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THE CALCULUS
A SERIES OF MATHEMATICAL TEXTS
EDITED BT
EARLE RAYMOND HEDRICE
THE CALCULUS
By Elleby Williams Davis and William Chableb Bbenee.
ANALYTIC GEOMETRY AND ALGEBRA
By Alexander Ziwet and Louis Allen Hopkins. '
ELEMENTS OF ANALYTIC GEOMETRY
By Alexander Ziwet and Louis Allen Hopkins. >
PLANE AND SPHERICAL TRIGONOMETRY
By Alfred Monroe Kenton and Louis Ingold.
ELEMENTS OF PLANE TRIGONOMETRY
By Alfred Monroe Keityon and Louis Ingold. .
ELEMENTARY MATHEMATICAL ANALYSIS
By John Wesley Young and Frank Millett Morgan.
PLANE TRIGONOMETRY
By John Wesley Young and Frank Millett Morgan.
COLLEGE ALGEBRA
By Ernest Brown Skinner.
MATHEMATICS FOR STUDENTS OF AGRICULTURE AND
GENERAL SCIENCE
By Alfred Monroe Kenyon and William Vernon Lovitt.
MATHEMATICS FOR STUDENTS OF AGRICULTURE
By Samuel Eugene Rasor.
THE MACMILLAN TABLES
Prepared under the direction of Eable Raymond Hedrick.
PLANE AND SOLID GEOMETRY
By Wai/eer Burton Ford and Charles Ammerman.
CONSTRUCTIVE GEOMETRY
Prepared under the direction of Earle Raymond Hedrick.
JUNIOR HIGH SCHOOL MATHEMATICS
By William Ledley Vosburgh and Frederick William
Gentleman.
A BRIEF COURSE IN COLLEGE ALGEBRA
By Walter Burton Ford.
THE CALCULUS
/
ELLERY WILLIAMS DAVIS
It
PBOFESBOR OF BCATHEMATICS IN THE UNIYEBSITT OF NEBRASKA
AND
WILLIAM CHARLES BRENKE
PBOFE8BOB OF MATHEMATICS IN THE UNIVEBSITY OF NEBRASKA
REVISED EDITION
THE MACMILLAN COMPANY
1924
All rights reserved
PRINTED IN THE UNITED STATES OF AMERICA
Copyright, 1912 and 1922
By the MACMILLAN COMPANY
Set up and electrotyped
Revised Edition published September. 1922
'Sn. ,.:._:.. 638522 ••> : - -••;^'
^j
.T)^?2
PREFACE TO THE FIRST EDITION
The significance of the Calculus, the possibility of applying
it in other fields, its usefulness, ought to be kept constantly
and vividly before the student during his study of the subject,
rather than be deferred to an uncertain future.
Not only for students who intend to become engineers, but
also for those planning a profound study of other sciences, the
usefulness of the Calculus is universally recognized by teach-
ers; it should be consciously realized by the student himself.
It is obvious that students interested primarily in mathe-
o? matics, particularly if they expect to instruct others, should
recognize the same fact.
To all these, and even to the student who expects only gen-
^ eral culture, the use of certain types of applications tends to
make the subject more real and tangible, and offers a basis for
an interest that is not artificial. Such an interest is necessary
to secure proper attention and to insure any real grasp of the
essential ideas.
For this reason, the attempt is made in this book to present
as many and as varied applications of the Calculus as it is
possible to do without venturing into technical fields whose
subject matter is itself unknown and incomprehensible to the
student, and without abandoning an orderly presentation of
fundamental principles.
The same general tendency has led to the treatment of
topics with a view toward bringing out their essential useful-
ness. Thus the treatment of the logarithmic derivative is
^^
vi PREFACE
vitalized by its presentation as the relative rate of change of a
quantity; and it is fundamentally connected with the impor-
tant "compound interest law," which arises in any phe-
nomenon in which the relative rate of increase (logarithmic
derivative) is constant.
Another instance of the same tendency is the attempt, in
the introduction of the precise concept of curvature, to explain
the reason for the adoption of this, as opposed to other
simpler but cruder measures of bending. These are only
instances, of two typical kinds, of the way in which the effort
to bring out the usefulness of the subject has influenced the
presentation of even the traditional topics.
Rigorous forms of demonstration are not insisted upon, es-
pecially where the precisely rigorous proofs would be beyond
the present grasp of the student. Rather the stress is laid
upon the student's certain comprehension of that which is
done, and his conviction that the results obtained are both
reasonable and useful. At the same time, an effort has been
made to avoid those grosser errors and actual misstatements
of fact which have often offended the teacher in texts other-
wise attractive and teachable.
Thus a proof for the formula for differentiating a logarithm
is given which lays stress on the very meaning of logarithms;
while it is not absolutely rigorous, it is at least just as rigorous
as the more traditional proof which makes use of the limit of
(l+^/n)** as n becomes infinite, and it is far more convincing
and instructive. The proof used for the derivative of the sine
of an angle is quite as sound as the more traditional proof
(which is also indicated), and makes use of fundamentally use-
ful concrete concepts connected with circular motion. These
two proofs again illustrate the tendency to make the subject
PREFACE vii
vivid, tangible, and convincing to the student; this tendency
will be found to dominate, in so far as it was found possible,
every phase of every topic.
Many traditional theorems are omitted or reduced in im-
portance. In many cases, such theorems are reproduced in
exercises, with a sufficient hint to enable the student to
master them. Thus Taylor's Theorem in several variables,
for which wide applications are not apparent until further
study of mathematics and science, is presented in this manner.
On the other hand, many theorems of importance, both
from mathematical and scientific grounds, which have been
omitted traditionally, are included. Examples of this sort
are the brief treatment of simple harmonic motion, the wide
appUcation of Cavaheri's theorem and the prismoid formula,
other approximation formulas, the theory of least squares
(under the head of exercises in maxima and minima), and
many other topics.
The Exercises throughout are colored by the views ex-
pressed above, to bring out the usefulness of the subject and
to give tangible concrete meaning to the concepts involved.
Yet formal exercises are not at all avoided, nor is this neces-
sary if the student's interest has been secured through convic-
tion of the usefulness of the topics considered. Far more
exercises are stated than should be attempted by any one
student. This will lend variety, and will make possible the
assignment of different problems to different students and to
classes in successive years. It is urged that care be taken in
selecting from the exercises, since the lists are graded so that
certain groups of exercises prepare the student for other
groups which follow; but it is unnecessary that all of any
group be assigned, and it is urged that in general less than
viii ' PREFACE
half be used for any one student. Exercises that involve
practical applications and others that involve bits of theory
to be worked out by the student are of frequent occurrence.
These should not be avoided, for they are in tune with the
spirit of the whole book; great care has been taken to select
these exercises to avoid technical, concepts strange to the
student or proofs that are too difficult.
An effort is made to remove many technical difficulties by
the intelligent use of tables. Tables of Integrals and many
other useful tables are appended; it is hoped that these will
be found usable and helpful.
Parts of the book may be omitted without destroying the
essential unity of the whole. Thus the rather complete treat-
ment of Differential Equations (of the more elementary
types) can be omitted. Even the chapter on Functions of
Several Variables can be omitted, at least except for a few
paragraphs, without vital harm; and the same may be said
of the chapter on Approximations. The omission of entire
chapters, of course, would only be contemplated where the
pressure of time is unusual; but many paragraphs may be
omitted at the discretion of the teacher.
Although care has been exercised to secure a consistent
order of topics, some teachers may desire to alter it; for
example, an earlier introduction of transcendental functions
and of portions of the chapter on Approximations may be
desired, and is entirely feasible. But it is urged that the
comparatively early introduction of Integration as a summa-
tion process be retained, since this further impresses the
usefulness of the subject, and accustoms the student to the
ideas of derivative and integral before his attention is diverted
by a variety of formal rules.
PREFACE ix
Purely destructive criticism and abandonment of coherent
arrangement are just as dangerous as ultra-conservatism.
This book attempts to preserve the essential features of the
Calculus, to give the student a thorough training in mathe-
matical reasoning, to create in him a sure mathematical
imagination, and to meet fairly the reasonable demand for
enlivening and enriching the subject through applications
at the expense of piu-ely formal work that contains no essential
principle.
E. W. Davis,
W. C. Brbnke,
E. R. Hedrick, Editor.
June, 1912.
PREFACE TO THE REVISED EDITION
The Davis Calculus was very favorably received by the
mathematical world at the time of its original appearance
in 1912. The necessity for some revision arose from the
usual exhaustion of the old lists of exercises by repeated use
of them in class-rooms, and from suggestions of minor changes
of forms and of arrangement of the textual matter as a result
of actual experience in its use. Professor Davis was intend-
ing such a revision kt the time of his death, and it has re-
mained for Professor Brenke, in collaboration with Professor
E. R. Hedrick, to carry it out.
The lists of exercises have been thoroughly revised. Most
of the old formal exercises have been replaced by new ones,
the lists have been extended or shortened, as experience
indicated they should be, and some of the applications for-
merly contained in the lists of problems have been trans-
ferred to the body of the text.
The spirit of the original text was to bring out to the
student the real significance of the Calculus; and this was
accomplished in an unusually effective manner. In the
revision, every effort has been made to retain and to amplify
this spirit. The technique, and mechanical drill, have not
been neglected, but the reasons for learning this technique
have been demonstrated to the student unmistakably.
Some rearrangement of topics has occurred. Thus integra-
tion as a simmiation has been postponed until after the
technique of integration has been mastered. The latter half
of the book has been somewhat simplified, and a few more
xi
xii PREFACE
difficult topics that were not reached by many classes have
been omitted. It is hoped that the revision will appeal to
many and that it will do justice to the great teacher who
was the principal author of the original edition.
W. C. Brbnke,
E. R. Hedrick.
CONTENTS
(Page numbers in roman type refer to the body of the book; those in italic type
refer to pages of the Tables.)
CHAPTER PAGE
I. Functions — Slope — Speed 1
II. Limits — Derivatives 14
III. Differentiation of Algebraic Functions .... 24
IV. Implicit Functions — Differentials 39
V. Tangents — Extremes 49
VI. Successive Derivatives 60
VII. Reversal of Rates — Integration 79
VIII. Logarithms — Exponential Functions .... 99
IX. Trigonometric Functions 119
X. Applications to Curves — Length — Curvature. . 133
XL Polar Coordinates 147
XII. Technique of Integration 156
XIII. Integrals as Limits of Sums 192
XrV. Multiple Integrals — Applications 208
XV. Empirical Curves — Increments — Integrating Devices 234
XVI. Law of the Mean — Taylor's Formula — Series . . 247
XVII. Partial Derivatives — Applications 274
XVIII. Curved Surfaces — Curves in Space 289
XIX. Differential Equations 311
TABLES
TABLE
I. Signs and Abbreviations 1
II. Standard Formulas .......... 3
III. Standard Curves 19
IV. Standard Integrals 35
V. Numerical Tables 61
Index 61
xui
THE CALCULUS
CHAPTER I
FUNCTIONS — SLOPE — SPEED
1. Variables. Constants. Functions. A quantity which
changes is called a variable. The temperature at a given
place, the annual rainfall, the speed of a falling body, the dis-
tance from the earth to the sun, are variables.
A quantity that has a fixed value is called a constarrt.
Ordinary numbers, such as 2/3, n/2, — 7, tt, log 5, etc., are
constants.
If one variable, y, depends on another variable, x, in such
a way that y is determined when x is known, y is called a
function of z; the variable z is called the independent variable,
and y is called the dependent variable. Thus the area A of a
square is a function of the side a of the square, since A = s^.
The volume of a sphere is a function of the radius. In
general, a mathematical expression that involves a variable
a: is a function of that variable.
2. The Function Notation. A very useful abbreviation for a
function of a variable x consists in writing / (x) (read / of x)
in place of the given expression.
Thus if fix) = a;2 + 3x + 1, we may write /(2) = 2^ +
3- 2 + 1 = 11, that is, the value of x^ + 3 x + 1 when x = 2
is 11. Likewise /(3) = 19, /(- 1) = - 1, /(O) = 1, etc.
/(a) = a^ + 3 a + 1. f(u + v)==(u + vy + Siu + v) + 1.
Other letters than / are often used, to avoid confusion, but
/ is used most often, because it is the initial of the word func-
1
2 THE CALCULUS [I, § 2
Hon, Other letters than x are often used for the variable.
In any ease, given /(x), to find /(a), simply substitute a for
X in the given expression.
3. Graphs. In our study of variables and functions, much
use will be made of graphs. To draw the graph of an equa-
tion that contains two variables, we may determine by trial
several pairs of numbers which satisfy the equation and plot
these number pairs as points of the graph, as in elementary
analytic geometry.
Shorter methods for drawing certain graphs are indicated
in some of the following exercises. Thus to draw the graph
of the equation
y = amx-\- cos x,
first draw on the same sheet of paper the graphs of the two
equations
2/ = sin X and y = cos x,
and add the corresponding ordinates.
Certain standard graphs are shown in the tables at the
back of this book.
EXERCISES
Calculate the values of each of the following functions for a suitably
chosen set of values of x, and draw the graph. Estimate the values of
X and / (x) at points where the curve has a highest or a lowest point.
Also determine graphically the solutions of the equation f(x) = 0.
1. fix) = aj2 - 5a; 4- 2. 2. fix) ^ x^ -2x + ^.
3. /(x) = fr*-5x3 4-3a:2-2a:4-3. 4. fix) = ix + l)/i2x-S).
6. fix) = logioo;. 6. fix) = (logioaj)^.
7. fix) = sin X. 8. fix) = esc x,
9. fix) = cos X. 10. fix) = sin a; + cos a:.
11. fix) = a; + sin x. 12. fix) = sin 2 x.
I, § 3] FUNCTIONS — SLOPE — SPEED 3
13. If fix) = sin a; and <t>ix) = cosx, show that [f{x)\^ + [<l> (x)]2 = 1;
/(a:)-?-<A(x)=tanx; fix + y) ==f{x) <t>(y) +f(y) 4>{x); <t>(x + y)=7;
fix) = <^(x/2 - x); 4>ix) =/(x/2 - a;) = - <^ (x - x); /(- x) =
14. If fix) = logio Xf show that
fix) +f(y) ^fix-y); fix^) = 2 fix);
/(m/n)-/(n/m) =2/(m)-2/(n); /(m/n) +/(n/m)=0.
16. Jf fix) = tan a;, 4» ix) =» cos a;, draw the curves y — / (a;), y = <^ (x),
y —fix) — <t>ix). Mark the points where fix) = ^ (x) and estimate the
values of x and y there.
16. Taking /(x) =x2, compare the graph of 2/ =/(x) with that of
2/ = fix) + 1, and with that of 2/ = /(x + 1).
17. Taking any two curves y =/(x), y = <^(x), how can you most
easily draw y -fix) — <t>ix) ? 2/ = /(x) + <^ (x) ? Draw y - x^ -\- 1/x.
18. How can you most easily draw 2/=/(x)+5? y=/(x + 5)?
assuming that y = /(x) is drawn.
19. Draw y = x^ and show how to deduce from it the graph of
y — 2x^; the graph of y = — x^.
Assuming that y—fix) is drawn, show how to draw the graph of
y = 2/(x); that of y = —fix).
20. From the graph oi y — x^, show how to draw the graph of
2/ = (2 x)2; that of 2/ = x2 + 2; that of t/ = (x + 2)2; that of 2/ = (2x - 3)2.
21. What is the effect upon a curve if, in the equation, x and y are
interchanged? Compare the graphs of y =/(x), x ^fiy).
Plot each of the following curves:
22. 2/ + l = sin(3x-2). 23. 2^=2«+sinx.
24. y = 2* + 2-*. 26. 2^ = 2-» cos x.
Plot each of the following curves, using polar coordinates.
26. r = sin 0. 27. r = sin 2 0.
28. r = C06 3 ^. 29. r = 3 + 2 sin 0.
30. r = 2 + 3 sin e, 31. r = 1 + cos 0.
32. r = 0. 33. r = lA
34. r = 2». 36. r = 2-«.
THE CALCULUS
n, H
4. Rate of Increase. Slope. In the study of any quantity,
its rate of increase (or decrease), when some related quantity
changes, is a very important consideration.
Graphically, the rate of increase of y with respect to z is
shown by the rate of increase of the height
of a curve. If the curve ia very flat, there
is a small rate of increase; if steep, a lai^
j-ate.
The steepness, or slope, of a curve shows
the rate at which the dependent variable is
increasing with respect to the independent
variable. When we speak of the slope of
a curve at any point P, we mean the slope
of its tangent at that point. To find this,
we must start, as in analytic geometry, with
a secant through P. Let the equation of
the curve, Fig. 1, he y = 3^, and let the
point P at which the slope is to be found, be the point (2, 4).
Let Q be any other point on the curve, and let Ax represent
the difference of the values of x at the two points P and Q-*
Then in the figure, OA = 2, AB = iix, and OB =2 + Ai.
Moreover, since y = z" at every point, the value of y at Q is
BQ = (2 + Az)^. The slope of the secant PQ is the quo-
tient of the differences Ay and Ax:
_ A^ _ MQ _ (2J
Fig. 1.
tan Z MPQ = ^
Ax
MQ_
' PM~
Ax
= 4 + Aar.
The slope m trf the tangent at P, that is tan Z MPT, is the
* Ax may be regarded aa an abbreviation of the phrase, "difference of
,he x's," The quotient of two such differencea is called a difference ipio-
ienl. Notice particularly that Axdoeanol mean A X z. Inateod of "differ-
iiice of the x'a." the phrases "chaoge in x" and "increment of x" are ofteo
I, § 4] FUNCTIONS — SLOPE — SPEED 5
limit of the slope of the secant as Q approaches P. But it
is clear that this limit as Q approaches P is 4, since Ax
approaches zero when Q approaches P. Hence the slope
m of the curve is 4 at the point P. At any other point the
argument would be similar. If the coordinates of P are
(a, o*), those of Q would be [(a + Aa;), (a + Aa;)^]; and the
slope of the secant would be the difference quotient Ay -r- Ax:
Ay ^ (g + AxY - a^ ^ 2aAx + Ax^ = 2a + Ax
Ax Ax Ax
Hence the slope of the curve at the point (a, a^) is *
m = lim Ay /Ax = lim (2 a + Ax) = 2 a.
On the curve y = x^, the slope at any point is numerically
twice the value of x. When the slope can be found, as above,
the equation of the tangent at P can be written down at
once, by analytic geometry, since the slope m and a point
(a, b) on a line determine its- equation:
y —6 = m(x — a).
The normal to a curve is defined to be the line through a
point on the curve perpendicular to the tangent line at that
point. Hence, if the slope of the tangent at a point (a, b)
is m, the slope of the normal is -^ 1/m, and the equation of
the normal is
2/-6= --(x-a).
Thus, in the preceding example, at the point (2, 4), where we found
m=4, the equation of the tangent is
(2/ — 4) = 4 (x — 2), or 4 a; — y = 4.
The equation of the normal is
y - 4 = - J (x - 2), or a; + 4 2/ = 18.
* Read " Aa;-» 0'* "as Aa; approaches zero." A discussion of limits is given
in Chapter II.
6 THE CALCULUS [I, § 5
6. General Rules. A part of the preceding work holds true
for any curve, and all of the work is at least similar. Thus,
for any curve, the slope is
m = lim (Ay /Ax) ;
Ax— K)
tfiat is J the slope m of the curve is the limit of the difference quo-
tient Ay /Ax.
The changes in various examples arise in the calculation of
the difference quotient, Ay 4- Ax, and of its limit, m.
This difference quotient is always obtained, as above, by find-
ing the value of y at Q from the value of x ai Q, from the equa-
tion of the curve, then finding Ay by subtracting from this the
value of y at P, and finally forming the difference quotient by
dividing Ay by Ax.
6. Slope Negative or Zero. If the slope of the curve is
negative, the rate of increase in its height is negative, that is,
the height is really decreasing with respect to the independent
variable.*
If the slope is zero, the tangent to the curve is horizontal.
This is what happens ordinarily at a highest point (maximum)
or at a lowest point (mininaum) on a curve.
Example 1. Thus the curve 2/ = x^, as we have just seen, has, at any
point X = a,a. slope w = 2 a. Since m is positive when a is positive, the
curve is rising on the right of the origin; since m is negative when a is
negative, the curve is falling (that is, its height y decreases as x increases)
on the left of the origin. At the origin w = 0; the origin is the lowest
point (a minimum) on the curve, because the curve falls as we come
toward the origin and rises afterwards.
* Increase or decrease in the height is always measured as we go toward
the right, i.e. as the independent variable increases.
I, §61
FUNCTIONS — SLOPE — SPEED
Example 2. Find the slope of the curye
y— x* + 3x — 5
(1)
at the point where a; = — 2; also in general at a point a; = o. Use these
values to find the equation of the tangent
at a; = 2; the tangent at any point.
When X = — 2, we find y = — 7 (P in
Fig. 2); talcing any second point Q, (— 2 +
Ax, — 7 + Ay), its coordinates must satisfy
the given equation, therefore
-7+A2/ = (-2+Ax)2
+ 3 (- 2 + Ax) - 6,
or
Ay — — ^ Ax -j-Ax2 4- 3 Ax
= — Ax + Ax^,
where Ax^ means the square of Ax. Hence
the slope of the secant PQ is
Ay /Ax = — 1 + Ax.
The slope m of the curve is the limit of Ay /Ax
as Ax approaches zero; i.e. Fig. 2.
\
\
y=
X*
► sir-
5
\
L
"1
~b
/
i
/
\
A
\
/
\
\
<»
ll^v
\
V
/
1
1
\
/
1
L
p
X
Az
M
\
T
\
w
= lim ^ = lim (- 1+ Ax) = - 1.
Aaf->0 Ax
Ajp-»0
It follows that the equation of the tangent at (— 2, — 7) is
(y + 7) = - 1 (x + 2), or X + 2/ + 9 = 0,
Likewise, if we take the point P (a, b) in any position on the curve
whatsoever, the equation (1) gives
5 = o2 + 3 a - 5.
Any second point Q has coordinates (a + Ax, b + Ay) where Ax and
Ay are the differences in x and in y, respectively, between P and Q.
Since Q also lies on the curve, these coordinates satisfy (1) :
. b + Ay={a+ Ax)2 + 3 (a + Ax) - 5.
Subtracting the last equation from the preceding.
Ay = 2 o Ax + Ax^ + 3 Ax, whence Ay /Ax = (2 o +3) + Ax,
and
m = lim ^ = lim [(2 o + 3) + Ax] = 2 a + 3.
Ax— »0 Ax J^x-*Q '
Therefore the tangent at (o, b) is
y-(a2 + 3o-5) = (2a4-3) (x-o), or (2a + 3)x - y = a^ _|. 5.
8 THE CALCULUS [I, ! 6
The slope m is zero when 2 o + 3 = 0, i.e. when a - — 3/2. Hence
the tangent is horizontal at the point where x = — 3/2.
Example 3. Consider the curve y = x' — 12 1 + 7. If the value of
X at any point P is «, the value of y is o' — 12 o + 7. If the value of x
at Q is a + Ax, the value of ji at G is (a + Ax)' ~ 12 (a + Ax) + 7.
Hence
Ay_ [(o + Ax)^~12(a + A3:)+71-[<i'-12a+7]
= {3 a^ + 3 o Ax + Ax') - 12,
#F
-■/
^,, ,
/
-1-
\
\
:' i'.t
-^
'"
~--m
-1 1
1 ! ! ■
f
---
!
K
■^
4t
m
m
For example, if i — 1, y = ~ 4; at this pomt (1, — 4) the slope
is 3-1^ — 12 = —9; and the equation of the tangent is
(y + 4) = - 9 (X - 1), or 9 1 + y - 5 =0.
The points at which the slope is zero would be determined by the
equation w = 3 a* - 12 = 0, which gives a = ± 2. When x has either
of these values, the tangent is horizontal. From the equation of the
curve, it X = + 2, y ■■ — 9; when x = — 2, y — 23. Hence the horizontal
tangents are at the points (2, - 9) and (- 2, 23). (See Fig. 3.)
I, § 7] FUNCTIONS — SLOPE — SPEED 9
EXERCISES
Find the slope and the equation of the tangent line to each of the
following curves at the point indicated. Verify each result by drawing
the graph of the curve and the graph of the tangent line from their
equations.
1. 2/ = a;2 - 2; (1, - 1). 2. 2/ = x^l2\ (2, 2).
3. y = 2 x2 -3; (2, 5). 4. 2/ = a;2 - 4 a; + 3; (2, - 1).
6. y = rr3; (1, 1). 6. y = x3 - 9 x; (2, - 10).
7. 2/ = xs - 3 a; + 4; (0, 4). 8. 2/ = 2 a;3 - 3 x^\ (1, - 1).
Draw each of the following curves, using for greater accuracy the
precise values of x and y for which the tangent is horizont£d, and the
knowledge of the values of x for which the curve rises or falls.
9. 2^ = a;2 4- 5 a; — 5. \% y — 0^ ^ x^. 16. 2/ = ^^
10. y^Zx-Q^.j 13. 2/=a^-6a;. 16. y^x^ — 2x^.
11. y = aj3-3x4-2. 14. 2/ =25' -6 a; + 5. 17. 2^ = a;* __ 3^.
7. Speed. An important case of rate of change is the
rate at which a body moves, — its speed.
Consider the motion of a body falling from rest under the
influence of gravity. During the first second it passes over
16 ft., dining the next it passes over 48 ft., during the third
over 80 ft. In general, if i is the niunber of seconds, and s
the entire distance it has fallen, s = 16 /^ if the gravitational
constant g be taken as 32. The graph of this equation (see
Fig. 4) is a parabola with its vertex at the origin.
The speed, that is the rate of increase of the space passed
over, is the slope of this curve, i.e.
,. As
lim — •
This may be seen directly in another way. The average
speed for an interval of time A^ is found by dividing the dif-
ference between the space passed over at the beginning and at
10
THE CALCULUS
[I, §7
the end of that interval of time by the difference in time: i.e.
the average speed is the difference quotient As -7- A^ By the
speed at a given instant we mean the limit of the average
speed over an interval At beginning or ending at that instant
as that interval approaches zero, i,e,
speed =^ lim-r?-
Taking the equation s = 16 ^, if ^ = 1/2, 8 = 4 (see point P in Fig. 4).
After a lapse of time A^, the new
values are
t==l/2+At,
and « = 16 (1/2 + M^
(Q in Fig. 4).
Then
As = 16 (1/2 4- AO^ - 4
= 16 A< + 16 A?,
As /At = 16 + 16 At.
Whence
As
\
s
/
\
-AA-
//
-w
s-
= 16<*
■y/
\
/
A
i
k
n
L
J
V
/
\
1ft
1
7
V
T>
//
\
^'
\t-^
t
1
7
\
speed = lim ^ ^
Fig. 4.
= lim (16 + 16 AO = 16;
that is, the speed at the end of the first half second is 16 ft. per second.
Likewise, for any value of «, say « = T, s = 16 T^) while for
< = T + Af, « = 16 (T + At)^\ hence
awrages
peerf-^ = ^^^^ + ^\'-^'^-32r+16A«
A^
and
As
speed = lim —: = 32 T.
^-^At
Thus, at the end of two seconds, T = 2, and the speed is 32-2 = 64, in
feet per second.
8. Component Speeds. Any cm^e may be regarded as the
path of a moving point. If a point P does move along a
curve, both x and y are fixed when the time t is fixed. To
I, §91
FUNCTIONS — SLOPE — SPEED
11
specify the motion completely, we need equations which
give the values of x and y in terms of L
The horizontal speed is the rate of change of x with
respect to the time. This may be thought of as the speed
of the projection Af of P on the x-axis. Likewise, the
vertical speed is the rate of change of y with respect to
the time. It would be the speed of the projection of P on
the 2/-axis. Hence, by § 7, we have
and
Ax
horizontal speed = lim -r-t
At-^^t
Since
vertical speed ^ lim -Ji
Ay _^Ay . Ax
Ax" A/"^ A/'
it follows that
m = (vertical speed) 4- (horizontal speed) ;
that is, the slope of the curve is the ratio of the rate of increase
of y to the rate of increase of x.
9. Continuous Functions. In §§ 4-8, we have supposed that
the curves used were smooth. All of the functions which
we have used could be represented by smooth curves. Except
perhaps at isolated points, a small change in the value of
one coordinate has caused a small change in the value of
the other coordinate. Throughout this text, unless the
contrary is expressly stated, the functions dealt with will
be of the same sort. Such functions are called continuous.
(See § 10, p. 14.)
The curve y = 1/x is continuous except at the point
a? = 0. The curve y = tan x ig continuous except at the
12 THE CALCULUS [I, § 9
points X = =*= 7r/2, =t 3 7r/2, etc. Such exceptional points
occur frequently. We do not discard a curve because of
them, but it is understood that any of our results may fail at
such points.
EXERCISES
1. From the formula a = 16 ^, calculate the values of s when ^ = 1,
2, 1.1, 1.01, 1.001. From these values calculate the average speed between
^ = 1 and ^ = 2; between t = 1 and t = 1.1; between ^ == 1 and t = 1.01;
between ^ = 1 and t = 1.001. Show that these average speeds are
successively nearer to the speed at the instant ^ = 1.
2. Calculate as in Ex. 1 the average speed for smaller and smaller in-
tervals of time after t ^2; and show that these approach the speed at the
instant t ^=2.
3. A body thrown vertically downwards from any height with an
original velocity of 50 ft. per second, passes over in time t (in seconds) a
distance s (in feet) given by the equation « = 50 < + 16 ^ (if ^ = 32, as
in § 7). Find the speed v at the time ^ = 1; at the time t ^2; at the
time < = 4; at the time t = T.
4. In Ex. 3 calculate the average speeds for smaller and smaller m-
tervals of time after * = 0; and show that they approach the original
speed «o = 50. Repeat the calculations for intervals beginning with
« = 2.
Calculate the speed of a body at the times indicated in the follow-
ing possible relations between 8 and t:
6. a = f2j ^ = 1, 3, 20, T, 7. s = - 16 «« + 80 «; < = 0, 2, 5.
6. a = 16 f2 - 50 <; « = 0, 2, r. 8. « = ^ - 6 « + 4; < = 0, 1/2, 1.
9. The relation in Ex. 7 holds (approximately, since ^ = 32 ap-
proximately) for a body thrown upward with an initial speed of 80 ft.
per second, where s means the distance from the starting point coimted
positive upwards. Draw a graph which represents this relation between
the values of s and t
In this graph mark the greatest value of a. What is the value of v at
that point? Find exact values of s and t for this point.
10. A body thrown horizontally with an original speed of 8 ft. per
second falls in a vertical plane curved path so that the values of its hori-
I, § 9] FUNCTIONS — SLOPE — SPEED 13
zontal and its vertical distances from its original position are respec-
tively, X ==St,y == Wt^, where y is measured downwards. Show that the
vertical speed is 32 Tj and that the horizontal speed is 8, at the instant
t = T, EUminate t to show that the path is the curve 4 y = x^.
11. Find the component speeds and the resultant speed when the
path is given by the equations
x=t + l,y = fi'-l.
Calculate their numerical values when < = 1; when ^ = 0; when t = 2,
Plot the path.
12. Proceed as in Ex. 11 when the path is given by the equations
a; = 2 + <2, y = 2 - ^.
CHAPTER II
LIMITS — DERIVATIVES
10. Limits. Infinitesimals. We have been led in what pre-
cedes to make use of limits. Thus the tangent to a curve at
the point P is defined by saying that its slope is the limit of
the slope of a variable secant through P; the speed at a given
instant is the limit of the average speed; the difference of the
two values of x, Ax, was thought of as approaching zero; and
so on. To make these concepts clear, the following precise
statements are necessary and desirable.
When the difference between a variable x and a constant a he-
com£S and remains less, in absoliUe value* than any prea^signed
positive quantity y however smaUy then a is the limit of the
variable x.
We also use the expression "x approaches a as a limit,'' or,
more simply, "x approaches a." The symbol for limit is
Urn; the symbol for approaches is — >; thus we may write
lim X = a, or x— >a, or lim (a — x) =0, or a — a;— >0.t
When the limit of a variable is zero, the variable is called
an infinitesimal. Thus a — x above is an infinitesimal.
The difference between any variable and its limit is always
an infinitesimal. When a variable x approaches a limit a,
any continuous function f (x) approaches the limit f (a) :
* When dealing with real numbers, absolute value is the value without
regard to signs, so that the absolute value of —2 is 2, for example. A con-
venient symbol for it is two vertical lines; thus |3 — 7|=4.
t The symbol = is often used in place of ->.
14
II, § II] LIMITS — DERIVATIVES 15
thus, if y =/(x) and ft =/(«), we may write
lim y = b, or lim f{x) =f (a) .
This GonditioQ ia tho precise definition of continuity at the
point x=a. (See § 9, p. 11.)
11. Properties of Limits. The following properties of limits
will be assumed as self-evident. Some of them have already
been used in the article noted below.
Theorem A. The limit of the sum of two varuAlesis the sum
of the limits of the two variables. This is easily extended to the
case of more than two variables. (Used in §§ 4, 6, and 7.)
Theorem B. The limit of the jfroduct of two varieties is the
product of the limOs of the variables. (Used in §§ 4, 6, and 7.)
Theorem C. The limit of the quotient of one variable divided
by another is the gu(^ient of the limits of the variables, provided
the limit of the divisor is n,ot zero. (Used in § 8.)
The exceptional case in Theorem C ia really the most in-
teresting and important case of all. The exception arises
because when zero occurs as a denominator, the division can-
not be performed. In Boding the slope of a curve, we con-
sider lim (Aj//Ai) as Ax approaches zero; notice that this is
precisely the case ruled out in Theorem C, Again, fhe speed
is lim (As/M) as At approaches zero. The limit of any such
difference quotient is one of these exceptional cases.
Theorem D. The limit of the ratio of two infinitesimals de-
pends upon the law connecting them; otherwise it is quite inde-
terminate. Of this the student will see many instances; for
the Differential Calculus coTmsts of the consideration of just such
limits. In fact, the very reason for the existence of the Differ-
16 THE CALCULUS [H, § 11
ential Calculus is that the exceptional case of Theorem C is
important, and cannot be settled in an offhand manner.
The thing to be noted here is, that, no matter how small
two quantities may be, their ratio may be either small or
large; and that, if the two quantities are variables whose
limit is zero, the limit of their ratio may be either finite, zero,
or non-existent. In oiu* work with such forms we shall try to
substitute an equivalent form whose limit can be found.
Theorem D accounts for the case when the numerator
as well as the denominator in Theorem C is infinitesimal.
There remains the case when the denominator only is
infinitesimal. A variable whose reciprocal is infinitesimal is
said to became infinite as the reciprocal approaches zero.
Thus
y = lA
is a variable whose reciprocal is 2;. As x approaches zero, y
is said to become infinite. Notice, however, that y has no
value whatever when x == 0.
Likewise
y = sec X
is a variable whose reciprocal, cos x, is infinitesimal as x
approaches t/2. Hence we say that sec x becomes infinite
as X approaches t/2. In any case, it is clear that a
variable which becomes infinite becomes and remains larger
in absolute value than any preassigned positive number,
however large.
The student should carefully notice that infinity is not a
number. When we say that "sec x becomes infinite as x ap-
proaches V2/' * we do not mean that sec (t/2) has a value,
we merely tell what occurs when x approaches 7r/2.
♦ Or, as is stated in short form in many texts, "sec (ir/2) « 00."
II, § 11] LIMITS — DERIVATIVES 17
EXERCISES
1. Imagine a point traversing a line-segment in such fashion that it
traverses half the segment in the first second, half the remainder in the
next second, and so on; always half the remainder in the next following
second. Will it ever traverse the entire line? Show that the remainder
after t seconds is l/2<, if the total length of the segment is 1. Is this
infinitesimal? Why?
2. Show that the distance traversed by the point in Ex. Ivat seconds
is 1/2 + 1/22 + ... + 1/2*. Show that this sum is equal to 1 - 1/2'; hence
show that its limit is 1. Show that in any case the limit of the distance
traversed is the total distance, as t increases indefinitely.
3. Show that the limit of 3 — o^ as 2; approaches zero is 3. State this
result in the symbols used in § 10. Draw the graph of y = 3 — x^ and
show that y approaches 3 as 2; approaches zero.
Evaluate the following limits:
4.
lim (7-5a;+3a;2).
7.
,. 3-2a;2
i^24 + 2x2-
®
,. a;2 - 3 a; + 2
x^2X^+2x-\-3'
6.
lim(7-5a;+3a;2).
x-*2
8.
,. 2x
um 7 •
x-^ 4 — a;
11.
lim^J^.
6.
]im{k^+kx-2x^).
9.
,. 1 — X
lim •
X_>2 X
12.
y a + 6a;+ca;2
x-^om-\-nx-\-lx^'
If the numerator and denominator of a fraction contain a common
factor, that factor may be canceled in finding a limit, since the value of
the fraction which we use is not changed. Evaluate before and after
canceling a common factor:
^- ,. (x - 2) (x - 1) .^ ,. a;2(x-2)
-j^_i(2 x — Z)(x—l) x-*2 (x2 + 1) (x — 2)
a;2 lex ,. a;2-4a; + 3 />. ,. a;2+a;-2
15. linl?-^ % lim ^^7^+^ rf7) Um
x_>o X ^^* x-^1 a;2 — 1 v_/ x^i 2 x2 + X — 3
4oi 1- ^X-hl -ft r X2(X--1)2 X«
IBJ hm r^' 19. lim —7^ — jr-f-. 20. lim — =
x^-i x + 1 x-*ox^ — 2x^ x-^ X
0, n>l,
ll,n = l.
@ Show that
,. 3x2 + 5 ,
lim « , . ;— = = 3.
x-^ x2 + 4 X + 5
[Hint. Divide numerator and denominator by x*; then such terms as
5/x> approach zero as x becomes infinite.]
18
THE CALCULUS
[11, § 11
Evaluate:
2x
22
-1
-2
23. lim
5a;2-4
25. lim
2z
«— >«
ViT
26. lim
rr-Ko3 a;2 + 2
24. lim
aa; + 6
X'
X— MO Vaj^ — 4
27.
x—¥»mx -\-n
lim
px+g'
28. Let be the center of a circle of radius r = 05, and let a = Z C0J5
be an angle at the center (Fig. 6). Let BT be
perpendicular to 05, and let BF be perpendic-
* ular to OC, Show that OF approaches OC as a
approaches zero; likewise arc CB-*0, arc
DB^O, andFC-»0, asa-*0.
29. In Fig. 6, show that the obvious
Fig. 6. geometric inequality FB < arc CB <BT is
equivalent to r sin a<r'a<r tan a, if a is measured in circular measure.
Hence show that a/sin a lies between 1 and 1/cos a, and therefore
that lim (a/sin a) = 1 as «-> 0. (Cf. § 72.)
30. In Fig. 6, show that
,. FB ^ ,. OF , ,. BT f. y FC ^ ,. arc 05 ^
Imi — = 0: Imi — =1; lim — = 0: lim — = lim = 0.
12. Derivatives. While such illustrations as those in the pre-
ceding exercises are interesting and reasonably important,
by far the most important cases of the ratio of two infini-
tesimals are those of the type studied in §§ 4-8, in which each
of the infinitesimals is thq difference of two values of a varia-
ble, such as Aiz/Ax or As/ At, Such a difference quotient
Ay/AXj of y with respect to x, evidently represents the
average rate of increase of y with respect to x in the interval
Ax; if X represents time and y distance, then Ay /Ax is the
average speed over the interval Ax (§ 7, p. 9); if y =/(x)
is thought of as a curve, then Ay /Ax is the slope of a secant
or the average rate of rise of the curve in the interval Ax
(§ 4, p. 4).
II, § 13] LIMITS — DERIVATIVES 19
The limit obtained in such cases represents the instanta-
neous rate of increase of one variable with respect to the
other, — this may be the slope of a curve, or the speed of a
moving object, or some other ratej depending upon the
nature of the problem in which it arises.
In general, the limit of the quotient Aiz/Ax of two infinitesim/il
differences is called the derivative of y with respect to x; it is
represented by the symbol dy/dx:
JL s derivative of y with respect to x= lim = •
ax Ax-^ Ax
Henceforth we shall use this new symbol dy/dx or other
convenient abbreviations; * but the student must not forget
the real meaning: slope, in the case of curve; speed, in the
case of motion; some other tangible concept in any new
problem which we may undertake. In every case the meaning
is the rate of increase of y mth respect to x.
Any mathematical formulas we obtain will apply in any of
these cases. We shall use the letters x and y, the letters s
and t, and other suggestive combinations; but any formula
written in x and y also holds true, for example, with the
letters s and t, or for any other pair of letters.
13. Formula for Derivatives. If we are to find the value of
a derivative, as in §§ 4-7, we must have given one of the vari-
ables 2/ as a function of the other x:
(1) y=f(x).
If we think of (1) as a curve, we may take, as in § 4, any
* Often read "the x derivative of y." Other names frequently used are
differerUial coefficient and derived function. Other convenient notations are
Dxy, yx, fix) y', y; the last two are not safe unless it is otherwise clear
what the independent variable is.
20 THE CALCULUS [II, § 13
point P whose coordinates are x and y, and join it by a secant
PQ to any other point Q, whose coordinates are x + Ax, y + Ay.
Here x and y represent fixed values
of X and y. This will prove more
convenient than to use new letters
each time, as we did in §§ 4-7.
Since P lies on the curve (1), its
coordinates {x, y) satisfy the equa-
tion {l),y —f {x). Since Q lies on
(1), X + Ax and y + Ay satisfy the same equation; hence we
must have
(2) t/ + A2/=/(x + Ax).
Subtracting (1) from (2) we get
(3) Ai/=/(x + Ax)-/(x).
Whence the difference quotient is
fA^ ^y /(x + Ax) —fix) , „,^
(4) -^ = '^— — ' — -^ — ^-^^^ = average slope over PM,
and therefore the derivative is
(5) <k =lim^^lim ^(' + y--^('> =aopeatP*
ax az-»oAjc az->o Ax
14. Rule for Differentiation. The process of finding a de-
rivative is called differentiation. To apply formula (5) of § 13:
(A) Find (y + Ay) by substituting (x + Ax) for x in the given
function or equation; this gives y + Ay = f{x + Ax).
{B) Subtract y from y + Ay; this gives
Ay=f{x + Ax) -fix).
* Instead of slope, read speed in case the problem deals with a motion,
as in § 7. In general, Ay /Ax is the average rate of increase, and dy/dx is
the instantaneous rate.
II, § 14] LIMITS — DERIVATIVES 21
(C) Divide Ay by Ax to find the difference quotient Ay/Ax;
simplify this result.
(D) Find the limit of Ay /Ax as Ax approaches zero; this
result is the derivative, dy/dx.
Example 1. Given y ^f {x) ^x^^to find dy/dx,
(A) / (a; + Aa;) = (a; + Aa;)2.
(B) Ay=f(x+/^)-f{x) = (x + Aa;)2-a;2 =2xAx + Ix^,
(C) Ay/Ax = (2xAx + A?) -^ Aa; = 2 a; + Aaj.
(D) dy/dx = lim Ay /Ax = lim (2 x + Aa;) = 2 a;.
Aap— ►O Ax— ►0
Compare this work and the answer with the work of § 4, p. 5.
Example 2. Given y =f(x) ^ofi — 12 a: + 7, to find dy/dx,
}{x-YAx) = (a;+Aa;)3- 12 (a; + Ax) + 7.
{B) Ay =/(a; + Ax)-/(x) = 3 x^ Ax + 3 x A? + Aa:' - 12 Ax.
(C) Ay/Ax = 3 x2 + 3 X Ax + Ax^ - 12.
(D) dy/dx = lim Ay/Ax = lira (3 x2 + 3 x Ax + Ax^ - 12) = 3 x2 - 12.
A«— M) Ax— »0
Compare this work and the answer with the work of Example 3, § 6.
Example 3. Given y —fix) ^ l/x2, to find dy/dx,
1
(A) /(x+Ax) =
{B) Ay=/(x + Ax)-/(x) =
(x + Ax)2
1 _ 1_ 2xAx + Ax
(X + AX)2 X2 X2 (X + AX)2
rn ^ = _ 2X + AX
^^ AX X2(X+AX)2'
(Z» dy/dx =^lim^g = mnj~ /(^ J^) j = - |f = " |'
Example 4. Given y =/ (x) s Vx, to find dy/dx, or (^ {x)/dx.
(A) / (x + Ax) = Vx + Ax.
(5) A2/=/(x + Ax)-/(x) = Vx+Ax— y/x,
(C) ^ = ^^ + Ax — Vx _ Vx + Ax -- Vx Vx + Ax + Vx
Ax Ax "" Ax Vx + Ax + Vx
1
Vx + Ax + Vx
(D) ^=ii„,^=lto 1 ?_.
dx Ax-»o Ax Ax-»0 Vx + Ax + Vx 2Vx
22 THE CALCULUS [II, § 14
Example 5. Given ?/ = / (a;) ^ x^t ^ find ^/ {x)/dx.
(A) / (x + Ax) = (x + Ax)7 = x7 + 7 x^ Ax + (terms with a factor A?).
(5) Ay = / (x + Ax) — / (x) = 7 X® Ax+ (terms with a factor Ax^).
(C) Ay/Ax = 7 X® + (terms with a factor Ax).
(D) dy/cte = lim Ay /Ax = lim [7 x^ + (terms with a factor Ax)] = 7 x^.
EXERCISES ,
Find the derivative, with respect to x, of each of the following
functions:
1.
x2 - 4 X + 3.
5. 8x — X*.
2.
X8+2X2.
6. X4+X2+1.
3.
3x--x«.
X-- 1
4.
•
X4+2X2.
^' A ^ o'
10.
11.
X2
x-1
3x-2
x + 2
tt. VF=1.
13. Find the equation of the tangent to the curve y = 4/x at the point
where x = 3.
14. Determine the values of x for which the curve y = x'--12x + 4
rises, and those for which it falls. Draw the graph accurately.
Proceed as in Ex. 14 for each of the following curves:
16. x«-15x + 3. 17. 2x*--64x.
16. x8-3x2. 18. x4-32x2.
19. If a body moves so that its horizontal and its vertical distances
from a point are, respectively, x = 15^, y = — 16^ + 15 1, find its hori-
zontal speed and its vertical speed. Show that the path is
2/ = - 16x2/225+x,
and that the slope of this path is the ratio of the vertical speed to the
horizontal speed. [These equations represent, approximately, the mo-
tion of an object thrown upward at an angle of 45°, with a speed 15V2.]
20. A stone is dropped into still water. The circumference c of the
growing circular waves thus made, as a function of the radius r, is c = 2 irr.
Show that dc/dr = 2 x, i.e. that the circumference changes 2 x times as
fast as the radius.
II, § 14] LIMITS — DERIVATIVES 23
Let A be the area of the circle. Show ihsiidA/dr = 2 irr; i.e. the rate
at which the area is changing compared to the radius is numerically
equal to the circumference.
21. Determine the rates of change of the following variables:
(o) The surface of a sphere compared with its radius, as the sphere
expands.
(6) The volume of a cube compared with its edge, as the cube enlarges.
(c) The volume of a right circular cone compared with the radius of
its base (the height being fixed), as the base spreads out.
22. If a man 6 ft. tall is at a distance x from the base of an arc light
10 ft. high, and if the length of his shadow is s, show that «/6 = 05/4, or
« = 3 x/2. Find the rate (ds/dx) at which the length s of his shadow
increases as compared with his distance x from the lamp base.
CHAPTER III
DIFFERENTIATION OF ALGEBRAIC FUNCTIONS
16. Classification of Functions. For convenience it is usual
to classify functions into certain groups.
A function which can be expressed directly in terms of the
independent variable x by means of the three elementary
operations of multipUcation, addition, and subtraction is
called a polynomial in x.
Thus, 2x^ + 4:31^ — 7x + 3y x^ — 4x + 6, etc., are poly-
nomials. The most general polynomial is Oox" + aix^~^ +
. . . + On-iX + a», where the coefficients a©, ai, . . . , On are
constants, and the exponents are positive integers.
A function which can be expressed directly in terms of
the independent variable x by means of the four elementary
operations of multiplication, division, addition, and sub-
traction, is called a rational function oix. Thus, 1/x, {x^ —
3 x)/{2 x+7), etc., are rational. The most general rational
function is the quotient of two polynomials, since more than
one division can be reduced to a single division by the rules
for the combination of fractions. All polynomials are also
rational functions.
If, besides the four elementary operations, a function re-
quires for its direct expression in the independent variable x
at most the extraction of integral roots, it is called a simple
algebraic function * of x. Thus, Vx, (Vx^ + 1 - 2)/(3 - v^x),
* Since the expression "algebraic function" is used in the broader sense
of § 25 in advanced mathematics, we shall call these simple algebraic
functions.
24
Ill, § 16] ALGEBRAIC FUNCTIONS 25
etc., are simple algebraic functions. All rational functions
are also simple algebraic functions.
Simple algebraic functions which are not rational are
called irrational functions,
A fimction which is not an algebraic function is called a
transcendental function. Thus sinx, log x, e^, tan-^ (1 + x) + x^,
etc., are transcendental.
In this chapter we shall deal only with algebraic functions.
16. Differentiation of Polynomials. We have differentiated
a number of polynomials in Chapter I. To simplify the
work to a mere matter of routine, we need four rules:
The derivative of a constant is zero:
The derivative of a constant times a function is equal to the
constant times the derivative of the function:
[H] i^ = c.*f.
dx dx
The derivative of the sum of two functions is equal to the sum
of their derivatives:
[III] d{u + v) ^^ dv^
dx dx dx
The derivative of a power y x**, with respect to x is nx**""^:
[IV] ^ = nxn-K
dx
[We shall prove this at once in the case when n is a positive integer;
later we shall prove that it is true also for negative and fractional
values of n.]
Each of these rules was illustrated in Chapter II, § 14.
To prove them we use the rule of § 14.
26 THE CALCULUS [III, § 16
Proof of [I]. Iiy=Cj a. change in x produces no change in
y; hence Ay = 0. Therefore dy/dx = lim Ay /Ax = lim =
as Ax approaches zero. Geometrically, the slope of the
curve 2/ = c (a horizontal straight line) is everywhere zero.
Proof of [H]. If y = C'U where w is a function of x, a
change Ax in a: produces a change Au in u and a change Ay in
y; following the rule of § 14 we find:
(A) y + Ay = c (u + Au).
(B) Ay = c Au.
.-r,. dy ., r Aul ,. Au du
(D) :^= lim \c'-r- = clim t— = c-t"
ax Ax-»oL Ax J Aa^-♦oAa: dx
Thus d{7x^)/dx = 7 • dix^)/dx = 7 • 2 x = 14 x. (See §§ 4,
15.)
Proof of [III]. If y = u + Vf where u and v are functions of
X, a change Ax in x produces changes At/, Aiz, Ay in y, u, v,
respectively, hence
(A) y + Ay=(u + Au) + (v + Av);
(B) Ay = Au + Av;
^^Au Av_^
^^ Ax Ax "^ Ax '
(D) :j = li"^ T~ + li"^ T- = 3- + 3--
ax Aaj-*oAX Aa;-*oAx ttX OX
Thus
d{x^ - 12 X + 7) _ d(x3) _ d(12 x) ^ d{l) _ 3 ^ ^g
dx dx dx dx , '
by applying the preceding rules and noticing that dx^/dx
= 3x2.
