ ... + S^_,B. ^ (7)
The difference between the sum of the exterior and the
sum of the interior rectangles is
SR^ + ^A + - + ^n-iBn = rectangle ^^1^= TQ . ^x.
If the function /(a;) does not become infinite as x varies from
a to 5, TQ will be finite and hence TQ • ^x will approach zero
simultaneously with Aa;. Hence the limit of the sum of the
exterior rectangles equals the limit of the sum of the inte-
rior rectangles. From (7) it follows that the area is equal
to the common limit of these two sums.
To determine this sum observe that
Rectangle APR^A^ = AP - AA^ =/(a) • Ax.
Similarly A^P^R^A^ =/(« + Aa;) • Ax,
A„_,P„_,R„B =fia + ir=l. Ax) . Ax,
Adding,
sum of rectangles
= [/(«) +/(« + A^) + - +/(« + ^ir^Ax')-]Ax,
Hence, by requiring Ax to approach zero,
Area APQB
= Ai'So[/(«) +/(« + ^^) + - +/('' + ^^^^^ Aa;)]Aa:. (8)
149-150.]
INTEGRATION AS A SUMMATION
255
The expression just obtained for the area is identical with
that occurring in the right-hand member of (5), and affords
one of the simplest and most interesting of the geometrical
interpretations of that formula. Thus
Xb r*h
f(x)dx = I ydo,
(9)
150. Generalization of the area formula. Positive and
negative area. Instead of taking the limit of the sum of
the interior (or exterior) rectangles, a more general pro-
cedure would be to take a series of intermediate rectangles.
Fig. 62.
Let x-^ be any value of x between a and a + Aa;, x^ any value
between a 4- Aa; and a + 2 Lx, etc. Then f{x^Lx would
be the area of a rectangle KLA^A (Fig. 62) intermediate
between PA^ and SA^ ; that is,
Likewise
PA^<f{x^Lx<SA^. ,
256 INTEOUAL CALCULUS [Ch. VI.
Hence
Sum of interior rectangles < [/(i^{)+f(^2)-\ — ] Aa;
< sum of exterior rectangles,
and therefore (cf. Fig. 61),
Area APQB = ^^ 0 [/(^i) +/C^2) + - +/(^n)] Aa:. (10)
If the area to be found is entirely above the a^-axis, the
ordinates are all positive. If at the same time Ax be taken
positive (that is, if 6 > a), formula (8) or (10) gives a posi-
tive sign to the area. On the other hand, the area is nega-
tive if below the a;-axis.
If the curve i/ =f{x) is partly above and partly below the
jc-axis, the value of the definite integral (8) will be repre-
sented by the algebraic sum of the positive and negative
areas limited by this curve.
151. Certain properties of definite integrals. From the
definition of the definite integral I f(x) dx as the limit of a
particular sum [formula (5), p. 250], certain important
properties may be deduced.
(a) Interchanging the limits a and b changes the sign of the
definite integral.
For if X starts at the upper limit h and diminishes by the
addition of successive negative increments ( — Aa;), a change
of sign will occur in formula (5), giving
F(ia^-Fih) = Jj(ix)dx.
Hence j^f^^^ ^^ = "X'^^^^ ^^' ^^^^
(3) If c he a number between a and b (a<c<b'), then
j^V(2:) dx =£fix) dx + J /(a:) dx, (12)
160-152.]
INTEGBATION AS A SUMMATION
257
((?) The Mean Value Theorem.
The area APQB (Fig. 63), which represents the numerical
value of the definite integral may be determined as follows :
Let an ordinate JfiVbe drawn
in such a position that
area PSN = area NB Q.
If f denote the value of x cor-
responding to the point iV, then
illfiV^ = /(f), and
Area APQB = rectangle ASBB
= M]Sr'AB=fQ)(b-ay
Hence,
jr>(;r)cfo=/(?)(6-«), (13)
in which f is some value of x between a and 5. This result
is known as the Mean Value Theorem. (Compare Art. 45.)
The theorem may be expressed in words as follows :
The value of the definite integral
^Jix)dx
is equal to the product of the difference between the limits by
the value of the function f(x) corresponding to a certain value
x= ^ between the limits of integration.
152. Definition of the definite integral when /(a?) becomes
infinite. Infinite limits. In the preceding sections it has
been assumed that /(a?) is always finite so long as x remains
within the prescribed limits. It is now necessary to examine
the cases in which /(a;) is infinite.
Suppose, in the first place, that f(x^ becomes infinite at
the upper limit x=b, but is elsewhere finite. In that event,
258 INTEGRAL CALCULUS [Ch. VI.
take for upper limit a value x — x\ which is less than 5,
a<x'<h. Then, according to the preceding results,
F(x') - F(a) ^^J(x)dx,
Now let x^ increase and approach h as limit. If at the
same time the integral
£'f(x)dx (14)
approaches a definite, finite limit, that limit will be defined
as the value of the integral
in the case under consideration ; that is,
jj(x-)dx = )%jj(x)d^. a/<b.
On the other hand the integral (14) may increase without
limit. When that happens, the integral will be said to have
an infinite value, or
j f(x)dx = QO.
In a similar manner, if f(a) = oo, the value of j f{x)dx
will be defined to be the limit of the integral
Cf{x)dx a<x^<h
as a/ diminishes and approaches a as limit.
Finally, if /(<?) = oo, where c is any number between a and
6, it is necessary to determine the meaning of
Cy(x)dx, and i f(x)dx
by the method just suggested, and then add the two results
in accordance with formula (12).
162.] INTEGRATION AS A SUMMATION 259
Heretofore the limits a, h have been assumed to be finite.
The case in which one of the limits, say 6, is infinite, is
readily disposed of by integrating from a to a finite upper
limit x\ and then considering the limit which the integral
approaches as x' increases to infinity. This limit, when one
exists, will be defined as the value of the integral, so that kC
An exactly similar mode of procedure is to be followed if / j «
the lower limit a is — qo, or if both limits are infinite. V
EXERCISES
Ex. 1. Prove, without performing the integration, ttiat
J-a^ + x' ' ' ji
Ex. 2. Without integrating show that
C^ xdx ^ r^^ xdx
Ex.3. If 2/ = ^(a:) and y = if/(x) are the equations of two curves
which are continuous between x = a and x = b, and such that to each
value of X (a<a;<6) corresponds but one value of y, prove that the
area bounded by these curves and the two ordinates x = a, x = b is
numerically equal to
il/(x)'\dx.
s/
Clcl>(x)
•fa,
Ex. 4. Prove that the area of the circle {x - hy ■\- {%f — kY — r^ is
equal to /^A+r , _
^ * 2\/r2- (x-hydx.
Jn-r
Ex.
5.
Evaluate
♦'o <
dx
32 4. a;2*
F.TT
6.
Evaluate
dx
Jx-1
Ex.
7.
Evaluate
V-
dx
0 (x - l)t
CHAPTER VII
GEOMETRICAL APPLICATIONS
153. Areas. Rectangular coordinates. It was shown in
Art. 149 that the area bounded by the curve y =/(a;), the
a:-axis, and the two ordinates a; = a, a: = 5, is represented by
the definite integral
jj(x)dx^j^ydx. ■ (1)
In an exactly similar manner it can be shown that the area
limited by the curve, the y-axis, and the two abscissas «/ = «,
«/ = yS, is represented by
xdy, (2)
£•
It was remarked at the end of Art. 150 that when h is
greater than a the integral (1) gives a positive or negative
result according as the area is above or below the a;-axis.
Similarly, if /S>a, the integral (2) gives a positive or
negative result according as the area which it represents is
to the right or left of the y-axis.
Whenever it is required to determine the area of a figure
which is partly on one side and partly on the other side of
the coordinate axis, it is necessary to calculate the positive
and the negative areas separately and add the results, each
taken with a positive sign. [Cf. Ex. 5, p. 262.]
154. Second method. Another method of determining
the area is based on the result of Art. 10, p. 23. It
was there shown that if z represents the area measured
260
153-154.]
GEOMETRICAL APPLICATIONS
261
from a fixed ordinate AP (at a; = «) up to an ordinate MB'
corresponding to a variable abscissa x, then the deriva-
tive of area with respect to
X is equal to the function
f(x) ; that is
or, in the differential nota-
tion,
dz = i/dx =f(x)dx,'
The area z may accordingly
be found by integrating/(a;).
Hence z = \f(x)dx + C.
Fio.64.
The value of the constant of integration Q is determined
by the condition that when x = a^ z must be zero, since in
that event the ordinate MJSf coincides with the initial
position.
Ex. Find the area bounded by the curve y = log x, the ar-axis, and
the two ordinates a; = 2, a; = 3.
Axesi APNM= (log xdx-\- C
= x(\ogx-l)-\-C.
Since the area is zero when
a; = 2, it follows that
0 = 2 log2-2 + C,
■ whence
C = 2-2 1og2.
Accordingly
X (log a; - 1) + 2 - 2 log 2
represents the area measured
from the ordinate a: = 2 up
to the variable ordinate MN.
When ar = 3 the required area
1) + 2 - 2 log 2 = log -2;^ - 1.
Fig. 65.
is found to be 3 (log 3
262
INTEGRAL CALCULUS
[Ch. VII.
EXERCISES
1. Find the area bounded by the parabola y = 4 ax% the a:-axisj
and the ordinate x = h.
2. Find the. area of the triangle formed by the line - + |= 2 and
the coordinate axes.
3. Find the area between the a:-axis and one semi-undulation of
the curve y = sin x.
4. Find the area bounded by the semi-cubical parabola y^ = ax^ and
the line x = 5.
5. Find the area between the curve y — sin^ x cos x and the a;-axis,
from the origin to the point at which a; = 2 tt.
Fig. 66.
An examination of the curve will show that the area is partly above
and partly below the z-axis. The curve crosses the axis at x = -, and
at a: =
2
The first portion of area, which is positive, is obtained by integrat-
ing from 0 to ^. The result is \. The next two portions of area are
negative, and are calculated by integrating from - to — ^. The result ia
3^
|. The last portion, which is positive, is found, by integrating from
to 2 TT, to be \. Hence total area = |4-f + | = f.
6. Find the area between the a;-axis and the curve y — a sin 4 x,
from the origin to x = tt.
7. Find the area bounded by the cubical parabola y = x*, the ^-axis,
and the line ^ = 8.
154-165.] GEOMETBICAL APPLICATIONS 263
8. Find the area bounded by the parabola y = x^ and the line
y = x. [Cf . Ex. 3, p. 259.]
9. Find the area bounded by the parabola y = x^ and the two lines
y = X, and y — 2 x.
10. Find the area bounded by the parabola y"^ = 4:px and the line
x= a, and show that it is two thirds the area of the circumscribing
rectangle.
What is the area bounded by the curve and its latus rectum?
11. Find the area of the circle x^ + y"^ + 2 ax = 0.
12. Find the area bounded by the coordinate axes, the witch
y = — , and the ordinate x = Xy By increasing x^ without limit,
find the area between the curve and the ar-axis.
13. Find the area of the ellipse ^ + ^L = i.
'14. Find the area of the hypocycloid x'^ + ys z= a*.
15. Find the area bounded by the logarithmic curve y = a*, the
a:-axis, and the two ordinates x = x^ x = x^. Show that the result is
proportional to the difference between the ordinates.
. Precautions to be observed in evaluating definite
integrals. The two methods just given for determining
plane areas are essentially alike in the processes required,
namely :
(1) to find the integral of the given function f{x) ;
(2) to substitute for x the two limiting values a and 5,
and subtract the first result from the second.
Erroneous results may be reached, however, by an in-
cautious application of this process.
In practical problems, the case requiring special care is
that in which f(^x) becomes infinite for some value of x
between a and h. When that happens, a special investiga-
tion must be made after the manner of Art. 152.
264
INTEGRAL CALCULUS
[Ch. vn.
Ex. 1. Find the area bounded by the curve y(x- 1)^ = c, the coordi-
nate axes, and the ordinate a; = 2.
A direct application of the formula gives
C^ cdx c ~|2 ^
area = \ — ^-^ — = — = — 2 c,
Joix-iy x-Uo
]h
is a sign of substitution, indicating that the values
a
ft, a are to be inserted for x in the expression immediately preceding the
sign, and the second result subtracted from the first.
This result is incorrect. A glance at the equation of the curve shows
that/(a;)j ==—£—— becomes infinite for x = \. It is accordingly
£C=2
Fia. 67.
necessary to find the area OCPA (Fig. 67) bounded by an ordinate AP
corresponding to a value x = x' which is less than 1. For this portion
the area f(x) is finite and positive, and formula (1) can be immediately
applied, with the result
area
Jo(x-iy (x-l)Jo x'-l
If now x' be made to increase and approach 1 as a limit, the value of
the expression for the area will increase without limit.
A like result is obtained for the area included between the ordinates
X = 1 and X = 2. Hence the required area is infinite.
Ex.2. Find the area limited by the curve y^ (x^ - a^y = 8 x*, the
coordinate axes, and the ordinate a; = 8 a.
165.] GEOMETRICAL APPLICATIONS
2x
265
Since /(a:) :
. becomes infinite for x = a,it is necessary in
(a;2 - a^)^
the first place to consider the area OP A (Fig. 68) and determine what
B X
Fia. 68.
limit it approaches as ^P approaches coincidence with the ordinate
X = a. Accordingly
area OPA = C ^^^^ = d(x^ - a2)il"'
= 3(a:'2 _ a2)i ^ 3 al, 0<a/<a.
Whence
^'/^^[areaOP^] =3 J. .
In the same manner, the area A'P'QB has the value
p 2xdx ::,6at-3(a/2~a2)i, a<a:'<3a.
As a:' diminishes towards a, the area increases to the limiting value 6 a^'
Hence, by adding the two results, the required area is found to be
3 at + 6 af = 9 ai
The same result is found by a direct application of (1), viz. :
•^' (a:2-a2)f J'
80 that in this case an immediate use of the area-formula gives the correct
result.
266
INTEGRAL CALCULUS
[Ch. VII.
Ex. 3. Find the area bounded by curve y — tan-^ar, the coordinate
axes, and the line x = 1.
In this problem we have to deal with a
many-valued function of x. In fact, to
each value of x corresponds an infinite
number of values of tan-^a:. The problem
accordingly has an indefiniteness which
must be removed by making some addi-
tional assumption.
The cm've y = tan-^ x consists of an in-
finite number of branches, corresponding
ordinates of which differ by integer multi-
ples of TT. Each branch is continuous for
all finite values of x (see Fig* 69). It is
evidently necessary to select one of these
branches for the boundary of the proposed
area, and discard all the others. Suppose, for example, the branch ^5 is
selected. The ordinate to this branch has the value tt when x is zero,
Y
^
A
B
r^
0
C
X
-=^^^-
X
-1
Fig. 69.
and increases continuously to tt +
to 1. Hence the required area is
as X increases continuously
f tan-i xdx = [a: tan-^a: - \\og (x^ + 1)]J
Ex. 4. Find the area of the parallelogram strip ABCO (Fig. 69).
Ex. 5. Find the area between the cissoid 2/^
2u-x
and its asymp-
tote x — la.
Ex. 6. Find the area inclosed by the curve xhp' - a^ (y^ _ x") and its
asymptotes.
Ex. 7. Find the area bounded by the curve aH — y{x - a), the x-axis,
and the asymptote x—a.
Ex. 8. Find the area included between the curve (2-x)y«=a:'(a:-l)*
and its asymptote.
Ex. 9. What restriction must be placed on the exponent k in order
that the area bounded by the curve (1 - a:)*y = 1, its asymptote a; = 1,
and the coordinate axes may be finite?
/
155-156.] GEOMETRICAL APPLICATIONS 267
156. Areas. Polar Coordinates. Let FQ be an arc of
a curve whose equation in polar coordinates is
P=/W- (3)
Let it be required to find the area bounded by this curve
and the two radii OP and OQ,
Draw from the origin a series
of radii OP^, OF^, .••, OP„_i at
equal angles A^. Let the coor-
dinates of the points P, Pj, Pg,
..., Q be («, a), (pi, ^i), (/32, l^a),
• ••, (5, /8). Draw the circle
arcs FB^R^, F^R^R^, .-. In
the circular sector FORi^
radius OF = a,
arc Pi^j = a • A^ ;
hence area P Oi^^ = |^ a^ A^.
Similarly area P^ Oi^g = 2 Pi^ ^^»
area P2 07^3 =ip2^A(9,
Fig. 70.
area F^_^OR„ = | pn-i^^O,
The sum of these sectorial areas is
(4)
This is an approximate value for the required area FOQ^
which is less than the true value by the amounts contained
in the neglected triangular portions FR^F^, F^R^F^, etc.
Suppose the figure FR^F^ revolved about 0 until it occupies
the position F'R^R^^ and similarly with F^R^P^t etc. Then
the sum of all the parts neglected is evidently less than the
268 INTEGRAL CALCULUS [Ch. VIL
strip P' R^' Ii„Pn-i, the area of which approaches zero as the
sectorial angle A6 is made to approach zero.
Hence area P0§ = ^^ oK«' + Pi" + P2^ + - + Pn-f)^^
-£Wd6.
Another method of procedure is illustrated in Ex. 1
immediately following.
Ex. 1. Find the area of the lemniscate p^ = a^ cos 2 0.
Let A denote the area of the sector
POQ measured from the polar axis to
an arbitrary radius vector OQ. The dif-
ferential of area is (Art. 88, p. 142)
dA = ip'^dO = ia^cos2 6 dO,
whence, by integration,
A =^a^(cos2edB
Fig. 71. =^%in2^+C.
4
If 6 were zero, the line OQ would occupy the initial position OP, and
the area would be zero. That is
^ = 0 when ^ = 0.
The substitution of this result in the preceding formula gives
0 = 0 + C.
Hence C = 0,
and A=^sm2e,
4
In order to find the total area of the figure put 0 = j. In this case
OQ will be tangent to the lemniscate at 0. On account of the symmetry
of the curve, the result obtained will be one fourth the total area, and
Ex. 2. Find the area of the cardioid p = a (1 — cos 0). .
Ex. 3. Find the area of the three loops of the curve p = a sin 3 ft
156-157.]
GEOMETRICAL APPLICATIONS
269
Ex. 4. Find the area bounded by the hyperbolic spiral pO = a and
the two radii p^, p^- Show that the area is proportional to the difference
between the radii.
Q
Ex. 5. Find the area limited by the parabola p = a sec^- and its
latus rectum. .
Ex. 6. Find the area of the circle p = 2a cos $.
Ex. 7. Find the area of the four loops of the curve p = a sin 2 ^. «/
Ex. 8. ^ AB - H^ i^ positive, the equation Ax^ ■{■ 2 H xy + By^= 1
represents an ellipse. By transforming to polar coordinates find its area.
kK.
157. Length of curves. Rectangular coordinates. Let it
f be required to determine the length of a continuous arc PQ
of a curve whose equation is written in rectangular coordi-
nates (x^ y).
It is first necessary to define what is meant by the length
of a curve. For this purpose,
suppose a series of points P^,
Pgi •••' ^n-i taken on the arc
PQ (Fig. 72), and imagine
the lengths of the chords
PPj, P^P^i,^ •••to have been
determined. The limit of
the sum of these chords as
the length of each chord
approaches the limit zero
will be defined, in accord-
ance with accepted usage, as the length of the arc PQ;*
that is,
arc P 5 = Lt (chord PP^ + chord P1P2 + • • • + chord P„_i 0.(5)
Fig. 72
♦ That this limit is always the same no matter how the points P< are chosen,
so long as the curve has a continuously turning tangent, and the distances
Pi-iPi are all made to tend towards zero, admits of rigorous proof. The proof
is, however, unsuitable for an elementary text-book. [See " Rouch^ et Com-
berousse. Traits de g^om^trie." Paris, 1891, part I, p. 189.]
270 INTEGRAL CALCULUS [Ch. VII.
This definition is immediately convertible into a formula
suitable for direct application.
For, let the points Pj, Pg' "* ^^ ^^ chosen that
the lines P-Bj, etc. being drawn parallel to the a;-axis.
Denote by A^ the increment MtPi of i/. Then the chord
P,_iPj has the length
It is clear that — ^ is the value of -^ corresponding to
Ax dx ^ ^
some point of the curve between P^.j and P,. [Cf. Art. 45.]
Hence, by substituting in (5) and using the principle
employed in deriving the area-formula (10), Art. 150,
in which (x\ t/'} and (x", y'') are the coordinates of P and
Q respectively.
The same result would also be obtained by integrating the
expressions for the derivative of arc, (1) and (2), p. 139.
Ex. 1. Find the length of arc of the parabola y^ = ipx measured
from the vertex to one extremity of the latus rectum.
In this case -^ = \^,
dx ^x
and hence length of arc = J "Vl + — rfx.
Jo X
Ex. 2. Find the length of arc of the semi-cubical parabola ay^ = r*
from the origin to the point whose abscissa is ^^
157-158.] GEOMETRICAL APPLICATIONS 271
Ex. 3. Find the entire length of the hypocycloid xt + 3/3 = a^.
Ex. 4. Find the length of arc of the circle (x-hy + (y-ky = r\
X X
Ex. 5. Find the length of arc of the catenary ?/ = ^ (e* + e «) from
the vertex to the point {x^, y^).
Ex. 6. Find the length of the logarithmic curve y = log x from a; = 1
to a; = \/3.
Ex. 7. Find the length of arc of the evolute of the ellipse
jAv^
158. Length of curves. Polar coordinates. When the
equation of the curve is given in polar coordinates, let the
points Pj, Pg' •**' -^^-i ^® ®^ chosen that the vectorial angle 6
Fig. 73.
increases by equal increments A^ in passing from a point P^
on the curve to the next succeeding point P^+j. Draw the
lines PiRi^i perpendicular to the radii OPi+i. Then
272 INTEGRAL CALCULUS [Ch. VII.
chord P,P,^,= ^P,R,J + R,^,P,^? [Cf. p. 141.]
= VO sin A^)2 + (^ + A/5 - /o cos M)^
The limit of the sum of all such chords will be, according
to definition, the length of the arc PQ. Hence
in which (/a', ^'), (^p"^ 6") are the coordinates of P and Q
respectively.
Ex. 1. Find the length of arc of the logarithmic spiral p = e"^ be-
tween the two points (p^, O^) and (p2, 6^'
Since -^ = ae«*,
it follows that p — = -,
dp a
and length of arc = y'\\ -\-ldp = Va-^ + 1 (pj - p,).
Ex. 2. Find the length of arc of the cardioid p = a (1 - cos ^).
Ex. 3. Find the length of the cissoid p = 2 a tan 6 sin 0 from 0 = 0
to 6 = '^-
Ex. 4. Find the entire length of the curve p = 2a sin 9. ^
Ex. 5. Find the length of the parabola pz=i a sec^ ^ between the
points (p,, d,) and (pg, 6.^).
Ex. 6. Find tlie length of arc of the hyperbolic spiral pd = a between
the points (p,, ${) and (p.^, $^).
V^
158-159.] GEOMETRICAL APPLICATIONS 273
159. Measurement of arcs by the aid of parametric repre-
sentation. When the coordinates of a point on a given
curve can be conveniently expressed in terms of a variable
parameter, the problem of calculating its length is often
simplified.
Suppose a curve has its rectangular coordinates expressible
in the form
in which <^(t)^ "^(f) ^^^ single-valued functions of the vari-
able U Then
dl
dy _dt -J _d^ ;,x
dx dx^ "" dt '
dt
and formula (6), p. 270, becomes
rv(i
in which t\ t'^ are the values of t corresponding to the ex-
tremities of the arc whose length is to be found.
In like manner, if (p, 6) are expressible in terms of a third
variable f, formula (7), p. 272, becomes
j:m'<ih
/ Ex. 1. Find the length of a complete arch of the cycloid
X = a{0 — sin ^),
y = a(l — cos 6).
V Ex. 2. Find the length of the epicycloid
X = a(m cos t — cos mt), y = a(m sin t — sin m{)
from t = 0 to t= -?^!^.
m — 1
^ Ex. 3. Find the length of the hypocycloid x^ -\- yf = as by expressing
X and y in the form x = a cos^O, y = a sin^ 0.