Ill, § 16] ALGEBRAIC FUNCTIONS 27
Proof of [IV]. If 2/ = .r", we proceed as in Example 5, § 14.
(A) y + Ay = {x + Ax)** = a:** + nx"~^Ax + (terms whicli
liave a common factor Ax^).
(B) Ay = nx'*~^Ax+ (terms with a common factor Ax^).
(C) Ay /Ax = nx**"^ + (terms wliicli liave a factor Ax).
(D) dy/dx = lim (AyJAx) = nx^~^.
Tliis proof liolds good only for positive integral values of n.
For negative and fractional values of n, see §§ 17, 21.
Example 1. d {x^)/dx = 9 x^.
Example 2. dx/dr = 1 • x° = 1, since a;® = 1.
This is also evident directly: dx/dx = lim Ax/Ax = 1. Notice how-
ever that no new rule is necessary. ^^»-»o
Examples. $-(x*-7 x^ + 3x- 5) =4a;8- 14a; +3.
dx
Example 4. ^ (Ax» + Bx^ + C) = wAa;w-i + nBa;»-i.
dx
EXERCISES
Calculate the derivative of each of the following expressions with re-
spect to the independent variable it contains (x or r or 8 or t or y or u).
In this list, the first letters of the alphabet, down to n, inclusive, repre-
sent constants.
1. y = Sx*. 5. s = 10t^-{-50t 9. g = 6<« + 3<«.
2. y = a;V5. 6. « = - 4« + 2 «». 10. q = tT-8t^ + 10.
3. y = 4a:5 + 5. 7. 8=Z{t*'-2fi). 11. g = 6(r8 + r2 + 1).
4. y = 5(a;« - 3). 8. « = ??i<2 + nt\ 12. g = Ario + ^^6.
13. u = v{v- 1). 14. w = (1 - f;2) (2 -f- ^2).
16. w = v\v5 + 2). 16. M = (t;« + 2) (t;6 - 3).
17. 3 = 02/5(2/8 -6t/4). (^ a = t*io(2 m7 - 5 1*9 + 8).
19. 2 = A;a;» + ^». (^ r =cx»» -- (ia;2».
21. r-«»(«2»_^2). 22. «=A-52/*"'.
28 THE CALCULUS [III, § 16
Locate the vertex of each of the following parabolas by finding the
point at which the slope is zero.
23. y = x^+4x + 4. 25. y = 5 + 2x-a:2.
24. y =x^-6x + 2, 26. y = ax^+2bx+c,
27. What is the slope of the curve j/ = 2 a;' — 3 x^ -|- 4 at a: = 0, ±2,
± 4? Where is the slope 9/2? - 3/2? Where is the tangent hori-
zontal? Draw the graph.
28. What is the slope of the curve y = x*/4 — 2 x' + 4 a:^ at a: = 0,
1,-1,-2? Where is the tangent horizontal? Where is the slope
equal to eight times the value of x?
29. Show that the function a;' — 3x2-f3a: — 1 alwa3rs increases
with X. Where is the tangent horizontal?
30. Find the angle between the curve y ^x^ and the straight line
2/ = 9 a; at each of their points of intersection.
17. Differentiation of Rational Functions. In order to dif-
ferentiate all rational functions, we need only one more rule,
— that for differentiating a fraction.
The derivative of a quotient NJD of two functions N and D
is equal to the denominator times the derivative of the numerator
minus the numerator times the derivative of the denominator, aU
divided by the square of the denominator:
rvi VD/ _ dx dx
^^ dx D2
To prove this rule, let y = N/D, where N and D are func-
tions of x; then a change Ax in x produces changes Ai/, AN,
AD in y, N, and D, respectively. Hence, by the rule of § 14 :
fB\ A ^ ^+^ N_ D'AN-N^D
Ill, i 171 ALGEBRAIC FUNCTIONS 29
(O
(D)
Ay__ Ax Ax ,
Ax"" D{D + AD) '
— = lim — = ^^ ^ .
, ,_^3 (3.-7)|(x2 + 3)-(x^4-3)|(3.-7)
^^^^^- &{.z^7) (S^T^^
(3a; -7)2 . (3a;-7)a
Example 2.
5 ^C^V
(x2)2 a:* aft
(Compare Example 3, § 14.)
Examples. -^ (x-*) «^ (^^ j = -^^ =^, = -A:x-»-i.
Note that fonnula IV holds also when n is a negative
integer, for if n = — fc, formula IV gives the result we have
just proved.
EXERCISES
Calculate the derivative of each of the following functions.
1. y-iz^. e. i,-2x-»(=J).
«• V-J+l- 8- *-— p
A _ 3a!-4 - t«
" x^ + % t^—1
3
6. y = -3 . 10. « = ^« - ktr^
30 THE CALCULUS [III, § 17
11 2 ^* + l 01 2 2/ + 1
11. t; = 1*2 5— ^« 21. X = f .
t*2-l •*• •*' 2/3-1 y24.2,4_i-
M ^2 r2/7
13. t; = !^^jj2 23. 8 = («-»4-4)(«-«-5).
14. 5 = ar^ + 6f-«. (M)r-a + &-5-(w + ^)-
16. t. = -^. 2g-^° .;>^ '^'u'^a-M
os^ ^ c (6w — an) (av + o)
i.r 2,3 ^,- 2 ox + 6
17. t^ = -5 + ir-rT- 27. 2/ =
2S-r^2^.i- -• y 2 a2(ax 4-6)2*
18. r.^(.2 + ;).(l-^). @^, = (^3 + 3'x4-l)^ -
r2 — r + 1 3(2/ — 1)'
20. .=4±l^^. 30..= ^^
2/2-22/4-6 (5-7a:5)2
31. Compare the slopes of the family of curves y = a;», where n = 0,
4- 1, 4- 2, etc., — 1,-2, etc., at the common point (1, 1). What is the
angle between y = x^ and 2/ = aJ-i? See Tables ^ III, A,
18. Derivative of a Product. The following rule is often
useful in simplifying differentiations:
The derivative of the product of two functions is equal to the
first factor times the derivative of the second plus the second
faxAor times the derivative of the first:
rtm d(U ' V) dV . dU
If y=u ' v where u and v are functions of x, a change Ax in
X produces changes Ai/, Aw, Av in i/, w, and t;, respectively:
Ill, § 191 ALGEBRAIC FUNCTIONS 31
(A) y+Ay= (u + Au) (v + Av);
(B) Ay= (u + Au) (v + Av) —u • v = uAv'\-vAu + AuAv\
Av
(C) Ay /Ax = u{Av/Ax) + v{Au/Ax) + Au— ;
(D) dy/dz = lim {Ay /Ax) = u{dv/dx) + v{du/dx).
Example 1. To find the derivative of y = (x^ + 3)(x8 -f- 4).
Meifwd 1. We may perform the indicated multiplication and write:
^=^[(a;2+3)(a;3 + 4)] = —[0:5 + 3x8 + 4x2 + 12] =5x4 + 9a;2-f-8a:.
dx dx ax
Method 2. Using the new rule, we write:
g-(x2 + 3)^(x8 + 4) + (x3 + 4)^(x2 + 3)
= (x^ + 3)S x'^ + (x^ + 4)2 X -= 5 x^ + 9 x^ -{- S X.
In other examples which we shall soon meet, the saving in labor due
to the new rule is even greater than in this example.
19. The Derivative of a Function of a Function. Another
convenient rule is the following:
The derivative of a function of a variable u, which itself is a
function of another variable x, is found by multiplying the deriva-
tive of the original function with respect to u by the derivative of
u with respect to x.
[vn] 3^ = 5^ •$^'
dx du dx
If 2/ is a function of w, and w is a function of x, a change Ax
in x produces a change Au m u\ that in turn produces a
change Ay my] hence:
Ay _ Ay Alt
Ax Au Ax
Taking limits on both sides, we have formula [VII].
'32 • THE CALCULUS [III, § 19
Example 1. To find the derivative of y = {x^ + 2)3.
Method 1. We may expand the cube and write:
^"S'^^^"*"^^^' =^(x« + 6a;4+12a;2+8) = 6a;5 + 24x8 + 24a;.
Method 2. Using the new rule, we may simplify this work: let
w = a;2 + 2, then y = {x^-\- 2)' = u^) rule [VI] gives
dy^dydu^ d(v^ d(x^ + 2) ^ ^ ^j .
dx du dx du dx
= 3(x2 + 2)2 -(2 0;)= 3(a:4 _(_ 4 a;2 4. 4) . (2a;) = 6x5 + 24 x' + 24x.
20. Parameter Forms. If x and y are given as functions
of a third variable t, in the form
the variable t is called a parameter , and the two given
equations are called parameter equations. Elimination of t
between these equations would give an equation connecting
X and y, from which dy/dx might be found. But it is often
desirable to find dy/dx without elimination of t. Since
Ax "" !"« • M
we have, in the limit as Ax approaches zero.
This formula is essentially the same as the result of § 8, p. 11.
If we replace thy y m [VII a], we obtain the following
important special case:
[VIIW *? = 1 ^ ^.
^ ^ dx ^ ' dy
Example, liy ^t^-\-2 and x = 3 < + 4, to find dy/dx.
Method 1. We may solve the equation a; = 3 < + 4 for t and sub-
stitute this value of t in the first equation:
/x - 4\2 , ^ a:2 8 , 34
Ill, § 20]
ALGEBRAIC FUNCTIONS
33
dy 2
^-2. 8__2«
^ = ^(3^ + 4)-^=-^
Method 2. Using the new rule (with letters as used in §8, p. 11),
we write:
dy dy . dx _ d(fi + 2) . d&t + 4) ^. . o_2,
dx~ dt ' dt ^ dt " dt -^« • ^-3'.
1. y
2. y
8. y
4. y
EXERCISES
2a;(a;2-l). 15. s
xHa^ + 3). 16. li
(2a: -3) (a: + 3). 17. m
(l+a:) (l-a;.H-a:2). 18. m
5. y = (4-aj2)(l+ir3).
19. 2/ =
= («»-/- 4)«.
= (7-5t; + 2t;8)*.
= (t;*+3t;2--2)*.
= (a + &v + ct;^)®.
1
6. 2/ = (2 a; - a;2) (2 - 3 aj - x^). 20. 2/ =
7. y = (a;2 - 1)».
8. y = (2 a;2 + 3)2.
21. y =
22. 2/
9. 2/ = (a;»-2)«. 23. r
10. y = (x« + 2)8. (fi) r
11. y = {3x^ + 5)*. 25. m
12. y = (5a;8-7)5. 26. u
13. 8 =(l+2«-3«2)2. (27) «
14. « = (<2 + 3 « + 7)8. 28. 8
Determine dy/dx in each of the following
6r-42«
31.
I"
(«8 + l)2*
1
(<2 + ^ + l)t'
a;« + 3
(a:2 + 2)8'
_ (2 x2 - 5)8
(x8 + 3)* '
= (2s«-3)-«.
= (l-«2 4.«4)-8.
= («^ + 2)8(3t;-5)2
= (^ _ 2)2 (2v- 1)8.
= «(<* + 3) (<8 4. 4).
= (l-0(l~5«2)(3-4^),
pairs of equations:
2, = 6w2-7i*- 1,
li = x2 - 1/2.
1-s
{
5z2
« = 2 — 4 X.
y =
2 =
z
1-x
1+x*
34 THE CALCULUS [III, § 20
Draw each of the curves represented by the following pairs of
parameter equations and determine dy/dx.
as l^ = ^'» 34 [^ = 2< + 3/2,
^' U = 3< + 2. ^- U = 2« + 4.
What is the slope in each case when < = 1? Show this in your graphs.
Find the value of the slope in each case at a point where th3 param-
eter has the value 2.
35. Draw the graph of the function y — (2 a: — 1)2(3 a; + 4)2.
Determine its horizontal tangents.
36. Proceed as in Ex. 35, for the function
2/ = (2 a; - 1)2 -h (3 a; + 4)2.
21. Differentiation of Irrational Functions. In order to
differentiate irrational expressions, we proceed to prove that
the formula for the derivative of a power (Rule [IV]) holds
true for all fractional powers:
[IVa] g = nx''-i, n=±^,
where n is any positive or negative integer or fraction.
The formula has been proved in § 17 for the case when n
is a negative integer. Suppose next that
(1) y = x^/^
where p and q are any positive or negative integers. If we
set
(2) X = ^, y = tP,
which together are equivalent to i/ = ojp/*, and apply formula
[Vila], we find:
but since t = x^'^, substitution for t gives
dx q q
This proves [IV] for all fractional values of n.
Ill, § 22] ALGEBRAIC FUNCTIONS 35
The rule also holds when n is incommensurable; for
example, given y = x^^y it is true that dy/dx = V2x ^^;
we shall postpone the proof of this until § 85, p. 85.
22. Collection of Formulas. Any formula may be combined
with [VII], for in any example, any convenient part may be
denoted by a new letter, as in § 20. For example. Rule [IV]
may be written
dw" du^ du u rTrxTi n-i du i rTTri
^=d^-di'^yt^^J' =n«-.-,by[IV].
The formulas we have proved are collected below:
m % - 0.
inj **<' • "> - e . *!.
dx dx
[III] . ^ W'^d" ^^^^^ ^^^ subtraction also.
[IV] ^ = nii"-i^.
^ ^ dx dx
./N\ dN dD j<^ ^W
dx dy
These formulas enable us to differentiate any simple alge-
braic function.
36 THE CALCULUS [III, § 23
23. Illustrative Examples of Irrational Functions. In this
article the preceding formulas are applied to examples.
Example 1. -i — = — ^ — = ^x^/^-^ = -x~^/^ =
dx dx 2 2 2Vx*
(See Ex. 4, p. 21.)
Example 2. Given y = Vs x^ + 4, to find dy/dx.
Method 1. Set w = 3 a;2 + 4, then y * Vw ; by Rule [VII],
dy _dy du _ 1 _ 6a; ^ 3 a; V3 o;^ + 4
dx du' dx 2 Vw 2V3a;2^4 3 x^ -f- 4
Method 2. Square both sides, and take the derivative of each side of
the resulting equation with respect to x:
d{y^) _ rf(3 x^ + 4) _ ^ ^^
dx dx
But by Rule [IV],
<^ W _ d{y^) ^dy _^ dy
dx dy dx dx*
hence,
^ dy ^ dy Zx Sx 3 .t V3 x^ + 4
dx dx y \/3 x^ + 4 3 x^ + 4
This method, which is excellent when it can be applied, can be used to
give a third proof of the Rule [IV] for fractional powers. The next
example is one in which this method cannot be applied directly.
Example 3. Given 2/ == x^ — 2 V3 x^ + 4, to find dy/dx.
dx dx dx 3 x^ 4- 4
Example 4. Given y = (x^ — 2)V3x2 + 4, to find dy/dx,
J = V3x2 + 4£ (x3- 2) + (x3-2)£(V3x2 + 4) [by Rule VI]
= V3^H:^.3x2+(x3-2)?^^^^ [byExample2]
12x^-t-12x2-6x
3x^ + 4
= V3x« + 4 [3 x* + (a:^ - 2) . ^^^1 = \/3F+4
Ill, § 23] ALGEBRAIC FUNCTIONS 37
y/x 4-1 — v^x
Example 6. Given y = . rr, to find dy/dx,
V x -f 1 -f V a:
First reduce 2/ to its simplest form:
= "^x-\-l — Vx ^ y/x + l — y/x ^ 2x + l-'2Vx^-{'X
^ " V^T^ + Vi VT+l-Vi (x + l)-x
= 2 X + 1 - 2 Va^ + x.
Then
wherein = x^ + a?; hence
f?=2-24.f^=2-— L=(2x + 1).
cte 2ViiCto Vx^ + x
This example may be done also by first appl3dng the rule for the
derivative of a fraction [Rule V] ; but the work is usually simpler, as in
this example, if the given expression is first simplified.
EXERCISES
Calculate the derivatives of
1. 2/ = x*/». 3. « = 2 -Jx2. 5. 1/ = Vx Vx.
« 6 10x» f^^ ,
(gp « = VS < - 4.
15. V = 1/ V2 -f- 3 1*.
17. « = Vf2 __ 3 ^.
2 xVx8 . 3--. /^-v _ _ 5 + 3«
'. J/
^ X — = ~" =r •
Vx* Vx*»
(5> "
6 2
9. 8
= «»(2 (2/8 4- 3^-2/3).
®? V
= 2 ■^'SCxVs + xB/8).
13. 2^ = V2+3x. "^0, y= \2x» + 4x.
38 THE CALCULUS [III, § 23
21. y « xWz x-4. 27. y = ^^^ ""
22. 2/ = (5 + 3a;)V6a;-4. 28. y = (9 - 6 a: + 5 a;2) ^^^Tf^.
23. t; = Vl-x + xK 29. s = (1 + ^2) vT^T^.
(First rationalize the denominator.)
25. y = Vl + Va; • 31. 2/ =
26. 8
vT+x2-
- V iZTfi' 32. 2/ =
X
Draw the graphs of the equations below, and determine the tan-
gent at the point mentioned in each case.
33. 2/ = Vn^, (x = f). 36. y = V(l + x) (2 + 3 a;), (a; = 2).
34. 2/ = VTT^, (a; = t). 37. y = xVTTx, {x = 1).
36. y = Vx, (x = 2). 38. 2/ = xi/2 - a;i/3,(x = 1).
39. Find the angle between the curves y = x^^^ and y = ar3/2 at (1, 1).
40. Find the angle between the curves y = x^^^ and y = x^^^ at (1, 1).
41. In compressing air, if no heat escapes, the pressure and volume
of the gas are connected by the relation pv^'*^ = const. Find the rate of
change of the pressure with respect to the volume, dp/dv.
42. In compressing air, if the temperature of the air is constant, the
pressure and the volume are connected by the relation pv = const.
Find dp/dv J and compare this result with that of Ex. 41.
CHAPTER IV
IMPLICIT FUNCTIONS — DIFFERENTIALS
24. Equations in Unsolved Form. An equation in two vari-
ables X and y is often given in unsolved form; i.e. neither
variable is expressed directly in terms of the other. Thus
(1) , a:* + j/^ = l
represents a circle of unit radius about the origin.
Such an equation often can be solved for one variable in
terms of the other; thus (1) gives
(2) y = Vl-a^, or 2/ = - Vl - xK
The first solution represents the upper half of the circle, the
second the lower half. Now we can find dy/dx as in § 23:
(3) ^^ -^ ,nr^y- +^
2
dx Vl — x^ d^ Vl — X
where the first holds true on the upper half, the second on
the lower half, of the circle.
By Rule [VII] such a derivative may be found directly
without solving the equation. From (1)
ax ax
hence
(4) 2x + 2y^ = 0,
39
40 THE CALCULUS [IV, § 24
or
(5) ^=-5.
dx y
This result agrees with (3), since y = =^ Vl — x^.
This method is the same as that used in the second solution
of Ex. 2, p. 36. It may be used whenever the given
equation really has any solution, without actually getting
that solution.
Such a formula as (4) is much more convenient than (3),
since it is more compact, and is stated in one formula instead
of in two. But the student must never use (5) for values of x
and y without substituting those values in (1) to make sure
that the point (x, y) actually lies on the curve; and he must
never use (5) when (5) does not give a definite value for
dy/dx* Thus it would be very unwise to use (4) at the
point x= 1, i/=2, for that point does not lie on the curve (1) ;
it would be equally unwise to try to substitute x = 1, y = 0,
since that would lead to a division by zero, which is impos-
sible.
26. Explicit and Implicit Functions. If one variable y is
expressed directly in terms of another variable x, we say
that y is an explicit function of x.
If, as in § 24, the two variables are related to each other by
means of an equation which is not solved explicitly for y, then
y is called an implicit function of x. Thus, (1) in § 24 gives
y as an implicit function of x; but either part of (2) gives y as
an explicit function of x.
Definition. If the original equation is a simple polynomial
♦ These, precautions, which are quite easy to remember, are really suffi-
cient to avoid all errors for all curves mentioned in this book, at least pro-
vided the equation like (4) [not (5)] is used in its original form, before any
cancelation has been performed.
IV, §25] IMPLICIT FUNCTIONS — DIFFERENTIALS 41
in X and y equated to zero, any explicit function of x obtained
by solving it for y is called an algebraic function. See § 15.
Example, x^ +y^ — Sxy = 0. (Folium of Descartes: Tables, III,
/6.)
This equation is diflScult to solve directly for y. Hence, as in § 24,
we find dy/dx by Rule [VII] ; differentiating both sides with respect to
X, we find:
3^+3y«g-3y-3^g=0;
dy' y — x^
wnence -r- ~ ~o •
dx y^ — X
At the point (2/3, 4/3), for example, dy/dx =4/5; hence the equation
of the tangent at (2/3, 4/3) is {y - 4/3) = (4/5) (x - 2/3) or 4 a; -
5 1/ + 4 = 0. Verify the fact that the point (2/3, 4/3) really lies on
the curve. Note that this formula is useless at the point (0, 0)
although that point lies on the curve.
EXERCISES
In each of these exercises the student should take some point on the
curve, and find the equation of the tangent there.
1. From the equation x^y = 1 find dy/dx by the two methods of
§ 24, first solving for y, then ,v^ithout solving for y. Write the result in
terms of x and y; and also in terms of x alone, when possible.
Find dy/dx in the following examples by the two methods of § 24.
2, s^y^lQ, 7. x8 -|_ y3 = o«.
Z. x^ — xy ^ 5, 8. X* — 4 y2 = 4.
4. 2xy +x + y =^0, 9. 3^+y^ — 3x=^0,
5. x2 _ 4 2,2 = 36. 10. (x + y)2 - 2 X = 4.
6. x^-y^^l. 11. xy^ -x^+y^=0.
Find dy/dx in the following examples without solving for y: check
the answers when possible by the other method of § 24.
12. x'^+2xy + y^=2, 16. ax^ + 2 hxy + hy^ =- k.
13. x^y^-Sxy + 7 = 0, 16. y*-2yH +x^ ==0,
14. ax2 +2bxy+cy^+2dx + 2ey+f = 0,
17. >/?+ V^ = Va. 18. ^3/2 _(. y3/2 == ^8/2.
42
THE CALCULUS
[IV, § 25
In the following pairs of parameter equations, find dyldx by § 20 :
when possible eliminate i to find the ordinary equation, and show that
the derivative found is correct. Regarding each pair of equations as
defining the position of a point (x, y) at the time t, find the horizontal
speed and the vertical speed at the time t, (See § 8.) Find also the
total speed from the relation
totoZ %'peQd == V (horizontal speed)^ -f- (vertical speed)^.
19.
a; = 4 <,
, y = 8 <2.
20. I^^^f/o' 21. j^ =
U=4«-2. \y =
= 3^ + 1,
2^3.
22.
X =
y =
3t
1 +t^'
23.
z =
y =
t^ + V
-2t
«2 + l*
24.
y =
2t
«2-l
26. On a circle of unit radius about the origin dyldx — —x/y\ this is
positive when x and y have different signs, negative when a; and y have
the same sign. Show that this agrees with the fact that the circle rises
in the second and fourth quadrants and falls in the first and third quad-
rants as x increases.
26. Show that the curve xy = 1 is falling at all its points.
27. Show that the curve xh^ = 1 is rising in the second quadrant and
falling in the first quadrant.
28. The equation xV2 -f- y^/^ = 1 is the equivalent of the equation
^.2 — 2 X2/ -t- y^ —2x — 2 2/4-1 = 0, if the radicals x^/^ and y^/^ be taken
with both signs. Show that the values of dy/dx calculated from the two
equations agree. By methods of analytic geometry, it is easy to see that
the curve is a parabola whose axis is the line y ^ x, with its vertex at
(1/4, 1/4).
29. The curve of Ex. 28 is also represented by the parameter equa-
tions 4 a; = (1 + 1)^, 4 2/ = (1 — t)^. Test this fact by substitution,
and show that the value of dy/dx obtained from these equations agrees
with the value obtained in Ex. 28. [The curve is most easily drawn
from the parameter equations.]
If t denotes the time in seconds since a particle moving on this curve
passed the point (1/4, 1/4), find the total speed of the particle at any
time.
IV, §26] IMPLICIT FUNCTIONS — DIFFERENTIALS 43
26. Differentials. Let the curve PQ (Fig. 8) be a part of
the graph of the equation y =f(x). Let P be any point
(x, y) on the curve, and let Q be a second point (x + Ax,
y + Ay) on it. The change Ax = PM in x causes a change
Ay = MQ in y.
The slope of the tangent
PT at the point P is given
by the formula
(1) m = tan a
Ax-»oAx dx
If this slope were main-
tained over the interval
Ax, the change produced in y would be
(2) MK = m • Ax.
When Ax is small, the change in y, MQ, will usually be
nearly equal to MK. In many problems, it is a sufficiently
exact approximation to Ay. Moreover, its value may be
found readily by (2), whereas the actual calculation of Ay
itself might be tedious or impracticable.
This quantity MK, which is an approximation to Ay, is
called the differential of y, and is denoted by the symbol dy.
Hence we may write
(3) dy
m
Ax = |.Ax.
In particular, if the curve is the straight line y = x, we find
m = 1; hence the differential of x is
(4) c/x = 1 • Ax.
If we divide (3) by (4) we find
(5) dy -r- dx = m,
where dy -r- dx now denotes a real division, since dy and dx
44 THE CALCULUS [IV, § 26
are actual quantities defined by the equations (3) and (4),
and dx (= Ax) is not zero.
Since m stands for the derivative of y with respect to x, it
follows that that derivative is equal to the quotient of dy by
dx,
(6) g=di,^dx;^
this fact is the reason for our use of the symbol dy/dx to repre-
sent a derivative originally.
In the figure all quantities here mentioned are shown:
cto = Aa; = AB, dy = MK, Ay = MQ,^ =. tan /3,^ = tan a.
Ax dx
The quantities dx(= Ax) ,dy(= mAx) , Ay, Ay — dy{= KQ) , are
infinitesimal when Ax approaches zero, i.e, they approach
zero as Ax approaches zero.
27. Diflferential Formulas. For any given function y = f{x),
dy can he computed in terms of dx(= Ax), by computing the
derivative and multiplying it by dx.
Every formula for diflferentiation can therefore be written
as a differential formula; the first six in the list in § 22, p. 35,
become after multiplication hy dx:
[I] dc = 0. (The diflferential of a constant is zero.)
[II] d{c -u) = c ' du.
[III] d(u + v)-=du + dv.
[IV] d (W) = nw^^du.
[VI] d{U'V) = udv + vdu.
IV, §271 IMPLICIT FUNCTIONS — DIFFERENTIALS 45
Rules [VII], [Vila], and [VII6], of § 22, p. 35, appear as
identities, since the derivatives may actually be used as
quotients of the diflferentials. From the point of view of
the differential notation Rule [VII] merely shows that we
may use algebraic cancelation in products or quotients
which contain differentials.
Rules [I]-[VI] are sufficient to express all differentials of
simple algebraic functions. A great advantage occurs in the
case of equations not in explicit form, since all applications of
Rule [VII] reduce to algebraic cancelation of differentials.
Example 1. Given y = x^ __ 12 x + 7, to find dy and m.
dy==d(xi-12x + 7) = d(2^) - d{l2x) +d(7) =3x^dx-12dXy
whence m = dy-^dx = Sx^ — 12 aain Example 2, p. 21.
a;2-f-3
Example 2. Given y =
3x-7
, to find dy (Example 1, p. 29).
(3x-7)d(x2 + 3)~(x2 + 3)d(3a;-7)
^^ = (3x-7)2 :
_ 3x2-14a;-9 ^
" (3x-7)2 ^'
Example 3. Given y = (x^ + 2)3, to find dy (Example 1, p. 32).
dy =d[(x2+2)3] =3(a;2+2)2d(a;2+2)
= 3(x2+ 2)2.2 x-dx.
Example 4. Given y — x^ — 2Vs x^ + 4, to find dy (Example 3
p. 36).
dy =d{3^)-2dVSx^+4:
1
= 3a;2da;-2
2 V3 x2 + 4
= ^"3x2 — L= 6 x) dx.
V Va a:2 ^_ 4 /
d (3x2 + 4)
V3 x2 + 4
Example 5. Given x^ + y^ = 1, to find dy in terms of dx (§ 24, p. 39).
dix^+y^) =d(l) = 0; but d (x2 + 2^2) = d (x2) + d (j^2)
= 2xdx + 2ydy;
hence 2 x dx + 2 y dy = 0, or dy = — (x/y) dx, or m = dy/dx = — x/y.
46 THE CALCULUS [IV, § 27
Example 6. To Qnddy and m when a^ -{-y^ — Sxy =0. (Example,
p. 41.) d (x3) + d (2/3) - 3 d (x2/) = 0,
or 3x2 dx -{-3y^dy — Sxdy — 3y dx = 0,
or (x2 —2/) dx + (2/^ — x) dy ^ 0,
whence rfy = ^ dXyOrm = -f == ^ — - .
2/2 — X dx y^ — X
Example 7. To find dy in terms of dx when x = 3< + 4, y = ^ + 2.
(Example, p. 32.)
We find dx =^ d (S t + ^) = S dt; dy ^ d {fi + 2) = 2 t dt;
hence w - dy -^ dx — (2/3) <, or d^ = (2/3) t dx;
but since t — (x — 4)/3, this may be written:
dy = (2/9) (X - 4)dx, or m = g = (2/9) (x - 4).
EXERCISES
[These exercises may be used for further drill in differentiation, and
for reviews. It is scarcely advisable that all of them should be solved
on first reading.]
Calculate the differentials of the following expressions.
1. y =^ax^ + bx+c. 15. u = l/V2v-\-v^.
2. 2/ = (o2 + x2)2. 16. w = (1 - 2 t;2)/(2 - v^).
3. 2/ = (ax^+bx+c)^, 17. u = '^^^\"^^ -
4., = (a -6x2)5. 18. . = J^2^.
3
5. y=^l/(ax + b), 19. z = 1/V(2 ~ 2/3)4.
6. y = l/(ax + 6)2. 20. s = l/\/(2/ - a)^.
7. s = (1 + 2 /) (1 - 3 0. 21. 2 = 2//VT+7.
8. s = (2 - 3 0M3 + 2 <2). 22. « = Va + by/y.
9. s = <2 (o - ^)3. 23. r = {a + b^)P.
10. s = «4 (3 - 2 «3)2 24. r = v^a -f 6s«.
11. s = V3 t - t\ 25. r = l/(a + 6s'»)p.
12. 8 = V<"4^n". 26. r = \/\/a + 6s».
1 + Vx
13. s = V(<3 - 3 0^ 27. 2/ =
V:
X
14. 5 = v^(2<2 + l)2. 28. 2/ = |/^
bx
bx
IV, § 27] IMPLICIT FUNCTIONS — DIFFERENTIALS 47
29.
xy y -4..
30.
x2- 2x2/- 32/2 = 0.
31.
x2-y2-^-
Determine dy in terms of dx from the equations below.
33. (1 - ax) (a;2 + y^) = 4.
34. x2 + y2 = (ax + 6)2.
* x2 a — hx
32. 2/4 _ 2 2/2x - 1 = 0. ^36. (x + 2/)^^^ + (a; - 2/)^/^ = o^/^.
37. Obtain the equation of the tangent at (2, — 1) to the curve
4x2-2a:2/-5 2/2-6x-42/-7 = 0.
^8. Obtain the equation of the tangent at (2, 1) to the curve
x^ - 7 x^ - 5y^ + ^ x^- 10 xy + Sx-5y-\- 18=^0.
Obtain the equation of the tangent at (xq, yo) to each of the following
curves:
CuBVE Tangent
39. y^ =4ax; yyo'=2a(x+ Xq),
40. x^-\r y^ = a2; xxq + yyo = a2.
41 ?'+?^' = l-
*^- 02 - 62 ^'
^±^0 = 1
a'
62
Find the derivative dy/dx for the curves defined by each of the
pairs of parameter equations given below.
S a — 2t
at '
_4(a-t)^
42.^
X —
-^6.
y =
x =
2t
1+^'
1-t
1-ht'
43. 4
a; = I V2 «3,
44.
X =
e
1 +0'
[2/ = 0-1 + 0-2.
46.
X = 4 irr2,
2/ =girr3.
47.
y =
u-
2/ =
02^2
4irr2»
3
4ir7^
48. Calculate the x and y components of the speed (% and Vy) at any
time tf and the resultant speed "^Vx^ + /^^ for the motion
2< _ l-/2
49. If a particle moves so that its coordinates in terms of time are
x = l-t + t\ 2/ = l+« + ^,
show that its path is a parabola. Show that from the moment < = its
speed steadily increases.
48 THE CALCULUS [IV, § 27
60. The electrical resistance of a platinum wire varies with the tem-
perature, according to the equation
calculate dRm terms of dd. What is the meaning of dR/dd?
61. Van der Waal's equation giving the relation between the
pressure and volume of a gas at constant temperature is
(p+^) (P-^) =c.
Draw the graph when a = .0087, 6 = .0023, c = 1.1. Express dv in
terms of dp. What is the meaning of dv/dp?
62. The crushing strength of a hollow cast iron column of length I,
inner diameter d, and outer diameter D, is
T = 46.65 (^:?i^)
Calculate the rate of change qt T with respect to £>, d, and I, when
each of these alone varies.
CHAPTER V
TANGENTS — EXTREMES
28. Tangents and Normals. We have seen in § 4, p. 4, that
if the equation of a curve C is given in explicit form:
(1) y=f{x),
the derivative at any point P on C represents the rate of rise,
or slope, of C a.t P:
(2) L^Jai p= t^^P^ ^f ^at p = ^^^P^ ^fP"^ = tow a = mp,
where a is the angle XHT, counted from the positive direc-
tion of the X-axis to the tangent PT, and where nip denotes
the slope of C at P.
Hence (§4, p. 5) the equation of
the tangent is
(3) iy-yp) = \_fX{x-xp),
where the subscript P indicates that
the quantity affected is taken with
the value which it has at P,
If the slope rrip is podtive, the curve is rising Sbt P; if mp is
negative, the curve is falling; if nip is zero, the tangent is hori-
zontal (§ 6, p. 6). Points where the slope has any desired
value can be found by setting the derivative equal to the
given number, and solving the resulting equation for x.
Since, by analytic geometry, the slope n of the normal PN
is the negative reciprocal of the slope of the tangent, we have,
(4) np = slope of PN = = — fj-rrr'
mp [ay/dx]p
49
r
\
y
\ /
^T
J^
«
,^
:^
\
X
/fl-
A
\N\
Fig. 9.
50 THE CALCULUS [V, § 28
hence the equation of the normal is:
(5) ^y-y'^'-g^^^ -=""'>•
29. Tangents and Normals for Curves not in Explicit Form.
The equation of the curve may be given in the implicit
form
(1) F (x, y) = 0,
as in §§24-25, pp. 39-41; or the equations in parameter
form may be given:
(2) x=f(t),y = <l>(t),
as in § 20, p. 32. In either of these cases, dy/dx can be
found, and this value may be used in the formulas of § 28.
No new formulas are necessary.
30. Secondary Quantities. In Fig. 9, § 28, since
tan a (= nip = [dy/dx]p)f and AP (= i/p),
are supposed to be known, the right triangles HAP and PAN
can both be solved by trigonometry, and the lengths HA,
AN, HP, PN can be found in terms of mp and ypi
[Subtangent]p = HA = AP -r- tan a = yp-i- mp = [y/m]p.
[Subnormal] p = AN = AP • tan a = [t/ • m]p, since a = Z APN,
[Length of tangent]p = HP = '^Ap + W^
= ^y% + \ylm\\ = \y Vl + {Mmf\p.
[Length of normaljp = PAT = ^ AP" + AN"
= ^^yp+iymyp = [yVl + m^]p.
It is usual to give these lengths the names indicated above;
and to calculate the numerical magnitudes without regard to
sigijs, unless the contrary is explicitly stated.
V, 1 31] TANGENTS — EXTREMES 51
31. Illustrative Examples. In this article, a few typical
examples are solved.
Example 1. Given the curve y = x^— 12x-\-7 (Ex. 2, p. 21), we
have m = dy/dz = 3 a;^ — 12.
(1) The tangent (T) and the normal (iV) at a point where x = a are
(T)y- (a3-12o + 7) = (3 a2 - 12) (x - a),
(iNr)2/-(a3-12a + 7)=j^^^(x-a);
thus, at X = 3, the tangent and normal are
(T) 2/ + 2 = 15(x-3), (N) 2/ + 2 = -A(a:-3).
(2) The tangent has a given slope A; at points where
3x2 - 12 = A;, i^e. x = ± ^ ^^^ ;
there are always two points where the slope is the same, if A; > — 12;
thus if A; = 0, a; = ± 2; if A; = - 9, x = ± 1; if A; = - 12, x = 0; if
A; < — 12, no real value for x exists (see Fig. 15, p. 68).
(3) The secondary quantities of § 30 may be calculated without
using the formulas of § 30. Thus, at the point where x = 3, the
tangent (T) cuts the x-axis where x = 47/15; the normal (iV) cuts the
X-axis where x = — 27. If the student will draw a figure showing
these points and lines, he will observe directly that t he subtange nt is
2/15, the subnormal 30 , the len gth of the tangent V22 + (2/15)2, the
length of the normal V302 + 22. These values agree with those given
by §30.
Example 2. Given the circle x2 + y2 = 1^ we have m = dy/dx =
— x/y [see § 24].
(1) The tangent (T) and normal (N) at a point (xq, yo) are
iT){y-yo)=-^{x-xo), (N) (y - yo) ^f {x- xo);
yo *o
or, since xo2 +yo^ ~ 1,
(T) xxo + yyo = 1, i^) V^o = 2/oa;;
thus, at the point (3/5, 4/5), which lies on the circle, we have
(r) 3 X + 4 2/ = 5, (iV) 3 j^ = 4 X.
(2) The tangent has a given slope A; at points where
^ = A;, i.e. xq + kyo ~ 0.
2/0
52 THE CALCULUS [V, § 31
The coordinates (xq, ^o) can be found by solving this equation simul-
taneously with the equation of the circle, or by actually drawing the line
xq + kyo = 0. Thus the points where the slope is -f 1 lie on the straight
line X + y ^0; hence, solving x +j/ = and x^ -^y^ = 1, the co-
ordinates are found to bea? = ± l/v^2, y — T 1/"^; but these points
are most readily located in a figure by actually drawing the line
x+y ^0.
EXERCISES
Find the equation of the tangent and that of the normal, and find
the four quantities defined in § 30, for each of the ft^owing curves at
the point indicated:
1. y=xi-12x + 7; (1,-4). 6. x = j^«- 32/' + 5; (3,1).
2- ^ = iSi^ (- ^^ ^)- «• {^ :(!;?}'• (^'^)-
3. 9x^+2/2=25; (1,4). rx = <» + 4/-n
4. xy + 2/2-2a;=5; (-4, 1). '* ly = ^ - 3 « + 5j ' ^' ^^•
8. The curves of Exs. 1 and 3 pass through the point (1, — 4);
at what angle do they cross?
Determine the equation of the tangent and that of the normal to
each of the following curves at any point (xo, 2/0) on it.
9. j^ = kxK 13. 62a;2 ± a2y2 = ^262.
10. y^ = 2px. 14. ax^-\-2bxy-{-cy^ =f.
11. x2 + 2/2 = o2. 15. (m;2 + 2 6x2/ + C2/2 + 2 da; + 2 P2/ +/ = 0.
12. y = kx^, 16. y = (ax-h h)/{cx + d).
32. Extremes. In § 6, and in numerous examples, we have
found points on a curve at which the tangent is horizontal,
i,e, at which the slope is zero. If the slope of the curve
y = f(x) is zero at the point where x = a, the curve may go
through the point in any one of the ways illustrated in
Fig. 10.
In case (a),f(a) is called a maximum otf(x).
In case (f)),f(a) is called a minimum oif(x).
V, § 32]
TANGENTS — EXTREMES
53
The value of f(x) at a point where x = a is a 4 . . I
'' ^ '' ^ {minimum J
value if it is i? ., l any other value of / (x) for values
of X sufficiently near to x = a,
A maximum or a minimum is called an extreme value,
or an extreme oif(x).
A value of x for which the slope m is zero is called a
critical value. The corresponding point on the curve is
y^^
yj^
^)
fia)
a
X
v^
(X)
J(<t)
a
X
(a) (6) (c)
Fig. 10.
called a critical point. At such a point, f(x) may be a
maximum or a minimum, but it is not necessarily either.
Thus, in cases (c) and (d), Fig. 10, the value of / (a) is neither
a maximum nor a minimum of / (x) .
On the other hand, extremes may also occur at points
where the derivative has no meaning, or at points where the
function becomes meaningless.
Thus, the curve y = x^/s gives
m = 2/(3 x^/^) : hence m is meaning-
less when X = 0; in fact, the curve
has a vertical tangent at that point.
It is easy to see that this is, however,
the lowest point on the curve.
Again, if a duplicating apparatus costs $150, and if the running
expenses are Ic. per sheet, the total costs of printing n sheets is / =
150 + .01 n. This equation represents a straight line; geometrically
there are no extreme values of t; but practically t is a. minimum when
n = 0, since negative values of n are meaningless. Such cases are
usually easy to observe.
54 THE CALCULUS [V, § 33
33. Fundamental Theorem. We proceed to show that a
function / (x) cannot have an extreme except at a critical
point; that is, assuming that/(a:) and its derivative have
definite meanings at a: = a and everywhere near x = a^ no
extreme can occur if the derivative is not zero atx = a.
We are supposing that all our functions are continuous;
if, then, the derivative m is positive at x = a, it cannot sud-
denly become negative or zero. Hence m is positive on both
sides of a: = a, and there can be no extreme there.
Likewise if m is negative, the curve is falling near x = a
on both sides of x = a; there can be no extreme.
34. Final Tests. It is not certain that/ {x) has an extreme
value at a critical point. To decide the matter, we proceed
to determine whether the curve rises or falls to the left and to
the right of the critical point: it rises if m > 0; it falls if
m < 0. (See Fig. 10 in § 32.)
Near a maximum, the curve rises on the left and falls on
the right.
Near a minimum, the curve falls on the left and rises on
the right.
If the curve rises on both sides, or falls on both sides, of the
critical point, there is no extreme at that point.
35. Illustrative Examples.
Example 1. To find the extreme values of the function y ^J{x) =
a;8 - 12 a; + 7. (See Ex. 3, p. 8.)
{A) To find the Critical Values, Set the derivative equal to zero and
solve for x\
w = ^ = 3 a;2 - 12; 3 a;2 - 12 = 0; X = 2 or a; = - 2.
ax
(B) Precautions, Notice that f(x) and its derivative each have a mean-
ing for every value of x; hence a; = + 2 and x = — 2 are the only critical
values.
V,5 35]
TANGENTS — EXTREMES
(C) Final Tests, m - 3 1^* - 12 = 3 (i« -
than 2, negative if x is slightly leas than 2;
hence the curve rises on the right and fails
ontheleftof3;-2, theretore/(2)=-9iBn
TOiiumuQiof/(2). The student maj show
that /(- 2) = 23 ia a maximum of f(x).
(See Fig. 3, p. 8.)
Example 2. To find the extremes of
the fimction
)/-/(i) =3H-12a^ + 50.
(A) CTilical Values. Settiog dy/dx - 0,
and solving, we find:
^ = 12^-
dx
r*;12a^
-36a^ =
= 0,01
) iB positive if X
s greater
A
t
-- t
. ^^
y-
t:
1
i
i
±
FiQ. 12.
1 the right, negative o
(B) Precaviums. y and dy/dx have a
meaning everj^vhere; the only critical
values are and 3.
<C) Fined Tests. Near i = 0, m -
12i*(i — 3) is negative on both sides;
hence there is no extreme there, though
the tangent is horizontal.
Near i = 3, m = 12 x'ix - 3) is positive o
the left; hence /(3) = - 31 is a minimum.
The information given above is of great assistance in accurate drawing.
ElxAUPLE 3. Two raihiiad tracks cross at right angles; on one of
them an eastbound train going 15 mi. per
hour clears the crossing one minute before
the engine of a southbound train running
at 20 mi. per hour reaches the crossing.
Find when the trains were closest together.
Let X and y be the distance in miles of
the rear end of the first train and the en-
gine of the second one from the crossing,
respectively, at a time t meaaured in min-
utes beginning with the instant the first
Fig. 13.
train clears the crossing; then
where D is the distance between the trains in miles.
25,
56 THE CALCULUS [V, § 35
Since D is a positive quantity, it is a minimum whenever D^ is a mini--
mum; hence we write:
_ d{D^) _ 2 , 25, _2,25. ^ .16
"^^ dt " 9 "'"72^' 9 ■^72^"""' ^""25'
when t <.16/25, m < 0; if /> 16/25, w> 0; hence D^ is diminishing
before t = 16/25 and increasing afterwards. It follows that D is a
minimum when t == 16/25. Substituting this value for <, we find the
values X — 4/25, y = 3/25, D^ = 1/25; hence the minimum distance
between the trains is 1/5 of a mile, and this occurs 16/25 of a minute
after the first train clears the crossing.
Example 4. To find the most economical shape for a pan with a
square bottom and vertical sides, if it is to hold 4 cu. ft.
Let X be the length of one side of the base, and let h be the height.
Let V be the volume and A the total area. Then V = hx^ = 4, whence
h = 4/a;2; and
A =x2+4Aa;=a;2 + ^;
X
whence we find
d A f^ 16. rt 16n«o o
When a; < 2, m = 2(x^ — S)/x^ is negative; when x>2, m is positive;
hence A is decreasing when x is increasing toward 2, and A is increasing
as a; is increasing past 2; therefore x = 2 gives the minimum total area
A = 12. Notice that the height is A=4/a;2=l. The correct
dimensions are x = 2, A = 1 (in feet).
EXERCISES
Determine the maximum and minimum values of the following
functions and draw the graphs, choosing suitable scales.
1. j^ = a^ - 6 a;2 + 2. 2. s = 2 <3 - 6 ^ - 18 « + 15.
3. p = g3 - 6 ^2 - 15 g. A. y = X!^ -{- 2 OX^ + G^X.
6. x = 2/4 — 4 2/2 _f. 2. 6. t; = w4 - 4 w3 _j_ 4 ^2 _|. 3.
7. m pn5-5w* + 5n8 + l. 8. A=^r8-6r4+4r3H-9r2-12r+4.
9. s = (2^-l)(l-02. 10. V = h(h-1)K
11. r=(s2-l)(a2-4). 12. a; = (2/ - 2)3 (2/ + 3)8.
(a; + 2)2 ^. , a*
V, § 35] TANGENTS — EXTREMES 57
a;2-2a;+4 o/i^ + ft^ + c
17. 2/ = ^!!^2 + r ^®- o = ^ + ^^r^-
19. D = r VS^^. 20. 22 = ^^ + 6 - x.
21. What is the largest rectangular area that can be inclosed by a line
80 feet long?
22. What must be the ratio of the sides of a right triangle to make its
area a maximum, if the hypothenuse is constant?
23. Determine two possible numbers whose product is a maximum
if the sum of their squares is 98. Is there any minimum?
24. Determine two numbers whose product is 100 and such that the
sum of their squares is a minimum. Is there any maximum? Did you
account for negative possible values of the two numbers?
25. What are the most economical proportions for a cylindrical can?
Is there any most extravagant type? Mention other considerations
which affect the actual design of a tomato can. Is an ordinary flour
barrel this shape? What considerations enter in making a barrel?
26. What are the most economical proportions for a cylindrical pint
cup? (1 pint = 28J cu. in.) Mention considerations of design.
27. Determine the best proportions for a square tank with vertical
sides, without a top. Is there any most extravagant shape?
28. The strength of a rectangular beam varies as the product of the
breadth by the square of the depth. What is the form of the strongest
beam that can be cut from a given circular log? Mention some other
practical considerations which affect actual sawing of timber.
29. The stiffness of a rectangular beam varies as the product of the
breadth by the cube of the depth. What are the dimensions of the
stiffest beam that can be cut from a circular log?
30. Is a beam of the commercial size 3" X 8*^ stronger (or stiffer)
than the size 2* X 12*^ (1) when on edge, (2) when lying flat?
[Conmiercial sizes of lumber are always a little short.]
31. What line through the point (3, 4) will form the smallest triangle
with the coordinate axes? Is there any other mininauin? maximum?
58 THE CALCULUS [V, § 35
32. Determine the shortest distance from the point (0, 3) to a point
on the hyperbola x^ — y^ = 16. Show that it lies on the normal.
[Hint. Use the square of the distance.]
33. The distance D from the point (2, 0) to any point of the circle
a;2 _j_ y2 — 1 ig given by the equation Z)^ = 5 — 4 a?. Discover the
maximum and minimum values of Z)2, and show why the rule fails.
34. Show that the maximum and minimum on the cubic y = x^ — ax
-j- h are at equal distances from the y-axis. Compute y at these points.
36. Show that the cubic x^ — ax -^b = has fhree real roots if the
extreme values of the left-hand side (Ex. 34) have different signs.
Express this condition algebraically by an inequality which states that
the product of the two extreme values is negative.
[Any cubic can be reduced to this form by the substitution x = a?' -j- A;;
hence this test may be applied to any cubic]
36. Show that if the equation x^ — ax -{-b = has two real roots, the
derivative of the left-hand side (i.e. S x^ — a) must vanish somewhere
between the two roots. Show that the convert is not true.
37. The line y = mx passes through the origin for any value of m.
The points (1, 2.4), (3, 7.6), (10, 25) do not lie on any one such line:
the values of y found from the equation y = mx at a; = 1, 3, 10 are m,
3 m, 10 w; the differences between these and the given values of y are
(rn — 2.4), (3 m— 7.6), (10 m — 25). It is usual to assume that that
line for which the sum of the squares of these differences
S= (m- 2.4)2 -^ (Sm- 7.6)2 -f (lo m - 25)2
is least is the best compromise. Show that this would give m = 2.50
(nearly). Draw the figure.
38. In an experiment on an iron rod the amount of stretching s (in
thousandths of an inch) and the pull p (in hundreds of pounds) were
found to be (p = 5, s = 4), (p = 10, s = 8), (p = 20, s = 17). Find
the best compromise value for m in the equation s = m-p, under the
assumption of Ex. 37. Ans. About 5/6.-
39. A city's bids for laying cement sidewalks of uniform width and
specifications are as follows: Job No. 1: length = 250 ft., cost, $110;
Job No. 2: length, 600 ft., cost, $250; Job No. 3: 1500 ft., cost, $630.
Find the price per foot for such walks, under the assumption of Ex. 37.
How much does this differ from the arithmetic average of the price per
foot in the three separate jobs?
V, § 35] TANGENTS — EXTREMES 59
40. The amount of water in a standpipe reaches 2000 gal. in 250 sec,
5000 gal. in 610 sec. From this information (which may be slightly
faulty). find the rate at which water was flowing into the tank, under
assumption of Ex. 37.
41. The values 1 in. = 2.5 cm., 1 ft. = 30.5 cm. are frequently
quoted, but they do not agree precisely. The number of centimeters
c, and the number of inches t, in a given length are surely connected by
an equation of the form c = ki. Show that the assumptions of Ex. 37
give k = 2.541. Is this the same as the average of the values in the
two cases? Which result is more accurate?
42. In experiments on the velocity of sound, it was found that sound
travels 1 mi. in 5 sec, 3 mi. in 14.5 sec. These measurements do not
agree precisely. Show that the compromise of Ex. 37 gives the velocity
of sound 1088 ft. per second. How does this compare with the average
of the two velocities found in the separate experiments?
43. A quantity of water which at 0° C. occupies a volume vq, at 0° C.
occupies a volume
r = yo (1 - 10-4 X .5758^ + 10-6 X .756 6^ - 10-7 X .351 0^).
Show that the volume is least (density greatest), at 4° C. (nearly).
44. Determine the rectangle of greatest perimeter that can be in-
scribed in a given circle. Is there any minimum?
46. What is the largest rectangle that can be inscribed in an isosceles
triangle? Is there any minimum?
46. Find the area of the largest rectangle that can be inscribed in a
segment of the parabola y^ = 4iax cut off by the line x == h.
47. Determine the cylinder of greatest volume that can be inscribed
in a given sphere. Is there also a minimum?
48. Determine the cylinder of greatest convex surface that can be
inscribed in a sphere. Is there a minimum?
49. Determine the cylinder of greatest total surface (including the
area of the bases) that can be inscribed in a given sphere.
50. What is the volume of the largest cone that can be inscribed in a
given sphere?
61. What is the area of the maximum rectangle that can be inscribed
in the ellipse x^/a^ -f- y^/b^ = 1?
CHAPTER VI
SUCCESSIVE DERIVATIVES
36. Time-rates. In all the applications, derivatives are
rates of increase (or decrease) of some quantity with respect
to some other quantity which is taken as the standard of
comparison, or independent variable.
Among all rates, those which occur most frequently are
time-rates, that is, rate of change of a quantity with respect
to the time.
37. Speed. Thus the speed of a moving bodv is the time-
rate of increase of the distance it has traveled:
(1) V = speed* = lim -r: = 37 ,
A<-^o A< at
as in § 7, p. 9, and in numerous examples.
38. Tangential Acceleration. The speed itself may change;
the time-raie of change of speed is called the acceleration along
the path, or the tangential acceleration^.
A I? dv
(2) Jt = tangential acceleration f = lim -r.= -y.'
At— »o ^t at
* The speed v is distinguished from the velocity t by the fact that the
speed does not depend on the direction ; when we speak of velocity we shall
always denote it by t (in black-faced type) and we shall specify the direc-
tion.
t The general acceleration J is also a directed quantity; when we speak
of the acceleration J (not tangential acceleration jj,) we shall denote it by J,
and give its direction. As in the case of speed, the letter j, in italic type,
denotes the value of y without its direction.
60
VI, § 39] SUCCESSIVE DERIVATIVES 61
Thus for a body falling from rest, if g represents the gravitational
constant,
hence
da :
and
dv
it follows that the tangential acceleration of a body falling from rest is
constant; that constant is precisely the gravitational constant g.*
In obtaining the tangential acceleration, we actually dififer-
entiate the distance s twice, once to get v, and again to get
dv/di or Jt, hence the tangential acceleration is also said to be
the second derivative of the distance s passed over.
39. Second Derivatives, Flexion. It often happens, as in
§ 38, that we wish to differentiate a function twice. In any
case, given y =f{x), the slope of the graph is
ax Aa;-»0^
The slope itself may change (and it always does except on a
straight line) ; the rate of change of the slope with respect to x
will be called the flexion f of the curve:
, n . dm ,. Am
= flexion = -3- = lim -— »
ax Aa;->0 AOJ
and will be denoted by b, the initial letter of the word bend.
Thus for y = x^jWe find m = 2x, 6 = 2;$
* The value of g is approximately 32.2 ft. per second per second = 981
cm. per second per second.
t The word curvature is used in a somewhat different sense. See § 86,
p. 139.
t The flexion for this parabola is constant; note that this means the
rate of change of m per unit increase in x, not per unit increase in length
along the curve.
62 THE CALCULUS [VI, § 39
for 2/ = x^, m = 3 x^, 6 = 6 x;
for y = 3i^-l2x + 7, m = 3a^-12y 6 = 6a;;
for any straight line y = fcx + c, m = fc, 6 ='0.
The value of 6 is obtained by differentiating the given func-
tion twice; the result is called a second derivative, and is
represented by the symbol:
cPy _ d^ /d^\ __ dm __ ,
dx^ dx \dx/ dx
Likewise, the tangential acceleration in a motion is
d^ _ d /ds\ _dv _ ,
de~dt\dt)''dt~^'^'
If the relation between s and t is represented graphically,
the speed is represented by the slope, the tangential acceleration
m
by the flexion, of the graph. Thus if s = gt^/2 be represented
graphically, as in Fig. 4, p. 10, the slope of the (t, s) curve is
m = slope — ~Jf — Q^ — speed = v,
and the flexion of the (t, s) curve is
, ^ . dm d!^s dv ^ ^-77^.
b = flexion — ~17 — ~j^ — jf — 9"^ tangential acceleration =Jt'
40. Speed and Acceleration. Parameter Forms. Let the
equations
x=f{t), y = 4>{t)
represent the position of a moving point P in terms of time
t as variable. Then, as in § 8,
dx
(1) ^x = -T. = horizontal component of the speed of P;
dv
(2) ^v= jf — vertical component of the speed of P;
VI, S 401 SUCCESSIVE DERIVATIVES
63
(3)
-wv.
' + 0^ = total speed of P,
in the direction tan-' (v^/vx).
Further,
<Px
(4) jx= ;p- = horizontal acceleration of P;
<Pv
(5) jV = j2 = vertical acceleration of P;
(6) j = Vj/ + j^ = total acceleration of P,
in the direction tan-' (jg/jx).
Note that the direction of v is aloi^ the tangent to the
path of P, since v^/^ is equal to dy/dx, or m. But the total
acceleration j m not, in general, in the direction of the path
of P, since jy/j^ is ordinarily quite different from vjvx.
Hence j and jV are usually different.* To get jr, calculate
dv/dt from (3).
Example. Let the parametric equations of the patb be
a-=(*, !, = l/P(=(-s).
To plot the path take a series of values
of (:
= 0,
h 1, 3/2
2,
= o;
h 1, 9/1
4,
■ ■;
= OO
4, 1, 4/9
i-
= 2
Ji
= 2,
= -
2H,
jV
=6H,
-2v
P+H.
j
=.2VH-9H.
_dv
2 i - 6 H
(IIV), 1 22.)
VJ+W >'■■"■"■' Fi«.14.
• The reaaoQ for this difference is not difficult : jj, is the acceleration in
the path itself; j ia the lolal acceleration, part of its effect being precisely
to make the pa(A curved; heoce a part of / ia expended not t« increase the
speed, but to change the direction of the speed, i.e. to bend the path. Notice
that Ex. 33, p. 65, represents a straight line path; on itjj. = j; thia holds
only on straight line paths. In uniform motion on a circle, tor eiample.
64 THE CALCULUS [VI, § 40
EXERCISES
[In addition to this list, the second derivatives of some of the functions
in the preceding exercises may be calculated.]
Calculate the first and second derivatives in the following exercises.
Interpret these exercises geometrically, and also as problems in motion,
with 8 and t in place of y and x.
1. 2/ = x2 + 5 a; - 4. 11. 2/ = >/F+ Vx^ + 1.
2. 2/ = - x2 +4x - 4. 12. 2/ = (2 - 3x)2 (3 +x).
3. 2/ = 2 x2 - X - 15. IZ. y = (x + 2)3 (x^ - 1).
4. 2/ = — 5 x2 — X — 15. /14. y = Vl +x -r Vl — x.
^6. y = x2— jx — 21. 16. y=^ax-\-b.
6. y = — a;' + 3x2 _|_ i, 16. y = c (a constant).
7. 2/ = 2 x8 + 3 x2 - 36 X - 20. 17. 2/ = oa;2 + 6x + c.
8. 2/ = - a:^ + 8 x2 + 2. 18. 2/ = c (x - o)».
9. 2/ = a:* - 2 x3 + 5 x2 + 2. 19. 2/ = (a; - o)"» (x - 6)».
10. 2/ = (1 + a;) -^ (1 - x). 20. 2/ = Axr-K
21. Show that the flexion of a straight line is everywhere zero.
22. Show that if the distance passed over by a body is proportional
to the time the tangential acceleration is zero. What is the speed?
23. Show that the flexion of the curve y = ax2 + 6x + c is every-
where the same, and equal to twice the coefficient of x'.
24. Show that if the space-time equation is « = a^ + 6i + c, the
acceleration is always the same and equal to twice the coefficient of <2.
Is such a motion at all liable to occur in nature?
26. Find the flexion of the curve y = 1/x. Show that it resembles y
itself in some ways. Does the slope also resemble y?
-*26. Can you interpret Ex. 25 as a motion problem? What is true
at the beginning of the motion (t = 0)? Can a curve with a vertical
asymptote represent a motion? Can a piece of such a curve?
J27. Find the flexion of the curve y = (x- 2y (x + 3)« (x - 4).
Show that the flexion has a factor (x — 2), while the slope has a factor
(X - 2)2 (x + 3).
VI, § 41] SUCCESSIVE DERIVATIVES 65
28. Show that the flexion of the curve y =^^{x-\-aY (x^ -\-3) has a
factor {x-\-a).
29. If the function y = / (a;) , where / (x) is a polynomial, has a
factor (x — a)3, show that' dy/dx has a factor {x — a)2, and d^/dx^ has
a factor (x — a).
30. If the equation x^ + ax^ -i- hx^ -^ cx^ -^ dx -\- e =0 has a triple
root x — at show that the equation 20x^-{-12ax^-^Qbx-h2c=0
has a factor x — a.
^1. Show how to find the double and triple roots of any algebraic
equation by the Highest Common Divisor process.
C&2. If the equations of the curve in parameter form are x = fi,
y = ^j find the slope m and the flexion b in terms of t.
L * dx dt ' dt* dx dt ' dt J
For each of the following curves, proceed as in Ex. 32. Calculate
also the values, in terms of t, of each of the six quantities mentioned in
§ 40, and the value of jr. Compare j and jt-
S3. x = a + bt,y =^c + dt. ' 34. re = <2^ ^ = ^3.
36. a; = <, y = <-2; < = 1 and 2. 36. x^l+t,y = ^^^ ; < = ± 2.
41. Concavity. Points of Inflexion. If the flexion b =
dm/dx is positive, the slope is increasing, and the curve turns
upwards, or is concave upwards; if the flexion is negative, the
slope is decreasing, and the curve is concave downwards.
Thus y = x^ ia concave upwards everywhere, since 6 = 2 is positive.
For y = x^ we find 6 = 6 x, which is positive when x is positive, and
negative when x is negative; hence y — x^ ia concave upwards at the
right, and concave downwards at the left of the origin.
A point at which the curve changes from being concave up-
wards to being concave downwards, or conversely, is called a
point of inflexion.
The value of the flexion 6 changes from positive to negative,
66 THE CALCULUS [VI, § 41
or conversely, in passing such a point; hence, the value of b at
a paint of inflexion is zero, if it has any value there.*
Thus the origin is a point of inflexion on the curve y =*= s^j for the
curve is concave downwards on the left, concave upwards on the right,
of the origin.
42. Second Test for Extremes. In seeking the extreme
values of a function y = /(x), we find first the critical points,
i.e. the points at which the tangent is horizontal.
If, at a critical point, b = (Py/dx^ > 0, the curve is also con-
cave upwards,^ and the function has a minimum there; if
6 < 0,the curve is concave downwards, and/ (x) has a maximum;
that is,
vr dy ^ , , cPv
ifm = j^ = and b = -7-|
> Ol . -, V . (minimum
<0
atx = a,f{a)isa
maximum
Whenever the flexion is not zero at a critical point, this
method usually furnishes an easy final test for extremes. If
the flexion is zero, no conclusion can be drawn directly by this
method.J (See, however, § 34.)
43. Illustrative Examples.
Example 1. Consider the function y —3^ — I2x + 7. See Ex. 3,
p. 8, and Ex. 1, p. 54. The slope and the flexion are, respectively,
ax dx^ ax
* Points where the tangent is vertical, for example, may be points of
inflexion.
t The curve is then also concave upwards on both sides of the point ; if
the curve is concave upwards on one side and downwards on the othet*, b
must be zero if it exists at the point.
X Even in this case one may decide by determining whether the curve is
concave upwards or downwards on both sides of the point; but the method
of § 34 is usually superior.
VI, § 43] SUCCESSIVE DERIVATIVES 67
The critical points are x = ± 2. Since 6 a; is positive when x is
positive, h is positive for x > 0; lilcewise 6 < when a; < 0. Hence
the curve is concave upwards when a? > 0, and concave downwards
when a; < 0. At a; = + 2, 6 > 0, hence by § 42, y has a minimum at
X = + 2; at a; = — 2, 6 < 0, hence y has a maximum (compare p. 8
and p. 54).
To find a point of inflexion first set 6 = 0;
h — -T- — T^ = 6 a; = 0, i,e. a; = 0.
ax ax''
Since dm/dx is negative for x < and positive for a; > 0, the given
curve is concave downwards on the left and concave upwards on the
right of this point; hence a; = 0, y = 7 is a point of inflexion. (See
Fig. 15, and § 44, p. 68.)
Example 2. Consider the function y = 3a;*— 12a:» + 50 (Ex. 2,
p. 55).
The slope and the flexion are, respectively,
m = ^ = 12a:3-36a:2; 6 = ^ = ^ = 36a:2 - 72a;.
dx dx dx^
The critical points are x = 0, a; = 3. Ata; = 3, 6 = 108 > 0, hence
y is a minimum there. At a; = 0, 6 = 0, and no conclusion is reached
by this method (compare, however, p. 55). To find points of inflexion,
first set 6=0;
6 = ^ = f^ = 36x2 - 72x = 0, i^e. x = Cora; = 2.
dx dx^
Near x = 0, at the left, dm/dx = 36 x(x — 2) is positive; at the right,
negative; the given curve is concave upwards on the left, downwards
on the right, and (x = 2, y = 2) is a point of inflexion. (See Fig. 13.)
Example 3. For a body thrown vertically upwards, the distance s
from the earth is:
where vq is the speed with which it is thrown.
The speed and the tangential acceleration are, respectively,
ds ^ . . d^s dv
If we draw a graph of the values of s and <, the speed v (slope of the
graph) is zero when
V = —gt + VQ-Oy i.e. t - vq/q,
THE CALCULUS
[VI, § 43
I a critical point on the graph. The tangential
of the graph) is negative everywhere, hence the
that is, the point
acceleration (Qexio
graph is em.
In particular, at the critical point just found, b is negative; hence
has a maxim urn there;
1
Fig. 17 is drawn for the special values f o = 64 and g "
44. Derived Curves. It is very instructive to draw in the
same figure graphs which give the
values of the original function, its
derivative, and its second deriva-
tive.
These graphs of the derivatives
are called the derived oaves; they
represent the slope {or speed in
case of a motion) and the flexion
(or tangential acceleration).
• .^-^--i -\- \ . I-,-- The figures for the curves of Era, 1
and 2 of I 43 are appended. The stu-
dent should show that each statement
made in { 43 and each statement made
on p. 67, for each of the examples, is illustrated and verified in these
figures.
The similar curves for space, speed, and acceleration are drawn in
Fig. 17, for the motion of a body thrown upwards:
a =• ~ i gP + V(,t Sor g = S2, v^ '• Qi.
Verify the s4at«U)ents made in Ex. 3, { 43.
In drawing such cur^•es, the second derivative should be
drawn first of all; the information it gives should be used in
drawing the graph of the first derivative, which in turn should
be used in drawing the graph of the original function.
Fio. 15.
VI, i 441 SUCCESSIVE DERIVATIVES 69
rim
111!
1 1
' 1 1 1 ' M
Is
kP^.
^
mm
-
d:^^.-
zif-rWiT^
:
TT'i-
T^:^-
^m
:
-^4
M
:
/^'vX
Eli^:
-
:/ \
\-;.:/
.X / :JJ~_ _
-
. K.^
-
f:
J
i : ;" i ■
' . j --
^ ~r ; ^-^"^ I 1
;_:^ rr.: i ■■-^\-r\ , ^-y\-t
70 THE CALCULUS [VI, § 44
EXERCISES
1. Draw, in the order just indicated, the first and second derived
curves in Ex. 1, p. 56, and show that each step of your work in that
example is exhibited by these figures.
2. Draw the derived curves for Exs. 2, 4, 6, 14, p. 56; and show
their connection with your previous work.
3. Draw the original and the derived curves for the function
y =0:' — 6x2 — 15x — 6. Find the extreme values of y, and explain
the figures. For what value of x is the flexion zero? Does this give a
point of inflexion on the original curve?
Find the extreme values of y and the points of inflexion on the fol-
lowing curves; in each case draw complete figures:
4. 2/ = 2 x8 - 3 x2 - 72 x. 9. y ^ Ax^ -\-Bx + C.
6. y = 4 a:3 _j_ g a;2 _ 24 x. 10. y = mx-\-n,
6. 2/ = x8_|.a;2. 11. 2/=Vx.
7. 2/ = a:* — 6 a;2 — 40. 12. 2/ = a^ — px + 5.
8. 2/ = x{x + 2)». 13. y = a;2 - 16/x.
14. Show that the flexion of the h3rperbola xy ^ a^ varies inversely
as the cube of the abscissa x,
16. Show that the flexion of the conic Ax^ + By^ = 1 (ellipse or
hjrperbola) varies inversely as the cube of the ordinate y,
16. What is the effect upon the flexion of changing the sign of a in
the equation y — ax^ -{■ hx -{■ cl
17. A beam of uniform depth is said to be of ''uniform strength"
(in resisting a given load) if the actual shape of its upper surface under
the load is of the form y = ax^ + 6a; + c, where x and y represent
horizontal and vertical distances measured from the middle point of
the beam's surface in its original (unbent) position. Show that the
flexion of such a beam is constant.
18. Show that the addition of a constant to the value of y does not
affect the slope or the flexion.
19. Show that the addition of a term of the form A;a; + c to the
value of y does not affect the flexion. What effect does it have upon
the slope?
VI, § 44l SUCCESSIVE DERIVATIVES 71
20. Show, by means of Exs. 18 and 19, that any beam in which the
flexion is constant has the form specified in Ex. 17.
21. Show, by a process precisely similar to that of Ex. 20 that a
motion in which the tangential acceleration is constant is defined by an
equation of the form s = a^ -i-ht -^ c.
22. Find, by the methods of Exs. 18-21, what the form of y must be
if the slope is:
«.)|=0; (5)|=-3; (c)|=6.; (^g=a. + 6.
23. What is the form of y if the flexion is 6? if the flexion is 2 x + 3?
if the flexion js zero?
24. If a beam of length I is supported only at both ends, and loaded
by a weight at its middle point, its deflection y at a distance x from one
end is y = k (Sl^x — 4:X^)y provided the cross-section of the beam is
constant. Find the flexion and show that there are no points of in-
flexion between the supports.
26. If the beam of Ex. 24 is rigidly fixed at both ends, and loaded
at its middle point, the deflection of each half of the beam ia y = k
(3 fa;* — 4 x^), where x is measured from either end. Show that there
is a point of inflexion at a distance Z/4 from the end, and that the greatest
deflection is at the middle point.
Find the points of inflexion and the point of maximum deflection
of a uniform beam of length I whose deflection is:
26. 2/ = A;(3fa;2-a;8).
[Beam rigidly embedded at one end, loaded at other end. Origin at
fixed end.]
27. y=k{3x^P-2x^),
PBeam freely supported at both ends, loaded uniformly. Origin at
lowest point.]
28. y = A;(6Px2-4te3+x4).
[Beam embedded at one end only; loaded uniformly. Origin at fixed
end.]
29. y^k(l^x-3lx^+2x^).
[Beam embedded at one end, supported at the other end; loaded uni-
formly. Origin at free end.]
72 THE CALCULUS [VI, S 45
46. Angular Speed. If a wheel turns, the angle 6 which a
given spoke makes with its original position changes with
the time, i.e. 5 is a function of the time:
e = /(0-
The time-rate of change of the angle is called the angular
speed; it is dejwted by u:
lu = anffular speed =
~ dt "
4 At
46. Angular AcceleratioiL The angular speed may change;
the time-rate of change of the angular speed is called the angular
acceleration; it is denoted by a:
a = angular acceleration = lim -r-r = 37 = t^ ■
iu-K> it at dr
Example 1. A flywheel of an engine atarte from rest, and moves
for 30 seconds according to the law
1 1
1
1
]
-
/
-h --
.
' ^'1
4.
___
■„•;:,.. 1
1
•^
-tu
0"-
(* +
where 8 is measured in de-
grees, after which it rotates
uniformly.
Then
dl '
dt
1
160
10
This example furnishes
an instaace id which the de-
rived eta ma, i.e. the graphs
which show the values of ui
Fia. 18. and of a are more imparUmt
than the original curve; for
the total angle described is relatively unimportant.
In the figure a scale is chosen which shows particularly well the
VI, § 46] SUCCESSIVE DERIVATIVES 73
variation of «; Bis allowed to run ofF of the figure completely, since its
values are uninteresting.
The acceleration a is so arranged that it does not suddenly drop to
zero when the flywheel is allowed to run uniformly; and the values of
a are never large. Something resembling this figure is what actually
occurs in starting a large fl3rwheel.
In actual practice with various machines, curves of this tj^pe are
often drawn experimentally. The equationis serve only as approximations
to the reality, but they are often indispensable in calculating other
related quantities, such as the acceleration in this example.
Curves which resemble the graph of w in this example occur fre-
quently. (See §§ 87, 134.)
EXERCISES
1. A fljrwheel rotates so that ^ t^ -h 1000, where 6 is the angle of
rotation (in degrees) and t is the time (in seconds). Calculate the
angular speed and acceleration, and draw a figure to represent each of
them.
2. Suppose that a wheel rotates so that = fi -i- 1000 where 6 is
measured in radians [1 radian = 180°/7r]. Is its speed greater than or
less than that of the wheel in Ex. 1? What is the ratio of the speeds
in the two cases?
3. If a wheel moves so that 6 = — t^/lQ — </32, where 6 is measured
in radians and t in minutes, find the angular speed and acceleration in
terms of radians and minutes; in terms of revolutions and minutes; in
terms of radians and seconds (of time).
4. If a Ferris wheel turtis so that = 20 <2 while changing from rest
to full speed, where d is in degrees and t in minutes, when will the speed
reach 20 revolutions per hour?
5. If the angular speed is co = A;<, as in Ex. 4, show that the accelera-
tion a is constant. Conversely, show that if a = A;, and if < is the time
since starting, co = kt.
6. Express the linear speed of a point on the rim of a wheel 10 ft.
in diameter when the angular speed is 4 R. P. M.
7. Find the linear speed and the tangential acceleration of a point
on the rim of the wheel of Ex. 1, § 46, if the wheel is 5 ft. in diameter.
What are they when t = 30 sec?
74 THE CALCULUS [VI, § 47
47. Related Rates. If a relation between two quantities is
known, the time-rate of change of one of them can be ex-
pressed in terms of the time-rate of change of the other.
Thus, in a spreading circular wave caused by throwing a
stone into a still pond, the circumference of the wave is
(1) * C = 2 TIT,
where r is the radius of the circle. Hence
® %-'"%■'
or, the time-rate at which the circumference is increasing is
2 TT times the time-rate at which the radius is mcreasing.
Dividing both sides by dr/dt, we find
dc dr ^ dc , ,
-t; -^ Ti = 2 TT = 3- = ac -^ ar:
at dt dr '
that is, the ratio of the time-^ates is the derivative of c with re-
spect to r; or, the ratio of the time-rates is equal to the ratio of
the differentials.
The fact just mentioned is true in general; if y and x are
any two related variables which change with the time, it is
true (Rule [Vila], p. 32) that:
dy dx dy , ,
that is, the ratio of the time-rates of y and x is equal to the ratio
of their differentials, i.e. to the derivative dy/dx.
Example 1. Water is flowing into a cylindrical tank. Compare the
rates of increase of the total volume and the increase in height of the
water in the tanlc, if the radius of the base of the tank is 10 ft. Hence
find the rate of inflow which causes a rise of 2 in. per second; and find
Ibhe increase in height due to an inflow of 10 cu. ft. per second. Consider
the same problem for a conical tank.
VI, § 47]
SUCCESSIVE DERIVATIVES
75
(A) 'Bhe volume F is given in terms of the height h by the formula:
hence
dV
dt
100
IT
dt'
or, the rate of increase in volume (in cubic feet per second) is 100 ir times
the rate of increase in height (in feet per second).
If dh/dt = 1/6 (measured in feet per second), dV/dt = 100 7r/6 =
(roughly) 52.3 (cubic feet per second). If dV/dt - 10, dh/dt = 10 -:-
100 TT = (rou^ly) .031 (in feet per second) = 22.3 (in inches per
minute).
{B) If the reservoir is conical^ we have
F = J Ttr^h = J 7r/i3tan2 «,
where r is the radius of the water surface, h is the
height of the water, and a is the half -angle of the
cone; forr = h tan a. In this case
dV I.04. o dh
-TT — ir/i^tan'' a -j7 ,
dt dt
which varies with h. If a = 45** (tana = 1), at a
height of 10 ft., a rise 1/6 (feet per second) would
mean an inflow of 7r/i2 x (1/6) = 100 7r/6 = 52.3
(cubic feet per second). At a height of 15 feet, a rise
of 1/6 (feet per second) would mean an inflow of
225 7r/6 = (roughly) 117.8 (cubic feet per second).
An inflow of 100 (cubic feet per second) means a rise
in height of 100/ tt^^^ which varies with the height; at
a height of 5 ft., the rate of rise is 4/7r = 1.28 (feet/
second).
FiQ. 19.
Example 2. A body thrown upward at an angle of 45**, with an
initial speed of 100 ft. per second, neglecting the air resistance, etc.,
travels in the parabolic path
r2
2/ = -
gz'
10000
+ x,
where x and y mean the horizontal and vertical distances from the start-
ing point, respectively; g is the gravitational constant = 32.2 (about);
and the horizontal speed has the constant value 100/^2. Find the ver-
tical speed at any time ^, and find a point where it is zero.
76 THE CALCULUS [VI, § 47
The horizontal speed and the vertical speed, i.e. the time-rate of
change of x and y^ respectively, are connected by the relation (see
§§8,20).
^ _:- ^ = ^ = — ^ 4-1.
dt ' dt dx 5000"^ '
, dy , gx , ^.dx gx , 100
hence J = (- ^^ + i)_ = - ^ + _.
This vertical speed is zero where
gx .100 ^ . 5000 ,„o / u *^
— H 7= = 0, I.e. X = = 155.3 (about),
50 v^ V2 ' ff
which corresponds to y = 2500/^ = 77.7 (about). At this point the
vertical speed is zero; just before this it is positive, just afterwards it is
negative. When x — the value of dy/dt is 100/v^; when x =
2500/^, dy/dt = 50/V2; when x = 7500/^, dy/dt = - 6O/V2.
EXERCISES
y^. Water is flowing into a tank of cylindrical shape at the rate of
100 gal. per minute. If the tank is 8 ft. in diameter, find the rate of in-
crease in the height of the water in the tank.
^. A fimnel 12 in. across the top and 9 iA. deep is being emptied at
the rate of 2 cu. in. per minute. How fast does the surface of the liquid
fall?
3. If water flows from a hole in the bottom of a cylindrical can of
radius r into another can of radius r', compare the vertical rates of rise
and fall of the two water surfaces.
—4. If a fimnel is 12 in. wide and 9 in. deep and liquid flows from it at
the rate of 5 cu. in. per minute, determine the time-rate of fall of the
surface of the liquid.
5. Compare the vertical rates of the two liquid surfaces when water
drains from a conical funnel into a cylindrical bottle. Compare the
time-rate of flow from the funnel with the time-rate of the decrease of
the wet perimeter.
6. If a wheel of radius R is turned by rolling contact with another
wheel of radius R'y compare their angular speeds and accelerations.
VI, § 47] SUCCESSIVE DERIVATIVES 77
7. If the surface s of a cube increases at a given rate k (in square
inches per second), what is the rate of increase of the volume?
8. If a point moves on a circle so that the arc described in time t is
ff = ^ — 1/^2 _j_ 1^ find the angular speed and acceleration of the radius
drawn to the moving point.
9. A point moves along the parabola y = 2x^—xin such a manner
that the speed of the abscissa x is 4 ft. /sec. Find the general expression
for the speed of y; and find its value when x = 1; when a; = 3.
10. In Ex. 9, find the horizontal and vertical accelerations, the total
speed, the tangential acceleration, and the total acceleration.
11. A point moves on the cubical parabola y = x' in such a way that
the horizontal speed is 10 ft. /sec. Find the vertical speed when a; = 6.
12. In Ex. 11, find the horizontal, vertical, tangential, and total
accelerations.
13. If a person walks along a sidewalk at the rate of 4 ft. per sec.
toward the gate of a yard, how fast is he approaching a house in the yard
which is 50 ft. from the gate in a line perpendicular to the walk, when
he is 100 ft. from the gate? When 10 ft. from the gate?
14. Two ships start from the same point at the same time, one sailing
due east at 10 knots an hour, the other due northwest at 12 knots an
hour. How fast are they separating at any time? How fast, if the first
ship starts an hour before the other?
^ 15. If a ladder 13 ft. long rests against the side of a room, and its foot
moves along the floor at a uniform rate of 2 ft. /sec, how fast is the top
descending when it is 5 ft. above the floor? When the top is 1 inch
from the floor?
16. If the radius of a sphere increases as the square root of the time,
determine the time-rate of change of the surface and that of the volume;
the acceleration of the surface and that of the volume.
17. If a projectile is fired at an angle of elevation a and with muzzle
velocity vo, its path (neglecting the resistance of the air) is the parabola
y = X tan a —
2 v^ cos2 a *
X being the horizontal distance and y the vertical distance from the point
of discharge. Draw the graph, taking gr = 32, a = 20°, Vq = 2000 ft. /sec.
78 THE CALCULUS [VI, § 47
Calculate dy in terms of dx. In what direction is the projectile mov-
ing when X = 5000 ft., 10,000 ft., 20,000 ft.? How high will it rise?
18. If p'V — kj compare dp/dt and dv/dt in general; compare
d^V/d^ and cPv/d^.
19. If p • «» = ky compare dp/dt and dv/dt. [For air, in rapid com-
pression, n = 1.41, nearly.]
20. If 9 is the quantity of one product formed in a certain chemical
reaction in time t, it is known that q = ckH/{l + ckt). The time-rate
of change of g is called the speed v of the reaction. Show that
" = (tSo^ = "(* - «^*-
Show also that the acceleration a of the reaction is
CHAPTER VII
REVERSAL OF RATES — INTEGRATION
48. Reversal of Rates. Up to this point, we have been
engaged in finding rates of change of given functions. Often,
the rate of change is known and the values of the quantity
which changes are unknown; this leads to the problem of this
chapter: to find the amount of a quantity whose rate of change is
known.
Simple instances of this occur in every one's daily experience. Thus,
if the rate r (in cubic feet per second) at which water is flowing into a
tank is known, the total amount A (in cubic feet) of water in the tanlc
at any time can be computed readily, — at least if the amount originally
in the tank is known:
A = r . < + C,
where t is the time (in seconds) the water has run, and C is the amount
originally in the tank, i.e. C is the value of A at the time when < = 0.
If a train runs at 30 miles per hour, its total distance d, from a given
point on the track, is
d = SO 't + Cy
where t is the time (in hours) the train has run, and C is the original
distance of the train from that point, i.e. C is the value of d when < = 0.
(Notice that by regarding d as negative in one direction, this result is
perfectly general; C may also be negative.)
If a man is saving $100 a month, his total means is 100 'TI + Cj where
n is the number of months counted, and C is his means at the beginning;
i.e. C is his means when n = 0.
If the cost for operating a printing press is 0.01 ct. per sheet the total
expense of printing is
T = 0.01-n + C
where n is the number of copies printed, and where C is the first cost of
the machine; t.e. C is the value of T when n = 0.
79
80 THE CALCULUS [VII, § 49
49. Principle Involved. Such simple examples require no
new methods; they illustrate excellently the following fact:
The total amount of a variable quantity y at any stage is
determined when its rate of increase and its original value C
are known.
We shall see that this remains true even when the rate
itself is variable.
50. Illustrative Examples. The rate R{x) at which any
variable y increases with respect to an independent variable
X is the derivative dy/dx] hence the general problem of §§ 48-
49 may be stated as follows: given the derivative dy/dx, to find
y in terms of x.
In many instances our famiUarity with' the rules for obtain-
ing rates of increase (differentiation) Qnsblo;^ us to set down at
once a function which has a given rate of increase.
Example 1. Thus, in each of the examples given in § 48, the rate is
constant; using the letters of this article:
where /: is a known fixed number; it is obVious that a function which
has this derivative is
(A) y = kx+C,
where C is any constant chosen at pleasure.
While the examples of § 48 can all be solved very easily without this
new method, for those which follow it is at least very convenient. The
value of C in any given example is found as in § 48; it represents the
value of y when a; = 0.
Example 2. Given dy/dx — x^, to find y in terms of x.
Since we know that d (x^)/dx — 3 x^, and since multiplying a function
by a number multiplies its derivative by the same number, we should
evidently take:
X X r ^x \ T
y = -g , or else y =-^ +C; I check: ^ ( 3 + ^ ) = ^^cb\ »
VII, § 501 REVERSAL OF RATES — INTEGRATION 81
where C is some constant. As in § 48, some additional information
must be given to determine C. In a practical problem, such as Ex. 3,
below, information of this kind is usually known.
Example 3. A body falls from a height 100 ft. above the earth's sur-
face; given that the speed is v = — gt, find its distance from the earth
in terms of the time t.
Let s denote the distance (in feet) of the body from the earth; we are
given that
ds
(1) V — -r. = — gt, or ds = vdt "= — gt dt,
which is negative since s is decreasing. We know that d (^) =2tdt]
hence it is evident that we should take:
(2) s = - I <2 + C; [check: ds = - gt dt].
As the body starts to fall, ^ = and s = 100; substituting these
values in (2) we find 100 = + C, or C = 100. Hence we have
s = - I ^ + 100.
Example 4. Given dy/dx — ic«, to find y in terms of x.
Since we know that d(x«+i) = (n + 1) a;»»cte, we should take
(B) y == —^x^+i + C; [check: dy = x^dx].
Since the rule for differentiation of a power is valid (§21, p. 34)
for all positive and negative values of n, the formula (B) holds for all
these values of n except n == — 1 ; when n — — I the formula (B) can
not be used because the denominator n + 1 becomes zero.
Special cases:
dx
w = 1, 5l = ^» y = 2 ^^ "^ ^' check: d f ^ ^^) =xdx.
n = 0, -^ = 1, y =^x + C; check: d{x) = 1 -dx.
n = ^» ^ = xi/2, y = 1^3/2 + C; check: d (l^^^^ = x^/^dx.
n = - 2, ^ = a^^ y = - x-i + C; check: d (-1 a;-i) - x-^dx.
Notice that these include Vx ( = xV^), l/a;2 ( = a;-2), etc. ; other special
cases are left to the student.
82 THE CALCULUS [VII, § 50
Example 5. Given dyldx = x^ + 2 x^, to find y in terms of x.
Since d {x^)/dx = 4 x^ and d {x^)/dx = 3 x^, and since the derivative of
a sum of two functions is equal to the sum of their derivatives, it is
evident that we should write *
The check is
X4 2x3 ^
S-e(?+¥+<^)=-+^-*^
such a check on the answer should be made in every exercise.
In general, as in this example, if the given rate of increase (derivative)
is the sum of two parts, the answer is found by adding the answers
which would arise from the parts taken separately, since the sum of the
derivatives of two variables is always the derivative of their sum.
EXERCISES
Determine functions whose derivatives are given below; do not
forget the additive constant; check each answer.
1. ^^ = 4x. 2.f = -5x. 3. ^=3x^ 4.^ = 2.
dx dx dx dx
6. ^^=-6x^.6. ^^ = -10x5. 7. f = -x*. 8. ^ = .01 a*.
dx dx dx dx
In the following exercises, remember that the derivative of a sum
is the sum of the derivatives of the several terms; proceed as above.
9. ^ = 4 + 5x2. 10. ^ = 4x2-2x + 3. 11. ^ = t^-4t + 7.
dx dx dt
12. 1 = 3x5-8x4.18. | = «x+6. 14. |=a^ + M + c.
16. ^ = . 006 x2-. 004 x' + . 015x4. 16. ^ =■ - /» + St* - 6<« + 2
ox at
20. ^ = x2/8. 21. ^ = 2x1/2 - 3a;-i/2. 22. ^ = kv^-^\
dx dx dv
* In all the Examples of this paragraph, we have had an equation which
involves dy/dx; such an equation is often called a differential equation^ be-
cause it contains differentials. See also Chapter XIX.
VII, §511 REVERSAL OF RATES — INTEGRATION 83
61. Integral Notation. If the rate of increase dyjdx = R (x)
of one variable-!/ with respect to another variable x is given, a
function y=I(x) which has precisely this given rate of increase
is called an indefinite integral * of the rate R (x), and is repre-
sented by the symbol t
(1) I(x)^fR(x)dx;
that is,
(2) if ^[/(x)] = 2?W, thennx) = fR(x)dx,
or, what amounts to the same thing,
(3) if d [I (x)] = R (x) dx, then I (x) ^Jr (x) dx.
The results of Examples 1, 2, 3, § 50, written with the new
symbol, are, respectively,
[A] Ck dx^kx + C.
fa^dx^x^S + C.
s =/« dt + C =/- gtdt + C = - gt^/2 + C.
The first equation of Example 3 holds in general:
[I] s= Cvdt + C, since ~=i;.
•/ 4 at
* The common English meaning of the word integrate is "to make whole
again," "to restore to its entirety," "to give the sum or total." See any
dictionary, and compare §§ 48-49.
To integrate a rate R{x) is to find its integral; the process is called inte-
gration. Often the rate function R(x) which is integrated is called the in-
tegrand; thus the first part of equation (2) may be read: *'the derivative oj
the integral is the integrand.'* This is the property used in checking answers.
The first equation in (2) and the first in (3) are differential equations.
t Note that dx is part of the symbol. As a blank symbol, it is J* (blank)
dx; the function R{x) to be integrated {i.e. the integrand) is inserted in
place of the blank. The origin of this ssrmbol is explained in § 120.
84 THE CALCULUS [VII, § 51
The result obtained in (B), Example 4, § 50, gives
IB] rxndx=-^+C, HT^-l,
%/ n "T 1
for all positive and negative integral and fractional values of
n except n = — 1, for which see § 65.
As examples of the many special cases, we write:
n = 1, J'x dx = -^ + C.
n
= 0, fx^dx =fldx =fdx =x + C.
n=-2, fx-^dx =f^dx = -xr^ + C=---+C.
From Example 5:
/(a;3 + 2T')dx =fx^dx +f2x^dx =^ + ^ + C.
The general principle used in this example is that the inte-
gral of a sum of two functions is the sum of their integrals:
[C\ ^[R (X) + S (X)] dx = J/2 {X) dx + Js (x) dx,
which is true because the derivative of the sum R{x)+S (x)
is the sum of the derivatives:
d[R + S]/dx = dR/dx + dS/dx.
The rules (A), (B), (C) are sufficient to integrate a large
number of functions, including certainly all polynomials in x.
VII, § 51] REVERSAL OF RATES — INTEGRATION 85
EXERCISES
1. Express the value of y if dy/dx = 4x^+ 3x by means of the
new sign y (— ) <&. Then find y. Check the answer by differentiation.
Proceed as in Ex. 1 if y dy/dx has one of the following values:
2. 3^. 4. x-K 6. x^-2, 8. x^+3x-4, 10. mx - n.
3. x-K 6. 4 X + 5. 7. 9. 9. x^ - x\ 11. a? + V5.
In the following exercises express the given function as a sum of
powers of x; then proceed as above.
12. x^ (1 + x). 16. (1 + x^) (1 - x3). 20. (1 - x2) (1 + x-^).
13. (x3 + 5 x2) -f- a;2. 17. (3 - x) (5 + 2 x). 21. x^Hx-x^).
14. 3(1-2)2. 18. a;V2(2-x). 22. (x^-2x^+x)VK
16. 3 a:8 (4 _ 3 ^)^ 19. (3 - 2 x) Vx, 23. (1 -|- \(x)2.
Evaluate the following integrals:
24. fx'^dx, 29. y(^H-^)<^«'- ^4. f 12 f^/^dt,
26. ffi/^dt 80. f((ou-^-7u-^)du. 36. fsylr^dy,
26. fba-^ds, 31. f{^/^-^z-^/^)dz. 36. f(5/Vu^)du,
27. y (2 i^w - i*-i/2)dw. 32. /{yly^-^ y'^) dy. 37. f{y[u)^du.
28. y3 r-2/3 dr. 33. / (j/^o/^ + 2 2^/*) rfy.
Integrate the following expressions, making use of the principle of
Exs. 12-23.
38. y (1 - 0^ dt 44. y v^ (a + hx) dx.
39. y^ (1 + y/x) dx. 46. y^" (a + 6a;) (te.
40. fs (1 - >/«)2(fo. 46. f(a + &a;)2cte.
41. y^ (1 - fi) dt, 47. y (^-5 - 2) r6 d<.
42. yx-4(l4-x+x^)(^. 48.y.i-4(l+x2)2^.
43. ya? (a + hx) dx. 49. fVt(l +2f)^dt.
86 THE CALCULUS [VII, § 52
62. Fundamental Theorem. If dyldx = 2 a;, the answers
2/ = x^, 1/ = x^ + 5, 2/ = x^ + C are all correct. , To decide which
one is wanted, additional information is needed. However,
except for the additive constant C, all answers coincide.
For practical purposes, there is but one answer. Stated
precisely, this is the fundamental theorem of integral calculus:
If the rate of increase
(1) I = Ri^)
of a variable quantity y which depends on x is given, then y is
determined as a function of x,' I (x), except for a constant term:
(2) y=fRix)dx + C = I (x) + C.
63. Calculation by Integrals. In applications, we often care
little about the actual total; it is rather the difference
between two values which is important.
Thus, in a motion, we care little about the real total
distance a body has traveled; it is rather the distance it has
traveled between two given instants.
If a body falls from any height, the distance it falls is (Ex. 3, p. 81)
s = vdt + C =fgtdt + C =-^ + C,
where s is counted downwards.
The value of s when < = is s]i=o = C; the value of s when < = 1 is
s]i = i = g/2 + C. The distance traversed m^^^rs^ second is found by
subtracting these values:
where «]JlJ means the space passed over between the times < = and
< = 1.
In this calculation, we care little about where « is counted from; or
its toted value. The result is the same for all bodies dropped from any
height.
VII, §54] REVERSAL OF RATES — INTEGRATION 87
Likewise, the space passed over between the times ^ = 2 and ^ » 5 is
.2|5.^.,.|.338(,u.
In general the distance traversed between the times < = a and < = 6 is
•]::-],..-'l-.-(^+'^)-(4+-)-4-4-i<'--)-
64. Definite Integrals. The advantage realized in the
example of § 53 in elinunating C can be gained in all problems.
The numerical value of the total change in a quantity between
two values of x, x = a and a; = 6, can be found if the rate of
change dy/dx = R (x) is given. For, if
y = I{x)=fR{x)dx + Cy
the value of y for a; = a is
and the value of y for x = 6 is
The total change in y between the values x^ a and x = bis
= [fR ix) dx\- [fR (x) dxl
This difference, found by subtracting the values of the in-
definite integral at x = a from its value at x = b, is called the
definite integral of R (x) between x = a and x = 6; and is
denoted by the symbol:
r'^R{x)dx^\CR{x)dx\ - [ f^W^^l .
lz»a
88 THE CALCULUS [VII, § 54
It should be noticed that, in subtracting, the unknown con-
stant C has disappeared completely; this is the reason for
calling this form definite.
Example 1. Given dy/dx = x*, find the total change in y from
X = 1 to a; = 3.
Since
y=^f2^dx =x4/4 + C,
it follows that
,?-'=,] -y\ =J] _J] =20.
J«-l Jx-3 Jx-1 4Jx-3 4Jx-l
Interpreted as a problem in motion, where x means time and y means
distance, this would mean: the total distance traveled by a body be-
tween the end of the first second and the end of the third second, if its
speed is the cube of the time, is twenty units.
Interpreted graphically, a curve whoise slope m is given by the
equation m = rc^, rises 20 units between re = 1 and x = 3. The equa-
tion of the curve is 2/ = ^/^ + C,
EXERCISES
1. If 'water pours into a tank at the rate of 300 gal. per minute, how
much enters in the first ten minutes? how much from the beginning of
the fifth minute to the beginning of the tenth minute?
2. If a train is moving at a speed of 30 mi. per hour, how far does it
go in two hours? Does this necessarily mean the distance from iis
last stop?
3. If a train leaves a station with a variable speed v = t/2 (ft./sec),
find s in terms of t. How far does the train go in the first ten seconds?
How far from the beginning of the fifth to the beginning of the tenth
second?
4. A falling body has a speed v = gt, where t is measiu^d from the
instant it starts. How far does it go in the first four seconds? How far
between the times t = 3 and < = 9?
6. A wheel rotates with a variable speed (radians/sec.) <a = ^/lOO.
How many revolutions does it make in the first fifteen seconds? How
many between the times t = 1 and t = 10?
VII, §54] REVERSAL OF RATES — INTEGRATION 89
From the following rates of change determine the total change in
the functions between the limits indicated for the independent variable.
Interpret each result geometrically and as a problem in motion, and
write your work in the notation used in the text.
6. ^ = re, a; = 2 to X = 4. 11.^= ^^^ < = 1 to < = 3.
dx at v^
7. f^ = ^x«,x = -2tox = 2.12. | = (l + ^,« = 4to« = 9.
dx b at <»/2 '
8. T^ = rTi, a; = — 4 to a; = 4. 13. ^r = ^s , t = 0.1 to 1.
dx 12 dt ^
9. ^ = 2-a^,a; = Otox = 10. 14. ^ = VtVl; < = 1 to « = 16.
Determine the values of the following definite integrals. [In cases
where no misunderstanding could possibly arise, only the numerical
values of the limits are given. In every stich case, the numbers stated as
limits are values of the variable whose differential appears in the integral,]
Sxdx. 21. f x2/3 dx. 26. f sV} («2 -2s)ds.
^xdx, 22. / iX + t)di. 27. / ^.
18. r Zx^dx, 23. / 3(^-1) (ft. 28. / (.01 + .02^)<i&.
19. J^^l^x^ dx. 24./_*^(H-a + 52)ds. ^ f^ (yie + 4^) dd.
20. f^Vidx. 26. r^^J^ds. 30. f^'^ir-^^^ dv,
•/o ^1 s^ J2.3
31. A stone falls with a speed v = (7< + 10. Find « in terms of t and
find the distance passed over between the times t ==2 and t = 7,
32. A bullet is fired vertically with a speed v = — gt -\- 1500. How
far does it go in ten seconds? How high does it rise? How long is it in
the air? Make rough estimates of the answers in advance.
90
THE CALCULUS
[VII, j 55
, For any falling body, j = acceleration — ff = eonat. Find the
speed in ten seconds. Does it matter what particular ten
seconds are chosen?
34. If, in Ex. 33, the speed is 100 ft./sec. when ( = 5, what is the
speed when ( = 15? When will the speed be 250 ft./sec? Express v
in t«nus of I.
66. Area under a Curve, We saw in § 54 that the value of
any quantity could be computed if its rate of change could
be found. We shall proceed to illustrate this principle by
showing how to find the area bounded by any given curve,
the i-axis, and any two ordinates of the curve.
Let the equation of the given curve be
(1) S-/W,
and let A denote the area FMQP between this curve, the
X-axis, a fixed ordinate FP
(Fig. 20), and a variable
ordinate MQ. Since A will
vary as the value of x at ilf
changes, A is a function of x.
If X changes by an amount
ix = MN, the area A will
change by an amount Jtf^fiQ,
which we shall call AA. Then
^Ay
Fia.20.
it is evident from the figure that if the curve rises from Q
to R we shall have
(2) redangU MNSQ<AA< rectangle MNRT.
If the curve falls from Q to B, the inequalities would be
reversed. From (2) we have
(3) 1/ ■ Ax <a4 <(y + Ay) ■ Hx.
Dividing by Ax, we find
(4)
VII, §55] REVERSAL OF RATES — INTEGRATION 91
If the curve falls from Q to R, these inequaUties would
simply be reversed. If we now let Ax approach zero, Ay will
also approach zero, and we shall have, in either case,
(5)
dA ,. AA .. V
It follows, by § 54, that the area under the curve (1), between
any two fixed ordinates x = a and x = b, is given by the
formula
= 1 ydx=i f{x)dx.
Example 1. To find the area under the
curve* y =^x^ between the points where
a; = and a; = 2.
We have, by (2)
A--fydx+C -=fx^dx +C^^+C,
where A is counted from any fixed back
boundary x = A; we please to assume, up to
a movable boundary x = x.
The area between a; = and a; = 2 is
given by subtracting the value of A for yiq 21
a; = from the value for A for a; = 2:
aT'^a] -a] =/-%3^ = f.] -f] =|.
Jx-O Ja;-2 Jx-0 •^ x^O oJ«-2 «>J«-0 «>
Likewise the area under the curve between a; = 1 and a; = 3 is
= / x^dx = — — — = 8f •
aj-l -/x-i 3jx-3 3j«-i '
and the area under the curve between any two vertical lines x == a and
a; = 6is
AT-* = r-w=^i -I'l
♦The phrase "the area under the curve" is understood in the sense
used in § 55. When the curve is below the a;-axis, this area is counted as
negative.
92 THE CALCULUS [VII, § 55
If the equation of the curve is given in parameter form
i«=/(0,
<" !::
«w,
the equation (5) may be replaced by the equation
, . dA _ dA dx _ dx
^^^ dl~lx"M~'^' dA'
or
^^^ it-'^^^^'~dr'
and the formula (6) takes the form
or
which gives the area above the x-axis, below the curve (7),
and between the ordinates of the points at which t has the
values ti and <2, respectively.
Example 2. To find the area under the curve whose equations are
between the ordinates of the points where t ^2 and < » 3.
By (11), the required area is
-31+<
T" =/
.2 t
3
2tdt
^ f (2 + 2t)dt=^7,
J 2
[Cautign. By calculating in a similar manner the area under the
curve from ( = to ( = 1 we would find -AJJ"^ = 3. But this result
would require justification by the considerations of § 115, p. 188 J
VII, §55] REVERSAL OP RATES — INTEGRATION 93
EXERCISES
Find the area under each of the following curves between the ordi-
nates x =0 and x — 1; between x = 1 and a; = 4. Draw the graph
and estimate the answer in advance.