274
INTEGRAL CALCULUS
[Ch. VII.
' Ex. 4. Find the length of the involute of the circle
X = a(cos 0 -\-$sin6)f y = a(sin ^ — ^ cos 0)
. from ^ = 0 to ^ = ^1-
y^ Ex. 5. Find the length of arc of the curve x^ — y^ = a^, from (a, 0)
to i^v Vi)'
Assume a; = asec^^, ?/ = a tan^^.
\^Ex. 6. Find the length of arc of the ellipse ^ + 3^ = 1.
Putting X = acos<f>, y = b sin <j>,
complete arc = J Vl — e^ cos^ ^ d<^
= 2 ATT [1 - I
.4_ ...
64
],
by expanding v 1 — e^ cos^ </> into a series and integrating term by term.
y Ex. 7. Find the length of arc of the curve
x — e^ sin 6, y = e^ cos 6
from ^ = 0 to ^ = ^1.
fi^
160. Area of surface of revolution. Let ^^ be a con-
tinuous arc of a curve whose equa-
tion is expressed in rectangular
coordinates x and y. It is required
to determine a formula for the
area of the surface generated by
revolving the arc AQ about the
~ a;-axis.
It has been shown in Art. 86,
p. 140, that if 8 denotes the area
of the surface generated by the rotation of AP (P being
a variable point with coordinates (rr, y)), then
Fig. 74.
dS
ds
= 2 7r^.
from which
f— W..(|T.
and
f-'W.*(|)-
159-1*60.] GEOMETBICAL APPLICATIONS 275
Hence, by integrating these two expressions,
surface = 2 rr,j^y^^lj^(^Jdx
the limiting values of x and y being the coordinates of the
points A and Q.
That the result of integration is to be evaluated between
the limits a and h (or a and y8) is readily seen by following
the suggestions made in Art. 154. For, denoting the
indefinite integral
by (f> (x) + (7, since the area is evidently zero when a: = a
(^.e., when the point P coincides with A) it follows that
</,(«) +(7=0,
whence C = — ^(a).
Moreover, when P coincides with Q the required surface is
determined, and therefore
surface generated hy AQ = <^ (J)) -\- C= (f)(b}— ^(<«).
But according to Art. 148, <^(5) — <^(«) is the definite inte-
gral obtained by evaluating (10) between the limits a and b.
In like manner it is found that the area of the surface
obtained by revolving AQ about the 2/-axis is
In each of the above cases there is a choice of two formu-
las for the area. That one should be selected which can
most easily be integrated.
276 INTEGRAL CALCULUS [Ch. VII.
Ex. 1. Find the surface of the catenoid obtained by revolving the
X X
catenary y = ^ (e* + e **) about the y-axis, from x = Otox = a,
Since ^ = Ke*-e~«),
dx
it follows that
\dxl 4
and hence, by using the second formula "of (11), the required surface has
the area
2 IT J a;(e« + e <^)dx.
Jo
Ex. 2. Find the surface obtained by revolving about the y-axis the
quarter of the circle x^ + y^+ 2 x + 2y -h 1 = 0 contained between the
points where it touches the coordinate axes.
Ex. 3. Find the surface generated by revolving the parabola y^ = ipx
about the a:-axis from the origin to the point (p, 2p).
Ex. 4. Find the surface generated by the revolution about the y-axis
of the same arc as in Ex. 3.
Ex. 5. Find the surface generated by the revolution of the ellipse
(a) about its major axis (the prolate spheroid) ;
(6) about its minor axis (the oblate spheroid).
Ex. 6. Find the surface generated by the revolution of the cardioid
p = a(l 4- cos 6) about the polar axis.
Regarding the figure as referred in the first place to rectangular axes
such that x — p cos B, y = p sin 0 we have
surface = 2 tt f y ds = 2 tt f "psin^-ypHf-^Vrf^,
since ds =^l^ + (^^dO by Art. 87.
Ex. 7. Find the surface of the cone obtained by revolving that portion
of the line - + ^ = 1 which is intercepted by the coordinate axes,
a b
(a) about the X-axis;
(^) about the ^-axis.
160-161.]
GEOMETRICAL APPLICATIONS
/
277
Ex. 8. Find the surface of the sphere obtained by revolving the circle
p = 2a cos 0 about the polar axis. [Cf . Ex. 6.]
Ex. 9. Find the surface generated by the revolution of a complete arch
of the cycloid x = a{$— sinO), ^/ = a(l — cos^) about the a:-axis.
Fig. 75.
161. Volume of solid of revolution. Let the plane area,
bounded by an arc FQ of a given curve (referred to rectangular
axes) and the ordinates at
the extremities P and Q, be
revolved about the a;-axis.
It is required to find the
volume of the solid so
generated.
Let the figure AFQB
be divided into n strips of
width Ax by means of the
ordinates A^P^, -^2^2' ***'
^n-iPn-i' In revolving
about the a;-axis, the rec-
tangle APR^A^ generates a cylinder of altitude Ax, the area
of whose base is ir . AJ^. Hence
volume of cylinder = tt • AP • Ax.
The volume of this cylinder is less than that generated by
the strip APP^A^ by the amount contained in the ring gen-
erated by the triangular piece PR^P^ Imagine this ring
pushed in the direction parallel to the ic-axis until it occupies
the position of the ring generated by QBE. If every other
neglected portion (such as is generated by Pi_iPiRi) is
treated in like manner, it is evident that their sum is less
than the volume generated by the strip A„_iP„_iQB, and
hence has zero for limit as Ax approaches zero. Therefore
278 INTEGRAL CALCULUS [Ch. VII.
the sum of the n cylinders generated by the interior rectan-
gles of the plane, viz.,
ttCAP + A^^ + ... + A^^P,J)^x,
has for limit the volume required. But the limit of this sum
is [by formula (5), p. 250] the definite integral j iry^dx^ and
hence ^6
volume = TT I y^dx.
The same result is readily obtained by integrating the
expression for the derivative of volume. [Art. 85^ p. 140.]
The volume generated by revolution about the «^-axis is
found by a like process to be expressed by the definite
integral
in which a and fi are the values of y at the extremities of
the given arc.
Ex. 1. Find the volume of the oblate spheroid obtained by revolving
the ellipse -^ + ^ = 1 about its minor axis.
Ex. 2. Find the volume of the sphere obtained by revolving the
circle x^ ■\- {y — ky = r^ about the y-axis.
Ex. 3. The arc of the hyperbola xy = k^, extending from the vertex
to infinity is revolved about its asymptote. Find the volume generated.
What is the volume generated by revolving the same arc about the
other asymptote ?
Ex. 4. Find the entire volume obtained by rotating the hypocycloid
X* + y> = as about either axis.
Ex. 5. Find the volume obtained by the revolution of that part of
the parabola Vx -f- y/y = y/a intercepted by the coordinate axes about
one of those axes.
J Ex. 6. Find the volume generated by the revolution of the witch
y = 2^ — . about the x-axis.
^ a«+4a2
161-162.] GEOMETRICAL APPLICATIONS 279
Ex. 7. Find the volume generated by the revolution of the witch
about the ?/-axis, taking the portion of the curve from the vertex (x = 0)
to the point (x^, y^).
What is the limit of this volume as the point (x^, y^ moves tov^ard
infinity ?
Ex. 8. Find the volume obtained by revolving a complete arch of the
cycloid a; = a (^ — sin ^), y = a(l — cos &) about the ar-axis.
Volume = TT f ''''y^dx = Tra^ f ''(I - cos dydO.
Ex. 9. Find the volume obtained by revolving the cardioid
p = a{l — cos 6) about the polar axis.
Assume a; = p cos ^, ^ = p sin 6.
Then dx=: d(p cos 6) = d[a{l — cos 0) cos $] "
= asin^(-l + 2cos^)<f^.
Hence
Volume = 7r( y^dx = - 7ra^Csin»0(l - cos $y(l - 2 cos e)d$.
Ex. 10. A quadrant of a circle revolves about its chord. Find the
volume of the spindle so generated.
The equation of the circle being taken in the form
the a:-axis can be assumed as the axis of rotation. The ordinates of the
rotated arc are determined by the formula
^ 162. Miscellaneous applications. In the preceding article
the volume of the solid of revolution is shown to be the
limit of the sum of the volumes of a series of cylindrical
plates of thickness Lx. The notion here involved is, with
suitable modifications, applicable to a variety of problems.
The following examples (excepting Exs. 6, 9, 10) are illus-
trations of this principle.
280 INTEGRAL CALCULUS [Ch. VII.
Ex. 1. Find the volume of the ellipsoid ^ + ^ + 5? = 1.
^ a^ h^ c^
Imagine the solid divided into a number of thin plates by means of
planes perpendicular to the x-axis and at equal distances Aa; from each
other. Regard the volume of each plate as approximately that of an
elliptic cylinder of altitude Aa;. The base of the cylinder will be the
ellipse in which the ellipsoid is intersected by one of the cutting planes.
If the equation of this plane be denoted by a: = A, the equation of the
elliptic base of the cylinder is (in y, z coordinates)
or j^—^ + -7—^ =1.
The semi-axes of the ellipse are
-v^^^rx"2 and ^v/^TTlp.
a a
Since the area of the ellipse is the product of the semi-axes multiplied
by TT (Ex. 13, p. 263), it follows that
area of elliptic base = tt . - Va^ - A^ • - Va^ - A*
a a
= ^(a'»-A2),
and volume of elliptic cylinder = ^^ (a^ — A') AA,
(AA being used in place of Aar since x = A).
The result of summing all such terms and taking the limit as AA ap-
proaches zero is equivalent to the definite integral
£^ l±^(a^ _ Xi)dX = 2^' ^ (a« - A'') dX,
On account of the ellipsoid being symmetrical with respect to the
plane x = 0, the limits 0 and a include one half the required volume
and hence instead of using limits —a and -f- a it is more convenient to
write the definite integral in the above form.
Ex. 2. Find by the method of Ex. 1 the volume of the elliptic cone
1)'.
measured from the yz-plane as base to the vertex (1, 0, 0).
Ex. 3. Find the volume of a pyramid of altitude h and of base-area A,
162.]
GEOMETRICAL APPLICATIONS
281
Ex. 4. Given an ellipse ^ + ^ = 1. On the major axis a plane
rectangle A BCD is con-
structed perpendicular to
the plane of the ellipse.
Through any point P of
the line CD a plane is con-
structed perpendicular to
CD. The two points R
and S in which, the latter
plane meets the ellipse are
joined to P by straight
lines. The totality of all
lines so determined form
a ruled surface called a ^^^' '^^'
conoid. Given AC =pj find the volume of the above conoid.
Ex. 5. A rectangle moves from a fixed point P parallel to itself, one
side varying as the distance from P, and the other as the square of this
distance. At the distance of 2 ft., the rectangle becomes a square of 3
ft. on each side. What is the volume generated ?
Ex. 6. A string ^J3 of length a has a weight attached at B. The
other extremity A moves along a straight Une OX drawing the weight
r
A
Fig. 77.
in a rough horizontal plane XOY. The path traced by the point B is
called the tractrix. What is its equation ?
282 INTEGRAL CALCULUS [Ch. VII.
Let OF be the initial position of the string and AB any intermediate
position. Since at every instant the force is exerted on the weight B in
the direction of the string BA, the motion of the point must be in the
same direction ; that is, the direction of the tractrix at B is the same as
that of the line BA and hence BA is a tangent to the curve. The
expression for the tangent length is (Art. 79, p. 130)
dx
Solving for -—,
dy
dx
= V"-
dy ^ 2/^
Integrating with respect to y,
=1
^ dy = V a^ - y'^ — a log — + C.
The constant of integration is determined by the assumption that (0, a)
is the starting point of the curve. Substituting these coordinates in the
above equation we find C = 0.
Ex. 7. A woodman fells a tree 2 feet in diameter, cutting halfway
through on each side. The lower face of each cut is horizontal, and the
upper face makes an angle of 60° with the lower. How much wood does
he cut out?
Ex. 8. The center of a square moves along a diameter of a given circle
of radius a, the plane of the square being perpendicular to that of the
circle, and its magnitude varying in such a way that two opposite vertices
move on the circumference of the circle. Find the volume of the solid
generated.