1. 2/ = a;2. 4. 1/ = xVs. 7, y = VT+~x,
2. 2/ = Vx, 6. 2/ = 1 — a;2. 8. y = x^l — x).
3. 2/ = x3/2. 6. y = (I- x)2. 9. 2/ = xil - x^).
Find the area under each of the following curves and check graphi-
cally when possible.
10. 2/ = i^ + 6 a;2 + 15 X, (x = to 2; a; = - 2 to + 2).
11. y — x2/3, (x = — 1 to + 1; X = — a to + a).
12. y =x^ + l/x2, (x = 1 to 3; X = 2 to 5; x = a to 6).
Find the area under each of the following curves between the ordi-
nates determined by the indicated values of t, and check graphically.
13. X = < + 1, 2/ = < - 1; « = to 5.
14. X = (< - 1)A y = fi/S; « = 2 to 4.
16. X = 2 «, y = 3 VF; < = to 4.
16. X = 1 + V?, 2/ =2 ^; < = to 9.
17. X = (1+ 0^ 2/ = (1 - 0*; « = 1 to 2.
18. X = 1 - «, 2/ = vT+1; < = - 1 to 3.
19. X = VT+l, 2/ = ^^"^=1; « = 1 to 5.
20. Show the area A bounded by a curve x =<l>(y), the 2/-axis>
and the two lines 2^ = a and y = 6 is
^= ] <t>(y)dy.
21. Calculate the area between the 2/-axis, the curve x = y^, and the
lines y = and y = 1. Compare this answer with that of Ex. 14.
22. Find the area between the curve y — o^ and each of the axes
separately, from the origin to a point (A;, W), Show that their sum is k^.
d4
THE CALCULUS
[VII, § 56
66. Volume of a Solid of Revolution. Let us next consider
the volume of the solid of revolution which is described when
the area MNLK (Fig. 22) under the curve- y = f{x) from
X = a to X = b 18 revolved about the x-axis. Any section of
this solid perpendicular to the x-axis will be a circle.
Let us denote by V the volume from x = a to x = x, and
by AV the increase in this volume as x increases to x + Ax.
The radius of the circular section at any point is the ordinate
Fig. 22.
y of the curve. Hence the area of the section is wy^. If
the curve is rising steadily from P to Q, it is evident that
(1) 7ri/2.Ax<AF< TT (y + Ay)2 . Ax.
Dividing by Ax, we have
AV
(2)
Try < ^"^'^^y'^^y^ '
If the curve is falling, these inequalities are reversed. If we
now let Ax approach zero, Ay will approach zero, and we
shall have, in either case,
(3)
dv ,. ^v ,
-7- = lim -r- = 1^J/^
VII, §571 REVERSAL OF RATES — INTEGRATION 95
It follows, by § 54, that the volume of the solid of revolution
from X = a to X = fe, is given by the formula
(4)
]x^b /•x*6 /•x-6
= 1 iry'dx= \ ir\fix)}»dx.
x^a J xr»a J x» a
Similarly, the volume F of a solid of revolution formed by
revolving a curve x=<f>(y) around the y-axis satisfies the
relations
dV -^^"^ '•^"'^
(5)
dy
= TTX^,
yJ-.J
TX^dy.
y— c
Example. Find the volume generated when the area under the
curve 2/ = 1 — x^ from a; = — ltoa; = +l revolves about the rc-axis.
From the symmetry of the figure, we see that the total volimie re-
quired is twice the volume generated by the area from x = to a; = 1.
Hence
= 2 / iry^dx
J
=-2jr(x-
y-l-a;2
2 0^ a;5\-[i
3 "^5; Jo
16^
[Fig. 23.
57. Volume of a Frustum of a Solid. A frustum of a solid
is the portion of that solid contained between two parallel
planes. The solid itself, between the limiting parallel
planes, may be thought of as generated by the motion of the
cross-section parallel to these planes from one extremity to
the. other, if the shape of the cross-section is supposed to
vary in the correct manner during the motion. Thus, a
frustum of a circular cone may be generated by the motion
of a circle which remains parallel to its origiual position, if
96
THE CALCULUS
[VII, § 57
18^8
the radius of the ckcle steadily increases (or diminishes)
during the motion.
In Fig. 24, let s denote the distance along a Une AB
perpendicular to the variable cross-section, measured from
some fixed point A.
Let V denote the
volume of the solid
from 5 == a to s = s,
-«+^« and let AV denote
the increase in vol-
ume as 5 changes to
8 + As, Let As de-
note the area of the
cross-section at the
position s = s, and
let AA denote the increase in As as s changes from s to
5 + As. Then, if As increases as s increases, we shall have
(1) A,- As<AF<(il, + Ail) -As.
This inequality will be reversed if As decreases. Dividing
by As, and then allowing As to approach zero, AA will also
approach zero, and we shall have
Fig. 24.
(2)
^^ = lim^ = A„
ds A«-»o As
whence, by § 54, the volume of the entire frustum from
a; = a to a; = 6 is given by the formula
(3) ^ = I '^» <'*
The formulas (2) and (3) show that the rate of change of
the volume with respect to* the distance s is equal to the
area of the variable cross-section, and that the volume of
VII, §57] REVERSAL OF RATES — INTEGRATION 97
the entire frustum is obtained by integrating the area of
the cross-section with respect to s. The formulas of § 56
are special cases of the formula (3).
Example. A circle moves with its center on a given straight Une,
and its plane perpendicular to that hne. Its radius is proportional to
the square of the distances of its center from a fixed point of the line.
Find the volume of the frustum of the sohd generated as the circle
moves from « = a to « = 6.
If 8 denotes the distance from the fixed point to the center of the
circle, the radius of the circle, which is to vary as the square of a, must
be r = ks^f where A; is a constant. Hence the area of the circle is
Then the volimie of the frustum from 8 = ato s = bia
b Jb b
V
1 = / irA;2«4ds = irA;2 r 8*d8
EXERCISES
Find the volumes formed by revolving each of the following curves
about the x-axis, between a; = to a; = 2; between a; = — ltoa;=+l.
1. y = a^. 3. y = a^ — X. 5, y^ ^^ x +2y,
2. y ^x^-l. 4. 2/ = (1 + a;)2 6. V^+l + Vy = 4.
Proceed similarly for each of the following curves, between a? = 1
and X = 3; between x = a and x = 6.
7. 2/ = ^-^' 8. xy = 1+ a;2. 9. x* - x2y2 = 1.
Find the volimies formed by revolving each of the following curves
about the y-ajdSf between 2/ = and y = 2,
10. X = ^. 12. X = 4 2/2 — 2^. 14. aj = ^2 — y^
11. x2 = 2/8. 13. x2 + 2/4 = 81. 16. X = 2/^/2 + 2/V4.
16. Find by integration the volume of a frustum of a cone of height
hf if the radii of the two bases are, respectively, r and R.
98 THE CALCULUS [VII, § 57
17. Find the volume of the paraboloid of revolution formed by
revolving y^ = 4 ^ about the a;-axis, between x = and a; = 4; between
X = 1 and a; = 5; between x = a and x = 6.
18. Find the volume of a sphere by the formula of § 56.
19. Find the volume of the ellipsoid of revolution formed by revolv-
ing an ellipse (1) about its major axis; (2) about its minor axis.
20. Find the volume of the portion of the hyperboloid of revolution
formed by revolving about the 2/-axis the portion of the hyperbola
ic2 _ y2 =s 1 between 2/ = and y = 2.
21. Find the volume of the portion of the hyperboloid of revolution
formed by revolving x^ — y^ — 1 about the rc-axis, between x = 1 and
a; = 3.
22. Find the voliune generated by a square of variable size perpen-
dicular to the X-axis, which moves from x = Otox = 5, if the length of
the side of the square is (1) proportional to x; (2) equal to x^,
23. Find the volume generated by a variable equilateral triangle per-
pendicular to the X-axis, which moves from x=0tox = 2, ifa side of
the triangle is (1) equal to x^; (2) proportional to 2 — x.
24. Find the volume generated by a variable circle which moves in a
direction perpendicular to its own plane through a distance 10, if the
radius varies as the cube of the distance from the original position.
26. Find the mass of a right circular cylinder of variable density, if
the density varies (1) directly as the distance from the base; (2) in-
versely as the square root of the distance from the base.
CHAPTER VIII
LOGARITHMS — EXPONENTIAL FUNCTIONS
58. Necessity of Operations on Logarithms. The necessity
for the introduction of logarithms in the Calculus depends
not only on their own general importance, but also upon the
fact that integrals of algebraic functions may involve logarithms.
Thus, in §51, in the case n = — 1 the integral yx^dx could
not be found, although the integrand 1/x is comparatively
simple. We shall see that this integral, J*x~^ dx, results in a
logarithm. We shall see also in § 68 that numerous cases
arise in science in which the rate of variation of a function
/ (x) is precisely 1/x.
69. Properties of Logarithms. The logarithm L of a num-
ber N to any base B is defined by the fact that the two
equations
(1) iV = B^ logBN = L
are equivalent. Thus if L = logBiV and l^logsn, the
identity B^B^ = B^^^ is equivalent to the rule
(2) logs (N-n) == logs N + logs n,
where n and N are any two numbers. Likewise B^ -^B^ =^
B^'^ gives
(3) logB (iV -^ w) = logs N - logs n;
and (B^y = B^ becomes
(4) logBiV" = nlogsiV,
where n may have any value whatever.
The relations (1), (2), (3), (4), are the fundamental rela-
tions for logarithms.
99
100
THE CALCULUS
[VIII, § 60
60. Computations. Graphs. To draw the graph of the
equation
(1) y^^logBX,
for any fixed value of B, we may write the equation in the
form
(2) z = B^.
To compute the value of x when y is given, we take the
common logarithm of both sides of (2) :
(3) logio X = logio B" -=y' logio B,
by (4) § 59. But since y = logs Xy we have
(4) logio X = logB X ' logio By
or
(5) logB X = logio X -4- logic B.
y
3^_
-
—
^
B
5
"^
^
-^
B
2
2.7
y
^
—
—
■D
'
e =
*■
/
A
^^
^
-^
'
. —
—
D
— ^
/
r
-— ■
— -
—
—
B
=
10
1
^
■ —
—
"
1
5
1
1
5
X
f
II
Fig. 25.
The relations (4) and (5) enable us to compute logarithms
to any base quickly by means of a table of common loga-
rithms. The graphs of (1) for several values of B are shown
in Fig. 25.
VIII, § 601 LOGARITHMS 101
The relation (3) enables us to compute fractional powers
of any base. For, if B and y are given, as in (2), x may be
found from (3) by means of a table of common logarithms.
Similarly, if we take the logarithms of both sides of (2) with
respect to any other base 6, we find the corresponding rela-
tions
(4) ' logs X = logB X ' logs B,
(5)' logsx = logbX -4- logbB.
If we set X = fe in (4)' and (5)', since log^fe = 1, we find
(6) logsfe-logftS = 1 and log^fe = 1 -^ logjB.
EXERCISES
1. Find the value of 10* when x = 3; 0; 1.6; 2.7; - 1; - 1.9; 0.43.
2. Plot the curve y = 10* carefully, using several fractional values
of x.
3. Plot the curve y = logio x by direct comparison with the figure of
Ex. 2. Plot it again by use of a table of logarithms.
Plot the graph of each of the following functions.
4. logioxS. 6. logio-^. 6. logio (l/a;2). 7. logjoa:*/^
Do any relations exist between these graphs?
Plot the graph of each of the following functions and explain its
relation to graphs already drawn above.
8. logio (1 + x)2. 9. logio yll+x, 10. logic (x ^1 + x).
Plot the graphs of each of the following functions and show the
relations between them.
11. logjx. 12. logjx. 13. logja;2. 14. 3*.
Show how to calculate most readily the values of the following ex-
pressions, and find the niunerical value of each one.
16. logii7. 17. V(5.4)6-2. 19. 100-5 H- 10-0-5. 21. logs 100.
16. 24-53. 18. logieS. 20. 21og»5. 22. 10 logio 9.
102 THE CALCULUS [VIII, § 61
61. Napierian Logarithms. Base e. A careful examination
of Fig. 25 will convince anyone that there must be a value of
B for which the curve y = logsx has a slope equal to 1 at
X = 1. Indeed, the equation (5), §60, shows that the
slope of the curve y = logsX can be found by dividing the
slope of the curve y = logiox by logioB. Hence if logio-B is
equal to the slope of the curve y = logiox at the point (1, 0),
the slope of the curve y = logsX will be 1 at (1, 0).
Let this value of B be denoted by the letter e. Logarithms
to the base e are called Napierian logarithms,* or natural
logarithmsy or hyperbolic logarithms, (See Table V, C.)
62. Differentiation of log« x. To find the derivative of log«a:,
* let us write
"^ y=log.x p^,^-^^" ^^^ ^ = ^^^^'^•
^ (2) y + Ay=-loge{x + Ax).
logiuh)
X Hence
Ay = loge (X + Ax)- logeX
-.«g.(i+f),
..d |.i,^(i+f).i.i,<^(i+f).
Now let u = Ax/Xf so that we may write
A^ ^ 1 loge (1 + ^)
Ax X u ^
But the fraction {loge(l + u)}/u is simply the slope of
the secant AP of the curve log«t^, and as Ax— ^0, so also
* Named for Lord Napier, the inventor of logarithms. The value of c is
stated below. No assumption is made at this point except that the logarithm
curve has a tangent at (1,0).
VIII, § 63] LOGARITHMS 103
w— >0; hence the secant AP becomes the tangent at A and
its slope has the limit 1, by the definition of e. Hence
Therefore lim ^ = i • lim ^-^^i^l+i!^ = 1 • 1,
or
[vm] ^^ = ^.
On account of the simplicity of this formula the base e will be
used henceforth in this book for all logarithms and exponentials
unless the contrary is explicitly stated.
63. Differentiation of log^x. Since we have, by (4)', § 60,
(1) y = logB X = loge X . logfi e,
the derivative of log^x is found by multiplying the derivative
of log«a; by log^e:
dx X
In particular, for common logarithms, since B = 10, we have
(2) -^^ = --log.oe.
The constant factor logio e is the value of the slope oi y =
logioa: at (1, 0). It is called the modtdus of the system of
common logarithms, and is denoted by the letter Af , that is
logio e = M. Hence the preceding equation becomes
[Vnib] dlogiox^M,
dx X
By means of formula VII, § 22, for change of variable,
the formula VIII becomes
,«s dlogeU _ 1 ^dw
dx u dx
The formulas Villa and Vlllb may be rewritten in a similar
manner.
104 THE CALCULUS [VIII, § 64
64. Values of M and of e. To compute approximately the
value of M, that is the slope of the curve y = logiox at (1, 0),
let us draw the secant connecting the points P (1, 0) and
Q (1 + Ax, + Ay) on that curve^ Let us denote the slope
of this secant PQ by Mpq, Then
Ay logio (1 + Ax)
"^^^^Ai^ Ai
If we choose for Ax a succession of smaller and smaller
values,
Ax = 0.1, 0.01, 0.001, ...,
we find a corresponding succession of values of rripQi
mpQ = 0.414, 0.432, 0.434, ....
For the last of these values, a six or seven place logarithm
table is required, while still higher place tables would be
required to get a more accurate answer. Since the slope at
(1, 0) is the limit of ttipq as Ax approaches zero, we have
M = lim mpQ = 0.434 ... (approximately).*
From this value of M, we can compute e, since
logioe = M = 0.434 ...
Hence, from a table of common logarithms,
e = 2.72 ... (approximately).
65. Illustrative Examples.
Example 1. Given y = logi© (2 x^ + 3), to find dy/dx.
Method 1. Derivative notation. Set u = 2 x^ + Sj then
dy _dy du _ d logip u d(2x^ -\- 3) _M . _ 4: Mx
dx" du dx du dx u "~2a;2-j-3
* An independent method of calculating the values of M and of e will
be given in §§ 147, 153. Logically, we might have waited until that time to
state the value of M, but it is much more convenient, practically, to have
an approximate value at once. To ten decimal places, the values are
j»f = 0.4342944819, l/ifcf =2.3025850930, 6 = 2.7182818285.
VIII, § 65] LOGARITHMS 105
Method 2. Differential notation.
dy = dlog,o(2x2 + 3) = 2^p^d(2x2 +3) = 2^%^^
Example 2. Find the area under the curve y = 1/x from a; = 1 to
X =10, using formula [VIII] inversely:
= / -dx=logex\ = logc 10 = ,—^ = j^ = 2.3026 *
x-i J x~\ X ^ J-p.i '^ logio e M
Example 3. If the rate of increase dy/dx of a quantity y with
respect to a; is 1/x, find y in terms of x.
Since dy/dr = l/x,
2/ = /-^ = logcx+c,
where c is a constant, — the value of y when x = 1. It should be noted
that logarithms to the base e occur here in a perfectly natural manner;
the same remark applies in Example 2. Note that
logc X = logioX -r M.
This case arises constantly in science. Thus, if a volume v of gas ex-
pands by an amount AVj and if the work done in the expansion is ATT,
the ratio aW/Av is approximately the pressure of the gas; and dW/dv
T= p exactly. If the temperature remains constant pv — & constant;
hence dW/dv = k/v. The general expression for W is therefore
W — J - dv = k logc V -\-Cf
and the work done in expanding from one volume vi to another volume
V2 is
TtT"*" = r""- dv^k \o^ vT = A; loge '^ = ^ logic ^•
♦The number log« 10 =1 4- Af = 2.302585 is important because common
logarithms (base 10) are reduced to natural logarithms (base e) by multi-
plsring by this number, since log. iV"= logic ^Xlog* 10. Similarly, natural
logarithms are reduced to common logarithms by multiplying by Af =
log X e; since log i o iV = ifcf • log« iV". It is easy to remember which of these two
multipliers should be used in transferring from one of these bases to the
other by remembering that logarithms of numbers above 1 are surely
greater when e is used as base than when 10 is used.
106 THE CALCULUS [VIII, § 65
EXERCISES
Calculate the derivative of each of the following functions; when
possible, simplify the given expression first.
1. logioa;3. 2. logio >/J. 3. logw (1+ 2 a;).
4. logio (1 + a^). 5. loge (1 + x)\ 6. loge V3 +-5x.
I
7. loge (l/x), 8. logio (x"^), 9. X loge x^,
'^' '^^ (m) • ''' ^^«^« {' - r^^) • ''• ^^^ vi=^-
13. ^^^ 14. logejlogex}. 15. (logcO*.
Evaluate each of the following integrals..
16. /"^ -dx. 17. /"* '^ — ^dx. 18. f^
Jl X Jz x Js
X3
^^l-\-x^^ «-, /•^^(2-O^j, «- r^Qt^-2t^-l
dx.
dt.
19. /•'" L±£!dx. 20. /""" ^^^<«. 21. r ^J^^
^10 a; •/lo 3^4 ^1 3/3
26. Calculate the area between the hyperbola xy == 1 and the x-axis,
from X = 1 to 10, 10 to 100, 100 to 1000; from x = 1 to a; = A;.
26. Show that the slope of the curve y = logio x is a constant times
the slope of the curve y = loge x. Determine this constant factor.
27. Find the flexion of the curve y = loge x, and show that there are
no points of inflexion on the curve.
28. Find the maxima and minima of the curve
y = logc(a;2-2« + 3).
Find the maxima and minima and the points of inflexion (if anj'
exist), on each of the following curves:
29. y = x- loge X. 30. 2/ = X - logc (1 + x^).
31. y = x2 - 4 loge x2. 32. t/ = (2 x + log x)2.
Find the areas under each of the following curves between x = 2
and X = 5:
33. y=x + l/x. 34. y = (x^ + l)/x3. 35. y = (xi/2 - x)/x2.
VIII, § 66] LOGARITHMS 107
36. Find the volume of the solid of revolution formed by revolving
that portion of the curve xy^ = 1 between a? = 1 and a? = 3 about the
X-axis.
37. If a body moves so that its speed v = < + 1/^, calculate the dis-
tance passed over between the times ^ = 2 and 2 = 4.
38. Find the work done in compressing 10 cu. ft. of a gas to 5 cu. ft.,
if po = .004.
39. Find the areas under the hyperbola xy = A* between a; = 1 and
X — CyC and c^, c^ and c^, & and c*.
66. Differentiation of Exponential Functions. Let us con-
sider first the function
(1) V = B\
Taking the logarithms of both sides of this equation with
respect to the base e,
(2) loge 2/ = X • loge B.
Differentiating both sides with respect to x, we have, by VIII,
Hence we have the formula
[IXa] ^^^B^log^B.
For the special cases B = e and B = 10, we have
[IX] — = e',
[1X6] ^ = \{^ loge 10 = ^ = 10'(2.302585 . . .).
If u denotes a function of x, we may combine any of these
formulas with formula VII, § 22; thus the formula IX,
which we shall use most often, becomes
,M\ de^ du
108" THE CALCULUS [VHI, § 67
67. Illustrative Examples.
Example 1. Given y = e^,to find dy/dx.
Method 1. Set x^ = u; then
dy dy du de^ d(a;2) « o ^
ax du dx du dx
Method 2. dy^d^^^ d{x^) = 2 xe*'(to.
Example 2. Find the slope of the curve
(1) 2^ 2~'
and determine its extreme values.
Since de~^/dx = — e-*, we have
(2) ^ = ^-^^
To determine the extreme values, first set dy/dx = 0:
— 2 — =0> or e' =e * = -.
Clearing of fractions,
e2* = 1, whence x = 0.
To determine whether y is really a maximum or a minimum at a; = 0,
we find
. d^y _ e^+e~^ .
hence d^/dx^ = 1 when x = 0. Consequently y is a minimum
(§ 42, p. 66) at a; = 0.
The curve (1) is called a catenary. This curve is very important
because it is the form taken by a perfect inelastic cord hung between
two points. The given function is often called the hyperbolic cosine
of Xf and is denoted by cosh x. The expression (e* — e-*)/2 in (2) is
called the hyperbolic sine of Xj and is denoted by the symbol sinh x:
U) ^Jihx = ^'^J~' , coshx = ?^!^t^.
^^ 2 2
The equations (2) and (3) show that
^^v cf cosh X . 1. dsinhx *
(5) « woix^ ^ gjjj^ ^^ *'°*~^ = cosh X.
^ ^ dx dx
VIII, § 67] LOGARITHMS 109
Example 3. If a quantity y has a rate of change dy/dz with respect
to X proportional to y itself, to find y in terms of x. Given
dx ^'
we may write
,dx 1
dy y
hence •
^ = fr.dy = *^8e y+Cf
•/ y
by § 65, Ex. 3. Transposing c, we have
loge y — kx — Cj or ^ = e**^ = e^e** = Ce**,
where C( = e^)is again an arbitrary constant.
TAe only quantity y whose rate of change is proportioned to itself is
Cc** where C and k are arbitrary , and k is the factor of proportumality.
This principle is of the greatest importance in science; a detailed dis-
cussion of concrete cases is taken up in § 68.
£X£RCIS£S
Find the derivative of each of the following functions:
1. c3x. 2. e^^+^\ 3. e^^^i+5. 4. c><«^.
5. a;2e*. 6. (l-x)^€^. 7. 103^^+4. 8. a^^+^K
9. loge*. 10. log (l+e*). 11. loge"**. 12.. {loge^^)*
13. (e- + l)2. 14. e:!L±±^. 16. ^^^"^"^. 16. ''-''\
2 2 e*+er*
17. Show that the slope of the curve y — e^ ia equal to its ordinate.
18. Show that the area under the curve ^ = e* between the ^-axis
and any value of a; is y — 1.
19. Find the area under the catenary from a; = to a? = 3; from
ir = — ltoa; = +l; from a; = to a; = a. [See Tables, V, C.]
20. Find the area under the curve y = sinh x from a; = to a; = 3;
from X = tox — a.
Find the maxima and minima and the points of inflexion (if any
exist) on each of the following curves:
21. y = x€^. 22. y = x^e^. 23. y = sinh x.
24. y = e-^. 26. y = xer^, 26. y = sech a; = 1 -^ cosh x.
110 THE CALCIJLUS [VIII, § 67
27. Show that the pair of parameter equations x — cosh t^y = sinh t
represent the rectangular hyperbola x^ — y2 = i. Hence show that the
area under the h3rperbola x^ — y^ = 1 from a; = ltoa; = ais represented
(see (9) § 55) by the integral
f '\mh^ tdt-- f f(cosh 2 t - 1)/21 dt,
*^ t-Q •' < =
whepe cosh c =^a. Hence show that this area is (sinh 2 c)/4 — c/2.
e^(te. 81. y sinh2a;<ia;. 34. J (e*H-l)2(te.
29. J e-^dx. 32. J* coahSxdx. 36. ^ (e*+3)r«dx.
30. ye2* da;. 33. f sinh2 x (te. 86. f (e^+^ + 1) dx.
68. Compound Interest Law. The fact proved in the Ex. 3
of § 67 is of great importance in science:
If a variable quantity y has a rate of increase
with respect to an independent variable x proportional to y
itself, then
(2) y = Ce^,
where C is an arbitrary constant.
The equation (2) between two variables x and y was
called by Lord Kelvin the ''Compound Interest Law"
on account of its crude analogy to compound interest on
money. For the larger the amount y (of principal and in-
terest) grows the faster the interest accumulates.
In science instances of a rate of growth which grows as the
total grows are frequent.
Example 1. Work in Expanding Gas, The example used to illus-
trate Ex. 3, § 67, can be put in this form. Since, in the work W done in
the expansion at constant temperature of a gas of volume v, we found
dW/dv = k/v, it follows that dv/dW = v/k; hence v = ile^/*, which
agrees with the result of § 67.
VIII, § 68] LOGARITHMS 111
Example 2. Cooling in a Moving Fluid, If a heated object is
cooled in running water or moving air, and if ^ is the varying difference,
in temperature between the heated object and the fluid, the rate of
change of (per second) is assumed to be proportional to 6:
de ,^
di^-^'^
where t is the time and where the negative sign indicates that is de-
creasing. It follows that ^ = C-e^**. [Newton's Law of Cooling.]
Such an equation may also be thrown in the form of § 67; in this
example, dt/dd = ^ l/(kd)y whence t = — {l/k)'\ogeO + c, and the
time taken to cool from one temperature ^i to another temperature O2 is
where $ is the temperature of the body above the temperature of the
surrounding fluid.
The law for the dying out of an electric current in a conductor when
the power is cut off is very similar to the law for cooling in this example.
See Ex. 19, p. 114.
Example 3. 'Bacterial Growth. If bacteria grow freely in the pres-
ence of unhmited food, the increase per second in the number in a cubic
inch of culture is proportional to the number present. Hence
^^kN,N^ Ce^, t = JlogeiV + c,
where N is the number of thousands per cubic inch, t is the time, and k
is the rate of increase shown by a colony of one thousand per cubic inch.
The time consumed in increase from one number Ni to another number
N2 is
JNi *^ Nik N k ^ Jati k ** Ni
If N2 = 10 ATi, the time consumed is (I/A;) log^ 10 = l/{kM). This
fact is used to determine ky since the time consumed in increasing N ten-
fold can be measured (approximately). If this time is T, then T =
l/(kM)y whence k = 1/{TM), where T is known and M = 0.43 (nearly).
Nmnerous instances similar to this occur in vegetable growth and in
organic chemistry. For this reason the equation (2) on p. 110 is often
callecf the ''law of organic growth, ' ' (See Exs. 20, 21, p. 114.)
{
112 THE CALCULUS [VIII, § 68
Example 4. Atmospheric Pressure. The air pressure near the sur-
face of the earth is due to the weight of the air above. The pressure at
the bottom of 1 eu. ft. of air exceeds that at the top by the weight of
that cubic foot of air. If we assume the temperature constant, the
volume of a given amount is inversely proportional to the pressure,
hence the amount of air in 1 cu. ft. is directly proportional to the pres-
sure, and therefore the weight of 1 cu. ft. is proportional to the pressure.
It follows that the rate of decrease of the pressure as we leave the earth 's
surface is proportional to the pressure itself.
^ = - fcp, p = Cer^y A = - ^loge p + c,
where h is the height above the earth, and, as in Exs. 2 and 3, the dif-
ference in the height which would change the pressure from pi to ps is
Since A]^, and pt and pi can be found by experiment, k is determined
by the last equation.
69. Percentage Rate of Increase. The principle stated in
§68 may be restated as follows: In the case of bacterial
growth, for example, while the total rate of increase is clearly-
proportional to the total nimiber in thousands to the cubic
inch of bacteria, the percentage rate of increase is constant.
In any case the percentage rate of increasej rp, is obtained
by dividing 100 times the total rate of increase by the total
amount of the quantity y 100 • {dy/dx) -r- y; and since the equa-
tion dy/dx = ky gives (dy/dx) -i- y = ky it is clear that the
percentage rate of increase in any of these problems is a constant.
The quotient (dy/dx) -r- y, that is, 1/100 of the percentage
rate of increase, will be called the relative rate of increasey and
will be denoted by rr.
In some of the exercises which follow, the statements are
phrased in terms of percentage rate of increase, r^, or the relar-
live rate of increase, r, = r^, -5- 100.
VIII, §69] LOGARITHMS 113
£X£RCIS£S
Find dyldx and {dy/dx) -5- y for each of the following functions:
1. 7 63*. 4. ^\ 7. ^ax + h)^.
2. 4er2.to 5. e4«+5. 8. {x^ + pz-\- q)^,
3. ice*. 6. (a;2 + 2)e«. 9. (3x + 2)e-*'.
10. If a body cools in moving air, according to Newton's law, dd/dt
= — kdj where t is the time (in seconds) and 6 is the difference in tem-
peratm-e between the body and the air, find A; if ^ falls from 40® C. to
30** C. in 200 seconds.
11. How soon will the difference in temperature in Ex. 10 fall to
10® C?
12. In measuring atmospheric pressure, it is usual to express the pres-
sure in millimeters (or in inches) of mercury in a barometer. Find C in
the formula of Ex. 4, § 68? if p == 762 mm. when h = (sea level). Find
C if p = 30 in. when A = 0.
13. Using the value of C found in Ex. 12, find k in the formula for at-
mospheric pressure if p = 24 in. when h = 5830 ft.; if p = 600 mm.
when h = 1909 m. Hence find the barometric reading at a height of
3000 ft. ; 1000 m. Find the height if the barometer reads 28 in. ; 650 mm.
[Note. Pressure in pounds per square inch = 0.4908 X barometer
reading in inches.]
14. If a rotating wheel is stopped by water friction, the rate of de-
crease of angular speed, dos/dt, is proportional to the speed. Find co in
terms of the time, and find the factor of proportionality if the speed of
the wheel diminishes 50% in one minute.
15. If a wheel stopped by water friction has its speed reduced at a
constant rate of 2% (in revolutions per second and seconds), how long
will it take to lose 50% of the speed?
16. The length Z of a rod when heated expands at a constant rate per
cent ( = 100 k). Show that dl/dd = kly where is the temperature; if the
percentage rate of increase is .001% (in feet and degrees C), how much
longer will it be when heated 200° C? At what temperature will the rod
be 1% longer than it was originally?
[Note. This value of A; is about correct for cast iron.]
114 THE CALCULUS [VIII, § 69
17. The coefficient of expansion of a metal rod is the increase in
length per degree rise in temperature of a rod of unit length. Show that
the coefficient of expansion of any rod is the relative rate of increase in
length with respect to the temperature.
18. When a belt passes around a pulley, if T is the tension (in
pounds) at a distance s (in feet) from the point where the belt leaves the
pulley, r the radius of the pulley, and /i the coefficient of friction, then
dt/ds = /iT/r, Express T in terms of s. If T ^ 30 lb. when a = 0,
what is T when « = 5 ft., if r = 7 ft., and /j. = 0.3?
19. When an electric circuit is cut off, the rate of decrease of the cur-
rent is proportional to the current C Show that C = Coer^t where Co
is the value of C when t — 0.
(Note. The assumption made is that the electric pressure, or electro-
motive force, suddenly becomes zero, the circuit remaining unbroken.
This is approximately realized in one portion of a circuit which is
short-circuited. The effect is due to self-induction: k = R/L, where R
is the resistance and L the self-induction of Uie circuit.]
20. Radium automatically decomposes at a constant (relative) rate.
Show that the quantity remaining after a time t is q = qoe'^, where
^0 is the original quantity. Find k from the fact that half the original
quantity disappears in 1800 yrs. How much disappears in 100 yrs.?
in one year?
21. Many other chemical reactions — for example, the formation of
invert sugar from sugar — proceed approximately in a manner similar
to that described in Ex. 20. Show that the quantity which remains is
q = qoer^ and that the amount transformed is A = qo — q — qQ(l ^ e"**).
Show that the quantities which remain after a series of equal intervals
of time are in geometric progression.
22. The amount of light which passes through a given thickness of
glass, or other absorbing material, is found from the fact that a fixed per
cent of the total is absorbed by any absorbing material. Express the
amount which will pass through a given thickness of glass.
70. Logarithmic Differentiation. Relative Increase. In § 69
we defined the relative rate of increase rr of a quantity y with
respect to x as the total rate of increase {dy/dx) divided by y.
If y is given as a function of x,
(1) y=f(x),
VIII, §70] LOGARITHMS 115
the relative rate of increase
dy
can be obtained by taking the logarithms of both sides
of (1),*
(2) \0gey = \0gef{x),
and then differentiating both sides with respect to x:
(3) y _1 <^y ^ dlogey __ dlogef(x)
^ ^ y' dx dx dx '
This process is often called logarithmic differentiation: the
logarithmic derivative of a function is its relative rate of in-
crease, rr, or 1/100 of its percentage rate of increase.
Example 1. Given y = Ce**, to find n = (dy/dx) -^ y. Taking log-
arithms on both sides:
loge y = loge C + kx;
differentiating both sides with respect to x,
^' dx ' ^ dx ~^'
The result of Ex. 3, § 68, may be restated as follows: the
only fmiction of x whose relative rate of change (logarithmic
derivative) is constant is Ce**.
Example 2. Given y = x^+3x + 2,to find rr.
Method2. rr-^^v ^ ^ x2+3x+2 '
* Since log N is defined only for positive values of N, aU that follows holds
only for positive values of the quantities whose logarithms are v^ed.
t Here and hereafter the symbol log will be used to mean a logarithm to
the base e.
116 THE CALCULUS [YIII, § 71
71. Logarifhmic Methods. The process of logarithmic dif-
ferentiation is often used apart from its meaning as a relative
rate, simply as a device for obtaining the usual derivative.
We shall first apply this method to prove the rule for dif-
ferentiating any constant power of a variable. The equation
gives
logy = n log X.
Differentiating with respect to x, we have
Idy 1
--f- = n •-
yax X
or
dy y x^
-f- = n '-^n- —
ax X X
In this proof, n may be any constant whatever. (See §§ 17,
21.)
The logarithmic method is useful also in such examples as
those that follow.
Example 1. Given y = {2x^+ 3)104»-i.
Method 1. Ordinary Differentiation.
I = (2x2 + 3)^(104-1) +104^-1^(2 x2 + 3)
= (2x2 + 3)-4.-^104»-i + 104«-i.4x
= 4 . 104*-i r(2 a;2 + S)/M + xl, where M = logioe = 0.434.
Method 2. Logarithmic Method.
Since log y = log (2 a;2 + 3) + (4 x — 1) log 10, we have
I dy 4x , ^ 1 in
VIII, §71] LOGARITHMS 117
or ^ = y[2^^ + 41ogl0] = 4.1()4i^-i[x + (2x2 + 3)logl0],
which agrees with the preceding result, since loge 10 =.l/logioe = 1/Af.
Example 2. Given y = {3 x^ -\- 1)^+^^ to find dy/dx. Since no rule
has been given for a variable to a variable power, ordinary differentiation
cannot be used advantageously. Taking logarithms, however, we find
logy = (2 a: + 4) log (3 a;2 + 1),
whence i • ^ = 21og (3x2 + D + -f^TT (2 a; + 4),
y ax o a;^ -f- 1
or
g=(3a:2 + l)2x+4J21og(3a:2 + l)+^j^(2x + 4)[.
The use of the logarithmic method is the only expeditious way to find
the derivative in this example.
EXERCISES
Find the logarithmic derivatives (relative rates of increase) of each
of the following functions, by each of the two methods of § 71.
1. ^2*. 6. 0.1ei«-5. 9. (r2 + i)e-r».
2. 4e4«. 6. 102aj+3. 10. (2 - 3 <2) eM»-i.
3. e»+2. 7. er^Vtof'. n. (1 - f2 _j- ^) ioi'+^.
4. ff^, 8. 2<2e-7<. 12. e^.
Find the derivative of each of the following functions by the loga-
rithmic method.
13. (1 + aj)i+*. 16. x^^ 17. (1+ x) (1+ 2 x) (1+ 3 x).
i4. («2 + 1)2*-H3. 16. if^, 18. \^rT^ ^ vTT^^.
19. If y = uvj show that dy -i- y — du ■¥ u -^ dv -^ v. In general
show that the relative rate of increase of a product is the sum of the
relative rates of increase of the factors.
20. If a rectangular sheet of metal is heated, show that the relative
rate of increase in its area is twice the coefficient of expansion of the
material [see Ex. 17, List XXXI].
21. Extend the rule of Ex. 19 to the case of any number of factors.
Apply this to the expansion of a heated block of metal.
22. Show directly, and also by use of Ex. 21, that the relative rate of
increase of x** with respect to x, where n is an integer, is n/x.
118 THE CALCULUS [VIII, § 71
23. Compare the functions e^ and e2*+3; compare their relative
rates of increase; compare their derivatives; compare their second
derivatives.
Compare the following pairs of functions, their logarithmic deriv-
atives, their ordinary derivatives, and their second derivatives.
24. ^ and 1(F. 27. e^ and e+<«.
25. ^ and eo*+*. 28. ff-^ and sech x,
26. ^ and 10**. 29. c-*' and 1 -^ (a + 6x2).
30. Can k be found so that hff^ and 10** coincide? Prove this by-
comparing their logarithmic derivatives, and find 6 in terms of a.
31. If the logarithmic derivative {dy/dx) -f- y is equal to 3 + 4 x,
show that log y = 3 x + 2x2 -f const., or y = A;e3a;+2x\
32. If (d2//(ir) ^y=f{x) show that y = a/^^"^^ *^-
Find y if the logarithmic derivative has any one of the following
values:
33. 1 — x. 36. n/x, 37. e*.
34. ax + bxK 86. a + n/x. 88. e* + n/x.
CHAPTER IX
TRIGONOMETRIC FUNCTIONS
■
72. Limit of (sdn B)IB as ^ approaches Zero. To find the de-
rivatives of sin X and cos x, we shall make use of the limit
sin B
lim
B
Let B be the angle A05, Fig. 27, and let us draw a circle
about as center with a radius r = OA,
cutting OB at P. Draw PP' and fi5'
perpendicular to OA and draw OP' B'.
Then
(1) PP'< arc Pi4P'<BB',
or
(2) 2 r sin ^ < 2 r • ^ < 2 r tan ^, Fia. 27.
since arc PAP' = 2 r • ^ if ^ is measured in circular measure.
Dividing by 2 r sin 6, we have
e ^ I
(3)
1<
sin B cos 6
But cos 6 approaches 1 as ^ approaches zero. Hence
^/sin 6 must also approach 1. It follows that
(4)
,. sin e ^
hm -— = 1,
provided, as above, that 6 is measured in circular measure.*
*0n account of the simplicity of this formula and those that result
from it, we shall assmne henceforth that all angles are measured m circular
measure.
119
120 THE CALCULUS [IX, § 73
73. Derivatives of sdn x and cos x. Given the equation
(1) y = sin Xy
we proceed to find dy/dx by the fundamental process of § 17.
We have, using the notation of § 17,
(A) 2/ + Ay = sin {x + Ax)
(B) Ay = sin {x + Ax) — sin a;
^ f , Ar\ . Ax
= 2 cosyx + y^ sin y ,
by formula 13, Tables, II, G. Dividing both sides by Ax,
(C) g — G + f)
. Ax
sm-
Ax '
2
.- Ax
or, if we put ^ — -K- i
Ay r X n\ sin B
- = cos(x + ^)-^.
Hence, by § 72,
(D) -^ = lim -^ = cos X,
ax tf-+o Ax
and we have the formula
[X] — -2 = COS X.
ax
Similarly, starting with y = cos x, we obtain the formula
r«rTi d COS X
[XI] — 5 — = — sm X.
dx
By means of formula VII, § 22, these formulas may be
rewritten in the form
.^v dsint^ dsint^ du du
.^x d cos u __ dcost/ ^ _ _ • ^
dx du dx dx
IX, § 74] TRIGONOMETRIC FUNCTIONS ' 121
Example 1. Differentiate y = sin vT+^.
dy dsinVl-f-a;2 r^— — grfVl+x2 x ^/3— : — 5
-—■ = 3 = cos V 1 -h x2 = —. cos V 1 4- x2.
dz dx dx Vl -\-x^
Example 2. Differentiate y = cos e^.
dw dcose** . ^jde^' o ^ • ♦«
3^ = — 1 = — sin e*' -J— - — 2x^ sin e**.
dx dx dx
74. Derivatives of tan x, ctn x, sec x, esc x. Given y = tan x,
we may write
jSina: dsinx . dcosa:
J , ^ a cos X — ; sin x — 7 — ^
ot/ a tan a: cosx dx dx 1
dx dx dx cos^a; cos^a:
[^^] ^^ =7::^ai: = sec'x.
Similarly,
(fx cos'x
jCOSX
a —
nrTTTi rfctnx sma: 1 ,^
rrrr^n d SeC X COS X Sm X
[XIV] — := — = — • — = — s— = secx tanx.
ax dx cos^ X
r^ri d CSC X sm X — cos X .
[XV] ^ = ; = — r-s = - CSC X Ctn X.
dx dx 8w?x
These may be combined with formula VII, § 22, as in § 73.
Example 1. Differentiate y = cos^a;.
Setting u = cos x, we have y = u^, and
^2/ rfw3 ^ ^du _ 5 d cos x ^ „
j^=-3-=3w2— =3 cos* x — J — = — 3 cos2 x sin x,
ax dx ax ax
Example 2. Differentiate y = cos^ (2 x^ + 1).
Setting u = cos (2 a;2 + 1), we have y = w', and
:^=-^ ^^^ :iZ =3C0S2(2X2 + I).;;r-.
dx dx dv dx
122 THE CALCULUS [IX, § 74
But
dx- di -sin (2x2 -hi) _
= - 4a;sin(2a;2 + l).
Hence
^ = [3 cos2 (2 a;2 + 1)] [- 4 x sin (2 x2 + 1)]
= - 12x cos2 (2x2 -h 1) sin (2x2 + 1).
EXERCISES
Find the derivatives of : , -
1. sin 4 X. 5. sin x^. 9. x coe x.
2. cos {e/Z), 6. tan (3 — 20. 10. e» ctn $,
3. tan (— 2 ^). 7. cos (— 3 e). 11. log sec x.
4. sin2x. 8. sec(x/2). 12. cose~*.
13. sin X — 4 cos 2 X. 17. c* cos2 (3 « — 1).
14. c'sin (ir/10 - 2 0- 18. ei+2< gin (3 t - ir/4).
16. (1 -|-x2)sin(3-2x). 19. e'/io (cos < - 4 sin 3 0.
16. logsec6s*»»*. 20. cos log tan f*.
21. Find the area under the curve y = sin x from x = to x = 7r/2;
test the correctness of your result by rough comparison with the cir-
cumscribed rectangle.
22. Find the area bounded by the two axes and the curve y = cos x,
in the first quadrant.
Find the maxima and minima, and the points of inflexion (if any
exist) on each of the following curves.
23. y = sin x. 26. y — x cos x. 29. y = e~* sin x.
24. 2/ = cos X. 27. y = 1 — sin 2 x. 30. y = e"2« sin x.
25. y = tan x. 28. y = sin x + cos x. 31. j/ = cos (2 x + ir/6).
Find the derivative of each of the following pairs of functions, and
draw conclusions concerning the functions.
32. cos X and sin (ir/2 — x). 35. sin 2 x and 2 sin x cos x.
33. cos2 X and 1 — sin2x. 36. cos 2 x and — 2 sin2 x,
94, s^c ;^ and gee (— x). 37. tan2 x and sec2 ;j;^
IX, § 74] TRIGONOMETRIC FUNCTIONS 123
Integrate the following expressions; in case the limits are stated,
evaluate the integrals, and represent them graphically as areas.
38. / sin X dx, 40. / sec^ x dx. 42. / cos (3 t + 7r/6) dt.
/-Hr/2 ^ ^
cos x dx, 41. / sin 2 a; dx. 43. / tan ^ sec ^ dU
ir/2 •/ J
44. f (1 4- sin a;) dx.
Jo
45. J(co8 X 4- 3 sin 2 a;) dx.
46. /cos^x da;.
. Hint. 2 cos^ a; = 1 + cos 2 a;.
47. y(cos 2 X - 1) dx. 48. /*' sin^ x dx.
49. Find the derivative of sin x by showing that
sin (x + Ax) — sin X = sin x (cos Ax — 1) + cos x • sin Ax
and remarking that, as Ax-»0,
lim[(cos Ax — 1) -5- Ax] = and Iim[(sin Ax) -5- Ax] = 1.
60. Find the derivative of cos x as in Ex. 49.
61. Find the derivatives of the two functions
(a) vers x = 1 — cos x. (6) exsec x = sec x — 1.
62. Differentiate some of the answers in the list of formulas, Tablesy
IV, Eo, Eft. What should the result of your differentiation be?
[The teacher will indicate which formulas should be thus tested.]
Find the speed of a moving particle whose motion is given in terms
of the time t by one of the pairs of parameter equations which follow;
and find the path in each case.
*• <a; = 3 cos 2^. __ (x = sm<
) 2/ = 3 sm 2 ^. (y = ami.
-. Jx = 2cos4<. _g (x=sec<.
* J y = 3 sin 4 <. (y = tan t
+ cos^
67. A flywheel 5 ft. in diameter makes 1 revolution per second.
Find the horizontal and the vertical speed of a point on its rim 1 ft.
above the center.
68. A point on the rim of a flywheel of radius 5 ft. which is 3 ft.
above the center has a horizontal speed of 20 ft. per second. Find the
angular speed, and the total linear speed of a point on the rim.
124
THE CALCULUS
[IX, § 75
76. Simple Harmonic Motion. If a point M moves with
constant speed in a circular path, the projection P of that
point on any straight line is said to be
in simple harmonic motion.
Let the circle have a radius a; let
the constant speed be v] and let the
straight line be taken as the x-axis.
We may suppose the center of the
circle lies on the straight line, since
the projection of the moving point on
either of two parallel straight lines
Let the center of the circle be the
Fig. 28.
has the same motion,
origin. Then we have
(1) X = OP = a cos ^, or x = a cos (s/^),
where s = arc AM, since 6 = s/a. Moreover, since the
speed V is constant, v = s/Tyii T is the time since M was at
A; or V = s/(t — U) if t is measured from any instant what-
ever, and to is the value of t when Af is at A. We have
therefore
(2)
X = a cos - = a cos - (i — fo) = a cos [W + c];
where k = v/a, and € = — fcfo = — vto/a.
From (2), the speed dx/dt of P along BA is
(3)
dx d[a co8(kt + e)] i * m t \
and the acceleration of P is
(4)
or,
(5)
3t
dfi
= — dk^ cos(kt + «) = — k^'Xf
cPx
jT-i-X — ^-i-X « ;
IX, § 76] TRIGONOMETRIC FUNCTIONS 125
thcU is, the acceleration of x divided by x, is a negative constant,
— fc2. We shall see that much of the unportance of sunple
harmonic motion arises from this fact.
It is important to notice that (2) may be written in the
form
x = a cos (kt + €) = a [cos e cos kt — sin € sin kt],
or
(6) x = A sin kt + B cos kt,
where A = — a sin € and S = + a cos e are both constants.
The form (6) may be used to derive (5) directly.
The simplest forms of the equation (6) result when fc = 1
and either A = and 5 = 1, or A = 1 and B = 0:
.^. fx = sin <; if fc = 1, A = 1, fi = 0, i.e. a = 1, € = 3 ir/2.
\x = cost; if fc = 1, A = 0, B = 1, i.e. a = 1, € = 0.
The formulas (2) and (6) are general formulas for simple har-
monic motion; (7) represent two specially simple cases.
76. Vibration. The importance of simple harmonic mo-
tion, based on its property (5) of § 75, is evident in vibrating
bodies, such as vibrating cords or wires, the prongs of a tuning
fork, the atoms of water in a wave, a weight suspended by a
spring.
In all such cases, it is natural to suppose that the force
which tends to restore the vibrating particle to its central
position increases with the distance from that central posi-
tion, and is proportional to that distance. (Compare
Hooke's law in Physics.)
It is a standard law of physics, equivalent to Newton's
second law of motion, that the acceleration of any particle
is proportional to the force acting upon it.
126 THE CALCULUS [IX, § 76
In the case of vibration, therefore, the acceleration, being
proportional to the force, is proportional to the distance,
Xy from the central position; it follows that, in ordinary
vibrations, the relative acceleration is a negative constant, —
negative, because the acceleration is opposite to the positive
direction of motion. For this reason, each particle of a
vibrating body is supposed to have a simple harmonic mo-
tion, unless disturbing causes, such as air friction, enter to
change the result. Neglecting such frictional effects tempo-
rarily, the distance x from the central position is, as in § 75,
X = a cos (kt + e) = A sin kt-\- B cos kt,
where t denotes the time measured from a starting time to
seconds before the particle is at a: = a, and where € = — tjc.
Moreover, from § 75 and also from what precedes,
dx^
dfi
= — k^x.
The quantity a is called the amplitude, 27r/fc is called the
period, and to= — e/k is called the phase, of the vibration.
EXERCISES
Find the speed and the acceleration of a particle whose displace-
ment X has one of the following values: compare the acceleration with
the original expression for the displacement.
1. a; = sin 2 ^ 6. a; = sin 2 ^ -|- 0.15 sin 6 t.
2. X = sin {t/2 — ir/4). 6. x = sin ^ — J sin 3 ^ + i sin 5 ^
3. a; = sin < — } sin 2 ^ 7. a; = a sin {kt + e).
4. X = cos t + icosSt. S. X = A cos kt -^ B sin kt
9. Determine the angular acceleration of a hair spring if it vibrates
according to the law 6 — .2 sin 10 irt; what is the amplitude of one
vibration, the period and the extreme value of the acceleration? ^^
/
IX, § 76] TRIGONOMETRIC FUNCTIONS 127
Show that each of the following functions satisfies an equation of
the form d^u/dl^ + Aj^w ^ or d^u/dP^ — k^u = 0; in each case determine
the value of k.
10. w = 10 sin 2 ^ 16. w = 6 cos (t/S - ir/12).
11. u = 0.7 cos 15 1. 16. M = 12 cos 3 ^ — 5 sin 3 <.
12. w = 3 e^K 17. w = 3 sin 6 / + 4 cos 6 «.
13. w = 20 e-^. 18. w = Ci sin 3 < + C2COS 3 /.
14. w = sin (3 « + ir/3). 19. w = Cie^' + C2e-7'.
20. Show that the function u = A sin kt-\-B cos kt always satisfies
the equation cPu/di^ + k^u = for any values of A and B. Check by
substituting various positive and negative values for ky Ay B.
21. Show that u = A^ + Ber^ always satisfies the equation
d&
k^u = 0.
22. When an electrical condenser discharges through a negligible re-
sistance the current C follows the law d^C/d^ = — a^Cy where a is a con-
stant. Express the current in terms of the time. When a = 1000, what
is the frequency (number of alternations) per second?
23. Any ordinary alternating electric current varies in intensity ac-
cording to the law C = a sin kt; find the maximum current and the
time-rate of change of the current.
24. When a pendulum of length I swings through a small angle 9,
its motion is represented by the equation d^d/dt^ = — gO/l, very nearly,
I being in feet, in radians, t in seconds. Show that
= Ci sin kt -h C2 cos kt,
where k = ^g/L Find Ci and C2 if ^ = a and the angular speed cj
= when t = 0; and find the time required for one full swing.
25. A needle is suspended in a horizontal position by a torsion fila-
ment. When the needle is turned through a small angle from its posi-
tion of equilibrium, the torsional restoring force produces an angular
acceleration nearly proportional to the angular displacement. Neglect-
ing resistances, what will be the nature of the motion?
128 THE CALCULUS [IX, § 77
77. Inverse Trigonometric Functions. The equation
(1) y = sm~^x, •
is equivalent to the equation
(2) sin y = X.
Differentiating each side with respect to x, we find
(3) co8j/f?=l,or^ = ^- = ^i=..
ax ax cos y y/\ _ ^
Hence we have the formula
[XVI] ^^-'^ 1
It is evident that the radical in these expressions should
have the same sign as cos y, i.e, plus when y is in the first or
in the fourth quadrant, minus when y is in the second or in
the third quadrant.
Combining XVI with VII, § 22, we may write
. . s d sin~^ u 1 du
^ dx " Vl — u^dx'
In a similar manner, we find from formulas XI, XII,
XIII, XIV, XV, the formulas
[XVII] ^ ^^^"'^ - 1
(radical + in 1st and 2d quadrants).
[XVIII] —^ — = (all quadrants).
'■^^^ — dx — ^ 1 I ya (^11 quadrants).
[XX]
d sec-^x 1
(radical + in 1st and 3d quadrants).
IX, § 79] TRIGONOMETRIC FUNCTIONS 129
(fcsc-ix -1
[XXX]
[xxn]
(radical + in 1st and 3d quadrants),
tfvers~^x 1
(radical + in 1st or 2d quadrants).
Each of these fonnulas may be combined with VII, § 22,
as in equation (4) above.
78. Illustrative Examples.
Example 1. 1 = . = : ^ = ,
dx Vl — (a;2)2 dx Vl — x*
Example 2.
da; 1 + (e^)2 <ic 1 + e
Example 3. ^^ ^ = 3 (sec-i a;)2 , «
<ic da;
=3(8ec-ia;)2 ^
.2x
a;Va;2 - 1
T,_ . d log (cos"i a;) Id cos~i x
Example 4. — ^-^ ^ = — -^ 3 .
da; cos ^x dx
1 - 1
cos~i X Vl — a;2
79. Integrals of Irrational Functions. By reversal of the
formulas for the derivatives of the inverse trigonometric
functions, XVI-XXII, we obtain the integrals of certain
important irrational functions.
dx . , . ^ . dsin~*x 1_
X'
[XVIji I .- -_ = sm^ X + C, smce — 3 = ,
J V 1 - JC cte VI —
dx ^ ;„ , ri -:>..- d tan-^ a; _ 1
[XVn],- r ^ . ^^a = ta»~* JC + C, since
2>
130 THE CALCULUS [IX, § 79
since -
[XX]/ J ^^l—-^ = sec-^x + C, si]
[xxn],J
(fo a; Vx^ — 1 '
</x , . ^ • d vers"* a: 1
/^ o = vers~i X + C, since — ^ = > ,
V2x-x» ctr V2x-a:2'
where C in each case denotes an arbitrary constant. Since
&in~* X + cos~* X = 7r/2, the student may show that [XVII]
leads to the same result as [XVI].
Example. To find the area under the curve j/ = 1/(1 + a;2) from
the point where x = to the point where a; = 1.
Since A =fydx,we have
= / T-r-^dx = tan-i x\ = -r/4 - = 7r/4.
x=-0 ^x=0 l+iC'* Jx=0
The fact that we are using radian measure far angles appears very
prominently here. Draw the curve (by first drawing y = 1 -|- a;2) on a
large scale on millimeter paper and actually count the gmall squares as a
check on this result.
EXERCISES
Differentiate each of the following functions.
1. sin~i x/^, 6. sin~i Vl — x^, 9. log tan~i x,
2. cos~i (1 — x). 6. a; sin~i x. 10. cos"i {xe^).
3. sin-i (i/a;). 7. tan"! (l/x^). 11. x^ tan'i 2Vi-
4. tan-i (3 a;). 8. c* cos~i x, 12. sec-^aJ^ + !)•
13. csc-i VrT^2. 17. cos-i Vl - x2. 21. (log tan-i a;)3.
14. ctn-i (|-i^y 18. sec-i Gog tan a;). 22. ^^j^-
16. cos-i (e"*"^). 19- e^^~^^. 23. ^"^ ^
a;
-1.
16. tan-i (log e*). 20. 10 ^ \
IX, § 79] TRIGONOMETRIC FUNCTIONS 131
Integrate the following functions; in case limits are staited, evaluate
the integral.
Jo l+a;2 '*'• Ji
eV^-l
dt ^^ r dx
26. f -j^=' 28. f:r^7-2' [Sett* = 2x.]
^1/2 Vl — ^ J l+4a;2
26. f^'r^' 29. r , ^ ♦ [Setti=2a:.]
Integrate after making the change of letters u = 1 — x.
dx ^^ r dx m^ r dx
«»• f^,-n-.W '*• f i + a-x)^ - **• /
Vl - (1 - a;)2 * J IH- (1 - a;)2 'J V2 x - x*
Find the areas between the x-axis and each of the following curves,
between the limits stated.
33. 2/2 = 1-1- a;2y2; a; == to X = 1/2; a; = - 1/2 to x = + 1/2.
34. 2/ + ^^y = l;ic = Otoa; = l;x=Otoa;=a.
36. 2/2 = 1 ^_ 4 x'iy^; a; = to x = 1/4; x = - 1/4 to x = + 1/4.
36. 4 x22/ + 2/ + 1 = 0; X = 1 to X = 2; X = - 1 to X = 4- 1.
37. Show that the derivative of tan-i[(c* — e-*)/2] is 2/(e* +6"*).
[Note. The function tan"i[(e* — e~*)/2], or tan"i (ginh x), is called
the Giidermannian of x and is denoted by j^cf x: gdx = tan'i (sinh x).
It follows from this exercise that d gd x/dx = sech x.]
38. From the fact that d (sinh x) = cosh x c2x, show that the deriv-
ative of the inverse hyperbolic sine (x = sinh"! t^ if t^ = ginh x) is
given by the equation d (sinh"! w) = ± du/ Vl + v^. [See foot of
p. 108.]
39. Show that d cosh"! u =± du/ \/w2 — 1.
40. Show that d tanh"! u = du/{l - w2).
132 THE CALCULUS [IX, § 80
80. Collection of Formulas for Differentiation. For convenience
in reference we shall restate all of the formulas for differen-
tiation, combining each of them with [VII] when it is de-
sirable to do so.
- (fc ^ „ dC'U du
dJU_+V)^dU dv^ dun^ du
dx dx dx dx dx
X*,, dx dx __- duv dv , du
V- -^ ^ — VI. _=„^+„^.
Vii. $^ = ^-^- Vila. ^ = $^^^
dx du dx dx dt dt
,^ dB" y> . n du --- rfe« du
n. _ = 5-logB.^. IXa. ^ = e.^^.
X. _^ = co8«^. XI. -^ = -8m«^.
XIL ^ = sec«ug. Xin. ^ = -csc««g.
XIV. — J — = seciitanii^-- XV. — ^ — = — csciictnu j--
dx dx dx dx
±1 du
XX XXI tfj8^c-'"U ±1
±1 du
~^dx
du
dx
XXII. jT vers * u = . 3-
dx V2u-u*o«
CHAPTER X
APPLICATIONS TO CURVES
LENGTH — CURVATURE
81. Introduction. The formulas obtained in Chapters VIII
and IX make possible many new applications to curves.
We shall treat some of these in this Chapter.
82. Length of an Arc of a Curve. Let s (x) denote the length
of the arc of a given curve y =/(x), from a fixed point F
to a variable point P.
When X increases by ^
an amount Ax, let
As = arc PQ be the
corresponding increase
in s, and let Ac = chord
PQ. Then
(1) Ac^ = Ax^ + A^^
whence
and
(3)
x^x^Lx
As
Ax
'^hM-t
We now require the following fundamental axiom, which
forms the basis of the mensuration of curved lines.
As the chord and its arc approach zero, their ratio approadies
1, i.e.
(4)
133
lim T- = 1.
Ac-*oAc
134 THE CALCULUS [X, § 82
Combining (3) and (4), and passing to the limit as Ax
approaches zero, we have
where m = dy/dx is the slope of the curve.
It follows that the total change in s between any two
fixed points x = a and x = 6, is
(6) Total length = si ' ° * = /" ° Vl + m* dx.
83. Parameter Forms. When the equation of a curve is
given in parameter form
(1) x=f(t), y = <t>(t),
we may square both sides of (5), § 82, and multiply by dx^.
This gives the formula
(2) c/s« = c/x» + dy*,
which is called the Pythagorean differential formula. It is
readily remembered by reference to the triangle PQR,
Fig. 29. If we divide both sides of (2) by dt^, we find *
») ©'- ey+ ©'
From (3) we have
whence
whicn gives the length of the curve (1) between any two of
its points.
* This expresses the fact that the square of the total speed ds/dt is the
sum of the squares of the horizontal speed dx/dt and the vertical speed dy/dt.
This fact, proved in § 40, might have been used as the point of departure,
and all of the formula? of %\ 82-83 might have been deduced from it.
X, § 84] APPLICATIONS TO CURVES 13 J
84. Illustrative Examples. While the square root which
occurs in the formulas of §§ 82-83 renders the integrations
rather difficult in general, the work is quite easy in some
examples, as illustrated below.
Example 1. Find the length of the curve y^ = x^ from the origin
to the point where x —h.
From 2^ = x3 we find y = »8/2^ whence
and
335
•]:::-jrV'^¥-i('+?-n::-
27
8
Example 2. Find the length of the catenary (§ 67)
» = — 2-
from the origin to the point where x ^\.
We find immediately
dx 2' da;VV2//'
which reduces algebraically to the form
ds ^ / ggg -f- 2 + e"^ V^^ _ e^ + g"^
dx \ ^ J 2 '
hence
^ (2.718-0,368) ^ ^^^^ (^^^^jy^
Compare § 67, and Tables III, E, and V, C.
Example 3. Find the length of one arch of the cycloid (Tables
III, G).
a; = a (< — sin 0, y = a (1 — cos Q.
We find
cte = (a — a cos ^i dy = o sin t dty
d8= Vdx'^ + dy^ = a V2 - 2 cos < d« = 2 a sin|d<,
whence
= 1 2 a sin 7i d< = — 4 a cos « I = — 4 a [cos x — cos 0]
i«o Jo 2 2jo
= -4a[-l-l] =8a.
136 THE CALCULUS [X, § 84
EXERCISES
Determine by integration the lengths of the following curves, each
between the limits x = ltoa; = 2, x=2tox=4, x=atoa; = 6.
Check the first three geometrically.
1. y = 3 X - 1. Z. y ^mx+c. 6. y = J (2 x - 1)V2.
2. y = 3 +2x. 4. 2/ = f (x - 1)V2. 6. y = i (4x - 1)V2.
Find da, the speed v, and the length s of the path of each of the follow-
ing motions, between the given limits.
7. x = l+^, y = l-^;^ = 0to< = 2.
8. X = (1 + 0V2, y = (1 - t)y^; t = to t ^ 1.
9. X = (1 - 0^ y = 8 ^/2/3; < = to < = 9.
10. X = 1 + ^2, y = < ~ ^/3; < = to < = 5.
11. X = 2A y = t + 1/(3 ^3); ^ = a to ^ = 6.
12. Find the length of the cycloid (Ex. 3, § 84) for half of one arch,
i.e., from ( = to < = ir; for the portion from < = to < = ir/2; from
< = to « = t/3.
18. Show that the element of length for the cardioid (Tables III, G4)
X = 2 a cos ^ — a cos 2 d, y = 2 a sin ^ — a sin 2 ^,
is
d« = 2a[2-2 (cos2 dcos $ + sin 2 dsin e)]^/^de = 2a (2-2 cos 6) ^^dB,
Hence show that the entire length of the cardioid, from ^ = to
d = 2 IT is 16 a. Show that the^length of the part of the cardioid from
d = 0tod = ir/2is4a(2~ V2).
14. The equation of a circle about the orjgin may be replaced by
the parameter equations
X = o cos ^, y = a sin ^,
where a is the radius. Hence find by integration the length of the entire
circumference.
16. Show that the element of length of the four-cusped hypocycloid
(Tables III, Ge)
X = a cos' By y = a sin' ^,
is (is == 3 a sin ^ cos 9 (29 = I a sin 2 d (29.
Hence show that the length of one quarter of this curve is 3 a/2.
X, §85]
APPLICATIONS TO CURVES
137
16. Show that the length of the general catenary
y =
2a
y
y'-fix)
from the origin to any point x = x is (e^ — e"**)/2 a.
17. Writing the equation of the simple catenary in the form used
in § 67, y = cosh a;, show that its length between the origin and any
point X = XyVa sinh x.
85. Areas of Surfaces of Revolution. Consider the surface
generated when the arc
KL of the curve
(1) 2/=/(x)
(Fig. 30) revolves about
the X-axis. Let us denote
by S{x) the area of the
surface generated by the
arc KP, and by AS the area
of the surface generated by the arc PQ = As. If the curve
rises from P to Q we may write, with entire accuracy,
(2) 2^^y^s<^S<2'K{y + ^y) As.
Hence, dividing by Ax and passing to the limit, we may write
Fia. 30.
(3)
S = '"S— 'Nl' + (g;
whence the area of the surface of revolution from x = a to
a; = 6 is
«' ^]:::=r-'.^-r'->Rir'"-
Similarly, if the curve is revolved about the 2/-axis, the
area between y = c and y = dis
ds
(5) -S
]:::-r-'i*-r-»>KW*-
138
THE CALCULUS
[X, §85
Finally, if the equation of the curve is written in parameter
form
a;=/(0, y = <t>(.t),
we divide (2) by At, and let At approach zero, obtaining
dS
(30
dt
-^'»|-^'W(I)'+(I)' «83.)
Hence the area of the surface of revolution formed by
revolving the curve about the x-axis, between points at
which t = ^1, and < = fe, is
(6)
S
];>r-»-=r-»^(S)'+©'-
A similar formula can be written if the revolution is about
the t/-axis.
Example 1. Find the surface of the cone generated when the seg-
ment of the straight line y = x — 2 from
^ a; = 2tox = 5 revolves about the x-axis.
S
dx
Fig. 31.
= y^ 2 IT (x - 2) VTTTdx
= 2 TrV2f (x — 2)dx
.2;V2(f-2x)J=9,v^
Qieck this solution by finding the area of this cone by elementary
geometry.
Example 2. Find the area of the surface generated when the seg-
ment of the curve whose parameter equations are
a:=i(4« + l)3/2, y = «2+5
between t » 1 and ^ = 2 is rotated about the o^-axis.
X, § 86] APPLICATIONS TO CURVES 139
(fo2 =(ir2H-d2/2 = (4 < + 1+4^) d^ = (2^ + 1)2 («2
2iryds =2ir f {ti-\-5)(2t + l)dt
1-1 •/!
"^'[^■^^■^^^^"^^ <]^= 2 T [40| - lOf] = 5»| T.
EXERCISES
Find the area of the surface generated by each of the following
lines when revolved about the x-axis, from x = 1 to x = 2; from
X = 2 to x = 4; from x = a to x = 6.
1. 2/ = 2x-l. 2. 2/ = 3+4x. 3. y = 3x+2.
Find the area of the surface generated by each of the following
curves when revolved about the x-axis, between the limits indicated.
4. y = Vl — x2; x = to x = 1; x = } to x = 1.
5. y = V4 x--x2; x=0tox=4; x = ltox = 3.
6. 2/ = V7 -f 6 X — x2; x = — ltox = 7; x = 2tox=5.
7. Find the area of the surface generated by rotating the arc of the
catenary
» = — 2—
about the x-axis, between the points where x = and x = 1.
8. Find the area of the siu^ace generated by rotating the arc of the
curve whose parameter equations are x = 8 ^/2/3, y = (i — ^)2 about
the X-axis, between the points where t = and t = 1,
9. Find the area of the surface generated by the arc of the curve
2
whose parameter equations are 3? = 7 , y = t-\-l/(Sfi) about the
ovaxis between the points where ^ = 1 and t = 2,
86. Curvature. A very important concept for any plane
curve is its rate of bending, or curvature.
The.^exion(§39,p. 61),
(1) b = -^^
140
THE CALCULUS
[X, § 86
is not a satisfactory measure of the bending; since it evidently
depends upon the choice of axes, and changes when the axes
are rotated, for example.
If we consider the rate of
change of the inclination of the
tangent, a = tan~im, with respect
to the length of arc s, that is,
AcK da
(2)
A9->oAs as
Fia. 32.
it is evident that we have a
measure of bending which does
not depend on the choice of axes, since Aa and As are the
same, even though the axes are moved about arbitrarily, or,
indeed, before any axes are drawn. The quantity da/ds is
called the curvature of the curve at the point P, and is de-
noted by the letter K: the curvature is the instantaneous rate
of change of a per unit length of arc.
Since a = tan~i ^^ gj^d since ds^ = dx^ + dy^ (§ 83, p. 134),
we have,
1
da=s d tan~i m =
dm, ds = Vl + m^ dx,
l + m2
where m = dy/dx; hence the curvature K is
1 , dm
(3) V _da 1 + m
T,dm
2
Vl+m^dx
dx
(1 + m^)«/^
(1 + m«)»/«'
where 6 = d^y/dx^ (= flexion), and m = dy/dx (= slope). It
appears therefore that the flexion 6 when multiplied by the
corrective factor 1/(1 + m2)3/2 gives a satisfactory measure
of the bending, since K is independent of the choice of
axes.
X, § 86]
APPLICATIONS TO CURVES
141
The reciprocal of K grows larger as the curve becomes
flatter; it is called the radius of curvaturey and is denoted
by the letter iZ:
(4) ^ = -K-t-^^^^'
It should be noticed that this concept agrees with the ele-
mentary concept of radius
in the case of
since As = rAa
circle of radius r. Hence
ds/da = r.
Substituting the values
of 6 and m, formulas (3)
and (4) may be written in
the forms
cPy
a circlej
in any
(5) K =
dx^
[
1 +
/dyY-|8/2'
\dx/ J
Fig. 33.
(6)
R =
b + m"
dx^
It is preferable, however, to calculate m and 6 first, and
then substitute these values in (3) and (4).
Since vT+rn^ = g^^ ^^ the formulas may also be written
in the form K = 1/R = b cos^ a.
It is usual to consider only the numerical values of K, that
is \K\, without regard to sign. Since K and b have the
same sign, the value of K given by (3) will be negative when
& is negative, i.e. when the curve is concave downwards
(§ 41, p. 65). The same remarks apply to R, since R = 1/K.
142
THE CALCULUS
K §87
87. Center of Curvature. Evolute of a Curve. The center of
curvature Q of a curve, corre-
sponding to a point P on that
curve, is obtained by drawing
the normal to the curve at P
and laying off the distance R
(the radius of curvature) along
this normal from P toward the
concave side of the curve. Thus,
in Fig. 34, denoting the co-
ordinates of P by (x, y), and
those of Q by (a, j8), we have
(1) a= OB = OA - BA = X - R sin 0,
(2) p = BQ = AP + CQ = y + /2cos0,
where <t> is the angle which the tangent PT makes with the
X-axis,
(3)
whence
(4)
_^ _dy
tan = m =
dx'
sm0 =
m
Vl + m2 '
It follows that we may write
(5) a = x-'"(*+'"*>
where
COS0 =
y/l + nfi
= y +
1 + m*
b ' ^ - ^ b '
m = dy/dx, h = d!^y/dx^.
As the point P moves along the given curve, the point Q
also describes a curve, which is called the evolute of the
given curve. The parameter equations of the evolute are
precisely the equations (5), in which (a, j8) are the variable
coordinates of the point on the evolute, x is a parameter, and
2/ is to be replaced by its value in terms of x from the equa-
tion of the given curve.
X, § 87]
APPLICATIONS TO CURVES
143
In particular examples, it is possible to eliminate the
parameter x between the
two equations (5) after
having substituted for i/,
/n, and 6 their values
found from the equation
of the given curve. This
gives the equation of the
evolute as a single equa-
tion between the variables
a and j3.
Example 1. Find the values of K, Rj a, /3, and find the equation
of the evolute for the curve y = x^/4.
Here m = x/2, b = 1/2. Hence
1/2 4
iC =
(1 H- a;2/4) 3/2 (4 + a;2)8/2 '
jy 1 (4 -f x2)3/2
^^K 4
a; 1 -f gV4
**"* 2 1/2 '
4 '
/5 = y +
1 + gV4 _a:2
:i. 4-2 4-— =
4 ^ ^2
x^ 3x2
+ 2.
1/2 4 • " • 2 4
EUminating x between the last two equations, we find the equation
of the evolute in the variables a and fi,
27 a2 =4(/3-2)3.
Example 2. Find the values of K^ R/a, fi, and the parameter
equations of the evolute for the cycloid
x = cr (^ — sin ^), y = a (1 — cos $),
dx dx/d e a (1
a sin d .6
cos d) 2 '
- 1
\K\ =
dx dx/d e o (1 - cos ^)2
-1
4 a sin^ ^
-o+^-r
sin^^
1
4 a Sin ^
144
THE CALCULUS
[X, § S7
4 a sm ^ ,
e
^l+ctn2^
sin e) + ctn = ^ = a (^ + sin e),
4 a sin* ^
l+ctn2|
fi = a(l — CO8 0) j = — a (1 — cos d). ,
4 a sin* ^
Cycloid
Fig. 36.
These equations for a and fi are the parameter equations of the
evolute; they represent a new cycloid, similar to the given one, but
situated as shown in Fig. 36.
EXERCISES
Calculate X, i2, a, fi for each of the following curves; sketch the curve
and its evolute:
, 1. y — x^, 2. y = sc^. 3. y2 = 4 qx,
4. xy — a2. 6. 2/ = sin X. 6. y = e^.
7. y = (e* -f 6-«)/2 = cosh x. 8. y = (e* - e^)/2 = sinh x.
9. x2/a2 4. y2/52 = 1. 10. x2/a2 - 2/2/62 = 1.
11. Vx + V^ = Vo. 12. x2/3 + 2/2/3 = ^2/3,
J- J -, = a cos' df
13.
16.
19.
j a? = a cos ^,
< y = a sin d.
(x = <2,
ly = t-fi/Z,
(x = 2 + 3^,
ly^fi-i.
14. j^^f ^' 16. j^ = «^!
/ 2/ = 2 cos ^. / y = a si
17. j* = y*' 18. ^^ =
20. j* = *'.'«' -^
(y = sm < —
sin' e.
= sec tf
tan ^.
+ < sin <,
t COS ^
X, § 89] APPLICATIONS TO CURVES 145
88. Properties of the Evolute. If the point P (x, y) lies
on a curve y = / (x), and the point Q (a, 0) is the correspond-
ing point of the evolute, we have
m + m^ ^ , l + rn?
a = x ^ — , j8 = yH ^
Let m' be the slope of the evolute at Q; then
da da/dx
But we have
dp , 6-2 w6 - (1 + m2) db/dx
di = ^ + p
3mb^-(l+ Tffi) db/dx
da __ b (1 + 3 nfi) & -- (m + m^) db/dx
dx~~ 62
__ ~ 3 m262 + (^ + ^^) db/dx
" P
It follows that
doL m
Hence the normal to the curve at P is the tangent to its
evolute at Q. Hence the radius of curvature of the curve
at C is tangent to the evolute at Q. (See Fig. 35.)
89. Length of the Evolute. As Q moves along on the evo-
lute, the rate of change of R is the rate of change of the arc
of the evolute. For, since we have, in Fig. 34,
(1) /22 = Pq2=(^_«)2+(2,_^)2,
it follows that
(2) fi d/2 = (x - a) (dx - del) + (y - fi) (dy - dfi).
But we have
m = ^ = - |3-|> or (x-a)dx+(y'- 0) dy = 0.
146 THE CALCULUS [X, § 89
Hence (2) may be written in the form
(3) dR = (^ -oQda+jy- jyd fi^
V(x - «)2 + (j/ - ^)2
By § 88, we have
da m X — a
Substituting this in (3), we find
da-\-y^dfi da + ^dfi
(4) dR , '^ " = I . - = Vdcfi+d0^.
>i'+c^fy >!'+©'
This result, however, is precisely ds, where s is the length of
the arc of the evolute. Hence we have
dR = \ ds, or R2 — Ri = S2 — Sii
R = Ri ./«-«i
that is: the rate of growth of the radius of curvature is equal
to the rate of growth of the arc of the evolute; and the
difference between two radii of curvature is the same as the
length of the arc of the evolute which separates them.
This fact gives rise to an interesting method of drawing
the original curve (the involute) from the evolute: Imagine
a string wound along the convex portion of the evolute,
fastened at some point (say Q, Fig. 35, p. 143) and then
stretched taut. If a pencil is inserted at any point (say P,
Fig. 35) in the string, the pencil will traverse the involute as
the string, still held taut, is unwound from the evolute.
As exercises, the lengths of the portions of the evolutes
of curves given in the preceding list of exercises may be
found.
CHAPTER XI
P =/(<?).
POLAR COORDINATES
90. Introduction. Since the equations of curves in polar
coordinates often involve trigonometric functions, the equa-
tions of curves have been written in rectangular coordinates
throughout the earlier part of this book. We shall now
show how to extend many of the results already foimd for
curves whose equations are written in rectangular coordinates
to curves whose equations are written in polar coordinates.
91. Ani^e between Radius Vector and Tangent Let C
(Fig. 37) be a curve whose equation in polar coordinates is
(1)
Consider the angle ^
between the radius
vector OPR and the
tangent PT to C at
P. Let Q be a
second point on C,
with coordinates
p + Ap> + A^, and ^^ ^^
let be the angle
between the radius vector OQD and the secant PQS. As Ad
approaches zero, i,e, as Q approaches P along C, we shall
have
(2)
}p = lim 0, and tan yp = lim tan 0.
147
148
THE CALCULUS
[XI, § 91
Draw AP perpendicular to OQ; then we shall have
AP AP psm^e
(3) tan 4> =
and
AQ OQ - OA
(4) tan \[/ = lim
p sin Ad
p + Ap — p cos A^
p sin A^
= lim
A^->op + Ap— pcosA^ A«->op(l — cos A^) +Ap
To evaluate this limit, divide both numerator and de-
nominator by A^ and note that
(5)
and
(6)
,. sin A^ ^
2 sm2 — -
,. 1 — cos A^ ,. ^
lim T-z = lim — T-^ —
^-»0 Au Ad-»0 i^"
lim
. A^"
. A^ ^^^ -2
«^^2'"Ar"
= 01 = 0.
It follows that
sin Ad
(7)
tan^ =
= lim —
P'
1 — cog
A^
Ap
A0
P
dp
d0
and therefore
(8)
tanyf/ =
p d^
dp ^dp
de
The angle a between the a>axis and the tangent PT can
be found, after ^ has been found, by means of the relation
(9)
a=« + ^.
XI, § 91] POLAR COORDINATES 149
ExASfPLE 1. Given the curve p = e*, to find tan ^, and ^ itself.
Since p = e^, dp Id = efiy and tan ^ = p -s- {dp/d d) = \. Hence
^ = tan~i 1 = ir/4 = 45°. It follows that this curve cuts every radius
vector at the fixed angle of 45°.
Example 2. Given the curve p = sin 2 ^, find ^ at the point where
e = gr/8.
tan ^ = p -r- {dp/d ^) = sin 2 ^ -r- 2 cos 2 d = (1/2) tan 2 6.
When e = gr/8, tan ^ = 1/2, and ^ = 26° 34', approximately.
EXERCISES
Plot each of the following curves in polar coordinates; find the value
of tan ^ in general, and the value of ^ in degrees when = 0, t/6,
x/4, ir/2, TT,
1. p = 4 sin 0. ft. p = 0. 11. p = sin 3 0.
2. p = 6 cos — 5. 7. p = ^. 12. p = 2 cos 3 0.
3. p=3 + 4cos0. 8. p = l/0. 13. p = 3sin(3d + 2T/3).
4. p = tan e, 9. p = e^^. 14. p = 3 cos d -|- 4 sin (?.
6. p = 2 + tan2 ^. 10. p = e^**. 16. p = 2/(1 - cos e),
16. Show that tan ^ is constant for the curve p = A;e^.
Find tan ^ for each of the following curves:
17. p = p/(l — e cos d) (conic). 19. p = a (1 + cos 6) (cardioid).
18. p = o sec d ± 6 (conchoid). 20. p2 = 2 a^ cos 2 ^ (lemniscate).
92. Areas in Polar Coordinates. Let KL be an arc of a
curve whose equation in polar coordinates is
(1) P=/W.
Consider first the area of the sector OKP bounded by the
radius vector OK, for which B = d\, any other radius vector
OP for which 6 =^ B, and the intercepted arc KP of the
curve. The area is a function of the angle 6) let us denote
itbyil(^).
150
THE CALCULUS
[XI, § 92
Now let increase by an amount A^ = Z POQ, and let
PS and TQ be circular arcs whose radii are p ( = OP) and
p + Ap (= OQ), respectively. Then we shall have
(2) sector POS < sector POQ < sector TOQ,
if p increases with 6. If p
decreases when increases,
the inequality signs must
be reversed in (2) and in
what follows.
The area of any circular
sector is equal to half the
product of the angle (in
circular measure) and the
square of the radius. The
sector POQ is the amount
of increase in the area A ;
let us call it A A (6) . Then
Fig. 38.
(2) becomes
(3)
p2A^
<Ail(^)<
(p + Ap)2A^
Dividing through by A^, we have
(4)
whence
(5)
P^ AAjS) (p + Ap)2
2 ^ A^ ^ 2
lim
AA(e) pf
2
dA
= TT , or
2
A^ 2 ' ^* do
It follows that the area of the sector bounded by the
curve and the radii vectores for which d = di, and d = $2,
respectively, is given by the formula
(6)
J<
P^dS.
XI, § 92]
POLAR COORDINATES
151
Example 1. Find the area of the sector bounded by the curve
p — \/.e and by the radii corresponding to 6 — t/3 and — t/2.
The curve may be plotted readily by taking corresponding values of
p and By as in the following table.
e
ir/6
ir/3
ir/2
2ir/3
5ir/6
IT
etc.
p
00
1.91
0.95
0.64
0.47
0.38
0.32
etc.
The required area is
Fig. 39.
-l-K
-c(-r-(i)"']
= ;5^ = 0.16+.
Example 2. Find the area
of the sector bounded by the
curve p = tan d, and by the
radii corresponding to ^ = 45°
and e = 60°.
netf* 1 yT/3 1 yT/3 1 yr/S , .
J45*> 2Jt/4 2Jt/4 2 Jt/4
=i (tan <? - (?) I^J' =i [(V3 - ir/3) - (1 - t/4)]
= 5 ( >/3 - 1 - x/12) = .2351 +
EXERCISES
Calculate the area formed by each of the following curves and tlie
indicated radii, and check graphically.
1. p = ^; d = to T. 2. p = 6/^; ^ = x/3 to gr/2.
3. p = VS; d = T to 2 T. 4. p = -j^?; « = to 2 T.
6. p = 4:/\l'e\ ^ = ir/8 to IT. 6. p = 1 + V^; = x/4 to ir.
7. p = Vr+7; d = 1 to 3. 8. p = Vl + ^; d = to 3.
152
THE CALCULUS
[XI, § 93
9. p = (^ - 1)2; ^ = 1 to 6. 10. p = (1 + e)l^\ e = 180° to 360**
11. p = sin ^; ^ = to x/2. 12. p = cos ^; ^ = x to 2 ir.
13. p = sec e\ e = 7r/4 to ir/3. 14. p = 1 + sin ^; ^ = to x/2.
Find the area bounded by each of the following curves:
16. p = 4 cos d. 16. p = 4 cos 2 d.
17. p2 = 4 cos 2e. 18. p = 1 — sin e.
93. Lengths of Curves in Polar Coordinates. Let the equa-
tion of a curve in polar coordinates b^
(1) p =/(«).
and let s denote the length of arc from a fixed point X to a
variable point P. Then s is a function of d. We shall show
AS
K{e=-k)
Fig. 40.
(2)
we find
(3)
As _ As
A^~ Ac
first how to obtain ds/d$,
whence we may proceed to
find s itself by an integration.
Let the coordinates of P be
(p, 6), and those of a second
point Q be (p + Ap, 6 + A^).
Denote the arc PQ by A^, and
the chord PQ by Ac. Then
from the identity
Ac
' A^
^ = lim^! = lim^.f^ = limf?. See (4), §82.
A^ A«-*o A^
dd Afl-ToA^ Ad^oAc
From the law of cosines,
Ac= Vp2 + (p + Ap)2 - 2p (p + Ap) cos A^
= •v^2(p2 + pA p) (1 - cos A^) + V.
It follows that
U^ ^^ L 2 _i_ A ^ 2( 1 - cos A^) /Ap\2
XI, § 93]
POLAR COORDINATES
153
But we have *
(5)
lim
2(1 - COS Ad)
1 and lim -A = -^
A->o A^ dS
Hence, by (3) and (4),
Integrating both sides of equation (6) with respect to 6,
we find the important formula
(7)
5
»;
Fig. 41
Equation (6) may be supplied
by squaring both sides and then
multiplying both sides by {dsy.
The resulting formula
(8) d^ = p^d0 + dp^
is the P3rthagorean differential
formula in polar coordinates.
(See § 83.)
Example. Find the length of the curve p = 1 — cos d.
dp = sin dd9.
ds/de = V((fp/(|gp + p^.
= Vsin^ 5 + (1 - cos ^)2.
= V2 (1 - cos e) = 2 sin {e/2).
The complete curve is traced when varies from to 2 t.
Hence
]2t y2T . , . ^ ^n2T
= / 2 sin {e/2) de = -4 cos^ = 8.
Jo ^Jo
. * See and compare (6), p. 148. In this case we have
2 (1 - cos Ag) 4 sin« Y / sm y
A9>
A0»
154 THE CALCULUS [XI, § 94
EXERCISES
Find the length of each of the following curves, or of the portion
specified.
1. p = 5 sin ^. 6. p = o CSC e; = 45** to 90**.
2. p = 3 cos 0. 7. p — Bin + COB 0; ^ to w,
3. p = 6«; ^ = to t/2. 8. log p = ^; ^ « 2 to 3.
4. p = ea«; ^ = to X. 9. log p = 3 + 2 d; tf = to 1.
6. p = sec ^; ^ = } to 1. 10. p = 1 + sin 0,
94. Curvature in Polar Coordinates. By the definition of
the cxirvature K of any curve (§86), we have
^^ Js" ds/do'
Since a = ^ + ^ (§ 91), we have da/dS =1 + #/d^. De-
noting dp/ do by p' and d^p/dS^ by p", we have
(2) ^^ = tan-i _e^ ^
P
/Q\ d^ __ ^ . ^^_i P^ 1 _d / P\
^^^ d^ "" de ^^"^ p' ~ 1 + (p/p02 ' ds \p7
_ 1 p'' - PP'' ^ p'' - PP"
i + (p/p02' p'^ f^+p'^ '
It follows that
m d« 1 , # 1 , P^^-P/^ p2 + 2p^^-pp^^
Also, from (6), §93,
(5) 1=^^+^
Hence we have
/AN j^_da/dg_ p» + 2p^»-pp"
^^^ ds/d^ "" [p* + p;*i'/* *
Example 1. Find the curvature for any point on the curve p = e^^.
Since p' = dp/d0 = 2 f2e, and p" = cPp/d^ = 4 e^», we have
K =
(e4<? 4- 4 e4<?)3/2 .^^2*
XI, § 94] POLAR COORDINATES 155
Example 2. Find the curvature at any point of the circle p = a sin ^.
Here we find p' =^ a cos e, and p" — — a sin B. Therefore
^ fl^ sin2 g -f 2 gg cos^ B -\- a^ sin^ g
(a2 sin2 ^ + a2 cos2 ^)V2
^ 2 02 (sin2 g -f C082 g) ^2
a3 (sin2 ^ + cos2 ^)3/2 a*
This is the reciprocal of the radius. (§ 86.)
EXERCISES
Find the radius of curvature of each of the following curves:
1. p = 6?. 4. p = cos B, 7. p = o(l -|- cos B),
2. p = aP, 6. p = sin 3 d. 8. p = 2/(1 -|- cos B),
3. pB = a. 6. p = a sec 2 9. 9. p » a + & cos 9.
CHAPTER XII
TECHNIQUE OF INTEGRATION
95. Question of Technique. Collection of Fonnulas. The
discovery of indefinite integrals as reversed differentials was
treated briefly, for certain algebraic functions, in Chap-
ter VII. We proceed to show how to integrate a variety of
functions, but the majority are referred to tables of integrals,
since no list can be exhaustive. See Tables, IV, A-H.
To every differential formula (pp. 44, 132) there corre-
sponds a formula of integration:
if d4> {x) =fix) dx, then f/ W dx = 4> W + C.
The numbers assigned to the following formulas corre-
spond to the number of the differential formula from which
they come. Certain omitted numbers correspond to rela-
tively unimportant formulas.
FUNDAMENTAL INTEGRALS
[I]i If ^ = 0, then y = constant. [See § 52, p. 86.]
[The arbitrary constant C in each of the other rules results from this rule.]
[II], fhf{x) dx = kff{x) dx.
[Ill], /{/ {x) +it>{x)]dx =ff (x) dx +f<t> (x) dx.
[TV]i (xn dx = 5^ + C, when n 5^ - 1. (See VIII.)
166
XII, § 95] TECHNIQUE OP INTEGRATION 157
[VI]i uv = Cd{uv)^ Cudv+Cv du. ["Parts"]
[The corresponding formula [V]i for quotients is seldom used. See § 103.]
[VlIji J/(ti) du]^ ^ ^^^^ =Jf [<!> (X)] d<t> (X)
[Substitution] = (f[<t>{x) ] ^^ dx.
[Vni]i r^ = log X + C. [IX]i C&^ dx = e* + c.
[X]i I cosxc/jc = sinjc + C. [XI]i | sinxdx = -cosx + C.
[Xn]i jsec* X c/x = tan X + C.
[Xinii fcsc* X dx = - ctn X + C.
[XlVji J sec X tan x dx = sec x + C.
[XV]i J CSC X ctn X dx = — esc x + C.
[XVIji J-7== = sin-i x + C= -cos-ix + C. [xvn]i
[XVra]i rj^ = tan-ix + C= -ctn-ix + C [XIX]i
[XX]i r--^== = sec-ix + C= -csc"-ix + C'. [XXI]i
[XXn]i r-7=^== = vers -ix + C
The remaining differential formulas referred to on p. 132
give rise to other integral formulas; these will be found in
the short table of integrals, Tables, IV, A-H.
158 THE CALCULUS [XII, § 96
96. Polynomials. Other Simple Forms. The rules [II],
[III], [IV] are evidently sufficient without further explana-
tion to integrate any polynomials and indeed many simple
radical expressions. This work has been practiced in
Chapter VII extensively.
Attention is called especially to the fact that the rules [II]
and [III] show that integration of a sum is in general simpler
than integration of a<product or a quotient. If it is possible,
a product or a quotient should be replaced by a sum unless
the integration can be performed easily otherwise. Thus
the integrand (1 + x^)lx should be written 1/x + a;; (1 + o^Y
should be written 1 + 2 a;^ + a;*; and so on. This principle
appears frequently in what follows.
97. Substitution. Use of [VII]. As we have already done
in simple cases in Chapters VII and VIII, substitution of a
new letter may be used extensively, based on Rule [VII].
dx
Va2 — x^
Set u = xla^ then du = 3x1 a, or dx = a dUy and
/dx r adu r du • _i , ^ • -i^c . ^
Va2 — x2 J Vo2 — a2w2 J Vl — w2 a
^ , ' ^x 1 J /x\ dx/a dx
Check. d sin"^ - = j i i _ —
/dx
-v//i2 —
a \ x^ \^/ I x^
\^~a2 \^"-a2
Vo2 - a;2
Example 2. To find f Bm2xdx.
Method 1. Direct Svbstitution,
rsin2xdx = jTsin (2 x) d (2 x) = - }[cos2a; + C]
= - J cos 2 a; + C.
Check, d (— J cos 2 a;) = — } d cos (2 x)
= + J sin (2 x) d{2x) = sin 2xdx,
Method 2. Trigonometric Transformation and Svbstitution,
fain 2xdx = y 2 sin x cos xdx = — 2 /^cos x d (cos x)
= - (cos x)^ + K =-coa^x+ K,
XII, § 98] TECHNIQUE OF INTEGRATION 159
Notice that cos^ x -^ K =^ (1/2) cos 2 a; + C since cos 2 a; = 2 cos^ x
— 1. Do not be discouraged if an answer obtained seems dififerent
from an answer given in some table or book ; two apparently quite
different answers both may be correct, as in this example, for they may
diif er only by some constant.*
Whenever a prominent part of an integral is accompanied
by its derivative as a coefficient of dx, there is a strong indication
of a desirable substitution; thus if sin x occurs prominently
and is accompanied by cos x dx, substitute m = sin x; if log
X is prominent and is accompanied by (l/x)dXy set u = log a;;
if any function / (x) occurs prominently and is accompanied
by df (x),setu=f (x) . This is further illustrated in exercises
below.
98. Substitutions in Definite Integrals. In evaluating defi-
nite integrals, the new letter introduced by a substitution may
either be replaced by the original one after integration, or the
values of the new letter which correspond to the given limits
of integration may be substituted directly without returning
to the original letter.
Example 1. Compute / sin a; cos x dx.
Method 1. Let u = sin x.
sin X cos X dx ^ I udu
fi-^x^T/2 sin2a;n*-»/* 1
Method 2.
_ w2-|a;-T/2_ gin2 a;-|«.
"■2Jx-o 2~Jx.
/ sm a; cos a; ox = / udu ^ -^\ = ^ ,
Jx»o Ju^o ^Jk-o ^
since w ( = sin x) =0 when x = 0, and w = 1 when x = ir/2.
Care must be exercised to avoid errors when double-valued functions
occur. The best precaution is to sketch a figure showing the relation
between the old letter and the new one. In case there seems to be any
doubt, it is safer to return to the original letter.
* Occasionally it is really difficult to show that two answers do actually
differ by a constant in any other way than to show that the work in each case
is correct and then appeal to the fundamental theorem ($ 52).
160 THE CALCULUS [XII, § 98
EXERCISES
Integrate each of the following expressions:
1. /(l-a;)(H-a;2)cte. 6. /(e* - e-«)2 cte.
2. / =-! dx. 6. / dx.
J x^ J X
3. J{a-\-hxYdx, 7. f{l-2x)'^Vidx.
4. /^-^^^ <^. 8. /(a^ - 2) (a;i/2 + a;2/8) <to.
In the following integrals, carry out the indicated substitution; in
answers, the arbitrary constant is here omitted for convenience in
printing.
9. y V3 a; + 2 cte; set w = 3 a; + 2. Ana, f ^3/2 = j(3a; + 2)3/2.
.0. f^^-2 = i log (3a; +2) = log ^3 a; +2.
^' /nR^ + 2)5 = i^«^"'(3x + 2).
,2. fx V4+a;2 cte; set w = 4 + a;2. Ans, w3/2/3 = (4 + a;2)3/2/3.
3- /^^2 = * log (4 +a:2) = log Vr+T^.
,4. /sin X Vcosx dx; set w = cos x. Ans. — 2 W8/2/3 - — 2{gos^^x)/S.
6. y cos a; Vsin a; dx = 2 (sin8/2a;)/3.
6. /cos X (1 + 4 sin X + 9 sin2 x) dx = sin x + 2 sin2x + 3 sin^ x.
,7. /sin3 X dx = /sin x(l — cos2x)dx = — cos x + (cos^ x)/3.
8. ycos (3 X - 2) sin (3 X - 2) dx; = J sin2 (3 a;- 2).
,9. ysin (1 - 3 x) cos5/2 (1 - 3 x) dx = A cos7/2 (i - 3 a;).
20. / o ■ o ; e^t tt = -• Ans, f - tan~^ w = - tan~i - •
•/ o2 -J- x2 ' ^ a I a a a
XII, § 98] TECHNIQUE OF INTEGRATION 161
In the following integrals, find a substitution by inspection and
complete the integration.
21. /4^73=5log(4a:+3)=log(4x+3)i/4.
22. y V3 - 4 a; dx = - (3 - 4 x)^/^/6.
23. ysin (4 a: - 3) cte = - } cos (4 a; - 3).
26. ycos8 a; da? = sin a; - (sin^ a;)/3. (See Ex. 17.)
26. J 008 X sin^ xdx — (sin^ x)/b.
27. ^2 a; cos (1 + x2) da; = sin (1+ a;2).
28. Aan3 x sec^ xdx=^ (tan*a;)/4. (« = tan ».)
29. fctn^ X csc2 a; dx = — (ctn* a;)/4. {u = ctn a;.)
30. ycos2 X dx =y [(1 + cos 2 a;)/2] da; = a;/2 + (sin 2 a;)/4.
31. ysin2 X dx ^f[{l - cos 2 a;)/2] da; = a;/2 - (sin 2 a;)/4.
32. Jcos^ xdx = sin a; — 2 (sin^ a;)/3 -|- (sin** x)/5.
33. y ctn X dx = Acos a;/sin a;) da? = log sin x, (Put t* = sin a;).
34. Ttan x dx = — log cos x = log sec x.
^^ r dx 1 r dx lx_i/a;\
•^ Vl2-4a;2 ^J y/3^^ 2®"^ \V3/
162 THE CALCULUS [XII, § 98
Compute the values of each of the following definite integrals.
* Jx-o 3+a;2 V3 *° VsJ^-o vl ^ V3 6V3'
39. / , = sin-i ( -F I = sin-i 1 = -5-
*0- T"^ ^^2 == i I0& (3 + ^')T"^= i (loge 12 - lofo 3) = log.2.
Jx-O o + X-* Jx-0
41. f'^-^^= ^ « V2ir72T"' = - (vT- V2) = .4142 +.
y^x^x/2 r nx-T/2
sin3 x da; = - cos a; + (cos' a;)/3 = 2/3.
z-O L Jx-O
43. f'^x^dx-- €^/2T"^ = e/2 - 1/2 = .8591 +.
Jx^O Jz-0
y.if-2 ^-1 ^
yr^-2 ^x-2 1 + 2a;
e-2«(to. 49. r ^-^5:^ da;.
a»» 1 •/ X" 1 X -J" X^
^-ir/3 ^x-t/4
46. / sin^ a; cos a; da;. 60. / cos^ a; dx.
*/6 /•*-ir/4
sin 3 a; (2a;. 61. / cos' a; da;.
Jx-^o
62. Find the area under the witch y = l/(o + 6a;2) for a = 9,
6 = 1, from a; = to a; = i; for o = 8, 6 = 2, from a; = 1 to a; = 10.
See Tables, III, J.
63. Find the volmne of the solid of revolution formed by revolving
one arch of the curve 2/ = sin a; about the a>-axis.
64. Find the area under the general catenary
y = a cosh (x/a) = a(e*/«+e-*/«)/2
from a; = to a; = a.
66. Find the area of one arch of the cycloid
a; = a (^ — sin ^), y == a (1 — cos 0),
66. Find the volume of the solid of revolution formed by revolving
one arch of the cycloid about the x-axis.
67. Compare the area of one arch of the curve y = sin a; with that of
one arch of the curve y = sin 2 a;; with that of one arch of y = sin^ x.
XII, § 99] TECHNIQUE OF INTEGRATION 163
68. Show how any odd power of sin x or of cos x can be integrated by
the device used in Ex. 17.
69. Show how any power of sin x multiplied by an odd power of cos x
can be integrated.
99. Integration by Parts. Use of Rule [VII. — One of the
most useful formulas in the reduction of an integral to a
known form is [VI], which we here rewrite in the form
[VI T J'u dv = uv —fv du
called the formula for integration by parts. Its use is illus-
trated sufficiently by the following examples:
Example 1. J'x Binxdx, Put u ^ x^ dv — sin x dx; then du = dx
and t; =y sin X cte = — cos x; hence,
Jx sin xdx = — x cos x + /cos xdx = — x cos x + sin x] {check).
Example 2. yiog x dx. Put log a; = w, dx = dv; then du = (l/x) dx,
V = Xf and
Jlogxdx = xlogx — fx'-'dx = a;logx — J dx
— X logx — X + C; {check).
Example 3. y Va^ — x^ dx. Put u = Vq2 — a:^, dv=dx; then v = Xj
du= ^ ^^ - dx.gmdfVa^-x^dx =xVa^-x^-^ r_ j^dx_
but, by Algebra, f f^ = - f\^a^ -x^dx + f-^^;
hence
(j^'dx
x^
J, . /» a^ ax
V a2 - x2 rfx = x Va2 - x^ + I ^-^--
= X Va2 - x^ + a2 sin-i (-^ + C.
This important' integral gives, for example, the area of the circle
J?' -\-y^ — a^f since one fourth of that area is
f^'^'^a^ - a;2 dx = i fx Va^ - x2 + a2 sin"! /5\T"''
21 2j 4
164 THE CALCULUS [XII, § 99
EXERCISES
Carry out each of the following integrations:
1. Jx cosxdx = X sin X + cos X + C.
2. J'xeF dx = eF (x — 1) + C. [Hint, u = x, dv = eF dx.]
3. fxlogxdx =- xV4 + (x2 log x)/2 + C.
4. fx^logxdx=-- x*/lQ + (x^ log x)/4 + C.
6. fx^dx = e» (x2 - 2x + 2) + C. [Hint. Use [VI] twice.]
6. ysin"! X dx = X sin~i x + Vl — a;2 + C. [Hint, u = sin-i ^jj
7. ytan-i X (ix = X tan-i x — log (1 + x2) V2 ^- c.
8. yx2 tan-i X dx = (x^ tan-i x)/3 - xV6 + log (1 + x2)V6 + c.
9. yx (e* — e~')/2 dx —Jx sinh x dx = x cosh x — sinh x + C.
10. fx^e^' dx=^(9x^-6x-h 2)/27 + C.
11. /e* sin X dx = e* (sin x — cos x)/2. [Set w = e*; use [VI] twice.]
12. fe^ cos 3 X dx = e2x (3 sin 3 X + 2 cos 3 x)/13.