Ex. 9. The equiangular spiral is a curve so constructed that the angle
between the radius vector to any point and the tangent at the same
point is constant. Find its equation.
Ex. 10. Determine the curve having the property that the line drawn
from the foot of any ordinate of the curve perpendicular to the cor-
responding tangent is of constant length a.
162.]
GEOMETRICAL APPLICATIONS
283
If the angle which the tangent makes with the a:-axis be denoted by
<^, it is at once evident that
y
= cos <f>
V1 + taii2</>
V
1+f^y
\dx,
From this follows
log(y+-\/y^-a^) + a
When the tangent is parallel
to the a:-axis the ordinate itself
is the perpendicular a. If this
ordinate be chosen for the y-axis
the point (0, a) is a point of the
curve, and hence
C = — log a.
The equation can accordingly
be written
Fig. 78.
(1)
y + y/f
From this follows, by taking the reciprocal of both members,
= e «>
or, rationalizing the denominator,
(2) ^ ^ = e «.
^ ^ a
2
Adding (1) and (2) and dividing by ->
* _x
y = ^(e'a + e «),
which is the equation of the catenary.
Ex. 11. A right circular cone having the angle 2 ^ at the vertex has
its vertex on the surface of a sphere of radius a and its axis passing
through the center of the sphere. Find the volume of the portion of the
sphere which is exterior to the cone.
X^ ^2
Ex. 12. Find the volume of the paraboloid —-\-^=z cut off by the
plane z = c. ^
284 INTEGRAL CALCULUS £Ch. VII.
EXERCISES ON CHAPTER VII
1. Find the area bounded by the hyperbola xy — a\ the a:-axis, and
the two ordinates x = a, x = na.
From the result obtained, prove that the area contained between an
infinite branch of the curve and its asymptote is infinite.
Cj 2. Find the area contained between the curves y^ = x and a^ = y,
3. Find the area of the evolute of the ellipse
(ax)^ + (by)^ = (a"^ - b^)l
4. Find the area bounded by the parabola Vx + y/y = Va and the
coordinate axes.
5. Find the area contained between the curve
^ a + x
and its asymptote x = — a.
[Hint. The integration may be facilitated by the substitution
X = a cos $.']
6. Find the area between the curve y\y^ — 2) = a: — 1 and the
coordinate axes»
V 7. Find the area common to the two ellipses
8. If (a, a) and (6, P) are two pairs of values of x and y, the
formula for integration by parts gives
J y dx = bp — aa ~ i xdy.
Interpret this result geometrically in terms of area.
9. Find the area bounded by the logarithmic (or equiangular) spiral
p = e«« and the two radii p^, p^
10. Find the length of an arc of the spiral of Archimedes p — aO
f between the points (pj, ^,), (pg, 6^.
J 11. Find the surface of the ring generated by revolving the circle
a:2 + (y - ky = a^ {k>d) about the a>axi8.
12. Find the volume of the ring defined in Ex. 11.
162.] GEOMETRICAL APPLICATIONS 285
13. Find the volume obtained by revolving about the a;-axis that
X X
portion of the catenary ^ ~^ (e" + e~*)
limited by the points (—x^, y^) and (xj, y^,
14. Find the entire volume generated by the revolution of the cissoid
a — X X /
about its asymptote. i \ *
[Hint. For the purpose of integration, assume tyt*
x = 2a sin2 6, whence y = 1^1^. 1
cos 6 J
15. Find the surface generated by the rotation of the involute of the
^ circle x = a(cos t + t sin t), y = a(sin t — t cos t)
about the a:-axis from t = 0\>o t = ty
16. Find the volume generated by the revolution of the tractrix (see
Ex. 6, p. 281) about the positive ar-axis.
17. Find the area of the surface of revolution described in Ex. 16.
, / 18. Find the length of the tractrix from the cusp (the point (0, o))
to the point (xj, y^).
CHAPTER VIII
SUCCESSIVE INTEGRATION
163. Functions of a single variable. Thus far we have
considered the problem of finding the function y oi x when
-^ only is given. It is now proposed to find y when its nth
derivative -^ is given.
The mode of procedure is evident. First find the func-
tion ^_^ which has -y^ for its derivative. Then, by inte-
d^'-^y
grating the result, determine ■, n_2, and so on until after n
successive integrations the required result is found. As an
arbitrary constant should be added after each integration in
order to obtain the most general solution, the function y will
contain n arbitrary constants.
Ex. 1. Given ^4 = -«» ^^^ V'
dx^ x^ ^
Integration of — with respect to x gives
3r
H^o+C'^
dx^~ 2x2
Integrating a second time,
dx 2x * *
and finally y = i log a: + i C^x^ + CgX + C^
The triple integration required in this example will be symbolized by
which will be called the triple integral of — with respect to x.
Ch. VIII. 163.] SUCCESSIVE INTEGRATION 287
Ex. 2. Determine the curves having the property that the radius of
curvature at any point P is proportional to the cube of the secant of the
angle which the tangent at P makes with a fixed line.
If a system of rectangular axes be chosen with the given line for
a;-axis, it follows from equation (6), p. 164, and from Art. 10, that
in which a is an arbitrary constant. This equation reduces to
d^ y
dx^
from which follows
:^a,
y = I Ja(rfx)2 =? a[|' + C^x + C2],
C, and C2 being constants of integration. Hence the required curves
are the parabolas having axes parallel to the y-axis.
The existence of the two arbitrary constants Cj, Cg in the preceding
equation makes it possible to impose further conditions. Suppose, for
example, it be required to determine the curve having the property
already specified, and having besides a maximum (or a minimum) point
at (1, 0).
Since at such a point -^ = 0, it follows that
dx
0 = a(l + Ci),
whence Cj = — 1.
Also, by substituting (1, 0) in the equation of the curve,
0 = a(i-l + C2),
from which Cg = i-
Accordingly the required curve is
Ex. 3. Find the equation (in rectangular coordinates) of the curves
having the property that the radius of curvature is equal to the cube of
the tangent length.
[Hint. Take y as the independent variable.]
288 INTEGRAL CALCULUS [Ch. Vlll.
Ex. 4. A particle moves along a path in a plane such that the slope
of the line tangent at the moving point changes at a rate proportional to
the reciprocal of the abscissa of that point. Find the equation of the
turve.
Ex. 5. A particle starting at rest from a point P moves under the
action of a force such that the acceleration (cf. Ex. 14, p. Ill) at each
instant of time is proportional to (is k times) the square root of the time.
How far will the particle move in the time t'i
164. Integration of functions of several variables. When
functions of two or more variables are under consideration,
the process of differentiation can in general be performed
with respect to any one of the variables, while the others
are treated as constant during the differentiation. A repe-
tition of this process gives rise to the notion of successive
partial differentiation with respect to one or several of the
variables involved in the given function. [Cf. Arts. 68, 72.]
The reverse process readily suggests itself, and presents
the problem : Griven a partial (^first^ or higher) derivative of a
function of several variables with respect to one or more of these
variables^ to find the original function.
This problem is solved by means of the ordinary processes
of integration, but the added constant of integration has a
new meaning. This can be made clear by an example.
Suppose u is an unknown function of x and y such that
dx
Integrate this with respect to x alone, treating y at the
same time as though it were constant. This gives
in which ^ is an added constant of integration. But since
y is regarded as constant during this integration there is
nothing to prevent <f> from depending on it. This depend-
163-165.] SUCCESSIVE INTEGRATION 289
ence may be indicated by writing <^(y) in the place of (j).
Hence the most general function having 2 a: 4- 2 ?/ for its
partial derivative with respect to x is
U = X^+2X7/ -^(l>(7/},
in which <^(y) is an entirely/ arbitrary function of y.
Again, suppose
dxdy
Integrating first with respect to y^ x being treated as
though it were constant during this integration, we find
where '>^(x) is an arbitrary function of x^ and is to be
regarded as an added constant for the integration with
respect to y.
Integrate the result with respect to a;, treating y as con-
stant. Then
Here <!>(«/), the constant of integration with respect to x^
is an arbitrary function of «/, while
'^(x)=^'>\r(x)dx.
Since '>^(x) is an arbitrary function of x^ so also is "^(x).
165. Integration of a total differential. The total differen-
tial of a function u depending on two more variables has
been defined (Art. 69) by the formula
du=^^dx^^-^dy.
dx dy ^
The question now presents itself: Given a differential
expression of the form
Fdx + Qdy, (1)
290 INTEGRAL CALCULUS [Ch. VIII.
wherein P and Q are functions of x and y^ does there exist
a function u of the same variables having (1) for its total
differential P
It is easy to see that in general such a function does not
exist. For, in order that (1) may be a total differential of a
function «*, it is evidently necessary that P and Q have the
form
P = ^, Q = ^. (2)
dx By
What relation, then, must exist between P and Q in order
that the conditions (2) may be satisfied ? This* is easily
found as follows : Differentiate the first equation of (2) with
respect to y^ and the second with respect to x. This gives
dP^^u_ dQ ^ d^u
dy dydx dx dxdy
from which follows
dP^8Q (g.
dy dx ^ ^
This Is the relation sought.
The next step is to find the function u by integration. It
is easier to make this process clear by an illustration.
Given (2x + 2y-\-2')dx+(2y + 2x-\-2)dy,
find the function u having this as its total differential.
Since P=2a; + 2y + 2, Q = 2y + 2x + 2,
it is found by differentiation that
^ = 2 and ^=2,
dy dx
and hence the necessary relation (3) is satisfied.
From (2) it follows that
|^=2a; + 2y + 2.
dx
165.] SUCCESSIVE INTEGRATION 291
Integrating this with respect to x alone,
u = x^-{-2xy + 2x + (l>(iy). (4)
It now remains to determine the function <^(^) so that
^(=(?)=^^ + 2^ + 2. (5)
Differentiate (4) with respect to y alone, whence
dy
where <j>\y) denotes the derivative of <^C^) with respect to
y. The comparison of this result with (5) gives
2y4-2r?:+2 = 2rr + </)'C«^),
or <^'(^)=2^ + 2, (6)
whence, by integrating with respect to y^
Ky^^y^+'iy + O,
in which O is an arbitrary constant with respect to both
X and y.
Hence u = x^-\-2xy-\-2x-iry'^ + 2y + Q.
It is to be remarked that in integrating (6) we integrate
exactly those terms in Q which do not contain x. Hence
the following rule may be formulated for integrating a total
differential :
Integrate P with respect to x alone^ treating y as constant.
Then integrate with respect to y those terms of Q which con-
tain y hut do not contain x^ and add the result^ together with
an arbitrary constant 0, to the terms already obtained.
It is evident that it would be equally well to first inte-
grate Q with respect to ?/, and then integrate those terms
of P which contain x alone with respect to x, and add the
two results.
^.
292 INTEGRAL CALCULUS [Ch. VIII.
EXERCISES
Determine in each of the following cases the function u having the
given expression for its total differential :
1. ydx + x dy.
2. sin X cos ydx+ cos a; sin y dy.
3. ydx — X dy.
ydx — xdy '
xy
5. (3a;2 - 3 ay) dx + (3y2 _ ^ax)dy.
ydx xdy
' ar2+3/2 y2+a.2*
7. {2x^-\-2xy-\-b)dx + {x^ + y^-y)dy.
8. (a:* + !/* + a;2 - y^)dx -\- (^y^x - 2xy + y - y^+ 2)dy.
166. Multiple integrals. The integration of was
considered in Art. 164. If F(^x, y) be written for the given
function, the required integration will be represented b}
the symbol
u = j JFQx, i/)dxdi/,
and the function sought will be called the double integral of
F(x^ y) with respect to x and y.
Likewise III -^(^' ^' z)dx dy dz
will be called the triple integral of F(x^ y» 2)' It represents
the function t* whose third partial derivative — — — -- is the
ox ay dz
given function F(x, y, «). It will be understood in what
follows that the order of integration is from right to left,
that is, we integrate first with respect to the right hand vari-
able 2, then with respect to y, and lastly with respect to x.
Such integrals (double, triple, etc.) will be referred to in
general as multiple integrals.
165-167.] SUCCESSIVE INTEGRATION 293
167. Definite multiple integrals. The idea of a multiple
integral may be further extended so as to include the notion
of a definite multiple integral in which limits of integration
may be assigned to each variable.
Thus the integral I j a^y dx dy will mean that ^y^ is to
be integrated first with respect to y between the limits 0
and 2. This gives
^y"^ dy — ^i7?.
X
The result so obtained is to be integrated with respect to x
between the limits a and 6, which leads to
^^^^dx = ^(b^-a^)
as the value of the given definite double integral.
In general the expression
££ Fix,y-)dxdy
will be used as the symbol of a definite double integral.
It will be understood that the integral signs with their
attached limits are always to be read from right to left, so
that in the above integral the limits for y are h and h', while
those for x are a and a'.
Since x is treated as constant in the integration with
respect to y, the limits for y may be functions of x. Con-
sider, for example, the integral I j xydxdy. The first
integration (with respect to y) gives
a? — a?
£.,i3/ = -[£,= <f-|) =
By integrating this result with respect to x between limits
0 and 1 the given integral is found to have the value — ^^.
/
294
INTEGRAL CALCULUS
[Ch. VIII.
EXERCISES
Evaluate the following integrals :
1. yj^''sec\xy)dxdy.
rn ra{l+coa0)
Jo Jo
r^ 8m edO dr.
rb /•lOy
• Jojy ^^y-y^dydx.
n' r^-^xdzdxdy
> Jo x^ + y^ '
168. Plane areas by double integration. The area bounded
by a plane curve (or by several curves) can be readily ex-
pressed in the form of a definite double integral. An illus-
trative example will explain the method.
Ex. 1. Find by double integration the area of the circle (a: — ay +
(y-6)2=r2.
Imagine the given area divided into rectangles by a series of lines
parallel to the y-axis at
equal distances Ax, and
a series of lines parallel
to the ar-axis at equal
distances Ay.
The area of one of
these rectangles is Ay •
Ax. This is called the
element of area. The
sum of all the rectangles
interior to the circle will
be less than the area
required by the amount
X contained in the small
subdivisions which bor-
der the circumference of
the circle. By a method exactly analogous to that used in Art. 149, it
is easy to show that the sum of these neglected portions has a zero
limit when Ax and Ay are both made to approach zero.
To find the value of the limit of the sum of all the rectangles within
the circle it is convenient to first add together all those which are con-
tained between two consecutive parallels. Let P1P2 be one of these
parallels having the direction of the x-axis. Then y remains constant
Fig. 79.
167-169.] SUCCESSIVE INTEGRATION 295
while X varies from a — Vr^ — {y — b)'^ (the value of the abscissa at P,)
to a + Vr- — (y — b)'^ (the value at Pg)- The limit as Ax approaches
zero of the sum of rectangles in the strip from PJ^^ i^ evidently
(1) Az/[limit of sum (Aa; + Aa; + •.•)] = A?/ C-+^^^^-^y-^ ^^^
Now find the limit of the sum of all such strips contained within the
circle. This requires the determination of the limit of the sum of terms
such as (1) for the different values of y corresponding to the different
strips. Since y begins at the lowest point A with the value 6 — r, and
increases to 6 + r, the value reached at B, the final expression for the
area is
\ dy ) , dx:=^ \ \ dy dx.
»'h-r •^a-Vr2-(y-6)2 ^h-r -^ a-Vr2-{y—b)2
Integrating first with respect to x,
•^a-v'r2-(y-6)2 Jo-s/r2-(y-6)2
This result is then integrated with respect to y, giving
C'^''2Vr^-(y-bydy = (y -b)Vr^ - (y - by + r^ sin-i^^l ''*"'= irr^.
Jh-r f- Jb-r
If the summation had begun by adding the rectangles in a strip paral-
lel to the 2/-axis, and then adding all of these strips, the expression for
the area would take the form
V
X
a+r rb+Vr2~(x-a)2
\ dxdy.
r •^6-V'r2-(x-a)2
It is seen from this last result that the order of integration in a double
integral can be changed if the limits of integration be properly modified
at the same time.
Ex. 2. Find the area which is included between the two parabolas
2^2 = 9 a: and 2/2 = 72 - 9 x. .
Ex. 3. Find the area common to the two circles
a;2 _ 8 a: + 2/2 - 8 2/ + 28 = 0,
a;2 - 8 X + 2/2 - 4 y + 16 = 0.
169. Volumes. The volume bounded by one or more
surfaces can be expressed as a triple integral when the
equations of the bounding surfaces are given.
296
INTEGRAL CALCULUS
[Ch. vni.
Let it be required to find the volume bounded by the
surface ABC (Fig. 80) whose equation is z=f(x^y^^ and
by the three coordinate planes.
Imagine the figure divided into small equal rectangular
parallelopipeds by means of three series of planes, the first
series parallel to the ^2-plane at equal distances A a;, the
FiQ. 80.
second parallel to the rca-plane at equal distances Ay, and
the third parallel to the iry-plane at equal distances A2.
The volume of such a rectangular solid is AxAi/Az; it is
called the element of volume. The limit of the sum of all
such elements contained in OABO is the volume required,
provided that the bounding surface ABQ is continuous.
169.] SUCCESSIVE INTEGBATION 297
(The reader can easily show that the sum of the neglected
portions is less than the volume of the largest plate formed
by two consecutive parallel planes and that its limit is
therefore zero.)
To effect this summation, add first all the elements in
a vertical column. This corresponds to integrating with
respect to z (x and y remaining constant) from zero to
f(x^ ^). Then add all such vertical columns contained
between two consecutive planes parallel to the ?/2!-plane (x
remaining constant), which corresponds to an integration
with respect to y from y = 0 to the value attained on the
boundary of the curve AB. This value of y is found by
solving the equation f(x^ ^) = 0. Finally, add all such
plates for values of x varying from zero to the value at A,
The final result is expressed by the integral
ax ay az.
in which ^{x) is the "result of solving the equation /(a;, ^) = 0
for y^ and a is the a;-coordinate of A.
Ex. 1. Find the volume of the sphere of radius a.
The equation of the sphere is
a;2 + y^ + 2:2 _ ^2^
or 2 = Va2 _ a;2 - y\
Since the codrdinate planes divide the volume into eight equal por-
tions, it is sufficient to find the volume in the first octant and multiply
the result by 8.
The volume being divided into equal rectangular solids as described
above, the integration with respect to z is equivalent to finding the limit
of the sum of all the elements contained in any vertical column. The
limits of the integration with respect to z are the values of z correspond-
ing to the bottom and the top of such a column, namely, 2 = 0, and
z — y/a^ — x^ — y\ since the point at the top is a point on the surface of
the sphere.
298 INTEGRAL CALCULUS [Ch. VIII. 169.
The limits of integration with respect to y are found to be y = 0 (the
value at the a;-axis), and y = Va^ — x^ (the value of y at the circumfer-
ence of the circle a'^ — x^ — y^ = 0, in which the sphere is cut by the
zy-plane).
Finally, the limiting values for x are zero and a, the latter being the
distance from the origin to the point in which the sphere intersects the
X-axis. Hence
F (= volume of sphere) = 8 \ I \ dxdydz.
Integrating first with respect to 2,
/•a /'VaZ— 12 ,
F=8j^j^ yJa^-x'^-y'^dxdy\
then with respect to y,
V = 8 i''dx\y. yJa^-x^-y^ + "' " ^' sin"! ^ l^"^^
47ra8
-r
|(a2-x2)rfx= g
Ex. 2. Find the volume of one of the wedges cut from the cylinder
a:^ 4- y2 _ ^2 ijy tiig planes 2 = 0 and z = mx.
/
Ex. 3. Find the volume common to two right circular cylinders of
the same radius a whose axes intersect at right angles.
Ex. 4. Find the volume of the cylinder (a: - l)^ + (y - 1)2 = 1 limited
by the plane 2 = 0, and the hyperbolic paraboloid xy = 2.
Ex. 5. Find the volume of the ellipsoid
a^ b^ c^
Ex. 6. Find the volume of that portion of the elliptic paraboloid
2=1-^^-2?
which is cut off by the plane 2 = 0.
ANSWERS
Page 20. Art. 9
6 a; -4. 3. -^.
4a;2
4. 4a:3_6
37*
Page 25. Art. 11
14«_4-33«2. 3. 12m2_2.
4. 4x-5.
1. 2 a; -2.
1. 162/2-2.
Page 28. Art. 13
2. Inc. from — co to ^ ; dec. from | to 1 ; inc. from 1 to + oo ; ^ and 1.
3. Two. +l3itx=l±V^; -I atx = l±V^. 4. ±tan-i^V
Page 29. Art. 14
2. (6u-4)6x2. 3. -^(lOx-2). 4. (g n - A_^ (a.2 - i).
Page 37. Art. 19
10. J4±A.
2Vx + 2
J, Va(Vx — Vq) ^
2\/x(Vx + a)(Va + \/^)2
12. 1
13
1.
10ic9.
2.
-8x-9.
3.
c
2Vx
4.
1 1
5.
-iV^.
6.
w(x + a)'*-i.
7.
MX«-i.
Vl - x2 (1 - a;)
1
8.
(a2
14.
2 xil - a;2) + VI - y/-^
ia^ + Sx^
iVx^a^ + x'^
2-6a;-a;2 15, ^^ .
(a;2 + 2)2 * ' a;Vl-a;2
299
300 AN 8 WEBS
16.
17.
(1 - aj2)i(l 4- a:2)f ' dx
26. (2M + 6a;M)^+3M2 + 4a:8.
(x«-l)2' wwn-i^
Oft dx nu^
18
wi(6 + a;) +
n{a-{-x)
(a + a;)'»+i.
(6 + a;)«+i
in
-2
x'^i:^ + 1)1
20.
66a;8(a;a + l)i
21.
dx
(a + x)'» (a + x)«+i
27. 2 wa;8w7 ^ + Wh^^ — + 3 waxa^;.
dx dx
jQ — &% __ 6x
«'2^ «\/a^ - x=^
82. (0,0), f-i-,~-§_V
^ V9a 27 ay*
22. 12(u2-t* + l)^. ^^" ^^«>'
^ ^ ^dx ^ (21t/8-i9zf)10x
23. muKl-^u^f^^ (7te2 + 5)t
dx 35. At right augles at (3, ± 6).
Page 43. Art. 24
1. -4- 12. Iogio6 2^ + 7
as + a x2 + 7a6
__« 18. 2^
ax + 6 xlogx
8x-7 14. ae*-.
4x2-7x + 2 15. 4e4«+«
2 _i_
r^* 16. -=-i^.
(H-a;)2
4x
1 - X* 17.
(1 + e*)2
6. logx + 1. 18 y_3a;2ex.
7. wx«-i logx + x«-i. 19. 1 - j/2.
8. 7ix"~^ log x« + mx""i. 20 e* + e~* ^
^^--1 21. ^^''^
1 x + e'
10.
2("v/x + 1) 22. wx"-! a*+ ai^a* log a.
11. log.6. 12xv/2T^-l . 23. ^:;«
2 V2 + x(3 X- - V2 + X) Vx(a - »)
ANSWERS
301
24. -
25.
27.
10.
11.
12.
13.
14.
15.
16.
17.
x([ogxy
2 log a;
X
xlogx
x^^Qogx^ 1).
X
Page 47.
7 cos 7 «.
— 5 sin 5 x.
2 X cos a;2.
2 cos 2 oj cos X — sin 2 a; sin x.
3 sin2 x cos X.
10 X cos 5 x^.
14 sin 7 a: cos 7 a;.
sec-^ a; (tan2 a; — 1).
3 sin2 x cos^ x — sin* x.
secx(tanx + secx).
- 6 X (1 - 2 x2) sin (1-2 x2)2
cos (1-2 x2)2.
-20 X (3 - 5 x2) sec2 (3 -5 x2)2.
2 tan X sec2 x — 2 tan x.
secx.
cot Vx
2Vx
1 ^ ^
— rloga • ««• sec2(a«).
X2
w sin**-! X sin (n + 1) x.
31.
Art.
19.
-(a;_l)f(7a;2 4.30x-97)
12(x-2)i(x-3)V-
2 + X - 5 x2
2Vl -X
1 + 3 x2 - 2 X*
(1 - x2)i
5x4(a + 3x)2(a-2x)
(a2 + 2ax-12x2).