15. y^^ sin 3 X dx = — e-* (3 cos 3 X + sin 3 x)/10.
14. fe^ cos nxdx = e^ (w sin nx + a cos nx)/(cfi + n^),
16. Show that yP (x) tan"^ x dx, where P (x) is any pol3momial, re-
duces to an algebraic integral by means of [VI]. Show how to integrate
the remaining integral.
16. Show that yP (x) log x dx, where P (x) is any polynomial, re-
duces to an algebraic integral by means of [VI]. Show how to integrate
the remaining integral.
17. Express J^x^ e^ dx in terms of fx^"^ 6"* dx. Hence show that
yP (x)e«* dx can be integrated, where P (x) is any polynomial.
XII, § 100] TECHNIQUE OF INTEGRATION 165
Carry out each of the following integrations:
18. f{2 - 4 a; + 3 a;2) log X dx. 20. fix^ - x)e-^' dx,
19. fi3 a;2 + 4 X + 1) tan-i x dx. 21. f{x^ - 6)e** dx,
22. From the rule foi^ the derivative of a quotient, derive the formula
J*(l/v)du = u/v +y (w/y2) dv. Show that this rule is equivalent to [VI]
if u and v in [VI] are replaced by 1/v and w, respectively.
23. Integrate ye" sin x dx by applying [VI] once with w = e*, then
with u^sinx, and adding.
24. Integrate J^e^ sin nxdx by the scheme of Ex. 23.
Find the values of each of the following definite integrals:
26. r^"*log X dx. 28. r" (1+3 x^) tan'i x dx.
y^-2 y^-T/2
26. / xcdx. 29. / e-^'cosZxdx.
Jx^-l J x^O
^-1/2 ^-2
27. / sin"i X dx. SO. / (e* — e"*) dx.
31. Find the area under the curve y = e"» sin x from x = to x = t.
Find the area under each of the following curves, from x = to x = 1.
32. y = X6-*. 33. y = x^e-*. 34. 2/ = x^e^.
36. Compare the area beneath the curve y = log x from x = 1 to
X = e with the area between this curve and the y-axis, and the lines
2/ =0 and y = 1.
36. Show that the sum of the area beneath the curve 2/ = sin x from
X = to X = A; and that beneath the curve y = sin"^ x from x =
to X = sin A; is the area of a rectangle whose diagonal joins (0,0) and
(k, sin k).
100. Rational Fractions. Method of Partial Fractions. A frac-
tion N/D, whose numerator and denominator are poly-
nomials, is called a rational fraction. If such a rational
fraction is to be integrated, we first note whether or not the
degree of N is less than the degree of D. If not, we divide
166 THE CALCULUS [XII, § 100
N by D, to obtain a quotient Q and a remainder R whose
degree is less than that of D.
Example, yr = ^— , = 2 x H =— ; .
D x^ -{-X . 3^ -\-x
The integration of Q can be carried out at once. It
remains only to consider the integration of rational fractions
of the form R/D, where the degree of R is less than that of
D, We shall proceed to discuss the integration of such
forms, and we shall divide the discussion into several cases.
101. Case I. Denominator Linear or a Power of a Linear Form.
If the denominator of the fraction R/D (§ 100) is linear, or
is a power of a linear form, i,e. if
D = dx + b, or D = (ax + 6)",
the substitution u = ax + b transforms the integral of R/D
into a new form which is readily integrated. For this reason,
we shall reduce other cases to this case whenever possible.
102. Case n. Denominator the Product of Several Linear
Factors. If the denominator of the fraction R/D (§ 100)
is the product of several linear factors, the fractions R/D
may be replaced by a sum of simpler fractions. The
simpler fractions which compose this sum are called partial
fractions. Each linear factor ax + b oi D gives rise to one
such partial fraction, whose denominator is ax + b and
whose numerator is a constant. Each of these partial
fractions can be integrated as in § 101. The process is
illustrated by the following example.
Example. Evaluate the integral / 7 777 — r— pr? — t-ft: ^•
J {x — l)(a; + l)(a; +2)
The numerator of the integrand is already of less degree than the
denominator. . We first set down the partial fraction sum just described:
6a:2+3a;-15 _ A B C
(x-l)(a;H-l)(a;+2) a; - 1 ' a; + 1 ' x + 2'
XII, § 103] TECHNIQUE OF INTEGRATION 167
where the constants A^ Bj C are as yet unknown. To find them, we
clear of fractions, obtaining the equation
6x2 + 3x - 15 = A(x + l)(x + 2) + J5(x-l)(x+2) + C(x-l)ix +1),
which must hold for every value of x. If any three values of x are sub-
stituted for X in turn, we obtain three equations that may be solved for
A, B, and C, Values that are particularly simple are x = 1, x = —1,
and X = — 2. Substituting each of these in turn, we get A = — 1,
B = Q, C = h It follows that we may write
Qx^ + Sx-15 _ 1 6 1 .
(x- l)(a; + l)(a;+2) x - 1 ' x + 1 ' a; +2*
Each of the fractions on the right may be integrated readily as in
§ 101. Hence
6x2+3x-15 ^ /---Ij ./• 6 J .rNl
^ (x"l)(xH-l)(x + 2)'^ ^x-l'^^^x + l'^^^
dx
x+2
= - log (x- 1) +61og (x + 1) +log (x +2)
(X + 1)6 (X + 2)
= log
X- 1
103. Case m. Repeated Linear Factors. If one of the linear
factors of the denominator D (§ 100) is repeated, i.e., if there
is a factor of D of the form (dx + 6)**, that factor gives rise
to several partial fractions of the form
ax + b ' (ax + by ' (ax + by ' ' {ax + by
Otherwise the process remains as in § 102. This case is
illustrated by the following example,
/3 /p2 — X 4- 1
( 4-2V — 2')3 ^'
According to the processes described in §§ 102-103, we first write
3x2-x + l _ A B C D
(x-|-2)(x-3)3 x-f-2 ' X--3 ' (x-3)2 ' (x - 3)3 *
To find AyByCy and D, we clear of fractions
3 x2 - X + 1 = A(x - 3)3 + B{x + 2)(x - 3)2 + C(x + 2)(x - 3)
+ D{x + 2)
168 THE CALCULUS [XII, § 103
and then substitute for x, in turn, any four values of x. If we take
a; = - 2, 0, 1, and 3, in turn, we find A = - 3/25, B = 3/25, C =
12/5, D = 5. Hence
r 3a;2--g + l ,
J (x + 2)(x-3)8^
/ — Sdx r Sdx r \2dx . r 5dx
25(a; + 2) "^^ 25(x - 3) "^ ^ 5(x - 3)2 "^ ^ (a; - 3)3
= --7^1og(a;+2)+llog(^-3)- ^^ ^
25 ^' . -/ . 25*^e»v-' -/ 5(2; _ 3) 2(a;-3)2
=:Ai a; — 3 24 a; — 47
25^^ a; + 2 10 (a: - 3)2 *
104. Case IV. Quadratic Factors. If the denominator D
(§ 100) contains a quadratic factor aa^ + bx + c which can-
not be factored into real linear factors, we insert in the sum
of partial fractions one fraction whose denominator is that
quadratic factor and whose numerator is a linear expression
Ax + B. The resulting fraction
Ax + B
as? + 6x + c
can always be integrated. The process is illustrated in the
example which follows, and in exercises in the following list.
The general case is treated in the Tables, IV, B, No. 21,
p. 38.
( — lU 2 4-4.\ ^'
We first write
10a: -5 _ A Bx -^ C
(x- l)(a;2+4) a;--l ' a;2+4 *
Clearing of fractions, we have
10 a; - 5 = A(a:2 + 4) + (x - 1) (Bx + C).
Setting a: = 1, 0, — 1, in turn, we find A = 1, 5 = — 1, C = 9. Hence
/ 10 a; — 5 ^ _ f dx f x — 9 ,
(a;- l)(a:2+4)^"^ a; - 1 J x^ + ^^'
XII, § 105] TECHNIQUE OF INTEGRATION 169
The first integral on the right is easily evaluated by § 101. The second
integral on the right may be broken up into two parts which can be
integrated separately:
fj^4, ^ = ^ *^"^ i (S®® ^- 20, p. 160.)
Collecting all these partial results, we have
106. Case V. Repeated Quadratic Factors. If the denominator
D (§ 100) contains a quadratic factor to a power, i.e., if D has
a factor of the form {cu? + 6x + c)**, we insert among the par-
tial fractions a set of fractions analogous to those of § 103,
but with numerators that are linear, and with denominators
that are successive powers of the quadratic:
Ax + B . Bx + D , , Lx + M
ax^ + bx + c {ax? + 6x + c)^ {ax? + bx H-c)" '
The determination of the unknown constants A, B, (7, etc.,
is performed as before. In general, the resulting partial
fractions can always be integrated, but the problems become
more and more difficult as the exponent n increases. The
method of integration essentially depends on the process of
§99 {integration by parts). Difficult examples should not
be attempted without the assistance of the Tables, where
the results of integration by parts are given in the general
formulas 23 and 25, p. 38.* These, together with formulas
* Even when it is desired to proceed without the use of tables, it is best,
for difficult examples, to derive such formulas as 23 and 25, p. 38 (Tables),
once for all, by integration by parts, and then to use these formulas instead
of repeating the work in each example.
170 THE CALCULUS [XII, § 105
21, 22, and 24, p. 38, are sufficient to integrate any such
expression. The process is illustrated by the following
example.
Example. Evaluate the integral f -. o\/ o i o\o <^'
J {x — 2) (a;2 -j- 3)2
As indicated above, we first set
10 3:3 4- 7 a; -f 4 A Bx + C Dx + E
(a:-2)(a;2H-3)2 a: - 2 "*" x'^+Z "^(2:2+3)2
Clearing of fractions and determining the constants as before, we find
^ = 2, -5 = - 2, C = 6, D = 6, and £? = - 11. It follows that
r 10xM-_7^jf4 , _ f 2 /' -2a; + 6 r Gx-ll
^ (a;-2)(x2-f 3)2'^"y a;--2'^"^^ x^ + 3 '^ "+" 7 (a;2 + 3)2 '^•
The first integral on the right can be evaluated by § 101. The
second one can be evaluated as shown in the example of § 104. The last
integral may be evaluated either by an integration by parts, or by use
of the formulas 22 and 23 on p. 38 of the Tables.
EXERCISES
Carry out each of the following integrations.
^ r dx 1 /•/ 1 1 \j 1, «-i
r dx \ y X + a
f dx _ f dx _ 1 _, a; + 1
*• J x'i+2x + 10 J {x + 1)^+9 3 3
»• / x^ + 6^ + 10 °^°"(-+ ^)-
r__dx___ _ f dx ^lif.«?ZlJ:
^- 7a:24.2x--3"^ (a; + l)2-4 4'''^a;+3'
/'___i£_-ll 2x-2 ^ 1 . x-1
^' •/4a;2+4x--8"l2 ^^2x + 4 12 ^x + 2*
XII, § 105] TECHNIQUE OF INTEGRATION 171
In each of the following integrals, first prepare the integrand for in-
tegration as in §§ 101-105 ; then complete the integrations.
9. / ^ai'^s'^ + G ^ =log (^-2) +2Iog (x-3) =log [(x- 2) (x-Sn
^ , , dx ^ x + tan~i x,
12. / » frV ,, (te = log Vx2 + 2a;H-2 - 2 tan"! (1 + x).
•/ x^ -|- J X -|- J
<• /• <^ 1 X _i a; 1, ,x
14. / 4:=f d^. 16. f- ' ^
J x^-'2x *^ X
2+3X+2
1A f__xdx__ ^- r x^dx
^^' J x'^-bx + % J x«--4*
x3+x ^^x3-7xH-6
^x3H-2x2* ^2x3H-7x2 + 6x'
^^- /(x--l)2(x--2)' ^'- /(X--1)(X2 + 1)"
« . /• X dx OK ^2x2 -f 1
^ 7 (x+2)(x+3)» - "• JirfW^-
^- J {x + l)(x-2)^'^- ^- yF^=T6'
28- J (a.2 _ 9)2- =*»• Ja;*-5x« + 4*'-
Derive each of the following formulas.
30 r ^ ^i— loK "^+".
•/ (oa; 4- 6) (ma +n) an — bm " <tx+b
•1 /• a;<fa: _ I . (x + o)»
~. /• « <ir 5 , Vji' + o J. Va . „_,
«*• / fx» + aHx + fe) °^+^^°gT+r^^q:^*^'
X
(x2 + a)(x + 6) a + 62"6 a;-|-6 a-|-62 ^
172 THE CALCULUS [XII, § 105
33. Derive each of the formulas Nos. 18-24, Tables , IV, A.
dx.
Evaluate each of the following definite integrals.
^«- 12 a; (a;- 4)
So. / -p jrr^ dx.
[Note. Further practice in definite integration may be had by insert-
ing various limits in the previous exercises.]
Carry out each of the following integrations after reducing them to
algebraic form by a proper substitution.
^_ r ainx , -,« /• cos a: , <,^ r ^ dx
37. / zr-y o- ^' 38. / -. ^-5- dx, 39. / :; 5- •
^ 1 + cos2 X J ^ — sm2 X •/ 1 — 62»
.- /• , /• cosa; , 1, 1 4-sin a; , . /x , xX
40. I secxdc - I z j-^— oc = olog:i -. — = log tan I -+- I •
J ~'l — sm^x 2 *=•! — sma; ® \4 2/
41. JcBCxdc, 42. jBechxdc. 43. J each xdc.
.. c sec* a; , a^ C^ ^ ^~* ^
J tan a; — tan^ x J eF +e~*
106. Rationalization of Linear Radicals. If the integrand
is rational except for a radical of the form Vox + 6, the sub-
stitution of a new letter for the radical
r = y/ax + h
renders the new integrand rational.
Example. Find / — t-j—: — cLc-
Setting r = Vx + 2, we have x = r2 — 2 and cte = 2 r dr; hence
= 2/(r + 1 - f:^)^^ = r^ +2r-- 21og(r + 1) + C
= a; + 2 + 2 VxT2 - 2 log (VJT2 + 1) + C.
XII, § 107] TECHNIQUE OF INTEGRATION 173
The same plan — substitiUion of a new letter for the essential
radical — is successful in a large number of cases, including
all those in which the radical is of one of the forms:
«.,,(^ + „v.,(^J-
+
where n is an integer. Integral powers of the essential radiqal
may also occur in the integrand.
107. Quadratic Irrationals: Va + 6x d= x^. If the integral
involves a quadratic irrational, either of several methods
may be successful, and at least one of the following always
succeeds:
{A) If the quadratic Q = a + 6x i x? can be factored
into real factors, we have
K a "T" iC
and the method of § 106 can be used. The resulting expres-
sions are sometimes not so simple, however, as those found by
one of the following processes.
{B) If the term in x^ is positive, either of the substitutions
VQ = t + x, \/Q = f-a:,
will be found advantageous. One of these substitutions may
lead to simpler forms than the other in a given example.
(C) Completing the square imder the radical sign throws
the radical in the form
VQ = Vztfc=fc(a:=tc)2;
the substitution x zt c = y certainly simplifies the integral,
and may throw it in a form which can be recognized instantly.
174 THE CALCULUS PQI, § 107
(Z>) After completing the square under the radical sign,
the radical will take one of the forms Vfc2 — a;^, Vfc^ + x^,
Vx^ — fc^. Then a trigonometric substitution often leads
to a simple form. Thus:
ii X ^ k sin 0, Vk'^ — x^ becomes k cos 0;
ii X = k tan 0, Vk^ + x^ becomes k sec 0;
if X = fc sec ^, Vx2 — fc2 becomes k tan 0,
Example. Let Vq = Vx^ ± a^; show the efifect of substituting
Vq-^t-x,
U Va;2 ± o2 = < - a;, we find
and the transformed integrand is surely rational. Carrying out these
transformations in the simple examples which follow, we find
(i)
(ii)
= log (x + Vq) + C, where Q = x^ ± aK
>s
'^±flog*=^±|log(x + v/Q) + C.
These integrals are important and are repeated in the Table of In-
tegrals, Tables, IV, C, 33, 45a. Many other integrals can be reduced
to these two or to that of Ex. 3, p. 163, or to Rules [XVI] or [XX] by
process (C) above.
EXERCISES
1. fx Vx^n. dx = (1/15) (6 » + 4) (a: - 1)8/2.
2. / ■ — dx -2 Vx + 1 + log , •
3. f — r^ — = 2 tan-i Vx - 1.
•^ xVx— 1
4. / J]^ dx = vT=T2 + sin-i X.
^ \1 -\-x
XII, § 107] TECHNIQUE OF INTEGRATION 175
6 f ^ ^ -J- f,,n-l \^Th
J (ox + 2 6) Vox -\-h aVh \ 6 '
/■ (to _ ViTl 1 v^Ti + 1
7. /(a + &a;)3/2 d^ = ^ (« + 5a;)6/2.
A f ^ ■= 2
''•^ (a + 6x)3/2 hVT+bi'
Carry out each of the following integratioDS.
9. fxVTTidx. 10. f ^L^
jj r dx ^2 r fr+
"•/>(^'^- "•/>i?r'*«-
17. /x ^r+^(to. 18. fx yl3x + 7dx.
dx
^ X
- dx.
X
^ (1 + x)3/2 ^0 1 — V5
+ y/T+~x
26. r'-^^dx, 26. r-
Carry out each of the following integrations by first making an ap-
propriate substitution.
„f_^osx^ 3^ r_smxvc
^ 3- Vsmx ^ 1-2V
X Vcosx
cosx
dx.
28. /^^3. 31. f'-±^^cosxdx.
QQ r sec2 X (to ^„ /• sin x (to
V2 + 3tanx ' -^ (2 + 3 cos x)3/2
176 THE CALCULUS [XII, § 107
Substitution of a new letter for the essential radical is immediately
successful in the following integrals.
S3. fxVl+x^dx, 86. f (^+^)/^ . 39. fx^Va + bx^dx.
V2 H- 2 a; + a;2
7^ dx ^^ r xdx ,^ r ^
36. /'x(H-x2)8/2dx.38. / , ff^wo ' *!• / r"'"^ '
•^ ^ (a + 6a:2)p/« y Va + te»
Carry out each of the following integrations.
/dx ____^^^
V^^n = log (* + VS?^^) .
dx.
s^ r dx .1, Va:2 — a^
43. / — , = - tan-i — =^
^ X V a;2 — a? a a
' -^ (X + a) Va2 - a;2 a \o + x
46. f ^ ^ = tan-i — ^=.
•^ (l+2a;2) Vl+a;2 V52T1
/3J "4" 1 ^——
^/r-^-^ dx = sin-ix - Vl - x\
4S. f , '^ ■ 49./ , '^ • 60./ '^
V4 x2 + 1 * "^ a; V4 a;2 + 1 * -^ a;2 V4a;2 + 1
61. / dx, 62. / 5 — dx. 63. / ■ , dx.
J X . J X^ J Vl — x2
The following integrations may be performed by the methods of
§ 107; note especially method (C), which consists in completing the
square under the radical.
/dfX _^_^^_^_^_
:^^=^==log(2x~H-2Va:2-a: + I).
56. /•-_^==sin-i2^.
66. f , = sin~^ ^ ^ = vers'^o [+ const.].
^ V4a;--a;2 2 2
XII, § 108] TECHNIQUE OF INTEGRATION 177
67. / — - == sm 1 7=r-
68. / — = sm 1 — -.
^ xy/lx^-\-Qx-l 4a;
69. /- , ^ . 60. r , ^ > 61. /-- ^
•^ v2»2H-a; + l •^vH-a;-2x2- -^ Vl--2a;-a;2
62. / ^ ^ 66. /'Vl4-xH-a;25i.
•^ V6a: — x2--5 "^
63. r , 66. rVSxa + lOx + Oda;.
•^ X Va;2 + 2 a; + 3 "^
64. f ^ 67. f X Vr^\^xTx^ dx,
•^ (xH-4)Va;2+3a;-4 -^
Integrate by parts, [VI], each of the following integrals.
68. /"x sin-i X dx. 71. J {Zx — 2) sin"! x (ix.
^- /•sin~ix, -„ /•2+x2
69. / 2~ ^- ^^* J — 2~ ^^ * ^•
70. jx cos~i X dx, 73. j (sin~i x + 2 x cos~i x) dx.
108. Trigonometric Integrals. A number of trigonometric
integrals have been evaluated in the preceding lists of exer-
cises. The processes explained in Exs. 14, 15, 17, 25, 26,
28, 29, 30, 31, pp. 160-61, may be generalized and stated in
the form of standard processes. They depend chiefly upon
the use of well known relations between the trigonometric
functions. (See Tables, pp. 12-13.)
Since it is desirable to avoid the introduction of radicals
that were not present in the original example, we do not
ordinarily use the trigonometric formulas that involve a
square root. But' it is desirable to notice that
(a) Any even power of sin x can be changed into a sum of
powers of cos x by the relation sin^x + cos^x = 1.
(6) Any even power of cos x can be changed into a sum of
powers of sin x by the same relation.
178 THE CALCULUS [XII, § 108
(c) Similarly, sums of even powers of tan x may be changed
into simis of even powers of sec x, and conversely, by the
relation sec^x = 1 + tan^x.
(d) All the trigonometric functions may be changed into
forms in sinx and cosx without introducing any new
radicals.
(e) The square of sin x and the square of cos x may be
expressed in terms of the first power of cos 2 a; by means of
the formula cos 2 x = cos^ x — sin^ x,
(/) All the trigonometric functions can be expressed
rationally in terms of tan (x/2).
109. Integration of Odd Powers of sin x or of cos-x. Any
odd power of sin x may be integrated, as in Ex. 17, p. 160,
by the substitution u = cos x.
Likewise, any odd power of cosx may be integrated by
means of the substitution u = sin x. (See Ex. 25, p. 161.)
More generally any integral of the form .
/
sin" X cos*" X dx
can be integrated if either m or n is an odd integer. If n is
odd, set u = cos x; if m is odd, set i* = sin x. This process
is illustrated by Exs. 14, 15, 16, 18, 19, p. 160; and by the
following example.
Example. Evaluate the integral J* sin3/2 x cos^ x dx.
Set w = sin a;; then
/ sin8/2a; cos^ xdx = /^sin3/2 x(l — sin^ a;)cos x dx
= y w3/2 (1 _ y2) du
= % w6/2 - ? w«/2 = ? gin6/2 x-\ sin9/2 X.
5 9 O U
XII, § no] TECHNIQUE OF INTEGRATION 179
110. Reduction Formulas. If the integrand is an even
power of sin x (or of cos x) we may use § 108 (e), as in Exs.
30, 31, p. 161, or we may proceed by integration by parts,
as follows. Let us first write
J sin" xdx = I sin"~^ a: sin a: dx
and then integrate by parts (§ 99) by taking
u = sin""^ Xy dv = sin x dx,
du = {n — 1) sin**"^ x cos xdx,v = — cos x.
Then we obtain
y^sin** X dx = — cos x sin"~^ x+ (n — 1) Jsin^-^x cos^ x dx.
Replacing cos^x by 1 — sin^x in the last integral and breaking
it up into two integrals, we have,
J&in^x dx = — cos x sin""^x + (n — 1) Jsin^^^^xdx
— (n —l)fsm^xdx.
Transposing the last integral and dividing by n, we find the
formula {Tables, IV, E, 57)
(1) fsin" xdx^- ^osx^n^-^x n-1 T .^„_,^ ^
J n n J
Repeated applications of this formula reduce the left-hand
side to integrals that involve sin'*"*^, sin""* ^, • • *, down to
sinx if n is odd, or to sin^x (= 1) if n is even. In either
case, the integration can be completed by the use of a stand-
ard formula.
Ixx a similar manner, we obtain the formula {Tables, IV, E,
60)
(2) fcos-x dx = ^^^^^^^""^ + ^^^ fcos-^xdx.
J n n J
180 THE CALCULUS [XII, § 110
By solving these formulas for the integral on the right-hand
side, we obtain
(3) fsin'^-'xdx = ""^^ »'"""' ^ + -?L_ fsin-a; dx;
J n — 1 n — IJ
and
(4) fcos"-'' X dx = - ^^^^0^]-'^ + _??_ fcos" X dx.
J n— 1 n — IJ
These formulas raise the exponent in the integrand. Hence
they are useful in integrating negative powers of sinx and
cos Xy i.e, positive powers of esc x and of sec x.
An analogous integration by parts leads to the formula
(Tables, IV, E, 64)
(5) fsin"
X Qo^^xdx
Sin""*"^ X COS*" ^X ^ m — I C ' n m-2 J
. ; — I sm"a: cos*" ^xdx
m + n m + nJ
sin"~^a; cos*""*"^x , n — \ C - n-2 m j
— . — I sm" ^x cos*" X dx,
m + nJ
m + n
The proof of this formula is left as an exercise for the student.
If either m or ti in (5) is an odd integer, the process of § 109
is usually quicker. But if both m and n are even integers,
the formula (5) is very useful.
Example 1. Evaluate J* cos* x dx.
/. , cos3 X8mx , S r „ ,
cos* xdx — -T — + 7 / cos2 X dx
cos3 a; sin a; , 3 fcos x sin a: , 1 f ■, "]
= i +iL 2—+2J'^j
cos^ re sin a; , 3 . ,3
= -: h Q cos xsmx -i-QX.
4 o o
XII, § 111] TECHNIQUE OF INTEGRATION 181
If, in this example, we should follow the method suggested under
(e) of § 108, we would write
+ COS 2x\^
A t 9 ^9 /l+oos2a:\'
= ^ (1+2 COS 2 a: + cos2 2 x)
1 /i I o o , 1 + cos 4 a;\
= I ^1 + 2 cos 2 a: H ^ )
Then
3 a; . sin 2 a: . sin 4 a:
/. J d a; , sm z a: ,
cos* xdx — -^ -\ -^ h
8 ' 4 ' 32
Example 2. Evaluate f sec^ x dx.
By (4), with n = — 1, we find
//* sin X cos"^ X ■"" 1 /*
sec' xdx = J cos""3 xdx = ^-^ f- —-^ J cos~i x dx
sin X sec^ x , 1 /*
2 "^2^
sec X dx.
The integration may now be completed by means of Ex. 40, p. 172,
which is essentially an application of § 109.
Example 3. Evaluate ysin^ x cos* x dx.
By the second part of (5), we have
/• 9 A J sinarcos^a; , 1 /• . ,
sm2 X cos* xdx — ^ \- ^1 cos* x dx.
The example may now be completed by following Example 1.
111. Powers of tan x and of sec x. Any even power of tan x
naay be reduced to a sum of even powers of sec a; (§ 108
(c) ). This may be integrated by the method mentioned in
§110.
Another method of integrating any even power of sec x
consists in making the substitution u = tan x. Since
du = sec2 X dxy and since even powers of sec x may be
reduced to a sum of even powers of tan x (§ 108 (c) ), the
integration becomes very simple, as is illustrated by Exam-
ple 1 below.
182 THE CALCULUS [XII, § 111
Odd powers of sec x can be most readily integrated by the
method of § 110, as shown in Example 2, § 110.
Any positive integral power of tan x may be integrated by
reducing it to one of the two integrals:
{\)fi2iiixdx= — log cos x= log sec x. (See Exs. 33, 34, p. 161.)
{2)ft2Ji^xdx =f{^Q? X — 1) dx = tan x — x,
by means of the reduction formula (see Tables IV, E, 70) :
(3) I tan" xdx= ^^_ ^ — | tan'*-^ x dx.
This reduction formula is obtained as follows:
Jtsin^ xdx = Jta,n^~^ x tan^ xdx= fidJi^'^ x (sec^ x — l)dx
= JtdM^'^x sec^xdx —JiBXi^'^x dx.
If we set u = tan x in the first integral on the right-hand
side, we obtain the formula (3).
Example 1. Evaluate the integral ysec* x dx.
Put u = tan x\ then du — sec^ x dx, and we may write
/ sec* xdx =J sec2 x sec^ x dx —JO- + tan^ x) sec^ x dx
= J (I +v^)du ^u + -^ = tan x-\ 5 — •
Example 2. Evaluate y tan* x dx.
As indicated above, we may write
/^tan* X dx = /^(sec^ x—l)^dx = /^(sec* x — 2 sec* x-^1) dx
= /^sec* X dx — 2 tan x-\-x.
The remaining integral may be integrated as in Example 1. The stu-
dent should also apply the reduction formula (3) to this integral.
Example 3. Evaluate y tan^ x dx.
Applying (3) we have
/tan' xdx ^ r — /tan xdx = 7 + log cos x.
n— I •^ n — 1°
XII, § 112] TECHNIQUE OF INTEGRATION 183
EXERCISES
Evaluate the following integrals.
1. / sin* X dx. 2. fsin^ x cos^ x dx. 3. /^tan* x dx.
4. / tan^ X dx. 6. Jsec^ x tan* x dx. 6. Jca(^ x dx.
7. Jta,n X sec* xdx. 8. / esc* x dx. 9. Jcsc^ x dx.
10. /^cot* X dx. 11. fcot^ X dx. 12. Ttan^ x csc^ a: da;.
13. fcSCXCOt^Xdx. 14. /-: • 16. /-: r— •
^ ^ sin a; cos x ^ sin a? cos^ x
le. / .^'fe ^ ■ 17. /?«L^±^dr. 18. /• ^ '^ ■
^ sin* a; cos* a; J suxx J Vsin x cos* x
//* /"COS^ X
sin2 a: cos* a: efcr. 20. / sin* x cos^ a: dx, 21. / . , dx.
J J sin^a;
22. f^^dx. 23. fsm^xco^xdx. 24. f . ^^ , -
J sin2 X •/ ^ sm2 a: cos* a:
^_ rsin* xdx g.^ r%\v^ x , «_ rcos^ x ,
25. / 26. / — ^-dx. 27. I . ^ dx.
J cos a; J cos^ a: ^ sin* x
28. /^sin* a; cos* a; da;. 29. /^sin 4a: cos Sa^dir. 30. fainSx cos 4a; dx.
31. Verify formula 111 of Table IV, E.
32. Verify formula 112 of Table IV, E.
112. Elliptic and Other Integrals. If the only irrationality
is VQ, where Q is a polynomial of the third or fourth de-
gree, the integral is called an elliptic integral. While no
treatment of these integrals is given here, they are treated
briefly in tables of integrals, and their values have been com-
puted in the form of tables.* See Tables, V, D, E.
* Some idea of these quantities may be obtained by imagining some per-
son ignorant of logarithms. Then y (l/x) dx would be beyond his powers,
and we should tell him ** values of the integral f (1/x) dk can he found taJtrw-
lated" which is precisely what is done in tables of Napierian logarithms.
Of course as little as possible is tabulated; other allied forms are reduced
to those tabulated by means of special formulas, given in the tables.
184 THE CALCULUS [XII, § 113
The discussion of such integrals, as well as of those in
which Q is of degree higher than four, is beyond the scope
of this book.
113. Binomial Differentials. Among the forms which are
shown in tables of integrals to be reducible to simpler ones are
the so-called binomial differentials:
f (ax"" + hy^xT dx.
It is shown by integration by parts that such forms can be
replaced by any one of the following combinations, where u
stands for (ax*^ + b) :
(1) /w^x*» dx = (Ai) u^x"^-^' + (Bi) fu^-'oT dx,
(2) /ii^x"» dx = (A2) i^^+'x^+^ + {B2) /w^+'x^ dx,
(3) / w^x"» dx = (A3) u^'-^'sT-^^ + (Bs) / w^x^+" dx,
(4) / w^x*" dx = (A4) i^^+^a:^-~+^ + (B^) /w^x"*-" dx,
where Ai, A2, Az, A^, Bi, B2, B3, B^, are certain constants.
These rules may be used either by direct substitution from
a table of integrals in which the values of the constants are
given in general * (see Tables, IV, D, 51-54), or we may
denote the unknown constants by letters and find their
values by differentiating both sides and comparing coeffi-
cients.
* Such formulas are called reduction formulas; many other such for-
mulas — notably for trigonometric functions — are given in tables of inte-
grals. (See Tables, IV, Ea, 57, 60, 64, etc.) It is strongly advised that no
effort be made to memorize any of these forms, — not even the skeleton
forms given above. A far more profitable effort is to grasp the essential
notion of the types of changes which can be made in these and other in-
tegrals, so that good judgment is formed concerning the possibility of
integrating given expressions. Then the actual integration is usually per-
formed by means of a table. See also Tables, IV, Ea, 78, 82 (&) ; Eft, 85,
86; Ec, 92-94; Ed, 98, 106; B, 17 (6), 25; etc.
XII, § 114] TECHNIQUE OF INTEGRATION 185
Example 1./^-^^-^^ = ^_^__ + B/_^__,by (2).
Differentiating and comparing coefficients of x'^ and a;0, we find 5=0
and A — 1/6; hence
/ (0x2+6)3/2 = 2,V(i+6) ' ^'^'*'
Example 2. J ^^^ ^ ^^3/^ = (^2+6)1/2 +^7 (00^2 +6)3/2' by (4).
Here A = 1/a, 5 = - 2 6/a,
J /• x^dx _ ax^ + 26
(0x2 + 5)3/2 o2V(ax2 + 6) *
114. General Remarks. The student will see that integra-
tion is largely a trial process, the success of which is dependent
upon a ready knowledge of algebraic and trigonometric trans-
formations. Skill will come only from constant practice.
A very considerable help in this practice is a table of integrals
(see Tables y IV, A-H). The student should apply his intelli-
gence in the use of such tables, testing the results there
given, endeavoring to see how they are obtained, studying
the classification of the table; in brief, mastering the table,
not becoming a slave to it.
In the list which follows, many examples can be done by
the processes mentioned above. The exercises 43-97 may
be reserved for practice in using a table of integrals.
REVIEW EXERCISES
^'J(x+2)^'^' ' J (2a: + 3)2 ^'
« f ^ A. r x2 + 1 J
^ J a:3+3x2* *• J i^^ '^^•
(a;-2)3 • J (x + l)(x-l)2
' a;3 + i /' a;3+2a;2.
;c2 - 3 X + 3 J X^-Ti'^
J
186 THE CALCULUS [XII, § 114
Q f ^ lA c ^ 11 r ^
J (1 - x2)2' "• ^ 4 a;4 + 5 x2 + 1* ^ ^ 16 - x^'
^A /* x^ + 3 X J ^^ /* x^rfa; ^. /*_^xMx_
"• y a:2 + 3a: + 4'^- ^^- 7 x4-2a:2 + l' ^*- ^ (5 - 7 x3)3-
xdx ^» r x^dx --/• dx
"• f^^- "• fwT2jn' "• Z:^
(ix.
a;2 + 2a; + 5 -^ a:2 + 4a; + 2
18. fxy/JT2dx. 19. f—%=^ 20. f-r^=^'
^ ^ x^x—\ ^ V ax + 6
21 r ^ ^ . 22 r /^ . 23. c^m^i^,
^ V(a + 6x)3 -^ V(a+x)3 -^ (x- 1)3/2
/' (2+x)cfo /- x2rfx /-(Ijfv^
2*- 7 ^3"^ • 2^- 7 (a + 6x3)3/2- 26. ^ ^/2
27. f , /^ w ' 28. / 3^!Ji » 29. fxylzlTpIdx.
'"• ^ (9x2-3)3/4* ^^- J Vf3^ ^ X(X2- 1)3/2*
^- J xHi- x2)3/2 ^- 7 vrT^+ ^rvx
35. f-fi%- 36. /-^. 37. Z-,^^^.
38. f l'^^ dx, 39. /"x ^3x + 7(Zx. 40. /x ^'a + 6x2 dx.
41. y x3 (a + x2)i/3 dx 42. ^ xS (1 + x3)i/3 cte.
In the following integrations, use Table IV freely.
f dx .. C dx -- f 3x + 2 ,
«• J p+l)2- **• 7 (x2 + 5)3* ***• 7 (a;2 + 3)3^-
AA r 5x-3 , -- r rfx .^ r a?<fa?
***• ^ (2 X2 - 1)2^^- *'• ^ (x2 + 2X + 5)2* ***• ^ (x + l)2(x+2)2
49. / . 60. / . 61. I .>- , . o^o/a '
y a;3 y/x^ - 4 -^ Vx2 - a2 ^ (7 + 4 x3)2/3
62. / (a2 - x2)3/2 dx, 63. / (x2 - a2)3/2 dx, 64. f —^^—^.
^^' J {a^ 6x2)3/2- °^- J (1 - a;2)3/2' "''7 (« + 6x2)6/2*
XII, § 114] TECHNIQUE OF INTEGRATION 187
68. f^^^-^dx, 59. f^^^-r—dx. 60. f Bin^ x cos^ x dx,
•/ sin a; •/ cos^ x J
^^' J 2 + sin^* ^^ J 2- 3 cos ^*' ^* ^ 2 + 5 sin ^'
64. f cos« a da. 65. ^ ctn2 Sxdx, 66. ^ sin8/2 ^ cos^ ^ d^.
^0 (ic — 2)2 Ji (2 X — 1)» Jo 2 a;2 + 3
70. / ,5—5 — xcte. 71. / — 7==- 72. / . dx.
Jo 2a;2 — 3 •^2 a;v2a; + 3 -'^ v a;2 — 3
^3^(l±£l^. 74Y-7^=- 76./^^=^.
Jo V4 — x2 Ji Va;2 + 1 Ji Va;2 — 1
^^' Jo (2 a: + 1) (0:2 + 2)* ^7- J _^ Vx^-Sx
Find the values of the following definite integrals by using the tabu-
lated numerical values: Tables^ V, A-H:
ei'dx. 84. / — 7^ — dx = I cosh x dx.
z-O J x=0 ^ •/sr-0
yr,2-2.5 ^=2.3
e2»(ir. 86. / sinha;cte.
x-1.2 ^x-o
y^x=1.4 >r=2 ^ -lx-2
e-^' (to. 86. / , = cosh-i x
x-O Jx«l Vx2— 1 J«-l
-dx. 87. / /^ =sinh-ia; .
x-i a? ^x-o va;2 + 1 Jx-o
— ^dx. 88. r ,Z1_ =co8h-i^
x-3.5 X — Z Jx-2 Va;2 — 4 2Jx-2
^x-3.2 1 ^-14.4 dx . , ,a;T-i4-4
83. / -s — 7(to. 89. / ■ . = sinh~^ »
Jx-2.1 a:^ — 1 Jx=o Vajz + Q ojx-o
a ^ 30** j/i
90. r = = /^ (J, 30°), Tables, V, D.
Je»o- Vl- (l/4)sin2^
91. r^'*^ Vl- (l/4)sin2^dd = E (J, 45°), Taftfes, V, E.
=■ 93. f Vl - .25 sin2 ^ dd.
5=16" Vl — .04sin2^ Jtf-30«»
y.=r-l/2 ^^ J^ ^-30- J^
./x-O Vl — ^2 Vl — 9K ^2 Jfl-0»
^y-O Vl-x2Vl-,25»2 J^-o' Vl-.25sin2^'
if X = sin d.
188
THE CALCULUS
[XII, § 114
- .36 a;2
a;2
(2x
(^x
.9-90<>
= /* Vl - .36sm2 ^ dd, if a; = sin tf.
^X-V2/2 \ 1 —
-.16x2
cte.
a;- 1/2 Vl — a:2 Vl — .49 a;2 J x^-nf^ \ i — x^
[Note. Many of the exercises in preceding lists may be used for
additional practice in use of the tables.]
115. Limits Infinite. Horizontal Asymptote. If a curve
approaches the x-axis as an asymptote, it is conceivable that
the total area between the x-axis, the curve, and a left-hand
vertical boundary may exist; by this total area we mean the
limit of the area from the left-hand
boundary out to any vertical line
X = m, as m becomes infinite.
Example 1. The area under the
curve y = e"* from the ^-axis to the
\
^
V
y
=
c-^
\
\
V
\
1
'^
i
^
^
^
z=
-.m
ordinate a; = m is
e"^ dx = 1 — e"».
Fig. 42.
-|a;=»oo ^-
*ao
>-»
dx
]X^m y,-U-
= f
As m becomes infinite e~"» approaches
zero; hence
e-* dx — lim (1 — 6~*») = 1,
... _. X^Q fflp-MO
and we say that the total area under the curve y = e~* from a; =
to x = + 00 is 1.
Example 2. The area imder the hyperbola y = 1/x from a; = 1 to
a: = m is
A \ = / — = log a: = log w.
Jx"! Jx'^l X Jx-i
As m becomes infinite, log m becomes infinite, and
lim ] A \ [ = lim log m
does not exist; hence we say that the total area between the a>axis and
the hyperbola from a; = 1 to a: = oo does not exist*
*This is the standard short expression to denote what is quite obvious, —
that the area up to a; =m becomes infinite as m becomes infinite. This result
makes any consideration of the area up to a; = x perfectly useless; hence the
expression "fails to exist," which is slightly more general-
XII, § 116] TECHNIQUE OF INTEGRATION
m
116. Integrand Infinite. Vertical Asymptotes. If the func-
tion to be integrated becomes infinite, the situation is pre-
cisely similar to that of § 115; graphically, the curve whose
area is represented by the integral has in this case a vertical
asymptote.
If / (x) becomes infinite at one of the limits of integration,
x = bywe define the integral, as in § 115, by a limit process:
J'*x = 6 rb-c
f (x) dx = lim I / (x) dx.
x=a c— >0 ^ a
A similar definition applies if / {x) becomes infinite at the
lower limit, as in the following example.
Example 1. The area between the curve y = 1/VS and the two
axes, from a; = to a; = 1, is
= / — p dx = lim / —7= dx
x=0 •/a;«0 y/x c->0 L*^x=c Va; J
= lim \2^/x\ '"' = lim [2 - 2Vcl = 2.
c— »0 L J x-c c-*0 L J
Example 2. The area between the
hyperbola y = 1/a;, the vertical line
x = 1, and the two axes, does not
exist For,
/ - cte = log a; = — log c,
but lim (— log c) as c-*0 does not
exist, for — log c becomes infinite as
c->0.
Example 3. The ar ea between
the curve 2/ = 1/ Va; — 1, its asymp-
tote a; = 1, and the line a; = 2 is
'
t
y
y
=
1
X
,
\
^
^\
1
^\
1
^^
S.
^
^^
^
^
.^.
^
C
1
(
X
^mmmi
Fig. 43.
/
'2 dx
= lim (^
dx
-^= - X.... . 3, = |lim (1 - c2/3) = ?
190
THE CALCULUS
[XII, S U6
£!xAMPLE 4. Show that f 1/x^ dz docs not exist. The ordinate
y '^ l/z^becomeainfiaiteaaxapproacheszero, i.e. the ^-axia is a vertical
asymptote. Hence to find the given integral we must l^roceed as
above, breaking the original integral
into two parts:
1-11-1
y ■ IJ
1
-
'
-
■
. -, -^
t
-T- ^
i
'l_
/
-
>
± j_
-a::H^
I.
The limit of neither exists since 1/c
becomes infinite as c = 0; hence the
given iat«gral does not exist.
Carelessness in such cases results In
absurdly false answers; thus if no atten-
tion were paid to the nature of the
''^' **■ curve, some person might write:
which is ridiculous (see Fig. 44).
The only general rule is to follow the principles of
§§ 115-116 in all cases of infinite hmits or discontinuous
integrands. Sucii integrals are called improper integrals.
EZERCISSS
-^i::!:-
Verify each of the following results.
'■/.TS'
non-exist«Dt.
t. r — ^^ =
Ji.sV2z-3
'■/,'.2
IS non-existent
! determinate if n < 1,
non-existent if n £ 1.
XII, § 116] TECHNIQUE OF INTEGRATION 191
dx
State a similar i*ule for / 77 — ri \
J a {hx + ky
9
Va2 — 0:2 2 * Jo Vox — x^ ^
^ xdx __ - -n /'^^ cte
Vl — a;2 "" ' * J-i a;2+5a:+4
10. /* , = 1. 12. /* o ■ ^ . ^ is non-existent.
Jo Vl — a:2 •/ -1
IT IT »
.2 ^2 ^2
13. Show that the integrals f tan x dx, f ctn a; (is, f sec x dx,
Jo Jo Jo
y— are all non-existent.
X
Verify each of the following results:
/•* dx 1 /•* dx
-a7=: is non-existent. 17. f 77-^ — ttt? is non-existent.
/** ^ ' j determinate if n > 1,
* Jo (1 -\-x)^ J non-existent if n ^ 1.
19. I ^rn — s = o* 22. / (ix is non-existent.
Jo l+aj2 2 Ji a;
20. r*T^2=J^- 23. f^e-^dx^l
Ja cfi +x^ 4a Jo
,r = 1. 24. r
1 a;2va;2 — 1 Jo
e2* do; is non-existent.
Determine the area between each of the following curves, the x-
ajda, and the ordinates at the values of x indicated.
25. 2/3(a:— 1)* = 1; a; = to9. Ans, 9.
26. aJ2/2 (1 +a:)2 = 4; a; = to 4. Ans, 4 tan-i 2.
27. 2/2a:4 (l+x) = 1; a; = to 3. Ans. 00.
28. x^y^ (a;2 - 1) = 9; a; = 1 to 2. Ans. 2 x.
29. 2/3 (a; _ 1)2 = 8 x3; a; = to 3. ilrw. 9 yl2 +9/2.
30. a;22/2 (a;2 + 9) = 1; a; = 4 to 00. Ans. | log 2.
31. j/2 (1 4. a;)4 = a?; a; = to 00 . Arw. x.
32. 2/3(a: + l)2 = l;a; = 0to «. Ans, 00.
CHAPTER XIII
INTEGRALS AS LIMITS OF SUMS
117. Step-by-step Process. The total amount of a variable
quantity whose rate of change (derivative) is given [i.e. the
integral of the rate] can be obtained in another way. The
method about to be explained has many theoretical ad-
vantages and one decidedly practical advantage, namely in
its application to the approximate evaluation of integrals
when the indicated integration cannot be carried out.
For example, imagine a train whose speed is increasing.
The distance it travels cannot be found by multiplying the
speed by the time; but we can get the total distance approxi-
mately by steps, computing (approximately) the distance
traveled in each second as if the train were actually going at
a constant speed during that second, and adding all these
results to form a total distance traveled.
If the speed increases steadily from zero to 30 mi. per hour,
in 44 sec, that is, from zero to 44 ft. per second in 44 sec,
the increase in speed each second (acceleration) is 1 ft. per
second. Hence the speeds at the beginnings of each of the
seconds are 0, 1, 2, 3, • • •, etc.
If we use as the speed during each second the speed at
the beginning of that second, we should find the total distance
(approximately)
s = + l + 2 + 3+--+42 + 43 = ^^-^ = 946,
which is evidently a little too low.
192
XIII, § 117] INTEGRALS AS LIMITS OF SUMS 193
If we use as the speed during each second the speed at the
end of that second, we should get (approximately)
44-45
s = l + 2 + 3 + 4+... +43 + 44 = — 2— = 990,
which is evidently too high. But these values differ only
by 44 ft. ; and we are sure that the desired distance is between
946 and 990 ft.
If we reduce the length of the intervals, the result will be
still more accurate; thus if, in the preceding example, the
distances be computed by half seconds, it is easily shown
that the distance is between 957 ft. and 979 ft.; if the
steps are taken 1/10 second each, the distance is foimd to
be between 965.8 ft. and 970.2 ft.
Evidently, the exact distance is the limit approached by
this step-by-step summation as the steps A^ approach zero:
= 1 vdt= \ tdt= ^\
t^o Jt=o Jt^o 2J<=o
968.
We note particularly that the two results for s are surely
equal; hence we obtain the important result:
r°^\df = limit;] -Ai + t;] -A^ + vl -A/H [•
118. Approximate Summation. This step-by-step process of
sununation to find a given total is of such general application,
and is so valuable even in cases where no limit is taken, that
we shall stop to consider a few examples, in which the methods
employed are either obvious or are indicated in the discussion
of the example.
Thus, areas are often computed approximately by dividing
them into convenient strips. We have seen in § 55 that
if A denotes the area under a curve between x = a and
194
THE CALCULUS
[XIII, § 118
x=b, then the rate of increase of A is the height h of the
curve:
dA
y=R{x)
dx
= h = R(x),
Fig. 46.
where R (x) is the rate
of increase of il, and is
also the height of the
curve.
For a parabola, /i =a^, we may find the area A approxi-
mately between x = — 1 and x = 2 by dividing that interval
into smaller pieces and computing
(approximately) the areas which
stand on those pieces as if the height
h were constant throughout each
piece. If, for example, we divide
the area A into six strips of equal
width, each 1/2 unit wide, and if we
take the height throughout each one
to be the height at the left-hand
corner, the total area is (approxi-
mately)
+ (f)*i = 19/8,
whereas, if we take the height equal to the height at the
right-hand comer we get 31/8. The area is really 3, as we
find by § 55. Taking still smaller pieces, the result is of
course better; thus with 30 pieces each yV ^^^^ wide, the
left-hand heights give 2.855, the right-hand heights 3.155.
With still more numerous (smaller) pieces these approximate
results approach the true value of the area. (See § 119, p. 196.)
/
\
y
r
y
1
\
o
y
c«
/
\
o
/
\
n
/
\
1
y
\
1
/
^
1
J
\
^
_^
/
—
I
•
4"
4
Fig. 46.
Xin, § 118] INTEGRALS AS LIMITS OF SUMS 195
EXERCISES
Approximate to about 1% the areas under the following curves, be-
tween the limits indicated. Estimate the answers roughly in advance.
Use judgment with regard to scales to gain accuracy by having the
figure as large as convenient. Express each area as a definite integral
and check by integration when possible.
1. y = z^ — 4:x; a; = 2 to 6. 2. y = 1/x; a; = 10 to 20.
8. i/ = »-2; a; = 1 to 5. 4. y = a;-3/2; a; = 2 to 4.
5. y = — ^^— ; a; = 2 to 4. 6. y = rX"^ ' a? = to 2.
7. y = 1/ Vl2 — x; x = 3 to 8. S. y = V9 — x; a; = to 5.
9. y = I/V9 - a;2; a; = Oto 1.5. 10. y = V9 + x^; a; = to 4.
11. y = ;— 1 — r-5— ; a; = Oto t/2. 12. y = — -, ; a; = to 1.
'^ i + sm2 X* ' ^ e» + e-* '
^' y = (4 + a;2)3/2; ^ = 0*<^2. 14. 2/ - ^^^==; a; =0 to .5.
15. y = V9+x4; a; = Oto 2. 16. y = (l-cosx)3/2;a;=0to2ir.
17. y = r ; a; = to ir/6. 18. 2/ = tan~i x\ a; = to 1.
V 4 — sin2 X
^^- y l+si^a; > a;=0toir/2.20. l/^^f^aJ a; = 0to.5.
21. y = e"^*; a; = to 1. 22. y = a; e~*'; a; = to 1.
Approximate to about 1% the distance passed over between the indi-
cated time limits, when the speed is as below; express each distance as a
definite integral, and check by integration when possible.
28. » = 4 < + «2; < = 1 to 3. 24. V = j^ ; < = to 50.
26. V = ^}, ^ ; < = 1 to 4. 26. v = — ^^ ; < = to 100.
2t + &' 1 + v^
27. V = o4-^ * < - 1 to 3. 28. V = ^r+l2; < = 10 to 20.
196
THE CALCULUS
pan, 5 118
29. Show how to lind the volume of a cone approximately, by adding
together hyen perpendicular to it£ axis.
30. Find the volume of a sphere by imagining it divided into small
pyriunids with their vertices at the center and their bases in the surface,
as in elementary geometry.
31. The volume of a ship is computed by means of the areas of cross
sections at small distances from each other; show how the result is cal-
culated. Show how to make a more accurate computation by the same
method.
119. Exact Results.
lummatioii Formula. Any definite
integral
CD §"j^'>'^
may be thought of as the
area under a curve
(2) l/=/{x)
between the ordinates x = o
and x = h.
If the interval AB from
x = aiox = bhe divided
into n parts, each of width
Ax, the whole area is approximately as ia § 118,
{Z)S = ^x■f{a) + ^x^f{a-\-Ax)+■■■+^x■S{a + {n~l)^x).
The term Ax-f(a) is the area of the rectangle ADiNiP,
since f(a) = AP. Likewise Ax-f{a + Ax) is the area of
DiD^sMi, etc. Hence the sum S represents the shaded
area in Fig. 47.
If the curve is rising from x = a io x = h, asm. Fig. 47,
S is smaller than the area under the curve. On the other
hand the similar sum
(4) R = Ax-/(a + Ax) + ix-/(o + 2 Ai) + • ■ ■ •
+ Ax-/(a + (n - 1) Ax) + Aj;-/(())
XIII, 1 120] INTEGRALS AS LIMITS OF SUMS 197
is represented by the shaded area in Fig. 48, and R is too
large if the curve is rising.
Hence
R>f''f(x)dx>S
if the curve is rising. But,
subtracting (3) from (4), we
have
(5) R~S =
Ax[f{b)-f{<i)],
and this approaches zero as
Ax approaches zero. It
follows that the true value of the int^ral is the limit of either
R OT S m Ax approaches zero, i.e., as n becomes infinite, and
we may write *
(6) J/(x) dx=lim ^AxJ(a)+Axf{a + Ax) +■■■
+ Axfia + {n-l)Ax)^,
at least if the curve rises from x = ato x = b.
Similarly, the formula (6) is true also if the curve falls from
X = a to X = b. Finally, if the curve alternately rises and
falls, we may prove (6) by separating the interval into several
parts, in some of which it rises and in some of which it falls.
The formula (6) is called the summation formula of the
integral calculus.
120. fotegrals as Limits of Sums. By far the greater num-
ber of integrations appear more naturally as limits of sums
than as reversed rates.
198 THE CALCULUS [XIII, § 120
Thus, as a matter of fact, even the area A under a curve,
treated in § 55 as a reversed rate, probably appears more
naturally as the limit of a siun, as in (6), § 119. Of course
the two are equivalent, since (6), § 119, is true; in any case
the results are calculated always either approximately, as
in the exercises under § 118, or else precisely by the methods
of §§ 52-54. Hence the method of § 54 was given first,
because it is used for each calculation even when the problem
arises by a smnmation process.
On account of the frequent occurrence of the sununation
process, we may say that an integral really means* a limit of a
sum, but when absolutely precise results are wanted it is
calculated as a reversed differentiation. The S5rmbol J* is
really a large S somewhat conventionalized, while the dx of
the symbol is to remind us of the Ax which occurs in the
step-by-step sunmiation.
EXERCISES
Express each of the following integrals as the limit of a sum, as in
(6), § 119. Find its approximate value to about 1%, and check by in-
tegration.
1. f^^^^dx. 2. f^^rr-odX' 3- f^ ^^Tx^dx.
J2 x—1 Jo l+x^ Jo
4. / yl+xdx. 6. / sin xdx. 6. / cos a; dx.
tgjixdx, 8. / sec xdx, 9. / logioxor.
Jo «/io
* It is really a waste of time to discuss at great length here which fact
about integrals is used as a definition, and which one is proved; to satisfy
the demand for formal definition, the integral may be defined in either way, —
as a limit of a sum, or as a reversed differentiation. The important fact is that
the two ideas coincide* which is the fact stated in the Summation Formula.
I
/
)
XIII, § 121] INTEGRALS AS LIMITS OF SUMS 199
Express each of the following quantities as the limit of a sum; find
its approximate value to about 1%; check by integration.
10. The area under the curve y = x2 from x == 1 to x = 3; from
a: = — 3toa;=3.
11. The area imder the curve y = 3^ from x =0 to x = 2; from
a; = — ltoa; = H-l.
12. The area under the curve x^y — 1 from x = 2 to a? = 6.
18. The distance passed over by a body whose speed is v = 4 i + 10
from < = to < = 3.
14. The distance passed over by a falling body (v = gt) from t = 2
to< =6.
15. The increase in speed of a falling body from the fact that the
acceleration is g = 32.2, from < = to < = 5.
16. The increase in th§ speed of a train which moves so that its accel-
eration isj = i/lQOj between the times < = and t — 6. The distance
passed over by the same train, starting from rest, during the same in-
terval of time.
17. The number of revolutions made in 5 min. by a wheel which
moves with an angular speed « = ^/lOOO (radians per second).
18. The time required by the wheel of Ex. 17 to malce the first ten
revolutions.
19. Repeat Ex. 18 for a wheel for which w = 100 — 10 1 (degrees per
second). Find the time required for the first revolution after < = 0;
note that the speed is decreasing.
20. The weight of a vertical column of air 1 sq. ft. in cross section and
1 mi. high, given that the weight of air per cubic foot at a height of h feet
is .0805 ~ .00000268 h pounds.
121. Volume of any Frastum. To illustrate the ease of
application of this process, consider again the volume of a
frustum of a solid. (See § 57.)
If such a frustiun be divided up into layers of thickness
A«, by planes parallel to the base, and if As represents the
area of any section at a distance s from the lower bounding
200
THE CALCULUS
[XIII, § 121
plane, the volume of each layer is, approximately, the
product of its thickness As times the area A, of the bottom
of the layer:
(1) Volume of one layer = AgASj approximately.
Now if we replace x in (3), § 119, by s, and if Ag = /(s), the
sum of all such layers would be, approximately,
AS'fia) + Ax'f(a + As) + + As'f(a+{n - l)As).
Hence, by (6), § 119, the exact total volume is
Asds= I f(s)ds.
This formula is the same as that derived in § 57. It may
be used to find the
volume of any solid,
if we know how to
Ing^g^Ag find the areas of
any such complete
set of parallel cross
sections.
In particular, if
the solid is a solid
of revolution, the
\8^S
xs^a
8^0
Fig. 49.
preceding formula reduces to formula (4) of § 56.
122. Surface of a Solid of Revolution. Similarly any such
formula is readily derived, and easily remembered by means
of this new process. Thus the formula for the surface of a
solid of revolution was derived in § 85. To obtain that
formula, or to remember it, we may remark that the curved
area of any short section is approximately
AA = 2 Try As,
since the ciured area is approximately the area of a section
XIII, § 123] INl^GRALS AS LIMITS OF SUMS
201
Fig. 50.
of a cone. It follows readily that the total area of such a
surface is
A = \2Tryds
123. Water Pres-
sure. As another
typical instance, con-
sider the water pres-
sure on a dam or on
any container. The
pressure in water in-
creases directly with
the depth A, and is equal in all directions at any point.
The pressure p on unit area is
(1) p = k'h
where h is the depth and k is the weight per cubic unit
(about 6.24 lb. per cubic foot).
Suppose water flowing in a paraboUc channel, Fig. 51,
whose vertical sec-
tion is the parabola
x^ = 225 y.
Let a be the depth
of the water in the
channel. If a cut-
off gate be placed
across the channel,
let it be required
-H 1 * — f^
-200 -160 -100 -W O
60
Fig. 51.
150 200 a;
to calculate the total pressure on the gate.
202 THE CALCULUS [XIII, § 123
Consider a horizontal strip of height Ah, whose upper
edge is h below the surface. The area of such a strip is its
width, Wy times its height, Ah, Hence the pressure on the
strip is, approximately,
pressure on horizontal strip = (k - h)w - Ah
In this example, w? = 2x = 2 V225y = SOy^^, h = a — y, and
Ay = Aft. Hence
pressure on horizontal strip = 30 fc (a — y)y^^^ Ay,
and the sum of the pressures on all such strips is the sum of
such terms as this one. Hence, by (6), § 119, the correct
total pressure is
P = 30 fc P*^"* (a - y) y'^Hy
= 3« ' [w - 5T2J0 = « *-'^'
where fc = 62.4 lb., and a is the total depth of the water.
EXERCISES
1. Find the volume generated by revolving Vx + Vy = Va about
the ic-axis, from x = to a; = a.
2. Find the volume of the paraboloid z = x^/a^ + y^/b, from 2 =
to h.
3. Find the volume of the cone z^ = x^/a^ -f y^/i^, from 2 = to ft.
4. Find the volume of a regular pjo-amid of base B and height ft.
6. On a system of parallel chords of a circle are constructed equi-
lateral triangles whose bases are those chords and whose planes are
perpendicular to the plane of the circle. Find the volume in which
all these triangles are contained.
6. On the double ordinates of an ellipse are constructed triangles
of fixed height ^, with planes perpendicular to the plane of the ellipse.
Find the volume contaming all these triangles.
7. Find the volume generated by a variable square whose center
moves along the x-axis from a; = to ir, the plane of the square being
perpendicular to the x-axis, and the side proportional to sin x.
XIII, § 124] INTEGRALS AS LIMITS OF SUMS 203
8. Find the surface of the spheroid generated by revolving the
ellipse y2 ~ (1 _ g2) ((,2 -_ a;2) about the x-axis.
9. Find the surface generated by revolving the catenary y =
(e» -\-e~*)/2 about the x-axis, from x = to a.
10. As in Ex. 9 for the hypocycloid x^/a + y2/8 = aP/8^ from x =
—a to a.
11. As in Ex. 9 for one arch of the cycloid.
12. Find the surface generated by revolving p = a cos B about the
initial line.
18. As in Ex. 12 for the cardioid p =» a(l + cos B),
Calculate the following pressures:
14. On one side of the gate of a dry dock, the wet area of the gate
being a rectangle 80 ft. long and 30 ft. deep.
15. * One side of a board 10 ft. long and 2 ft. wide, which is submerged
vertically in water with the upper end 10 ft. below the surface.
16. On an equilateral triangle 20 ft. on a side, submerged in water
with its plane vertical and one side in the surface.
17. On one side of a square tank 10 ft. high and 5 ft. on a side, the
tank being filled with a liquid of specific gravity .8.
18. On one face of a square 10 ft. on a side, submerged so that one
diagonal is vertical and one comer in the surface.
19. On one end of a parabolic trough filled with water, the depth
being 3 ft. and the width across the top 4 ft.
20. On one side of an isosceles trapezoid whose upper base is 10 ft.
long, parallel to the surface and 10 ft. below it, whose lower base is
20 ft. and altitude 12 ft.
21. What is the effect on the pressure if all dimensions given in Ex. 20
are multiplied by a constant c?
124. CavaUeri's Theorem. The Prismoid Formula. If two
solids contained between the same two parallel planes have
all their corresponding sections parallel to these planes equal,
i.e. if the area A\ of such a section for the first solid is the
same as the area A'\ of the second, it follows from § 57 that
their total volumes are equal, since the two volumes are
given by the same integral.
204 THE CALCULUS [XIII, § 124
This fact, known as Cavalieri's Theorem, is often useful in
finding the volumes of solids.
If the area Ag of any section of a frustum is a quadratic
function of s: *
(1) As = as^ + bs + c,
where, as in § 57, s represents the distance of the section Ag
from one of the two parallel truncating planes, the voliune is
= I (as^ + bs + c)ds^\a'- + b% + cs\
»-0 Je-0 L O Z J»-0
ah" , bh" , ,
where h is the total height of the frustum.
The area B of the base of the frustum, the area T of the top,
and the area Af of a section midway between the top and
bottom are
B =^]^_Q = [«s^ -\-bs + c^^ = c;
T =A,'\ = {a^ + bs + c\ = aA* + bh + c:
J|f=^l =ras2 + 6s + cl =a^+b^ + c.
If we take the average of 5, T, and 4 times M:
B + T + 4M _ah^ bh
6 -3^2"^^'
this average section multiplied by the total height h turns
out to be exactly the entire volume:
,^. B+T + 4.M ^. aK" bh^ . , n*-A
* It is shown in Ex. 3, p. 206, that the results of this section hold also
when Aa is any cubic function of «; A, = as* -}- fes^ + c« + d. Notice also
that any linear function &« -|~ c is a special case of (1), for a » 0.
XIII, § 124] INTEGRALS AS LIMITS OF SUMS
205
This fact is known as the prismoid formula. It is easy to
see by actually checking through the various formulas,
that this formula holds for every solid whose volume is given in
elementary geometry; the same formula holds for a great
variety of other solids* But the chief use to which the
formula is put is for practical approximate computation
of volumes of objects in nature: it is reasonably certain that
any hill, for example, can be approximated to rather closely
either by a frustum of a cone, or of a sphere, or of a cylinder,
or of a pyramid, or of a paraboloid; since the prismoid
formula holds for all these frusta, it is quite safe to use the
formula mihout even troubling to see which of these solids
actually approximates to the hill. Similar remarks apply
to many other solids, such as metal castings, though it may
be necessary to use the formula several times on separate
portions of such a complicated object as the pedestal of a
statue, or a large bell with attached support and tongue.
Example. The prismoid formula
applies to any frustum of an
ellipsoid of revolution cut off by
planes perpendicular to the axis
of revolution.
Let the origin be situated on one
of the truncating planes of the frus-
tum, and let the axis of x be the
axis of revolution. Then the equa- Fig. 52.
* The formula holds also, for example, for any prismoid, i.e. for a solid
with any base and top sections whatever, with sides formed by straight
lines joining points of the base to points of the top section. For example,
any wedge, even if the base be a polygon or a curve, is a prismoid. The
solids defined by (1) include all these and many others; for example, spheres
and paraboloids, which are not prismoids. The formula holds for all these sol-
ids and even (see Ex. 3, p. 206) for all cases where A, is any cubic function
of 8. One advantage of the formula is that it is easy to remember: even
the formula for the volume of a sphere is most readily remembered by re-
membering that the prismoid formula holds.
206 THE CALCULUS [XIII, § 124
tion of the generating ellipse is of the form Ax^ -f By^ -f Da; + F = 0.
The area A^ of a section parallel to the bases is inp'^ since the section is
a circle whose radius is y. Hence
which is a quadratic function of the distance x from one of the truncating
planes of the frustum. Therefore the prismoid formula holds.
Beware of applying the prismoid formula, as anything but an ai>-
proximation formula, without knowing that the area of a section is a
quadratic function of s, or (Ex. 3, p. 206) a cubic function of «.
EXERCISES
[This list includes a number of exercises which are intended for
reviews.]
1. Show that the prismoid formula holds for each of the following
elementary soUds; hence calculate the volume of each of them by that
formula: (a) sphere; (6) cone; (c) cylinder; (d) pyramid; (e) frustum
of a sphere; (/) frustum of a cone. See Tables, II, F.
2. Calculate the volume of the solid formed by revolving the area
between the curve y = x^ and the x-axis about the x-axis, between
X = 1 and a; = 3. Find the same volume (approximately) by the
prismoid formula, and show that the error is about 0.6%.
3. Calculate the volume of a frustum of a solid bounded by planes
^ = and h — HjH the area As of a parallel cross section is a cubic
function aA^ _^ 5^2 _j_ ^^ -f rf of the distance h from one base, first by
direct integration, then by the prismoid formula. Hence prove the
statement of the footnote, p. 205.
4. In which of the exercises relating to volumes on p. 97 does the
prismoid formula give a precise answer?
5. How much is the percentage error made in computing the volume
in Ex. 7, p. 97, from a; = 1 to a; = 3, by use of the prismoid formula?
6. Show, by analogy to § 64, that the area under any curve whose
ordinate y is any quadratic function (or any cubic function) of x, between
a; = a and a; = 6, is
^ [VA + ^VM + 2/bI,
where y^, 2/b, yM represent the values of 2/ at a; = a, a; = 6, a; = (a + 6) /2,
respectively.
XIII, § 124] INTEGRALS AS LIMITS OF SUMS 207
Calculate, first by direct integration, and then by the rule of Ex. 6,
the areas under each of the following curves:
7. y=x^ + 2x-\-S between x = 1 and a; = 5.
8. y=x^ — 5x + 4: between a; = and x = 5.
9. y =x^ + 5x between x = 2 and x — 4,
10. Calculate approximately the area under the curve j/ = x* between
ic = 1 and a; = 3 by the rule of Ex. 6. Show that the error is about .55%.
11. Show that any integral whose integrand/ (x) is a quadratic (or a
cubic) function of x, can be evaluated by a process analogous to the pris-
moid rule:
12. Evaluate the integral^ (l/x^) dx between x = 1 and a; = 5 approx-
imately, first by the rule of Ex. 11; then by applying the same rule
twice in intervals half as wide; then by applying the rule to intervals of
unit width.
18. Show that any integraiy/ (x) dx can be computed approximately
by using Ex. 11 with an even number of intervals of small width Ax:
y""" /(a;) dz =r/(a) +4/(a + Aa;) +2/(a + 2 Ax) +4/(a + 3 Aa;)
+ --+/(6)]f •
[This rule is called Simpson's Rule.]
Calculate the following integrals approximately by Simpson's Rule.
Notice that some of them cannot be evaluated otherwise at present.
14. f^T^dx, 16. f \/xdx. 18. C vT+^^ci
^0 •'0 •'0
(1/x) dx. 17. / V 1 4- X dx. 19. / sm x dx.
20. Find approximately the length of the arc of the curve y = x^ from
x = to X = J; from x = J to x = 1.
21. Find approximately the area of the convex surface of that portion
of the paraboloid formed by revolving the curve y = Vx about the x-axis
which is cut off by the planes x = and x = J; by x = J and x = 1,
CHAPTER XIV
MULTIPLE INTEGRALS —APPLICATIONS
126. Repeated Integration. Repeated integrations may be
performed with no new principles. Thus
\ -^dx = he; and If f- cjdx = — log x + cx+ c\
The final answer might be called the second integral of 1/x^.
Thus, in the case of a falling body, the tangential accelera-
tion is constant:
dv
where g is the constant; hence
^ = Sjrdt + const. = — gfi + c;
but since v = ds/dty
s =J'vdt + const, =y(— gt + c) cK + const. = —^ + ct + c\
If the body falls from a height of 100 ft., with an initial
speed zero, s = 100 and t; = when ^ = 0; hence c = and
c' = 100, whence we find
s = - gt^l2 + 100.
The equations s = fv dt + const., v = J'jr dt + const., just
obtained, apply in any motion problem, where jr is the
tangential accelerationj v is the speed, and s is the distance
passed over.
208
XIV, §126] MULTIPLE INTEGRALS— APPLICATIONS 209
126. Successive Integration in Two Letters. A distinctly
different case of successive integration which can be per-
formed without further rules is that in which the second
integration is performed with respect to a different letter.
Thus, the volume of any solid is (§ 57),
(1)
rh==b
As dh,
where Ag is the area of a section perpendicular to the direc-
tion in which h is measured, and where h = a and h = b de-
note planes which bound the solid.
In many cases it is convenient first to find A, by a first
integration, by the methods of § 55, and then integrate
As to find V by (1), this second
integration being with respect
to the height h.
Example 1. Find the volume of
the parabolic wedge
2/2 = x (1 - 2)2
between the planes 2 = and 2 = 1
and between the planes x = and
a; = 1.
The area A, of a section by any
plane z = h parallel to the xy-plsme is twice the area between the curve
y = (I — h) Vx and the a;-axis:
Fig. 63.
x=\
x^O
= 3(1-^),
hence this volume, by (1), is
]»=! ^A=i 4 /.»=! , „ 4/, A2\n*=i 2
Notice that A, during the first integration, was essentially constant.
210 THE CALCULUS [XIV, § 126
Combining the formulas used in this example, the volume
V may be written
This result is usually written without the brackets on the
right:
-[fc-i rh^i rx=i y-
V\ =2 {l-h)Vidxdh.
Such double integrals in two letters are very common in
applications.
Examples of triple integrals will be found further on in
this chapter.
127. Volumes. Double Integrals. Analogous to the problem
of finding the area under a given portion of a curve,
(§§ 55, 119), there is the problem of finding the volume under
a given portion of a surface.* This leads to double integrals.
Let A'B'C'D' be a portion of a curved surface whose
equation is
z = F(x, y).
Let ABCD be the projection of A'B'C'D' on the xy
plane, curve CD being the projection of curve CD',
The problem is to express the volume between the portion
A'B'C'D' of the surface and its projection on the a:2/-plane.
Divide this volume into slices of thickness PQ = Ax, by
planes parallel to the i/z-plane, and further subdivide the
slices into prismatic columns by planes parallel to the
an/-plane spaced at intervals Ay,
The area of the section PSS'P' depends on the position
* See Tables, I, b, for formulas from Solid Analytic Geometry.
XIV, 5127] MULTIPLE INTEGRALS— APPUCATIONS 211
of P, hence on x, and is therefore a function of x. Therefore
by the frustum formula {§ 57),
the required volume is
]x-b fi-b
^ ^= I ^A^dx.
But the area Ax is the area
under the curve P'S' whose
ordiuates (heights above xy-
plane) are 2 = F{x, y), where
X has the fixed value OP and
y alone varies from y = to
y = PS =f{x), this being Fw. 64.
the equation of curve DC, supposed given.
Hence
Then
<" ''J. " J... X-o f(^.l')*<^-
This is essentially the same sort of problem as that
discussed in § 126.
This double int^ral may be written as the limit of a
double sum. For if we consider the prismatic column on Ax Aj/
as base, its volume approximately is s ir Ay or F{x, y) Ay Ai.
This su^ests the double sum
or, as it is usually written,
212 THE CALCULUS [XIV, § 127
Here the limit of the inner summation is merely A, and then
the limit of the outer summation is the integral of A or the
volume. Hence we write the formula
(2) iimi:::i:ro'>(a:,j/)Aj/Ax
Ai/-»0
x=h >,y-/(x)
dx.
This is the fundamental summation formula for double
integrals.
Example. Determine the volume under the surface z = x^+y^
between the xa-plane, the planes x = 0, a; = 1, and y = 0, and the
cylinder y = Vx.
z=x^
z=
Fig. 55.
/ {x^+y^)dydx
J x=o L Jo
= y ra;5/2 + (1/3) a;3/2"| da; = 44/105.
XIV, § 127] MULTIPLE INTEGRALS— APPLICATIONS 213
EXERCISES
1. Determine a function y — f (x) whose second derivative d^/dx^
is 6 X. Ans. y = x^ + CiX + C2.
2. Determine the speed v and the distance s passed over by a particle
whose tangential acceleration cPs/dfi is 12 t. Find the values of the arbi-
trary constants if t; = and s = when ^ = 0; if v = 100 and « =
when ^ = 0.
Find the general expressions for functions whose derivatives have
the following values:
3. d^/dx^ = 6 x2. 6. dh/d0!^ = l/Vl - d. 9. d^/dx^ = ef.
4. d^a/dfi = 3 + 2 <. 7. dh/de^ ^6^-29, 10. d^s/dfi = sec2 1.
6. d^a/dfi = Vl-t. 8. dh/du^ = 1 - m2. 11. dh/du^ = l/w2.
Determine the speed v and distance s passed over in time t, when
the tangential acceleration jV and initial conditions are as below:
12. JT = sin <; t; = and 8=0 when ^ = 0.
13. ir = ^ + cos t; v = and « = when ^ = 0.
14. jj. = VY+1; v = 3 and 8 = when t = 0.
16. JT = t/Vl + ^2j t; = 1 and 8 = when < = 0.
Evaluate each of the following integrals, taking the inner integral
sign with the inner differential:
16. / / xydydx. 21. f f f -^drdsdt.
/ 6 a;2(2 - 2/)d2/ dx. 22. / / (x-{-y)dydx.
xasQ •^ y—1 •/a:=-l •/y=—i»
/ (x2 + l)(4-2/2)(i2/dx. 23. / / (x -{- y)^ dy dx.
x=*0 •/v«2 •/ x=0 •/y=l
/ Vu + vdudv. 24. / / / (x-{-y+2)dzdydx.
20. /^°' f^^'' ^^^ dy dx, 26. T' T'' r ^r^amdded<t>dr.
J x-\ Jv=o X Jo Jo Jo
26. Find the volume of the part of the elliptic paraboloid 4 x^ +
9 2^2 — 35 between the planes 2 = and 3 = 1; between the planes
a = a and 3 = 6.
214 THE CALCULUS [XIV, § 127
27. Find the volume of the part of the cone 3 x^ -f 9 2/^ = 27 a?
between the planes 2 = and 3 = 2; between 3 == a and z = b.
28. Find the volume of the part of the cylinder a;2 + 2/2 = 25 between
the planes 3 = and 3 = x; between the planes a = x/2 and 3 = 2 a;.
29. A parabola, in a plane perpendicular to the x-axis and with its
axis parallel to the 3-axis, moves with its vertex along the x-axis. Its
latus rectum is alwa3rs equal to the oK^oordinate of the vertex. Find
the volume inclosed by the surface so generated, from 3 = to s = 1
and from x = to x = 1.
30. Find the volume of the part of the cylinder a;* + j/* = 9 lying
within the sphere x^ -\- y^ ■}- z^ = 16.
31. For a beam of constant strength the deflection y is given by the
fact that the flexion is constant: b = cPy/da:^ = const, if the beam is of
uniform thickness. Find y in terms of x and determine the arbitrary
constants if y = when x = ± 1/2. [This will occur if the beam is of
length If and is supported freely at both ends.]
32. Determine the arbitrary constants in the case of the beam of
Ex. 31, if y = and dy/dx = when x =0. [This will occur if the beam
is rigidly embedded at one end.]
33. For a beam of uniform cross section loaded at one end and rigidly
embedded at the other, b = d^/dx^ = k(l — x) where I is the length of
the beam, x is the distance from one end, and A; is a known constant
which is determined by the load and the cross section of the beam. Find
y in terms of x, and determine the arbitrary constants.
Find y in terms of x in each of the following cases:
34. d^y/dx^ = A;(P - 2 te + x^); y = 0, dy/dx = when x-0.
[Beam rigidly embedded at one end, loaded uniformly.]
36. d^/dx^ = a + &r; 2/ = 0, dy/dx = when a; = 0.
[Beam of uniform strength of thickness proportional to (a + bx)^^,
embedded at one end.]
36. d^/dx^ = A;(P/8 - x*/2); y = when x = ± 1/2.
[Beam supported at both ends, loaded uniformly.]
37. d^/dx^ = k/x^; y = 0, dy/dx = at x = Z.
[Beam of uniform strength of thickness proportional to x^, embedded
at x = L]
88. Find the angular speed co and the total angle 6 through which a
wheel turns in time t, if the angular acceleration is « = d^O/dl^ = 2 <,
and if ^ = w = when < = 0.
XIV, § 129] MULTIPLE INTEGRALS— APPLICATIONS 215
128. Triple and Multiple Integrals. There is no difficulty
in extending the ideas of §§ 126-7 to threefold integrations
or to integrations of any order. Following the same reason-
ing, it is possible to show that, ii w = F{Xyy, z)
1«^ lilZ JZ'^ liZl w^xAyAz
Ax— *0
Az-*0
I I F (x, y, z) dx dy dz,
where the three integrations are to be carried out in succes-
sion, where the limits for x may depend on y and 2, and where
the limits for y may depend on z\ but the limits for z are, of
course, constants.
Thus it is readily seen that the volume mentioned in
§ 127 may be computed by dividing up the entire volume by
three sets of equally spaced planes parallel to the three coor-
dinate planes. Then the total volume is, approximately,
the sum of a large number of rectangular blocks, the volume
of each of which is Ax Ly Lz\ and its exact value is
Ax— >0
Air-^
A«->0
I I dzdy dx,
which reduces to the result of § 127, if we note that
dz = z\ = F{x, y).
129. Plane Areas by Double Integration. To find the area
bounded by a closed curve C, we divide it into small ele-
ments of area, either by lines drawn parallel to the coordinate
axes if the equation of C is given in rectangular coordinates.
216 THE CALCULUS [XIV, 1 129
or by a system of radial lines and concentric circles if the
equation of C is given in polar coordinates. (Figures 56, 57.)
75=^^--
w
\
: t
^y
lil J
J _
- 5^
_^^
t'''
n
In Fig. 56 let any ordinate whose abscissa is x meet the
boundary of the area in P and Q, the corresponding values
of 3/ being j/i and y-x- Let the extreme values of x between
which the oval lies be i = a and x = 6. Then the area will be
(1)
A = lim E^ „ E" "' Am Ai = I I dy dx.
This is merely a special case of (2), § 127, when Fix, y) = 1,
because the volume of a right cylinder of height 1 equals the
area of the cross section.
In Fig. 57 let any radius vector whose angle is 8 meet the
boundary of the area in P and Q, the corresponding values
of p being OP = pi, OQ = pj. Let the extreme values of
Ohe d = a and 6 = ^. Then the element of area is approxi-
mately pA6 ■ Ap and the total area is
(2) A - Iimy\*:fr'I"pApA(J= f'^" f''^'"pdpd$.
XIV, S129] MULTIPLE INTEGRALS— APPLICATIONS 217
In fonnulas (1) and (2) the first integration can be carried
out at once, giving
(1') A=J\y^-y,)dx (2') A = fj (p^- pv') dff.
The last results may also be derived from the formula of
§ 55 and § 92, so formulas (1) and (2) may be dispensed with
so far as the mere calculation of plane areas is concerned.
We shall, however, find the idea of a plane area as the limit
of a double sum very useful in the following sections.
Example 1. Find the area between the parabola j/' = 4 (x — 1) and
theline!/ = a;-l. {Fig. 58.)
The extreme values of x are found to be x = 1 and x = 5. Hence
^ = fll /"" _' dydx=f r2v7^n-(j;-I)"j(ir
= [4/3 {X - l)»/s - l(r - !)»]'= 8/3.
IE
Fro. 68.
ExAUPij; 2. Find the area between the circle p = cob 6 and the car-
dioid p - 1 + COB 8, from fl = to 45°. (Fig. 69.)
= f''*h [(1 + coa e)s - C03! fl] de
= if'^*il + 2 COB e) rffl = i {t/4 +V2).
218 THE CALCULUS [XIV, § 129
EXERCISES — DOUBLE INTEGRALS
1. Find the volume under the surface s = x^ -f ^2 between the x&-
plane, the planes a; = and x = 1, and the cylinder whose base is the
curve J/ — a;2.
Find the volume between the a:^-plane and each of the following
surfaces cut off by the planes and surfaces mentioned in each case:
2. « = X + J/ cut off by y = 0, X = 0, X = 1, J/ = Vx.
3. a = x2 H- y cut off by y = 0, X = 1, X = 3, y == x2.
4. 2 — x^ cut off by 2/ = 0, X = 2, X = 4, J/ = x2 -f 1,
6. a = xj/ + j/2 cut off by y = 0, X = 1, X = 5, J/ = x*.
6. 2 = y -\- y/x cut off by 2/ = 0, X = 0, X = 1, y = x^.
7. 3 = x2 + y3 cut off by X = 0, y = 1, y = 4, y2 =aj.
8. a = Vx +y cut off by X = 0, y = 2, y = 5, y = X.
9. s = x2 + 4 y2 cut off by y = and y = 1 — x^.
10. 2 = xy cut off by y = x^ and y = 1.
11. a = x2 — y2 cut off by y = x^ and y = x.
12. Find the volume of the portion of the paraboloid a == 1 — x^ —
4 2/2 which lies in the first octant.
13. If two plane cuts are made to the same point in the center of a
circular cylindrical log, one perpendicular to the axis and the other mak-
ing an angle of 46° with it, what is the volume of the wedge cut out?
14. Show that the volume common to two equal cylinders of radius a
which intersect centrally at right angles is 16 a^/S.
16. Show that the volume of the ellipsoid x2/16 + y2/9 + a?/4 = 1
is 32 IT.
16. How much of the ellipsoid in Ex. 15 lies within a cube whose
center is at the origin and whose edges are 6 units long and parallel to the
coordinate axes?
17. Where should a plane perpendicular to the x-axis be drawn so as
to divide the volume of the ellipsoid in Ex. 15 in the ratio 2:1?
Calculate by double integration the areas bounded by the following
curves:
18. 2/ = x2 and y = Vx. 22. x = 0, 2/ = sin x, and y = cos x.
19. 2/ = a;2 and y =x8. 23. 2/ = 0, 2/^ = x, and x2 — y2 = 2.
20. 2/ = a;2 and — x2 + 2/^ = 2. 24. y = 2 x, 2/ = 0, and y = 1 — x.
21. x2 + 2/2 = 12 and y == x2. 26. 2/2 = x, and y = 1 — x.
XIV, §130] MULTIPLE INTEGRALS— APPLICATION 219
Determine the entire area, or the specified portion of the area,
bounded by each of the following curves, whose equations are given in
polar coordinates:
26. p = 2 cos e. 27. One loop of p = sin 2 ^.
28. One loop of p = sin 3 ^. 29. The cardioid p = 1 — cos d.
30. The lenmiscate p^ = cos 2d. 31. The spiral p = 6, from ^ = to ir.
32. The spiral pd = 1, from 6 = ir/4 to ir/2.
33. p = 1 H- 2 cos dy from ^ = to ir.
34. p = tan 6, from ^ = to 45°.
36. p =a^, one turn. 36. p = 3 cos d + 2.
37. p = a sin ^ cos 9/(am^ d + cos^ 0); folium: the loop.
130. Moment of Inertia. When a particle of mass m re-
volves in a plane about a point 0,
with given angular speed w, its speed
is t; = rw, and its kinetic energy is
E = 1/2 mv^ = 1/2 mr'u)^ The mo-
ment of inertia of m about is
defined as the product of the mass m
by the square of its distance from 0, Fig. 60.
I = 7^m. Thus E is easily foimd where / is known.
Given now a thin plate of
metal of uniform density and
thickness, whose boundary C is
a given curve, let us divide the
plate into small squares by lines
equally spaced parallel to two
rectangular axes through 0.
Let P be a point in any one of
these squares and let OP = r =
V?+^. Then the mass of
V
^
^
^
/
>
J.
• ■
J
A
A^
f
Pj
f
i
!
f
/
I
i
/
J
V
J
r
y
f
>
\^
f
y
r
r
^
y
/
/
>
t
i
t^
c-^
X
Fig. 61.
220 THE CALCULUS [XIV, § 130
the square is k- Ay Ax where k denotes the constant surface
density {i.e. the mass per square unit) ; and the moment of
inertia of this square about is, approximately, k-r^ Ay Ax.
Hence the moment of inertia / of the entire plate about is:
Ax-»0
The limits of integration are to be taken so as to cover
the area, as in § 129.
Thus the moment of inertia of a plate bounded by the
two curves i/ = (1 — x^) and y ={x^ — 1), about the origin
(draw the figure) is:
I (x^ + y^)dydx
where k is the surface density.
The moment of inertia about an axis is defined similarly, the
distance r being replaced by the distance of P from the
given axis. Thus the moments of inertia of the plate in
the figure about the x- and i/-axes are, respectively,
Ix = kJJy^dydx; Iy = kj'j^x^dydx.
The radius of gyration of an area, whether about a point or
about an axis, is defined by
radius of gyration = K = Vl/M.
If the boundary of the plate is given in polar coordinateSj
the moment of inertia about is calculated by dividing the
area into elements rABAr. (See § 129.) Then
XIV, §131] MULTIPLE INTEGRALS— APPLICATIONS 221
Thus for a circle whose center is 0, r = / (6) = a, the
radius. Hence the moment of inertia of a circular disk
about its center is:
J 0=0 L4 Jp=o J tf=o 4 2 2
where fc is the surface density, and M = kra^ is the mass of
disk.
131. Moments of Inertia in General. The moment of inertia
of any mass about a given point may be defined as follows.
Divide the mass M into elements of mass Am; multiply
each element Am by the square of its distance, r^, from the
given point; the limit of the sum of these products is the
required moment of inertia:
(1) / = lim 2] ^Am = fr^dm,
Am->0
When M is a plate we take Am = kAA, where k is the
mass per unit area, and AA = Ay Ax or rA ^ Ar, as in § 129.
When Af is a thin wire bent into the form of a given
curve, Am = fcAs, where k is the mass per unit length and
As the element of length.
Then
(2) 7=2 kr^ As = kfr^ ds,
the limits being taken to cover the given curve. For ds we
use (1) of § 82 or (1) of § 93.
When ilf is a solid body, we take Am = kAV, where k is
the mass per unit volume and AF is the element of volume,
Ax Ay Az.
Then
(3) / = lim *^E2m r^AxAyAz = kfffr^ dx dy dz.
Ax—*0
Aa-»0
222
THE CALCULUS
[XIV, § 131
The limits of integration are to be taken to cover the given
volume.
For the moment of inertia about an axis we let r in forms
(1), (2), (3), be the distance of Am from that axis.
Example 1. Find / for a wire bent into the form of a circular
quadrant, about the center.
Using polar coordinates, ds —rddj and
I -^k f r^dd='kr^ I dd
JO Jo
= At3 t/2 = Mr^y since M = kv r/2.
/
^__i
Fig. 62.
Fig. 63.
Example 2. Find / for a circular cylinder of height H and radius
R about the center of one base.
The equation of the surface is a;2 + ^ = R^^ if the 2-axis is the axis
of the cylinder. Then, by symmetry,
/ / (x^ + y^ -h z^) dz dy dx,
x^O J v»0 J c=0
x—0 J I/-0
^4tkH f (x2 2/ + 2/8/3 + i^ 2//3) dx
•/ a:=0
^4tkH f'''\x^ Vi22-a;2 + (i22 - a;2)3/2/3 + [p VR^ - xVZ) dx.
XIV, §131] MULTIPLE INTEGRALS— APPLICATIONS 223
Let X = i2 sin ^. Then
ir/2
7=4 kHE^f (722 sinZ 6 cos2 (? + 722 cos^ B/Z + ff2 cos2 e/2) de
= 4 A;iyiE2 (722 ^/i6 + igs ^/i6 + 16 + fl2 ^/i2)
= A;^i22 (3 i22 + 2 iy«) (ir/6).
EXERCISES
Calculate / about the origin for the areas bounded by the following
curves (Density = A; = 1):
1. y = x^ and y = Vx, 6. a; = 0, 2/ = sin x, and y = coax
2. 2/ = x2 and 2/ = a:^. 6. 2/ = 0, 2/2 = x, and a;2 - 2/2 = 2.
3. 2/ = a;2 and — x^ + y^ = 2. 7. 2/ = 2 x, 2/ = 0, and 2/ = 1 — a;.
4. a;2 + 2/2 = 12 and 2/ = x^. 8. 2/^ = x, and 2/ = 1 — a;.
Find the moment of inertia and radius of gyration of each of the
following shapes of thin plate:
9. A square about a diagonal. About a comer.
10. A right triangle about a side. About the vertex of the right angle.
11. A circle about its center.
12. An ellipse about either axis. About the center.
13. A circle about a diameter.
14. A triangle of given base and height, about the base.
15. A trapezoid about one of its parallel sides.
16. A thin circular plate, about a point on the circumference.
17. A thin plate bounded by two concentric circles, about the center.
18. An equilateral triangle, about its center.
19. An equilateral triangle, about one vertex.
20. p = 2 cos d.
21. One loop of p = sin 2 0,
22. One loop of p = sin 3 6.
23. The cardioid p = 1 — cos 0.
24. The lemniscate p2 = cos 2 0,
224
THE CALCULUS
[XIV, § 131
26. The spiral p = 6, from ^ = to ir.
26. The spiral pd = 1, from d = ir/4 to t/2.
27. p = 1 + 2 cos ^, from ^ = to x.
28. p = tan ^, from ^=0 to 45^
29. Calculate the moment of inertia of a cube about a comer,
30. Calculate the moment of inertia of a rectangular parallelopiped
about a comer.
Calculate the moment of inertia about the origin of each of the
solids bounded by the following surface and lying above the xy-p]a.ne.
(SeeExs. 2-11, p. 218.)
31. z=x+y; y = 0; x = 0; x = 1; y = Vx.
32. z = xy; y = x^; y- 1.
33. 2 = a;2 + y; y = 0; x-1; x = S] y = x\
34. z=^xy'y 2/ = 0; a; = 1; a; = 2; j/ = aj2 + 1.
36. 3 = x2 — 2/^; y == a;2; y — x.
132. Average Value. Centroid. Let there be given a function
/ {x, y) of two independent
^ -^ — ■- variables. At any point of
the x2/-plane, that is, for a
given pair of values of x and
2/, this function will have a
definite value.
What will be the average
value of / (x, y) over a given
region fl? Divide the region
into small squares, note
the value of the function at some point of each square
(for convenience at the corner nearest the origin). Let
there be n complete squares in the region iZ, the fractional
parts around the border being disregarded.
O
x
Fig. 64.
XIV, §133] MULTIPLE INTEGRALS— APPLICATIONS 225
Then the average value of / (x, y) at the corners of the
squares will be
Sum of values of / (x, y) 2) / (x, y)
Number of these values n
Multiplying both sides by the element of area Ail, and
putting
n^A = AA + AA + AA + = S AA,
we have 2) / (x, y) A A
S AA
and the limit of this, as n increases and Ax and At/ decrease,
is called the average value of f(x, y) over the region R.
Thus
n^ 4« vnj r^fft^ .A/.«^rP / / {x, y) dA fff (x, y) dx dy
(1) Av. Val. of fix, y) over R = ^ ^^ — j^j^^^^^
The denominator is simply the area of R; the limits are to
be taken to cover the region R,
In polar coordinates, replace dA by rdddr, and /(x, y) by
133. Centroid or Center of Gravity of an Area. The centroid,
(Xyy), of a plane area, as R in the above figure, is the point
whose coordinates are the average values of the coordinates
of the points of 72, that is
^^^ '' = llfA'''''^^^JdA'
For a material plate of uniform thickness and of mass k
per unit area, the element of. mass is dM = kdA and its
centroid is defined by
(o\ • _ fxdM ^ _ fydM
^ ^ ^^ fdM ' ^" fdM
Here the density factor k may be a given function Oi x and y;
if k is constant it cancels out and we get the same result as
for a plane area
226
THE CALCULUS
[XIV, § 134
134. Centroid of a Curved Arc. Similarly, the centroid of a
curved arc is defined as the
point whose coordinates are
the average values of the co-
ordinates of the points of the
arc. Thus if AB is the arc,
we divide it into n equal
segments As, and note the
^ values of x and y for some
point of each As. The aver-
FiG. 65.
ages of these n values of x and y are, respectively,
a?! + a;2 + X3 + - - + a;„ ^ yi + ^2 + 1/3 + • " + l/i
n
n
Multiplying numerator and denominator by As and allowing
n to increase, we have
2xAs fxds
(3)
X = Urn _, .
fds'
^ fds
136. Centroid of a Volume. Similarly, by dividing a volume
into elements, and noting the coordinates (x, y, z) of a point
of each element of volimie, we obtain
^^^ ^ fxdV ^ fydV , fzdV
z =
fdV ' ^" fdV ' " fdV
In general we may take dV = dxdydz, and so express
(4) in terms of triple integrals. In numerous applications
it is simpler, however, to take for dV a slice of the volume;
thus in finding x cut the volume by a plane perpendicular
to the avaxis, forming a section of area A^; then take dV =
Axdx,
136. Definition of Centroid in Mechanics. From the stand-
point of mechanics the centroid is defined in a different
XIV, §136] MULTIPLE INTEGRALS— APPLICATIONS 227
manner that turns out to be equivalent to what we have
done. Suppose the xy-
plane to be horizontal and
a wire bent into the form
of any curve as AB to lie
in this plane and to be
balanced on a knife-edge
MNy falling on the ordi-
nate whose abscissa is j, ^^
X = X. Divide the wire
into elements of mass Am = k As, multiply each Am by its
distance from the knife-edge and form the sum of the
moments 2 A;(x — x)As, The limit of this sum must be zero
if the wire is balanced, so we have k J* (x ^ x) ds = 0,
Hence J*xds = J*xdSyOTy since x is a fixed value, x J* ds =
S X ds. Hence x = J* x ds/fds. Likewise we get y by sup-
posing the wire to be balanced on a knife-edge parallel to
the avaxis. The same considerations apply to a thin plate,
except that As is replaced by AA. For a solid we pass
planes parallel to the coordinate planes such that the sum
of the moments of the elements of mass, i,e,, the products
(i — a:) Am, (y — y)Am, (z — z) Am, shall have the limit
zero.
EXERCISES
Find the average value of each of the following functions, over the
area under the curve y = 1 — x*, from aj = to 1.
3. a;2.
4. 2/2
1. X, 2. y.
6. 1/(1+ x). 6. e*.
Find the average value of each of the following functions in the vol-
ume bounded by the coordinate planes and the plane x •}• y -{' z = 1,
7. / (Xy y, z) = X, or y, or z, 8. / (x, y, z) = xyz,
9. /(x, y, z) = xy. 10. /(x, y, z) ^x^ + y^ + sl^.
228 THE CALCULUS [XIV, § 136
11. Find the average ordinate of j/ = cos x, from a; = to ir/2.
12. Find the average ordinate of a semicircle.
13. Find the average distance of the points of the area of a circular
quadrant from the center. (Use polar coordinates.)
14. Find the average density of a rod in which the density varies as
the distance from one end.
Find the centroids of the following figures:
16. Of the segment of the parabola y2 = 4 ox, from x = Otoh,
16. Of a semicircle.
17. Of the first quadrant of an ellipse.
18. Of the area under y = cos x, from a; = to v/2,
19. Of a right circular cone.
20. Of a regular pyramid. (Use sections parallel to the base.)
21. Of a semiellipsoid of revolution.
22. Of a circular quadrantal arc.
23. Of a circular arc of angle 2 a.
24. Of the arc of a;2/3 -f- yVs = ^2/3^ in the first quadrant.
26. Of the arc of y == (e* + e^)/2, from a; = to 1.
Find the centroids (x, y) for the areas bounded by the following
curves:
26. y=x^ and y = Vx. 27. y=x'^ and y= a^,
28. a;2 + 2/2 = 12 and y = x\ 29. 2/2 = ^j and 2/ = 1 - a;.
30. 2/ = 2 x, 2/ = 0, and y = 1 — x.
31. p = 2 cos B. 32. One loop of p = sin 2 B,
33. The cardioid, p = 1 — cos B, 34. The lemniscate, p2 = cos 2 B.
36. The spiral, p = ^, from ^ = to t.
36. p = 1 + 2 cos ^, from ^ = to t.
XIV, §136] MULTIPLE INTEGRALS— APPLICATIONS 229
GENERAL REVIEW EXERCISES
Find the areas bounded by each of the following curves, or the part
specified:
1. p =^ a cos ^ + 6. 2. p = a cos 3 0,
3. fj^ia — x) = a^ [cissoid]; to its asymptote x = a.
4. y2 = a;2 (4 — a;) : the loop.
Find the volume generated by revolving each of the following curves
about the line specified:
. 6. y = 5 x/(2 + 3 x); about y = 0] x = to x = 1,
6. 2 a;2 + 5 y2 = 8; about 2/ = 0; total solid.
7. y = b sin (x/a); about y = 0; x = to x = w,
5. y = a cosh (x/a); about y = 0; x = i^ x = a,
9. (x — a)2 + y2 = r2; about a; = 0; total solid.
10. The cycloid 5 about base; one arch.
11. The cycloid; about tangent at maximum; one arch.
12. The tractrix; about asymptote; total.
13. X = a cos3 tj y = a sin' t; about y = 0; total solid.
Find the area, its centroid, and its moment of inertia about the
origin, for each of the following curves, between the limits indicated:
14. y == a (1— x2/62); 1st quadrant.
16. y = x/(l + x2); X = to a; = 1.
16. The sine curve; one arch.
17. The cycloid; one arch.
18. x2/8 + ^2/3 = a2/3[or X = a cos' t^ y = a sin' t]; first quadrant.
19. Between the two circles p = a cos d and p = 6 cos d;h > a.
20. X = 2 a sin2 0, y = 2a sin2 ^ tan 0; between the curve and its
asymptote.
Find the centroid of each of the following frusta:
21. Of the paraboloid x^ +y^ = ^ az by the plane a = c.
22. Of a hemisphere.
23. Of the upper half of the ellipsoid of revolution
4 a;2 + 4 2/2 -I- 9 22 = 36.
230 THE CALCULUS [XIV, § 136
24. Of the upper half of the ellipsoid x2 + 4 3/2 + 9 ^2 = 36.
26. Of the solid of revolution formed by revolving half of one arch
of a cycloid about its base.
26. Obtain a formula for the volume of a spherical segment of
height h.
27. Show that the volume of an elUpsoid of three unequal semiaxes,
a, 6, c, is 4 vabc/Z,
ff
28. Show that the volume bounded by the cylinder a;2 + y2 = ckC;
the paraboloid a;2 + 1/2 = bz, and the xy-plAue is (3/32) (va^/h),
29. Find the volume common to a sphere and a cone whose vertex
lies on the surface and whose axis coincides with a diameter of the
sphere.
Find the lengths of the arcs of each of the following curves, between
the points specified:
30. y = log x; X = a to X = h,
31. e" cos x = l;a; = Otoa; = a;.
32. x = fiyy = 2at (or 2/2 = 4 aH); « = <i to «=<^.
33. One arch of a cycloid.
34. p == a (1 + cos 0) [cardioid]; total length.
35. Calculate the moment of inertia I for a right circular cone about
its axis. Ans, (3/10) mass • square of radius.
36. Calculate the moment of inertia and the radius of gyration for
the rim of a flywheel about its axis, the inner and outer radii being
37. The moment of inertia of an ellipsoid about any one of its axes is
(1/5) (mass) (sum of the squares of the other two semi-axes).
38. Calculate the moment of inertia for a spherical segment about
the axis of the segment.
39. Show that, for any body, 2 /o •=/* + /» + /«, where /o, /*, /y,
Tg denote respectively its moments of inertia about a point and three
rectangular axes through that point.
40. Show that for any figure in the xy-plane, /, = /« + /y, where /„
Ipf It denote its moments of inertia about the three coordinate axes
respectively.
XIV, §136] MULTIPLE INTEGRALS— APPLICATIONS 231
41. Show that the total pressure on a rectangle of height h feet and
width h feet immersed vertically in water so that its upper edge is a feet
below the surface and parallel to it, is 62.4 hh (a + h/2). Show that the
depth of the center of pressure is at (6 a^ -|- 6 aA + 2 ^2)/(6 a _|_ 3 ^),
42. Show that the total pressure on a circle of radius r, immersed
vertically in water so that its center is at a depth a + ^, is 62.4 jtr^ {a-\'r).
Show that the depth of the center of pressure is a + r + r2/(4 r + 4 a) .
43. Show that the total pressure on a semicircle, immersed vertically
in water with its bounding diameter in the surface, is 41.6 r^. Show
that the depth of the center of pressure is 3 irr/16.