(x — 2 g) Vx + q ,
(x-a)t
31
?nn sin"*-i nx - cos (m — n) x
cos'*+i mx
2
1 + tanx
21. csc^
dx
22. cos (sin u) cos w
dx
23.
24.
25.
26.
27.
28.
29.
30.
31.
2 ae«* sin e"* • cos e«*.
e* • cos e* • log X + 5HLi!.
X cos x2
Vsin x2
'sin X
+ cosx
log x^
— 8 csc2 4 X cot 4 X.
8(4 X - 3) sec (4 x - 3)2
tan (4 X - 3)2.
3 x2 sin x^.
sec Vx tan y/x
18. cos 2 ?«
dx
- 2 X CSC2 X2 + '
y cos xy
1 — X COS xy
— CSC2 (X + y).
2Vx
4x
Vl-X2
Page 49. Art.
3.
33
V6 X - 9 x2
3
vr
302
ANSWERS
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
16.
16.
-2
1+X2
17. sin-ix+-
vT-^2
2 Vsin-i x Vl — a^
1
xVl -(logx)2
sec^x
Vl - tan2x
1
2
vT
xV^-1
2
l + a;2*
^ Vl + CSC X.
1ft
gtau z
l + X^
19
2
VI -x2
20.
-2
x2 + l
21.
1
2Vx(x+ 1)
22.
-2
e^ + e *
23.
n
C0S2x+ W2sftl2x
24.
2.
25
2sinx
Vl-4cos2x
26.
-1
2(1 + x2)
27.
-1
sec2 X • tan-i x +
tanx
1 +X2
Page 51.
6x + 15x2.
-6 15
x8 X**
3x-l
VT-i^
28. 0.
Exercises on Chapter 11
2Vir-~3
a2 - 2 x2
Va2 - x2
log sill X + X cot X.
9. — cotx.
2Vm
10. 1-loga.
X
11. -C3x + x8)
(1 + a;2)!
12. c*(cos X - sin «)
-1
xWa^ - y?
2xg-2g + l^
2rx-aJ2)*
18.
14.
xV2x-l
4
6 + 3 cos X
16. tan-i-^.
16.
2(1 + a^)
AN S WEBS 303
17. 4tan5a;. 26 2 xy^ + 3 x^_
\o<rx ' 2x'^2/ + 32/2
19.
(1 - xy
4
2/2+1
27.
5 + 3COSX -*'• i_2a;^_a;2
20. ^' 28. 4 cos (2 log a;2- 7)
" 1 — «* ' re
29. For all values.
21.
X ^x
22. 1.
_^x_+_a^ 32. a;, ?/ are determined from
^ ^~ ^ a'^y =z ±b'^x and equation of
curve.
2 tan a; + e ^^''^ • sec X tan x.
24 ?izL«l^. _
' ax -2/2 34. tan-i2V2»
Pages 54-55. Ezercises on Chapter III
^' '^^ ^' 19. TO" • cos ( mx + w - ).
2. 0. \ 2/
3. -^- 20 (-!)-• (m + n-D!
a^ " (w-l)!(a + x)'»+«
4. -|i. 21. TO(- 1)^-1. (n- 1)1.
•^ (a + x)«
5. e^.
6. e^\ogx + ^-^'
X X2
7. 2 log X + 3.
8. 8 tanxsec2x(3sec2x — 1).
9. 2 cot X csc"^ X.
10. 16 sin x cos x.
24
3j>3
— 2 g^xy
XX.
12.
(l-x)6
48
X
13.
sin X.
14.
15.
8(ex_e-x)
(ex + e-xy
8 x2e2x.
16.
4!
X2*
17.
a"e«^.
18.
r-l)«w!
(x - l)«+i
(2/2 _ ax)3
25 -yr(^-l)^ + (y-l)!I.
X2(?/ - 1)3
26. '^~ ^ . e^v.
(2 - 2/)3
32. (- l)«.2"-i.»il
[ 1 1
U2x-l)'*+i (2x+l)«+i
33. (^-^)^
X
34 2(-l)n.ri!
(l+.x)«+i
36. 2»»-icos(2x+— y
304 ANSWERS
Page 64. Art. 40
6. -8 + 4(y-3) + 3Cy-3)2.
Page 67. Art. 41
1. x-\-— + —x^+—x- + B. 8. 1-^-^+B.
3 15 315 2 8
« , , a;2 a;3 3x4_llx5 , „
"^^ I"^ 8 30 9- l+2x + 2a;2 + 2x8+i?.
7. ^ + |- + |'+^+^- 10. x4 + 7x3 +11x2 -16 a; -41.
Pages 75-76. Exercises on Chapter IV
hi /,3
1. COS X — ^ sin X cos X + — • sin X + ^.
^ o !
2. tan /i + X sec2 /^ + x^ sec2 ;^ tan A + — sec2 A (1 + 3 tan2 h) + B.
4. logx + ^-^ + ^-^ + i?. '
^ X 2X2 3X3 4a;4^
6. X6 + 5 X*y + 10 X3y2 + 10 x22/3 4. 5 xy4 + y6.
Pages
79-80. Art. 47
2.
2
a2
a2
+ 62
3.
Pages
83-84. Art. 49
*-^-
8.
f.
8.
-4.
12. 12&^.
16. 1.
log 6
4.
4.
9.
0.
"f-
17. 1.
6.
f
10.
2.
n
18. -f
7.
i.
11.
8.
15. 1.
19. h
Pages 87-88. Art. 52
1.
1.
6.
log a.
9. 0.
12. -i
t
0.
6.
-f
10. 0 or 00 according As n > 0.
8.
0.
7.
4a«
or<0.
18. h-
4.
6.
8.
1.
11. f
14. -1.
ANSWERS 305
Page 89. Art. 54
1. 1. 8. e«*. 6. 1. 7. 1. 9. e«.
4. 1. 8_ X «• 1-
Pages 89-90. Exercises on Chapter V
10. 0.
1.
1.
5.
0.
2.
0.
6.
0.
3.
00 if r>l,
Oif r<l,
7.
a
2*
jwe'««if r=l.
8.
1.
4.
1.
9.
aia2
11.
-1-
12.
a.
13.
1
y/2b
14.
e*.
15.
1
2\/2
16.
- 1.
17.
1.
18.
h
an-
Page 100. Art. 62
1. _ _i., max. ; J- , min. «• " 1' °^^^- ^ " *' ^i^"
v^ v^ 7.-2, min. ; 1, max.
2. 2, max.; 3, min.
o o • a 8. e, max.
3. 2, mm.; f, max. '
4. (2 w + i) TT, max. ; (2 w + |) tt, 9- 2 wtt, rain. ; also tan'i ± \/| for
min. for all integral values of n. angles in 2d and 3d quarter.
g a j^.j^ (2«+l)7r, tan-i± V|, Istand
4 4th quarter, max.
Pages 103-104. Exercises on Chapter VI
1. The line should be bisected at the given point.
2. The altitude is equal to the diameter of the base.
3. The side parallel to the wall is double each of the others.
4. The diameter of sphere = edge of cube.
5. Three-fourths slant height of cone.
6. Area is — . 7. 5 Vi inches. 8. 3 inches. 9. — •
2 ^ V3
10. (at + 6l)f. 11. JL. 12. Arc = 2 TrrCl - V|).
\/3
13. Circular arc is double the radius.
14. — ;- , D being the distance between the centers of the spheres.
rt + i?l
16. Angle at center of variable circle defined by ^ = cot d.
306 ANSWERS
Page 111. Exercises on Chapter VII
3.
.00145.
9.
2ab.
6.
miles per hour.
V2
10.
11.
±2.
5 IT.
7.
The point (3, 6).
12.
2.
8.
At 60°.
13.
1 and 5.
16. s = ^o, t = ^.
64' 32
17. ^.
16
19. ± 16, T 12 feet
per second.
Page 116. Art. 68
3. 1. 4. (x-\-y)coaxy. 6. 1.
Page 120. Art. 71
3 ax + hy + g
hx + by+f 2/3
Pages 127-128. Art. 76
^_2y^ = 0. 8. ^4-« =
dy'^ dy du^
I 1 +
3. 22 =
¥i — ^^'7
^y" 11. ^-f^+y=0.
6. ^ + M = 0. + (1 _ 2-2)2 2^ + ^4 = 0.
Pages 131-133. Art. 79
1 ?1^4.M=l 4. (a) X -I- 2 2/ = 4 a,
' a-^ &2 ' 2/-2a;4-3a = 0.
y_y, = «!yi(x_xO. (^)2y=±(x+l),
2. y = x. (7) y = ic +p, a; + y-3p = 0.
8. 2 y = 9 X - 3, 9 y + 2 X = 29. 6 3. 6. 4 Vl?.
7. («) Parallel at points of intersection with ax + hy = 0.
Perpendicular at points of intersection with hx -\- by = 0.
(/3) Parallel at (— ^, ^ap/2\ . perpendicular at x = 0.
(7) Parallel at ( i^ 2Via\ j perpendicular at (0, 0); (2 «, 0).
V 3' 3 1
ANSWERS 307
8. 1 — 1 = , i.e. they must be conf ocal.
a b a' b'
9. ^. 11. ^. 12. y^.
2.2 nx
13. ^<^ + «. 19. (2j9, ±2joV2).
o
Pages 136-137. Art. 82
1. rp = d.
2. Polar subtangent = ^, Polar normal = Va"-^ + p^, Polar subnormal = a.
a
3. ^ = - + 2 ^, Subtangent = - p cot 2 ^, Tangent = "^ ,
Subnormal = - «!j1H^, Normal = «^.
P P
6. -,2 a sin2 ^ . tan -•
2 2 2
7. They have a common tangent at the pole ; elsewhere, -•
Page 142. Exercises on Chapter XI
a a
1. A^-i-?. 2 Vox, 4 7rVa2 + ax, 4 Trax.- 2.
8. secx. 5. 7r-^(a2_a;2). 7- ^2 ap.
4. 7r(
6. /)Vl + (loga)2.
Page 151. Exercises on Chapter XII
1. y = 0, X = a, X = — a. 8. x = 0 twice ; one parabolic
2. x = 0, x = 2a, y = a, y = -a. branch.
- ^ . . 9. X = 0, « = 0, X + w = 0.
3. y = a, «/ = — a ; two imagmary. ,^ ' •'^ ;
10. 2/ = X ; two imagmary.
4. y = a ; X = c twice. ^^ x + ?/ + a = 0 ; two imaginary.
5. ?/ = - X + - ; two imaginary. 12. y + x = 0 ; two imaginary.
13. X = 0 twice ; x = y, x=:— y.
6. X = 1 ; one parabolic branch. 14, y-,^^y-_^. ^^q imaginary.
7. x = — a,y = — b,y = x + b — a. 15. x+2 2/=0, x+2/ = l, x— 2/= — 1.
Page 158. Exercises on Chapter XIII
1. An inflexion s^x = y = 2.
8. Point of inflexion at (a, f a) , tangent is x + y = -^ • Bending changes
from negative to positive.
308 ANSWERS
Pages 162-163. Art. 102
1. First. 5. y+l2x = 10.
2. They do not touch. 6. Second.
3. Third. 7. a=-l.
4. 3 x(x — a) = a(^y — a).
1. 2 a.
3. c».
4 (^i±i!)|.
Page 168. Art. 108
5 (c2a;2-a2)l
8.
9.
3(axy)i
ay/xi%a - 3 a;)*
3(2 a - xY
7 2(a: + y)i
Va
10.
a
Page 170. Art. 109
- -S-
e 4V^
3 '
7. «(^ +
<.)!
1. pVl + (loga)2.
2. «i.
3p
3 q (5 - 4 cos g)t
,9 - 6 cos ^
4 2p!.
Page 176. Art. HI
1. « = o, /3 = o. r? _?\
4. « = a;-^-\,e--c''j, /3 = 2y.
^a4 4-15y4 a4y-9y^ 6. (au)f - (6^)1 = (a^ + 6^)1
•" 6a^y ' ^ 2 a* * 7. (a + /S)^ (a - /3)^ = 2 ai
Page 186. Esercises on Chapter XV
1. (0, 0) ; ax±by = 0. 6. Two nodes at infinity ; the
« ^/^ /^^ *« *!• J A asymptotesare x = y ± l,x + y = ± 1.
2. (0,0): cusp of first k.nd,y = 0. ^ („, _„). (/„, q); (-«,0) =
8. Four cusps of first kind ; the tangents are, respectively,
(0, ±a),(±a,0); y = 0,x = 0. V3(y + a) = ± v^x ;
4. (0, 0); conjugate point with 2(x - a) = ± VSy '
real coincident tangents, y = 0. 8. (- a. 0) ; conjugate point.
6. (0, a); y=a4-a; ; cusp of second 9. (0, 0); x = 0, ?/ = 0.
kind. 10. (0, 0) is a tacnode ; y = 0.
ANSWERS 309
Pages 193-194. Exercises on Chapter XVI
3. x^ + y^ = ci 8. (c, c), (0, 0), and one at infinity
4. 4:xy = k^- on each axis.
Page
201. Art. 125
1.
ixf.
2.
a + 1
3.
f.i
4.
2 m m-i
2m- 1
6. 51ogx +
3
2x2
1
3x8
6.
aac - 1 at a;^ + § <
airci-^a;2. 7. \(x^ -^ a^y.
8.
(ax + 6)«+i
19.
(a + &)»«+»«
31.
\ sec8 M.
a(/i + l)
w log (a + 6)
32.
— log cosec M.
9.
log(x + a).
20.
1 (a; + sin x) .
cos (m 4- w)x
33.
— cosw.
10.
log V2 ax — x^.
21.
34.
sin-i -.
m + n
a
11.
log tan x.
22.
— ^ cos X2.
35.
isin-i2x.
12.
-l0g(l+C0SiC).
23.
sin X — ^ sin^ x.
ltan-i~.
a a
13.
log (log x).
24.
— cosx+^cos^x.
36.
14.
flog(x3 + l).
25.
^ X — :| sin 2 X.
37.
i-ton-i^.
26.
tan X — X.
a6 a
15.
log sec X.
16.
log sin X.
27.
\ tan8 X.
-icot(ax + 6).
38.
itan-%'-
28.
17.
le".
a
-f (cotx)i
39.
seo-i (« - 1).
a
29.
40.
Isec-'S.
18.
\e'\
30.
log tan X.
a a
Page 205. Art. 126
1. xsin-ix + Vinr^. 6. secx[logcosx+l].
2. ex tan-i e- - \ log (1 + e2x). 7. K(«^^ + 1) cofi x + x\
3. x2sinx4-2xcosx-2sinx. »• Ksi" 3x - 3 xcos3x].
9. icx(sinx + cosx).
4. _± f log x — \ •
w + 1 V w + 1 y 10. i e* (sin x - cos x).
6. ^[2x3tan-ix-x'''+log(l + x2)]. 11. fcosxsin 2x - ^cos2 xsin x.
310 ANSWERS
Pages 206-209. Art. 127
4. Ksin-ix)2. 14 logtan(^ + ^V
6. icos(x2+l)[l-logcos(x2+l)]. ^2 4/
1 .. 16. J-tan-i2£±i.
. x . .x-a 18. -llog?^^.
10. vers-i - , or sm-^ " » ^ x + 1
a' a
11. log(a; + \/x-^±a2).
19. |sin-i(3x-5).
20. icos(3x-2)
12. —log ^5 — -' +(a;-f)sin(3x-2).
2 0! X -\- d .—- —
Pages 213-214. Exercises on Chapter I
1. \/x2 + 2 X + 2 - log (X + 1 + Vx^ + 2 X + 2),
, 1 - 2 X + V5 x2 - 4 x+1
log ^
X
3. § V3x2 + X - 2 - -^log (X + ^ + Vx2 + ix - i).
^ 3V3
^^:iOg^.
x-l
4. _ V8 4- 4 X - 4 x2 + |sin-i
5. - V-x2 + 2x + 1 + sm-i^^.
>/2
2x — a
6. Vr^^ + 8in-ix. 7. v^^r=:¥2+|sin-i— ^
. ,, -, 1 , rV2+Vx2+2x4-3~l
8. Vx2+2x+3-log(x+l + Vx2+2x+3)--— log[ ^ ^_^^ J
9. Vx^ + x+l-^log[x + i + Vx'' + x + l]-log[^~'^'^^^'^'^^'''^^^
1 ^ 14. -log(c-* + v^2*-l).
10. log-(x2+l + Vx* + x« + l). rV2W^T2-|
16. log •
11. c< 2V2 I « -•
n. iiog^— ^.
18. ilog(6x» + 12x + 5). e* + l
18. ix5tan-ix-,«)yX* + TJff«a-Tifflog(x2 + l).
19. « - log (a? + 1) + 2 tan-» ».
ANSWERS 311
20. -^— -^[2 + 2xloga+(xloga)2].
a* (log a) 3
21. tan^-sec0. 23 1 iog(acos2a;+6sin2a;).
22 -cot^. 2<^^-«>
2 24. ^[x — log(sinx + cosx)].
25.' i(»^^ + 2)Vx2-l.
26. 6[^xH ix - ^x^ + -Jx* - 1 x^ + Ixi - x^ + log(xi + 1)].
1 /I i 30. tan-i(logx). '
27. -^vl -logx. 1
3j i
28. log Ve^' + 1. • 6 (a -ft tan x)'
.0. Sin-. (-11^). »- i--[^-^^^S^J-
33. 2 Vx vers-i- + 4\/2 a - x. \
34. -lV3x2 + 2x + l+logP + ^+^-^^^+-^^+l1. ^
X L a; J
85. -^.og(. + V^^-^r^+J^^.
Pages 219-222. E:sercises on Chapter II
2
Vx'-^ - 2 X - 3 - 2 log (x - 1 + Vx2-2x-3).
3. ^-^V2ax-x^-f«-sin-i^::i«.
2 2 a
5. The arrangements which can be used are [5], [C], [5], [O], and
[5], [^], [C], [CI
6. if— ^— +itan-i^l. 9. 5^'-v/^2Tr^ + ^sin-i?- ,
^ I a;2 + 4^ 2j 2 2 a.
7. _l^.IlJ_+-±_tan-i2^Ill. 10. -^Z^.
3(x2-x+l) 3V3 \/3 «'^
8. _^Va-rx-^ + f sin-i|. " "' 3 , ^.f^ ,)f + ^^^Sf^*
12. ^(2x2 + 5 a2) V^M^ +i|liog (X + Vx2 + «2).
8 o
13. ^ Vx2 + a + - log (x + Vx2 + a).
14. K2 aj2 - «a; - 3 a^) V2ax-x^ + ^ sin-i^^=-^.
15 (2 «x - x2)^ Ig vl + a;^
3ax8 ' "2x2
312 ANSWERS
J- 3(x + 2)8-5(x + 2) I 3 , x + 1
8(x2 + 4x + 3)2 ^16 ^x + 3
18. i(x+ l)Vl-2x-x2 + sin-i^^.
V2
Page 226. Art. 133
2a x+a 2 2^x+l
^ ^^g x'cx"^- 1 )• ^' x + \og(x-a)\x-b)\
6. ?-±^ log (X - 2 - v/3) - ?—2^ log (X - 2 + V3).
2>/3 2V3
^^ ^(2x + l)(x + 2) ° x-c
9. x + -^— [a2 1og(x + a)-62log(x + 6)].
6 — a
10. log[(x + 2)V2x-l]. 13. ^_7x + 641og(x + 4)
11. log(^-«)Cx+ft). !271og(x + 3).
X
12. ^log ^ -. • 14. JLlog^^.
* ^ (2 + X) (1 - x)6 2ab ax + b
15. sin-i-^ — sin-i^^~^)^^ - log (x - 1)"^(2 - x + V2^=^).
V2 v^ X - 2
16. log (X + 2 + y/x^ + 4 X + 7) - — log(x + 2)"\ V3 + Vx2 + 4x + 7)
V3
+ log (x + 3)"\l - X + 2Vx2 + 4 X + 7).
Page 227. Art. 134
2. +^log^^^±-l. 9. ax-i + log
2(x-l) X— 1 X x + a
8. x-51og(x-3)-p^. 10. x+-^-K281og(x+3)-logx].
^—^ 3x
2(a2-xa) 11- -a^og
-1
X
V2x(x + V2) l**- 2 log[x - 1 + Vx^-2x + 2]
«• f-2x + ^^«+.og.(.+l)'. _.„,[-lW.»_-2x^2-|
7. log(x3-a2)--,^^. +-^ Vx^-2x + 2.
x* — a* X — 1
8. —7-^^ 18. log(x-a) + i^5!_=_i«?.
4(>/2 + l-x)2 ^^'^ ^ 2(«-a)«
\
ANSWERS 313
Page 228. Art. 135
4. 3^,[log(a= + a) 7.
-^log(a;2-ax+a2)
tan-
V3 2 a;2 + 1
, /o. i2a^-an 8. l_tan-i-.
+ V3tan-i— — ^J. 2a{x-a) 2 a^ a
6. -Itan-i^ + ltan-i?. 9. x-log^'+^^ + ^
a a b b x—1
Page 230. Art. 136
2. tan-ia; + — ^^ • 6. _— ^— _ + log (x^ + a^)
x2 + l 2(aj2 4.a2; ^^ ^
3. Atan-i^+ ^-^^^ . -i-tan-i?.
2 a a 2 (x2 + a2) 2 a a
4. ilog ^^ + ^ + ^-1 .
* (x+l)2 2(a;2+l)
^_l±2^_3tan-ia;.
2(x2 + l)
Page 232. Art. 138
2. -2Vx-nx^ + 12 x^ + 6 log (x^ + l)-12tan-ixi
3. logX^LzJ:. 4. 2v^-3v^x4-6v^x-61og(v^+l).
V X + 1 + 1
5. 21og(Vx^4-l) =Ji
Vx - 1 + 1
6. 2tan-iV^r3^. 7. llog^^~^~A
^ Vx - a + 6
8. 14(xT^? _ 1 x7 + i a;T? - 1 x? + ^ xT?).
Page 236. Art. 139
1) .
3. 21ogri-At^l + ^
L ^l-xj x-l-\-
4. _21og[V2+V^ff3j.
\/l^^x2
314
ANSWERS
1.
3.
5.
6.
7.
8.
11.
12.
Pages 236-237. Exercises on Chapter IV
2. |(a;-a)7-
2(a;2+ l)\/x2 + 2
^[x2 - a;Vx2 - 1 + log (a; + Vx^ - 1)].
4V2 V2(x-a)i+l
^a;(Vx2 + 2-Vx2+ l)+^log
T3^(2x-3a)(a + x)i
61og(x3- 3x^ + 5).
X + Vx=2 + 2
9.
ti
p
-4
V Vx+ 1 +Vx-1
10.
log
Vx2 - a2 + a;V2
2v^a2 Vx2-a2-xV2
f x^ - 1 xs + f x^ + 2 x^ - 3 x^ - 6 x^ + 3 log (xi + 1) + 6 tan-i xi
Page 239. Art. 142
\ tan^ X + tan x. 5. f cosec^ x — cot x — § cot* x.
-^cot^x-cotx. 6. -64[cot4x + |cot84x].
tan X + I tan^ ic + i tan^ x.
- 128 [cot 2 X + cot8 2 X
+ |cot62x+ }cot7 2x].
-— + log tan X.
2 tan2 X
I C0t8 X — ^ COt^ X.
Page 240. Art. 143
\ sec* X -- I sec2 x. 6. ^ sec* x — sec^ x + log sec x.
I cosec'' X -\- 1 cosec^ x
^ cosec* X.
6.
sec^^x sec*» ^^
3. - {\ sec^ aa; — f sec* ax + sec ax).
a
4. — (sin X + cosec x).
- 1 « - 3
7. log sec X.
8. — log cosec a^
Page 241. Art. 144
1. — I cot' X + cot X + X.
2. — tan* ax log sec ax.
2a a
8. i(tan2x + cot2x)
+ 4 log (sin X cos x).
^ tan" ^x
71-1 *
6. \ tan' X - J tan» « + ^ tan» x
— tan» + «.
•1'-
ANSWERS 315
Pages 242-244. Art. 145
2. - cos X + ^ cos3 X. 16. j^s (5x4-1 sin^ 2 x — sin 4 x
3. - ^ cos^ X + ^ cos^ X. — |sin8x).
4. log sin X — sin2 x + ^ sin* x. 17. ^i^ (3 x — sin 4 x + i sin 8 x).
5. fcos4x + 3costx-|cosix. 18- i(3x + sin 4 x + |sin 8x).
^ ^ ,, 3 „ ,^ 5 19. — icotx — icot^x. u
6. 4(1 -cos x)^ - 1(1 -cos x) 2. 2 ^
'\ , ^ ^^ 20. logtan2x.
8. -icot^x. ^
21. tan X + ^ sin 2 X — f X,
22. 2cotx-^cot8x+|x+|sin2x.
9. — cot X — I cot^ X — I cot*^ X
10. - C0t5 X(i + } C0t2 X).
11. -4cot3x-2cotx + tanx. 23. L( ^^'^~^) .
12. I Vtan X (tan x — 3 cot x). ~ , «4 r
^ - , , ^ 24. ?(2x2-a2)Va2-x2+-sin-i?.
-„ tan»-^x tan"+^x 8 8 a
^ ^+^ o^ Vx2-a2 1 ^ a;
25. sec^—
15. I X - s\ sm 4 X. 2 a2 a;'2 2 a^ a
Pages 245-246. Art. 146
1. J_tan-if«i^Il^V ^tanfx-^V
Va2 + 62
2V3 V3tan^x-^)4-l
6tan?-a-\/a2 + 62 - ^ in^^^"^^-^-^
log
log ____. 2 VS tan X - 2 + V3
6 tan - - a + v'c2 + 62
3. itan-i(*-?|^y
4. itan-ihtan (7--)]"
a(a tan X + 6)
7. -L tan-i (*^2^V
V2 V V2 /
Page 247. Exercises on Chapter V
8. e^fsin ?+cos?V
9. — ^ e-*(sin x + cos x). 1
5. itan2x(2 + tan2x). !<>• i e2x(2 - sin 2x - cos2x).
1 /^ _.\ 11. ie*(sinx + cosx — |sin3x
«• -iif. + '^^'^d+i)- -icos3.).
^ , . „ . - - sin«+i X sin"+3 x
7. i tan2 X sin x 16. — •
n+1 n -{- S
4frsin«-logtanf- + -l . ,» „ 2 « /
^L \2 4yj 16. f tan5x-2Vcotx.
316
ANS WEBS
17. -32cot2a;(l + |cot22x+ ^cot*2x).
18. ^tania;(H- f tan-2^x + ^tan^^x).
Page 259. Art. 152
6. 2.
2a
4ct68
3
7. 00,
Pages 262-263. Art. 154
2. 2a6. 3. 2. 4. 20 V5^. 6. 2 a. 7. 12.
l 9. |. 10. faVop; |p2. n. ^a2. 12. 4a2tan-i-^; 4 Tra^.
^ a
13. 7ra6.
4. X
14. f7ra2.
Page 266. Art. 155
6. 3 7ra2. 6. 4a2. 7. 00.
Pages 268-269. Art. 156
Stto^^ 4. ^a(pi-p2).
15.
a"^ - a^i
log a
8. V- 9- ^<1'
5.
4a2
3
6. 7ra2.
Pages 270-271. Art. 157
1. i)[\^4-log(l+V2)].
61a
216*
6 a.
2irr.
6. |(e"-e '»).
6. 2 - \/2 + log
^ 4(a8-&8)
a6
l4-\^
V3 *
8 a.
2ira
Page 272. Art. 158
8.
. 2arV6-2-V31og ^ + ^ 1
L \/2(2+V3)J
aftan -sec- + logf tan - + secf^ 1*'.
L 2 2 *V 2 2yjtf,
a[-- vTT^ + log (d + >/rT^)T'.
Pages 273-274. Art. 159
8 a.
8 ma
Kxif + yit)i-|.
8. 6 a. 4. ia^i«.
7. >/2(«*>- 1).
ANSWERS 317
Pages 276-277. Art. 160
..(.-2)/^ 4.fC3V2-logCl.V2)3.
(a) 2 7r6f6+ C0S-1-).
(6) 2 ,ra2 + _^^^ log r«-+ ^«'-^''
Va2 _ 62.
4 7ra2.
9.
6.
7. (a) 7r6Va-^ + 62.
«i«^.
8.
(/3) 7raVa2 + 62.
3
Pages 278-279. Art. 161
1.
2.
4 7r«2 6
3
4 7rr3
3
3. TTk^; 00.
- 32 7ra3
• 105
5.
6.
7ra3^
15*
4 7r2a8.
7.
TrfSa^
log
2a_
-4a2(2a-yi)]; oo.
8.
5 7r2a3.
3 8.a«
10.
-'-^ fi.
3 6V2
Pages 280-283. Art. 162
1. f TT ahc. 2. 1 TT a6. 5. 41 cu. ft. 7. — cu. ft.
V3
3. \Ah. 4.
■n abp
0
9, p= ea'
■ c
12. ^Tahc^.
Pages 284-285. Exercises on Chapter VII
1. a21ogw. 2. 1. 5. a2(2+-j. 6. j%.
3. ^^i|!^l 4. |. 7. 4a6tan-^ 9. ^h!^
Sab 6 a 4 a
10. ^ r^ vT+T^ + log (e + vTT^)]^'.
13. 'La" [e a _ e a ] ^. ^ a2a;i. 14. 2 7r2a8.
11. 4 7r2aA;. 12. 2 7r2a2A;.
4
15. 2 IT a2 (3 sin h-Zh cos fi - h^ sin «i).
16. ^. 17. 2 7ra2. 18. alog^.
3 yi
318 ANSWERS
Pages 287-288. Art. 163
3. xy = C2/2 + C'y + J. ^. y = kx(logx-l) + CiX-^C2. b. ^kA
Page 292. Art. 165
1. xy + O. 2. - cos X cos 2/+ C. 6. st^ + y^ - S axy + C
X X
3. Impossible. 4. log-+C. 6. tan-i-.
7. ^oi^ + x^y + 5x + ^y^-iy^-\-C.
^
Page 294. Art. 167
1. 1. 2. fa3. 3. 66». .4.
Page 295. Art. 168
2. 64.. 8. ^-2 Vs.
o
Page 298. Art. 169
2. 2fl^. 3. Y^a. 4. ^. 6. iirabc. 6. ^^
INDEX
(The numbers refer to pages)
Absolute value, 59.
Absolutely convergent, 59.
Acceleration, 111.
Actual velocity, 105.
Arc, length of, 269.
Area, by double integra-
tion, 294.
derivative of, 23.
formula for, 255, 256.
in polar coordinates, 268.
in rectangular coordi-
nates, 260.
Asymptotes, 143.
Average curvature, 166.
Bending, direction of, 152.
Binomial theorem, 73.
Cardioid, area of, 268.
Catenary, 168, 283.
length of arc, 271.
volume of revolution,
285.
Catenoid, 276.
Cauchy's form of remain-
der, 71.
Center of curvature, 163.
Change of variable, 124.
Circle, area by double in-
tegration, 295.
of curvature, 163.
Cissoid, 168.
area of, 266.
Component velocity, 107.
Concave, 152.
toward axis, 157.
Conditionally convergent,
59.
Conditions for contact,
161.
Conjugate point, 184.
Conoid, 281.
Constant, 1.
factor, 31, 199.
of integration, 200.
Contact, 159.
of odd and even order,
161.
Continuity, 13, 113.
Continuous function, 13.
Convergence, 57.
Convex, 157.
to the axis, 157.
Critical values, 93.
Cubical parabola, 262.
Cusp, 182.
Cycloid, length of, 273.
surface of revolution,
277.
Decreasing function, 25.
Definite integral, 251.
geometric meaning of,
253.
multiple integral, 293.
Dependent variable, 1.
Derivative, 19, 20.
of arc, 138.
of area, 23, 142.
of surface, 140.
of volume, 140.
Determinate value, 78.
Development, 56, 80.
Differentials, 110, 196.
integration of, 289.
total, 117.
Differentiating operator,
24.
Differentiation, 24.
of elementary forms, 49.
Direction of curvature,
164.
Discontinuous function, 14.
Divergent series, 57.
319
Ellipse, area of, 263.
length of arc of, 274.
evolute of, 178, 284.
Ellipsoid, volume, 280.
Envelope, 187.
Epicycloid, length of, 273.
Equiangular spiral, 282,
284.
Evaluation, 80, 81.
Evolute, 170.
of ellipse, 176, 271, 284.
of parabola, 175.
Expansion of functions,
56.
Exterior rectangles, 254.
Family of curves, 187.
Formula for integration
by parts, 203.
Formulas of differentia-
tion, 49, 50.
of integration, 198, 210.
of reduction, 217, 218.
Function, 1.
Hyperbolic branches, 143.
spiral, area of, 269.
Hypocycloid, area of, 263.
length of arc of, 271, 273.
volume of revolution of,
278.
Implicit function, 120.
Impossibility of reduc-
tion, 218.
Increasing function, 25.
Increment, 13, 15.
Independent variable, 1.
Indeterminate form, 77.
Infinite, 2.
Infinite limits of integra-
tion, 257.
ordinates, 145.
320
INDEX
Infinitesimal, 2.
Integral, 195.
definite, 251.
double, 292.
multiple, 292.
of sum, 199.
triple, 286, 292.
Integration, 195.
by inspection, 197.
by parts, 203.
by rationalization, 231.
by substitution, 205, 238.
formulas of, 198, 210.
of rational fractions,
223.
of total differential, 289.
successive, 286.
summation, 248.
Interior rectangles, 251.
Interval of convergence,
57.
Involute, 170.
of circle, 274, 285.
Lagrange's form of re-
mainder, 70.
Lemniscate, area of, 268.
Length of arc, 269.
of evolute, 173.
polar coordinates, 271.
rectangular coordinates,
269.
limit, 1.
change of, in definite in-
tegral, 295.
Limits, infinite, for defi-
nite integral, 257.
Logarithm, derivative of,
39.
Logarithmic curve, 263.
spiral, length of arc, 272.
Maclaurin's series, 63.
Maximum, 91.
Mean value theorem, 75,
267.
Measure of curvature, 166.
Minimum, 91.
Multiple points, 181.
Natural logarithms, 40.
Non-unique derivative, 25.
Normal, 129.
Notation for rates, 108.
Oblique asymptotes, 147.
Order of contact, 160.
of differentiation, 121.
of infinitesimal, 8.
of magnitude, 7.
Osculating circle, 163.
Osgood, 57.
Parabola, 171.
semi-cubical, 262.
Parabolic branches, 143.
Paraboloid, 283.
Parallel curves, 175.
Parameter, 188.
Partial derivative, 114.
Point of inflexion, 153.
Polar coordinates, 133.
subnormal, 135.
subtangent, 135.
Problem of differential
calculus, 16.
of integral calculus, 195.
Radius of curvature, 164.
Rates, 105.
Rational fractions, inte-
gration of, 223.
Rationalization, 231, 233.
Rectangles, exterior and
interior, 254.
Reduction, cases of impos-
sibility of, 218.
formulae, 217-218.
Remainder, 61.
Rolle's theorem, 67.
Singular point, 179.
Slope, 21.
Solid of revolution, 140.
Sphere, volume by triple
integration, 297.
Spheroid, oblate, 276, 278.
prolate, 276.
Spiral, of Archimedes, 136.
equiangular, 137, 282,
284.
hyperbolic, 269.
logarithmic, 272.
Standard forms, 198, 210.
Stationary tangent, 153.
Steps in differentiation,
24.
Stirling, 62.
Subnormal, 130.
Subtangent, 130.
Summation, 251.
Surface of revolution, 140.
area of, 274.
Tacnode, 182.
Tangent, 21, 129.
Taylor, 62.
Taylor's series, 66.
Tests for convergence, 58.
Total curvature, 166.
differential, 117.
Tractrix, 281.
length of, 285.
surface of revolution of,
285.
volume of revolution of,
285.
Transcendental functions,
38.
Trigonometric functions,
integration of, 238.
Variable, 1.
Volume of solid of revolu-
tion, 277.
Volumes by triple inte-
gration, 295.
Witch, area of, 263.
volume of revolution
of, 278, 279.
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