44. Calculate the water pressure to within 1 % on a circular disk 10 ft.
in diameter, if its plane is vertical and center 10 feet below the surface.
45. Show that if a triangle is immersed in a liquid with its plane
vertical and one side in the surface, the center of pressiu^ is at the
middle of the median drawn to the lowest vertex.
46. Show that if a triangle is immersed in a liquid with its plane
vertical and one vertex in the surface, the opposite side being parallel
to the surface, the center of pressure divides the median drawn from the
highest vertex in the ratio 3:1.
47. Calculate the mean ordinate of one arch of a sine-curve. The
mean square ordinate. [Effective E. M. F. in an alternating electric
current.]
48. Calculate the average distance of the points of a square from one
corner.
49. What is the average distance of the points of a semicircular arc
from the bounding diameter?
60. When a liquid flows through a pipe of radius -R, the speed of
flow at a distance r from the center is proportional to i22 — r®. What is
the average speed over a cross section? What is the quantity of flow
per imit time across any section?
61. The kinetic energy ^ of a moving mass is Km S Am • i^/2, where
Am is the element of mass moving with speed v. Show that for a disk
rotating with angular speed w, ^ = (j^I/2, Calculate E for a solid car
wheel of steel, 30 in. in diameter and 4 in. thick when the car is going
20m./hr.
232
THE CALCULUS
[XIV, § 136
52. Show that the kinetic energy ^ of a sphere rotating about a
diameter with angular speed w is (1/5) (mass) r'^oy^,
63. Calculate the kinetic energy in foot-pounds of the rim of a fly-
wheel whose inner diameter is 3 ft., cross section a square 6 in. on a side,
if its angular speed is 100 R. P. M. and its density is 7.
54. The x-component of the attraction between two particles m and
m', separated by a distance r, is (fc • w • m^/r^) cos (r, x) where cos (r, x)
denotes the cosine of the angle between r and the x-axis. Hence the
a;-component of the attraction between two elementary parts of two
solids M and Af' is {k • AM • AM'/r^) cos(r, x). Show that the total
attraction between the two solids is expressible by a six-fold integral.
55. A uniform rod attracts an external particle m. Calculate the
components of the attraction parallel and perpendicular to the rod;
the resultant attraction and its direction.
[Hint. Let AM be an element of the rod; then aF = kAM • m/r^ is
the force due to AM acting on w, r being the distance from AM to m;
then the components of AF are AX = AF cos a and AY = AF sin o, where
a is the angle between r and the rod. Hence
X
-s
kmdM
cos a, and Y =
/
kmdM
sin a,]
56. Show that in spherical coordinates, (r, 6, <f>)j the volume of a
solid is given by an integral of
the form
ffff^ cos 4>d4>dd dr.
[Hint. Let P be a point on
a sphere of radius r, the longi-
tude of P being d and its
latitude <^. It is usual to let
the xjz-section of the sphere be
the equator and the xa-section
the prime meridian. Then P
is the point (r, dj 4>), PS and
QR are two adjacent meridians
and PQj SR are two adjacent
Fig. 67. parallels. Then
= (r, ^ -h AS, 4>), R -- (r, d + Ad, <l> + A<^), S = (r, d, <!> + A<l>).
Also PS = rA<l> and PQ = O'P • A^ = r cos <^ A^.
XIV, § 136] MULTIPLE INTEGRALS— APPLICATIONS 233
Suppose r to increase tor -{-Ar so that P, Q, R, S move out to P',
Q', R\ /S' respectively. In this way is formed the element of volume in
spherical coordinates. Its approximate volume is
PS PQ ' PP' = r A^ • r cos <^ A^ • Ar = r^ cos <^ A^ A<^ Ar.]
67. Calculate the volume of a sphere, using spherical coordinates.
68. Calculate the volume cut from a cone of angle 2 a by two con-
centric spheres with centers at the vertex of the cone.
69. Show that, in cylindrical coordinates (r, ^, 2), the volume of a
soUd is given by an integral of the
form
///'
rdddrdz.
Here r denotes distance from the
s-axis.
[Hint. The coordinates of P =
(r, e, z) are OM =r, xOM = e, MP=z.]
e = (r, ^ + A^,s);
R = (r, ^ + A^, s + As);
S = (r, ByS+Az).
Increase r to r + Ar, so that we
have an element of volume whose
approximate volume is
PQ'PS' PP' = r A^ • As • Ar.
60. Calculate the volume of a sphere using cylindrical coordinates.
61. Determine the part of the cylinder r = sin 2 ^ which lies between
the planes a = and z = y.
62. Determine the part of the cylinder r = sin 2 ^ which lies between
the planes 3 = and x -f y + s = V2.
Fig. 68.
CHAPTER XV
EMPIRICAL CURVES — INCREMENTS —
INTEGRATING DEVICES
137. Empirical Curves. Some of the methods used in
science to draw the curves which represent simultaneous
values of two related quantities and to obtain an equation
which represents that relation approximately are given in
Analytic Geometry. Usually the pairs of corresponding
values are plotted on squared .paper first; in all that follows
it is assumed that this has been done in each case.
'138. Polynomial Approximations. It is advantageous to
have equations which are as simple as possible. From
experimental results, it is not to be expected that absolutely
precise equations can be found, and the attempt is made to
get an equation of simple form which approximately repre-
sents the facts, in so far as the facts themselves are known.
One simple kind of function which often does approximately
express the facts is a polynomial:
(1) y = a + bx + cx^ + da^ + • • • + fcx**.
139. Logarithmic Plotting. The preceding forms of equa-
tions may not represent the facts very well imless a large
niunber of terms (1), § 138, are used.
If the first graph resembles one of the curves y = x^,
y = x?,y = x^y etc., or y = x^^^, y = x^^^, etc., or y = 1/x,
y = l/x2, etc., it is advantageous to plot the common
logarithms of the quantities measured instead of the actual
values of those quantities.
234
XV, § 139] EMPIRICAL CURVES ' 235
If X and y represent the quantities measured, and u =
logioo:, V = logioi/ are their common logarithms, the values
of u and v may lie very nearly on a straight line,
(1) v = a + few,
where a and 6 are found by drawing the straight line which
on the whole seems to approximate best to the points
(w, \)) and measuring its slope, 6, and the t;-intercept, a.
Then from (1), since u = logio^;, ^ = logioy,
(2) logio 2/ = a + 6 logio ^ = logio fc + logio ^ = logio Q^^) ,
where logio A: = «; hence
(3) y = W.
This f onn of equation is very convenient for computation and
is used in practice very extensively wherever the logarithmic
graph' is approximately a straight line.* This work applies
equally well for negative and fractional values of 6.
In many cases where the process just described fails, it is
sometimes advantageous to assiune that the equation has
the form (y — B) = k(x — A)^ which evidently has a
horizontal tangent at the point (A, B) if n > 1, or a vertical
tangent if n < 1. If the first graph (in x and y) shows such
a vertical or horizontal tangent, that point (A, B) may be
* To avoid the trouble of looking up the logarithms, a special paper
usually described in Analytic Geometry may be purchased which is ruled
with logarithmic intervals. No particular explanation of this paper is
necessary except to say that it is so made that if the values of x and y are
plotted directly, the graph is identical with that described above. To se-
cure this result the successive rulings are drawn at distances proportional
to log 1 (=0), log 2, log 3, • • • from one comer, both horizontally and ver-
tically.
Explanations and numerous figures are to be found in many books; see,
e,g., Kent, "Mechanical Engineers' Pocket Book" (Wiley, 1910), p. 85;
Trautwine, "Civil Engineers' Pocket Book" (Wiley), (Chapter on Hy-
draulics).
236 THE CALCULUS [XV, § 139
selected as a new origin, and the values x'=^x-tA and
y'= X — B should be used; thus we would plot the values of
u = logioa:' = logio ix-A), v = logioy' = logio iy-E),
in the manner described above. The values of A and B are
found from the first graph (in x and y) ; the values of k and
n are found from the logarithmic graph as above.
140. Semi-logaritfamic Plotting. Variations of this process
of § 139 are illustrated in the exercises below. In par-
ticular, if the quantities are supposed to follow a com/pound
interest laWy y = k^^, it is advantageous to take logarithms
of both sides:
logio y = logio k + bx logio e,
and then plot u = x, v = logio 2/; if the facts are approxi-
mately represented by any compound interest law, the
experimental graph (in u and v) should coincide (approxi-
mately) with the straight line ,
V = A + BUy
where A = logio k and B = b logio e. After A and B have
been measured, k and b [ = B log« 10 = 2.303 B] can be
found.
EXERCISES
1. Find the equation of a straight line through the points (— 1, 3)
and (2, 5); through (2, - 3) and (4, 5).
2. Plot the data of Exercises 37-42, page 68; draw a straight line
as closely as possible through all the points without giving preference
to any of them; determine the equation from this graph; compare with
former results.
I'
Plot each of the following curves logarithmically, — either by plot-
ting logic X and logic y, or else by using logarithmic paper:
3. 2/ = 2 0^3. 5. 2/ = .4 ofi-^. 7. 2/ = 5.7 x«.
XV, §140]
EMPIRICAL CURVES
237
In each of the following tables, the quantities are the results of
actual experiments; the two variables are supposed theoretically to be
connected by an equation of the form y = fcx". Draw a logarithmic
graph and determine k and n, approximately:
9. [Steam pressure; v — volume, p = pressure.] [Saxelby.]
V
2
4
6
8
10
p
68.7
31.3
19.8
14.3
J1.3
IC
\. [Gas
engine
mixture
i; notation as above.]
[Gibson.]
V
3.54
4.13
4.73
5.35
5.94
6.55
7.14
7.73
8.04
V
141.3
115
95
81.4
71.2
63.5
54.6
50.7
45
11. [Head of water A, and time t of discharge of a given amount.]
[Gibson.]
h
0.043
0.057
0.077
0.095
0.100
t
1260
540
275
170
138
12. [Heat conduction, asbestos; 6 = temperature (F.), C = coefE-
-'^ient of conductivity.] [Kent.]
32**
212**
392**
572**
752**
1112**
c
1.048
1.346
1.451
1.499
1.548
1.644
13.
[Track records: d = distance, t
= record time (intercollegiate).]
d
100 yd.
220 yd.
440 yd.
880 yd.
1 mi.
2 mi.
t
0:09t
0:21i
0:48t
1:56
4:17t
9:27f
[Note. See Kennelly, Fatigue, etc., Proc. Amer. Acad. Sc. XLII,
No. 15, Dec. 1906; and Popular Science Monthly , Nov. 1908.]
238 THE CALCULUS [XV, § 140
Plot the following cunres, using logarithmic values of one quantity
and natural values of the other:
14. y^€F. 16. j/ = 1063». le. y=:4e-«. 17. y = .le-*^.
Discover a formula of the t3rpe y=kef^ for each of the following sets
of data:
18.
x:
.2
.4
.6
.8
1.0
y-
4.5
*
6.6
9.9
15.0
22.2
19.
x:
.6
- 1.2
1.8
2.4
3.0
y-
1.5
2.2
3.3
6.0
7.4
20.
x:
.31
.63
.94
1.26
1.57
y-
2.44
2.98
3.64
4.46
5.44
21.
x:
.2
.8
2.0
4.0
y-
8.2
4.5
1.3
0.2
22.
x:
.63
1.26
2.51
3.77
5.03
y-
4.02
2.70
1.20
0.54
0.24
23.
x:
1
2
3
4
5
y-
3.26
2.68
2.16
1.80
1.46
24.
A:
is the amplitude of vibration of a long pendulum
, t is the time
since i
it was set swinging. Show that they are connected by a law of
the form A — ke~^.
A
in. =
= 10
4.97
2.47 1.22
.61
.30 .14
t min. =
1
2 3
4
5 6
141. Method of Increments. A method adapted to the
case where (1) of § 138 has the form
(1) y = a + bx + cx2,
is as follows. From two pairs of values of x and y, say
(x, y) and {x + Ax, 2/ + Ay) given by experiment, we should
have
(2) y = a + hx + cx^^y + Ay = a + b (x + Ax) + c (x + Ax)^,
whence
(3) Ay = bAx + 2cxAx + c A?.
XV, §141]
EMPIRICAL CURVES
239
If Ax is constant, i.e. if points axe selected at equal x-in-
tervals on the crudely sketched cursre drawn through the
experimental points, we might write
(4) y= Ay = (bh + chPj + 2 ch-x = A + Bx
where h = Ax. If we should actually plot this equation,
Y = A+ Bx, we would get (approximately) a straight
line. Now Ay = F is the difference of two values of y; it can
be f oimd for each of the values of x selected above, and the
(approximate) straight line can be drawn, so that A and B
can be measured.
We may repeat the preceding process; from (4) we obtain,
as above,
(5) AY = BAx = 2 cA2, (h = Ax),
whence AF is constant if h was taken constant. Now AY is
the difference between two values of Y; that is, AF is the
difference between two values of Ay:
AY=A{Ay)=A^y,
and for that reason is called a second difference, or a second
increment. If the second differences are reasonably con-
stant, we conclude that an equation of the form (1) will
reasonably represent the facts and we find c directly by
solving equation (5).
Example 1. With a certain crane it is found that the forces /
measured in pounds which will just overcome a weight w are
/
8.5
12.8
17.0
21.4
25.6
29.9
34.2
38.5
w
100
200
300
400
600
600
700
800
What is the law connecting force with the weight that it just overcomes?
[Perry.]
Plotting the values of / and w, it appears that the points are very
240
THE CALCULUS
[XV, § 141
nearly on a straight line f — a-\- bw. If they were on a straight line,
Af/Aw would be constant and equal to df/dw = b. As a matter of fact,
for each increase of weight, Af/Aw varies only from .042 to .044, its
average value being 30/700 = .0429 Taking this value for 6, one gets
for the equation of the line, and hence for the relation between force
and weight:
/ = 4.21 + .0429 w, 4.21 = 8.5 - 100 X .0429
Here 4.21 appears to be the force needed to start the crane if no load
were to be lifted.
Example 2. If is the melting point (Centigrade) of an alloy of
lead and zinc containing x% of lead, it is found that
a; = % lead
40
50
60
70
80
90
= melting point
186
205
226
250
276
304
/
Plotting the points (a;, 0) will show them not to lie in a straight line as
is also shown by the difference A0. But A(A^) or A*^ does run imi-
formly. Therefore one tries a quadratic function of z for 0, that is
= a-\-hx +cxK
It is evident that A0 = lOh + c(20x + 100),
and A2 ^ = 200 c.
The average value of A^ is 2.25 Hence c = .01 125 If we subtract
cx^ from 0j we find — ca^ = a + hx. These values can be calculated
from the data and from a = .01125; they will be found to lie on a straight
line; hence a and b can be found by any one of several preceding methods.
The student will readily obtain, approximately,
^ = 133 + .875 X + .01125 x\
a formula which represents reasonably the melting point of any zinc-
lead alloy. [Saxelbt.]
EXERCISES
1. Express f{x) as a quadratic function of x, when
x: 0.5 1.0 1.5 2.0 2.5 3.0
fix): 2.6 1.9 1.6 1.5 1.7 2.1 2.8
2. Express / (x) as a cubic function of x, when
x: .02 .04 .06 .08 .10 .12 .14
fix): .020 ,042 .064 .087 .111 .136 .163
XV, §141] EMPIRICAL CURVES 241
3. Express (m) as a cubic in m, when
m: .01 .02 .03 .04 .05 .06 .07 .08
<t>{m): .00010.00041 .00093 .00166 .00260 .00385 .00530 .00690
4. The specific heat S of water, at ^° C, is
^: 5 10 15 20 25 30
S: 1.0066 1.0038 1.0015 1.0000 0.9995 1.0000 1.002
Express S in terms of $.
6. Determine a relation between the vapor pressure P of mercury,
and the temperature C, from the data below:
^: 60 90 120 150 180 210 240
P:.03 .16 .78 2.93 9.23 25.12 58.8
6. The resistance R, in ohms per 1000 feet, of copper wire of diame-
ter D mils, is
D: 289 182 102 57 32 18 10
/J: .126 .317 1.010 3.234 10.26 32.8 105.1
Find a relation between R and D,
7. The Brown and Sharpe gauge numbers N of wire of diameter D
mils, are
N: 1 5 10 15 20 25 30
D: 289 182 102 57 32 18 10.
Express D in terms of N.
8. Find a relation between the speed S of a. train in kilometers per
hour, and the horse-power (H. P.) of the engine from the data below:
H.P.:550 650 750 850
S: 26 .35 52 70.
9. The energy consmned in overcoming molecular friction when iron
is magnetized and demagnetized (hysteresis, H, — measured in watts
per cycle per liter of iron) is given below in terms of the strength of the
magnetic field (B, — measured in lines per square centimeter). What
is the relation between them?
B: 2000 4000 6000 8000 10000 14000 16000 18000
H: .022 .048 .085 .138 .185 .320 .400 .475
10. Proceed as in Ex. 15, for cobalt, the hysteresis loss H being now
measured in ergs per cycle per second:
B: 900 2350 3100 4100 4600 5200 5850 6500
H: 450 2450 3950 6300 7400 8950 10950 13250.
242
THE CALCULUS
[XV, § 141
The table below contains some data on the comparison of a tung-
sten lamp with a tantalum lamp. The voltage or electrical pressure F,
is in volts, the resistance R, in ohms, the current consumed in watts per
candle power; C denotes candle power, and W watts per candle power.
11. Tungsten
12. Tantalum
Voltage
C. P.
Watts
Resistance
C.P.
Watts
Resistance
per C. P.
per C. P.
V
C
W
R
C
W
R
80
14
2.51
166
5
3.80
260
90
24
1.83
173
10
2.85
265
100
36
52
1.49
182
18
2.05
275
110
1.23
190
25
1.65
283
120
71
1.10
197
38
1.35
290
130
95
0.96
202
50
1.15
300
140
128
0.83
210
62
0.95
308
150
160
0.76
216
78
0.85
315
160
196
0.58
222
100
0.75
323
170
230
0.52
227
122
0.70
327
180
270
0.50
232
156
0.70
332
190
312
0.48
238
190
0.60
340
200
340
0.47
242
235
0.55
345
For each lamp, express each of the quantities C, W, R, in terms of V.
•
142. Integrating Devices. It is important in many prac-
tical cases to know approximately the areas of given closed
curves. Thus the volume of a ship is found by finding the
areas of cross sections at small intervals. Besides the methods
described above, the following devices are employed:
A. Counting squares on cross-section paper.
B. Weighing the figures cut from a heavy cardboard of
uniform known weight per square inch.
C. Integraphs. These are machines which draw the inte-
gral curve mechanically; from it values of the area may be
read off as heights.
XV, §142]
EMPIRICAL CURVES
243
y-fM
The simplest such machine is that invented by Abdank-Abakano-
wicz. A heavy carriage CDEF on large rough rollers, Ry R' is placed on
the pap)er so that CE is
parallel to the 2/-axis.
Two sliders S and /S'
move on the parallel
sides DF and CE\ to S
is attached a pointer P
which follows the curve
y ^ f(x). A grooved rod
AB slides over a pivot
at Af which lies on the
X-axis, and is fastened by
pivot B to the slider S,
A parallelogram mechan-
ism forces a sharp wheel
W attached to the slider
S^ to remain parallel to
AB, A marker Q draws
a new curve i = 0(x),
which obviously has a tangent parallel to W, that is, to AB, If AB
makes an angle a with Ox, tan a is the slope of the new curve; but
tan a is the height of S divided by the fixed horizontal distance h between
A andB:
d<t>(x) _,^ _ heighto{S _f(x),
dx "^^'^'' h "T
Fig. 69.
whence
1 ^«-x
» - io = ^ y ^ f(x)dx;
where a is the value of x at P when the machine starts, and io denotes
the vertical height of the new curve at the corresponding point.
D. Polar Planimeters. — There are machines which read
oflf the area directly (for any smooth closed curve of simple
shape) on a dial attached to a rolling wheel.
The simplest such machine is that invented by Amsler.
Let us first suppose that a moving rod ab of length I always remains
jjerpendicular to the path described by its center C. The path of C
may be regarded as the limit of an inscribed polygon, and the area
swept over by the rod may be thought of as the limit of the sum of small
244
THE CALCULUS
[XV, § 142
quadrilaterals, the area AA of each of which is lAp, approximately,
where Ap is the length of
the corresponding side of
the polygon inscribed in
the path of C. Hence the
total area A swept over
by the rod is evidently Zp,
where p is the total length
Pjq <7q of the path of C.
But if the rod does not
remain perpendicular to the path of C during the motion, and if ^ is the
angle between the rod and that path, the area AA becomes Z sin ^ • Ap,
approximately. The expression sin ^ • Ap
may be thought of as the component of Ap in
a direction perpendicular to the rod. Calling
thid component As, we have AA = lAs, ap-
proximately; and the total area A swept over
by the rod is precisely lim 2 AA = lim 2 1! As
=J*ld8 = lj*d8 = Is, where s =J*d8 is the
total motion of C in a direction perpendicular
to the rod.
The quantity s =J*d8 can be measured me-
chanically by means of a wheel of which the
rod is the axle, attached to the rod at C; for if
e is the total angle through which the wheel
turns during the motion, s = rd, where r is the
radius of the wheel, and d is measured in
radians. Hence A = fe = Ird; the value of d
is read off from a dial attached to the wheel;
I and r are known lengths.
In Am8ler*8 polar planimeter, one end h of the rod ab is forced to
trace once around a given closed curve whose area is desired; the other
end a is mechanically forced to move back and forth along a circular
arc by being hinged at a to another rod Oa, which in its turn is hinged
to a heavy metal block at O. As 6 describes that part of the given curve
which lies farthest from 0, the rod ah sweeps over an area between the
circular arc traced by a and the outer part of the given curve; as 6
describes the part of the curve nearest to 0, ab sweeps back over a
portion of the area covered before, between the circle and the inner
part of the given curve. This latter area does not count in the final
total, since it has been swept over twice in opposite directions. Hence
Fig. 71.
XV, S 142]
EMPIRICAL CURVES
the quantity A = IrB, given by the reading of the dial
is precisely the desired area al the given closed
curve, which has been swept over just once by
the moving rod ob.
In practicing with such a machine, begin
with curves of known area. The machine ia
useful not only in finding areafl of irregular
curves whose equations are not known, but
also in checking integrations performed by the
standard methods, and in giving at least
approximate values for integrals whose evalu-
ation is difficult or impossible.
For further information on int«grating
devices, see: Abdank-Abakanowicz, Les in-
legrajthes (Paris, Gautliier-Villars); Henrici,
Report on Planimeters (British Assoc. 189^
Mechanical Inlegrato?
Encyklopadie der Math. Wisa,, Vol. II. Catalogues of dealers
stnunenta also contain much really valuable information.
Fig. 72.
196-523); Shaw,
(Proc. Inst. Civ. Engs. 1885, pp. 76-143);
EXERCISES
1. Cottstruct a figure of each of the types mentioned below, with di-
mensions selected at random, and find their areas approximately by
counting squares; by Simpson's rule; by the planimeter, if one is avail-
able. (1) A right triangle; (2) An equilateral triangle; (3) A circle;
(4) An eUipse. (Draw it with a thread and two pins.) (5) An arch of
a sine curve; (6) An arch of a cycloid.
2. The figures below are repro-
ductions of indicator cards, taken
from three different types of en-
gines. The dotted curves are en-
tirely separate from the full lines.
The average pressure on the piston
is the area of one of these curves
divided by the length of stroke.
Find this value in each case, where
the stroke is 12 in. in the first figure, and 8 in. in each of the others.
(Unit of area = I large square.)
Pia. 73 (a).
THE CALCULUS
(Note. The uwfc done ia precisely the area in question, on a proper
scale, since the work is the average pieeaure times the length of stroke.]
:n
,l|:j
!-riir|
J.--, -+ ri
1:
uht^
=e*
W
--L
CHAPTER XVI
LAW OF THE MEAN — TAYLOR'S FORMULA — SERIES
143. RoUe's Theorem. Let us consider a curve
where / (x) is single-valued and continuous, and where the
curve has at every point
a tangent that is not ver-
tical. If such a curve
cuts the a;-axis twice, at
X = a and a; = 6, it surely
either has a maximum or
Fig. 74.
a minimum at at least one point x = c between a and b.
It was shown in § 33, p. 54, that the derivative at c is zero:
[A] If /(a) =/(&)= 0, then [^]^^^= 0, (fl<c<&);
this fact is known as Rolle's Theorem.
144. The Law of the Mean. Rollers Theorem is quite evi-
dent geometrically in the form: An arc of a simple smooth
curve cut off by the x-axis has at least one horizontal tangent.
The precise nature of the
necessary restrictions is
given in § 129.
Another similar state-
ment, which is true under
the same restrictions and
Fig. 75. is equally obvious geo-
metrically, is: An arc of a simple smooth curve cut off by any
secant has at least one tangent parallel to thai secant,
247
248 THE CALCULUS [XVI, § 144
If the curve is j/ = / (x) , and if the secant S cuts it at points
P:[a,/(a)] and Q: [6, /(&)], the slope of S is
Ay-hAx = [/(&) -f(a)] ^ (& - a).
The slope of the tangent CT at a; = c is equal to this:
This statement is called the law of the mean or the theorem
of finite differences.
It is easy to prove this statement algebraically from
RoUe's Theorem. For if we subtract the height of the secant
S from the height of the curve, we get a new curve whose
height is:
D{x) =/(x) - ['^^^{^ (x-a) +/(a)].
Now D(x) is zero when x = a and when x = b. It follows
by § 143 that d D {x)/dx = at x = c, (a < c < 6) :
which is nothing but a restatement of [B],
145. Increments. The law of the mean is used to detenninc
increments approximately, and to evaluate small errors.
If y =zf (x) is a given function, we have, by § 144,
In practice this law is used to estimate the extreme limit of
errors, that is, the extreme limit of the nimierical value of Ay,
It is evident that
[B*] |Ay|<Mi.|Ax|,
where Mi is the maximum of the numerical value of dy/dx be-
tween a and a + Ax. When Ax is very small, the slope dy/dx
XVI, § 145] LAW OF THE MEAN 249
is practically constant from a to a + Ax in most instances,
and Ml is practically the same as the value of dy/dx at any
point between a and a + Ax.
Example 1. To find the correct increments in a five-place table of
logarithms.
The usual logarithm table contains values of L = logio ^ at intervals
of size AN = .001. Hence
where iV<c<iV + . 001.
Logarithms are ordinarily given from N = I to N = 10. Hence AL
will vary from .00043 at the beginning of the table to .00004 at the end
of the table. This agrees with the " differences" column in an ordinary
logarithm table.
Example 2. The reading of a certain galvanometer is proportional
to the tangent of the angle through which the magnetic needle swings.
Find the effect of an error in reading the angle on the computed value of
the electric current measured. We have
C = A; tan $,
where C is the current and d the angle reading. Hence the error Ec in
the computed current is
where Ec is the error in the computed value of the current, and A^ is
the error made in reading the angle 6. Since A^ is very small,
Ec = k sec2 e • Ady approximately. The error Ec is extremely large if
e is near 90°, even if A^ is small; hence this form of galvanometer is not
used in accurate work.
EXERCISES
1. At what point on the parabola y — x^ \& the tangent parallel to
the secant drawn through the points where x = and a; = 1?
2. Proceed as in Ex. 1 for the curve y = sin x, and the points where
a; = 60° and a; = 75°.
3. Proceed as in Ex. 1 for the curve y = log (1 + x), for a; = 1 and
a; =3.
250 THE CALCULUS [XVI, § 145
4. Discuss the differences in a four-place table of natural sines, the
argument interval being 10'.
6. Proceed as in Ex. 4 for a similar table of natural cosines; of
natural tangents.
6. Discuss the differences in a four^place table of logarithmic sines,
the entries being given for intervals of 10'.
7. Proceed as in Ex. 6 for a table of logarithmic tangents.
8. Calculate the difference in a seven-place table of logio sin x at the
place where x = 30°; where x = 60°; where x = 85°.
9. Discuss the effect of a small change in x on the function
y = log (1 + 1/x).
10. If logio N = 1.2070 ± .0002, what is the uncertainty in N? [The
term ± .0002 indicates the uncertainty in the value 1.2070.]
11. If the angle of elevation of a mountain peak, as measured from
a point in the plain 5 mi. distant from it, is 5° 20' ± 5', what is the un-
certainty in the computed height of the peak?
12. The horizontal range of a gun is R = (V^/g) sin 2 a, where V is
the muzzle speed and a the angle of elevation of the gun. If F = 1200
ft. /sec, discuss the effect upon R of an error of 5' in the angle of ele-
vation.
13. T he distance to the sea horizon from a point h ft. above sea level
is D = V2 Rh + h^, where R is the radius of the earth. Discuss the
change in D due to a change of one foot in h. (Z), -B, and h are all to
be taken in the same units.) If D is tabulated for values of h at inter-
vals of one foot, what is the tabular difference at the place where h = 60?
14. If the boiling point of water at height H ft. above sea level is T,
H = 517 (212° - T) - (212° - T)^, T being the boUing temperature
in degrees F. Discuss the uncertainty m H^ \i T can be measured to
1°. If H be tabulated with argument T at intervals of 1°, what is the
tabular entry and the tabular difference when T — 200°?
16. When a pendulum of length I (feet) swings through a small angle
a (radians), the time (seconds) of one swing is T = vVlfg (1 + a2/16).
What is the effect on T of a change in a, say from 5° to 6°? Of a change
in I from 36 in. to 37 in.? Of a change in g from 32,16 to 3^.3?
XVI, § 146]
LAW OF THE MEAN
251
16. The viscosity of water at 0° C. is P = 1/(1 + .0337 d + .00022 ^).
Discuss the change in P due to a small change in d. What is the average
value of P from ^ = 20° to ^ = 30°?
17. The quantity of heat (measured in calories) required to raise one
kgm. of water from 0° C. to ^° C. is 27 = 94.21 (365 - ^)o-3i25 _}- k. How
much heat is required to raise the temperature of one kgm. of water 1° C.
when e = 10°? 20°? 30°? 70°? To find k, observe that H = when ^ = 0.
18. The coefficient of friction of water flowing through a pipe of
diameter D ^nches) with a speed V (ft. /sec.) is / = .0126 + (.0315
— .06 D)/VV. What is the effect on / of a small change in 7? in D?
146. Limit of Eiror. In using the formula [B] the uncer-
tainty in the value of c is troublesome. If the value of dy/dx
at a: = o is used in place of its value at x = c, the error made
in finding Ay by [B] can be expressed in terms of the second
derivative d!^y/doi^.
We shall use the convenient notation /'(x), f" (x), etc.,
for the derivatives of / (x) :
r (x) = ^ = ^ (the slope oiy= /(x) ).
r(x) =
dy(x) _ d^y _ df (x)
(the flexion).
dx^ dx!^ dx
Let Af 2 denote the maximimi of the numerical value of/" (x)
between two points V
X = a and x = 6, so
that
(1)' \f"(x)\<M,.
The area under the
curve y = /" (x) be-
tween X = a and any
-f(X)
Fig. 76 (a).
point X = X between a and h is evidently not greater than the
area under the horizontal line y = M2) that is, if a< x< 6,
(2)
I (' 7"(x)da;|< P "M-dx, or /'(a;)T"1<M2a;T ^
252
THE CALCULUS
[XVI, § 146
since df(x)/dx =/"(x), and M2 is a constant; whence, sub-
stituting the limits of integration in the usual manner,
(3) \r(x)^r(a)\<M,ix^a),
which is geometrically shown in Fig. 76 (b) . It follows that
the area under the curve
y =f (x) — f (a) is not
greater than that under
the line y = M2(x — a):
(4) I f'^lf'ix) -ria)]dx
^ I M2(x — a) dx;
or since /' (a) and Af 2 are
x^h
Fig. 76 (6).
constants and df{x)/dx =f(x),
r/(x)-/'(a)..ri<3f/-^r^
L Jx = a| — Z Jx=a
whence, substituting the limits in the usual manner,
[C] \f{x) -f{a) -/'(fl)(x- fl)|< M2 ^^ ,
which holds for all values of x between x = a and x = b.
This formula may be written even if x<a:
[C*] fix) ^f{a) +/'(fl)(x -fl)+£2,where|£2| <M% ^^^-^,
and E2 is the error made in using f\a) in place of /' (c) in
formula [C]; for (x — a)^ = | a; — a | ^.
It should be noticed that E2 is exactly the error made in
substituting the tangent at x = a for the curve, i.e. it is the
difference between At/[= f{x) — f(a)] and dy[= f{a) {x — a)]
mentioned in § 26, p. 43, and shown in Fig. 8.
The formula [B*] is exactly analogous to [C*]; since
Ay = /(x) — f(a) if Ax = X — a, [B *] may be written
[B *] fix) =/(a) +Ei, 1^1 1 < Ml . |x - a\.
XVI, § 147] TAYLOR'S FORMULA 253
Example 1. In Ex. 1, § 145, we found for L = logio AT, \
AL = •— ^ (nearly).
Applying [C*], with/(iV) = logio N.a^N.x^N + ANyZ-a^AN
= .001, we find
where M2 is the maximum value of | /" (N) \ = Gogio e)/N^ between iV = 1
and N = .10. Hence E2 < .00000022 The value of AL found before
was therefore quite accurate, — absolutely accurate as far as a five-place
table is concerned.
ExAitPLE 2. Apply [C*] to the function /(x) = sin x, with a = 0, and
show how nearly correct the values are for x < ir/90 = 2°.
Since /(x) = sin x, and a = 0, [C*] becomes
sin X = sin (0) + cos (0)- (x - 0) +E2 = x + ^2, \E2\ ^ M2^y
where Af 2 is the maximum of | /" (x) | = | — sin x | between and ir/90.
that is M2 = sin (ir/90) = sin 2° = .0349 Hence E2 < .0175 x2. Since
x<ir/90, x2<ir2/8100<.0013; hence ^2<. 000023, and sinx =x is
correct up to x 5 ir/90 within .000023
Similarly, for a = ir/4, we have, by [C*],
sinx = ^[l + (x-f)] + ^2, \E2\<M2^^^^\
where M2 < 1. If (x - ir/4) < ir/90, 1 E2I < (ir/90)2 -^ 2 = .0007
147. Extended Law of the Mean. Taylor's Theorem. The
formula [C*] can be extended very readily. Let/' (x), /" (x),
/'" (x), • • • Z^**^ (x) denote the first n successive derivatives of
^ ^^^~ dx*- " dx
and let the maximum of the numerical value of f^^ (x) from
X = a tQ X = fe be denoted by Mn- Then
I /*> (X) \<Mn,
and I n-'jin) (x)dx\^\ r"M„dx
254 THE CALCULUS [XVI, J 147
or !/<»-» (x) - /"-« (a) I ^ I Mn (x-a)\
for all values of x between a and b. Integrating again, we
obtain, as in § 146:
(x - ay
2
and, continuing this process by integrations until we reach
/(x), we find:
I /<"-'' (.x) -/<-*> (a) -/<"-» (a) (x - o) U
i»f.
[^1
/(x) -/(a) -/' (a) (X- a) --^ (X- a)»- ...
/(n-l) (a)
^M,'-"'"
nt
or.
[Z)*] / (X) =/ (a) +/' (a) (X - a) +^^ (x - a)» + . . .
/(n-l) (a)
where
and where Af^ is the maximum of 1/^**^ (re) | between x = a and
X = b.
This formula is known as the extended law of the mean, or
Taylor's Theorem, after Taylor, who first gave such approxi-
mations as it expresses. It is one of the more important for-
mulas of the Calculus.
In particular, if a = 0, the formula becomes
[D*] /(x)=/(0)+/'(0)x+^x«+...
/(n-l) (0) •
where \En\ ^ Mn\x^\ /n\ This special case of Taylor's
Theorem is often called Maclaurin's Theorem.
XVI, § 147] TAYLOR'S FORMULA 255
The formula [D*] replaces f (x) by a polynomial of the
nth degree, with an error En» These pol3aiomials are repre-
sented graphically by curves, which are usually close to the
curve which represents / (x) near x = a. See Tables, III, K.
Since the expression for En above contains n! in the
denominator, and since n! grows astoundingly large as n
grows larger, there is every prospect that En will become
smaller for larger n; hence, usually, the pol3aiomial curves
come closer and closer tof(x) as n increases, and the approxi-
mations are reasonably good farther and farther away from
X = a. But it is never safe to trust to chance in this matter,
and it is usually possible to see what does happen to En as
n grows, without excessive work.
Example L Find an approximating pol3momial of the third degree
to replace sin x near x = 0, and determine the error in using it up to
X = ir/18 = 10°.
Since/ (x) = sin x and a = 0, we have/' (x) = cos x, /" (x) = — sin x,
/'" (x) = - cos X, /»^ (x) = + sin x, whence/ (0) = 0, /' (0) = 1, /" (0) =
0,/'" (0) = - 1; and [Max. |/»v (x)\] = [Max.|sinxl] = sin 10° = .1736
between x = and x = ir/18 = 10°. Hence
sin X = + 1 • (X - 0) + + (- 1) . ^^^^ + ^4 = X - ^ + iEf4,
where \E^\< (.1736) • x4/4! ^ (.1736) (x/18)4 -^ 4! < .000007, when x
lies between and ir/18.
In general^ the approximation grows better as n grows larger, for
[/(«) (x) I is alwajrs either [ sin x [ or 1 cos x|; hence Mn < 1, and 1 ^n I ^
af^/nl which diminishes very rapidly as n increases, especially if x < 1 =
57°.3 For n = 7, the formula gives, for x > 0,
sinx=x- J+|]+E7, \E7Kxy7i
Example 2. Express Vl -f x as a quadratic in x and estimate the
error if x lies between and .2
Here /(O) ^ l;/'(0) = 1/2;/" (0) = - 1/4; M, > |/'" (©) | = 3/8.
Hence vTT^ = 1 + x/2 - xV8 + ^s) I ^3 1 < [(3/8)/(3!)] (.2)3 = .0005
Thus: VT2 = 1 + .1 - .005 + ^3 = 1.0950 + ^3; I ^8 1 <..0005
256 THE CALCULUS [XVI, § 147
EXERCISES
1. Apply the formula (C*) to obtain an approximating polynomial
of the first degree for tan x, with a = 0. Show that the error, when
I X 1 < t/90, is less than .00003. Draw a figure to show the comparison
between tan x and the approximating linear function.
2. Apply [/)*] to obtain an approximating quadratic for cos x, with
a = 0. Show that the error, when |a; | <ir/10 is less than (x/10)'-^3!
Draw a figure.
3. Apply [/)*]' to obtain an approximating cubic for cosx, near
x = 0. Hence show that the formula found in Ex. 2 is really correct,
when \x \ <t/10, to within (ir/lO)^ t- 4! Draw a figure.
4. Obtain an approximation of the third degree for sin x near
X = t/3. Show that it is correct to within (ir/10)* -J- 4! for angles
which differ from ir/3 by less than ir/10. Draw a figure.
Obtain an approximation of the first degree, one of the second
degree, one of the third degree, for each of the following functions near
the value of x mentioned; find an upper limit of the error in each case
for values of x which differ from the value of a by the amount specified;
draw a figure showing the three approximations in each case:
6. e», a = 0, I X - a I < .1 9. c-*, a = 2, | x - a | < .5
6. tan X, a = 0, 1 X — a| < ir/90. 10. sin x, a = ir/2, | x—a \ < t/45.
7. log(l+x), a=0, |x-a|<.2 11. tanx, a = V4, |a;— a | <t/90.
8. cosx,a = T/4, |x-a|<T/18. 12. x2+x+l,a = l, |x-al <l/5.
13. 2 x2 - X - 1, a = 1/2, I X - a I < 1.
14. x3-2x2-x + l, a = -2, |x-a| <.5
16. Find a polynomial which represents sin x to seven decimal places
(inclusive), forfx] <10°.
16. Proceed as in Ex. 15, for cos x; for e~*, when 0<x <1.
17. Show that x differs from sin x by less than .0001 for values of x
less than a certain amount; and estimate this amount as well as possible.
18. The quantity of current C (in watts) consumed per candle power
by a certain electric lamp in terms of voltage v is C = 2.7 + 108007-.0767».
Express C by a poljmomial in t; — 115 correct from » = 110 up to « =
120 to within .025 watt.
XVI, § 148] TAYLOR'S FORMULA 257
148. Application of Taylor's Theorem to Extremes. If a
function y = f(x) is given whose maxima and minima are to
be found, we may find the critical points where /' (x) = 0.
Let a be one solution of /'(x) = 0, that is, a critical value.
Then, since/' (a) = 0, we have, by [D*],
At/=/(x) -/(a) =0 + -^(a:-a)2+£;3, l^sl < ^3^^^'>
where ikf 3 ^ | /'" (x) \ . Hence the sign of Ay is determined by
the sign of /"(a) when (x — a) is sufficiently small.
If /"(a)>0, Ay>0, and/(x) is a minimum at x = a.
If /" (a) <0, Ai/ <0, and / (x) is a maximum at x = a. (See
§ 42, p. 66.)
If /"(a) = 0, the question is not decided.* But in that
case, by [D*] :
Ay=/(x)-/(a)=0 + 0+-^^(x-a)3+-^(x-a)4 + S^^
where|i?6| ^ Ms\x-a\^/5ly M^ ^ \r(x) | . From this we see
that if/'" (a) 7^ there is neither a maximum nor a minimum,
for (x — a)3 changes sign near x = a. But if/'" (a) = 0, then
/*^ (a) determines the sign of Ay, as in the case of/" (a) above.
In general, if /^*^ (a) is the first one of the successive deriva-
tives, /'(a), /"(a), • • •, which is not zero at x = a, then there
is:
no extreme if k is odd;
a maximum if k is even and /(*) (a) <0;
a minimum if k is even and /(*> (a) > 0.
♦ The methods which follow are logically sound and can always be car-
ried out when the derivatives can be found. But if several derivatives
vanish (or, what is worse, fail to exist), the method of § 34, p. 64, is better
in practice.
258 THE CALCULUS [XVI, § 148
Example 1. Find the extremee for y = x*.
Since/ (a;) = x^jf (x) = 4 x^j hence the critical values are solations of
the equation ^xf =0, and therefore a? = is the only such critical
value.
Since /" (x) = 12 x^, ]"' ^^) = 24 x, /»v (x) = 24, the first derivative
which does not vanish at x = is /*^ (x), and it is positive (= 24). It
follows that / (x) is a minimum when x = 0; this is borne out by the
famiUar graph of the given curve.
EXERCISES
Study the extremes in the following functions:
1. x«. 6. (x + 3)6. 9. x2sinx.
2. (x - 2)8. 6. x* (2 X - 1)3. 10. x4 cos X.
8. 4 x3 — 3 X*. 7. sin x^. 11. x^ tan x.
4. x3(H-x)3. 8. x-sinx. 12. e-i/^.
18. Discuss the extremes of the curves y == x", for all positive integral
values of n.
14. An open tank is to be constructed with square base and vertical
sides so as to contain 10 cu. ft. of water. Find the dimensions so that
the least possible quantity of material will be needed.
16. Show that the greatest rectangle that can be inscribed in a given
circle is a square.
[See Ex. 44, p. 59. Other exercises from § 35 may be resolved by
the process of § 148.]
16. What is the maximum contents of a cone that can be folded from
a filter paper of 8 in. diameter?
17. A gutter whose cross section is an arc of a circle is to be made by
bending into shape a strip of copper. If the width of the strip is a, show
that the radius of the cross section when the carrying capacity is a
maximum is a/ir. [Osgood.]
18. A battery of internal resistance r and E. IM. F. e sends a current
through an external resistance R. The power given to the external
circuit is
{R + r)2 •
If e = 3.3 and r = 1.5, with what value of R will the greatest power
be given to the external circuit? [Saxelby.]
XVI, §149] TAYLOR^S FORMULA 259
149. Indeterminate Forms. The quotient of two functions
is not defined at a point where the divisor is zero. Such
quotients / (x) -5- <^ (x) at x = a, where / (a) = <t> (a) = 0, are
called indeterminate forms* We may note that the graph
of
(1) ^=i§)' (/(«)=«(a)=0),
may be quite regular near x = a; hence it is natural to make
the definition:
(2) ,] ^m =lual^..
Jx^a <t> \X)jx^a x-*a 9 W
If we apply [D *], we obtain,
^f(x) _ 0+r(a)(x-a)+E2'
^ <t>{x) "0 + «'(a)(x-a) + i?2'"
where
\Ei'\< M2' {x - a)V2!, I £?2" I S M^" {x - a)V2!,
and
Mj' > |r(x) I, M2" > |«"(x) I, near X = a.
Hence
<^' (a) + p" M2" ^
where p' and p" are numbers between — 1 and +1. It
follows that
x->a x-^a <t> W <t> {a)
unless <^' (a) = 0. But if <!>' (a) = 0,q becomes infinite, and
the graph of (1) has a vertical asymptote at x = a imless
♦If 0(a) =0 but /(a) 5*^0 the quotient q evidently becomes infinite; in
that case the graph of (1) shows a vertical asymptote.
260 THE CALCULUS [XVI, § 149
/' (a) = also. If both /' (a) and 0' (a) are zero, it follows in
precisely the same manner as above, that
where either /^*^ (a) or 0^*^ (a) is not zero, but all preceding de-
rivatives of both / (x) and <t> (x) are zero at x = a; and where
MUi ^ 1/^*+'^ (x) I, Mi'+i > I «(*+^> (x) I near a: = a and where
p' and p" are numbers between — 1 and +1. It follows
that
,. ,. fix) /^*> (g)
Imi g = lim 77-^ = tott-^ ,
provided all previous derivatives of both f(x) and <t>{x) are
zero at x = a, and provided 0^*^ (a) ^ 0. If <^^*^ (a) = 0,
/^*^(a) 7^ 0, then q becomes infinite and the graph of (1) has a
vertical asymptote at x = a.
It should be noted that (3) is only a repetition of Rule [VII], p. 31.
For if w = / (x) and v = <t> (x), since f{a) = <t> (a) = 0,
= /W =^ fi^)—f (g) ^ 4!f =^ 4^ ^ 4?!
^ </> (x) (x) — <i>{a) Ay Ax ' Ax '
where Ax = x — a; and therefore
lim o = lim -— -^ lim — = -J- -^ 3- = iihA = -"^t^ ,
x->a Ai->oAx Ax-»oAx L«^ axjx=a L0 (^)J»-o 9(0)
provided 0' (a) is not zero (see Theorem D, p. 15).
Example 1. To find lim [(tan x) -r x].
Here / (x) = tan x, (x) = x; / (0) = (0) = 0; hence
,. tan X /' (0) [sec2 x]««o .
jr->0 X <t> W 1
Draw the graph q — (tan x) -^ x and notice that this value g = 1 fits
exactly where x = 0.
This limit can be found directly as follows:
,. tanfe _ .. tan {O + h) — tan (0) __ d tan x "| __ „ T __ 1
^Z h "Ho (o + A)-(o) ~ dx 1=0"^'' ''l^o"^-
XVI, § 150] TAYLOR'S FORMULA 261
Example 2. To find lim (1 — cos x)/a;2.
Here fix) = 1 - cos x, <^ (x) = x^;f(0) = <^ (0) = 0;/' (0) = sin (0) =
and 0' (0) = 0; /" (x) = cos x, 4>" (x) = 2; hence
,. 1 ~ cos X __ cos g "| _ 1
a^-*0 X^ 2 Jar=iO 2
Draw the graph of g = (1 — cos x)/x2, and note that (x = 0, g = 1/2)
fits it well.
150. Infinitesimals of Higher Order. When the quotient
approaches a finite number not zero when x is infinitesimal:
(2) lim g = lim^^ = fc 7^ 0,
1 hen / {x) is said to be an infinitesimal of order n with respect to x.
An infinitesimal whose order is greater than 1 is called an
infinitesimal of higher order.
The equation (2) may be reduced to the form
(3) lmi[/(a;)-fcx»] = 0,
a;-K)
or
(4) f(x) = (k + E)af^,
where lim -B = 0. The quantity fcx" is called the principal
part of the infinitesimal/ (x) . The difference/ (x) — fcx** = Eaf^
is evidently an infinitesimal whose order is greater than n, for
lim (Ex'' -r-x'')=limE = 0.
Thus by Example 2, § 149, 1 — cos x is an infinitesimal of the second
order with respect to x; its principal part is x'^/2. Note that
1 - cos a; = a:2/2+pa^/3 !,
by [D*], where — 1 ^ p ^ + 1; the principal part is the first term of
Taylor's Theorem that does not vanish.
In general, if we have /(O) =/' (0) =/" (0) = • • • =/*-i (0) = 0, but
/ *> (0) f^ 0, the formula [D*] gives, for a = 0,
/ (x) = /c*) (0) . X* A I + V Mk-\-i a;*+V(A; + 1) \
where Ma+i ^ |/ <*+i) {x) \ near x = 0, and — 1 ^ p ^ + 1. Hence / (x)
is an infinitesimal of order /c, and its principal part fe/^KO) x^/kL
262
THE CALCULUS
[XVI, § 150
EXERCISES
Evaluate the indeterminate forms below, in which the notation
<t» (x) lo means to determine the limit of 4> (x) when x = a. The vertical
bar applies to all that precedes it. Draw the graphs as in Exs. 1, 2, above.
1. sin x/x lo.
2. sin 2 x/sin 3 X |o.
.6^—1
7.
• x2-l
tan 2x
tan 3x
8.
^^ log(l-a;
)
Xv.
smo;
^^ x — sinx
19. .
X — tan X
11.
X
1 — cos 2 X
6
9.
12
8. tan3x/x|o.
X
o» — 6»
X
logx2
14
1 + cos y X
1 — X — log X
16.
19.
X cos X — sm X
x3
o'*>«» — X
logx
17.
sin~^ X
20
tan~^ X
log (x8 - 7)
• x2-5x + 6
15.
18.
log (1 + x)
■ I ■ ■ ■ — . — ^
X — sin X
21 s^P"^ (^ — 2)
' Vx2+x--6
22.
in-i Va2 - x2
sm
Va2 - x2
Determine the order of each of the quantities below when the vari-
able X is the standard infinitesimal :
23. X — sin X.
24. «• — «-».
26. x2 sin x>.
26. log(l+x)-».
27. «" — «^».
28. a»-L
. log[(a + x)/(a-x)].
80. X cos X — sin x.
81. sin 2 X — 2 sin X.
82. log cos X.
88. log (1+ e-i/»).
84. tan~^ x — sin"i -j.^
86. log cos X — sin2 x.
86. 2x — «» + c^.
87. QQs'^{l-x)-y/2x'-xK
151. O^er Indetenmnate Fonns. The numerator and de-
nominator can be replaced by their derivatives not only
when the fraction takes the form 0/0, but also when it takes
XVI, § 151] TAYLOR'S FORMULA 263
the form 00/ oo (see Pierpont, Functions of a Real Variable y
p. 305).
Since f(x)/4> {x) = [I/0 (x)] -r- [l//(a;)], any fraction that takes one
of the two forms 0/0, 00 -5- 00 , can also be put into the other form.
Thus, as X -♦ ir/2, tan x and sec x both become infinite, while ctn x and
cos X approach zero; hence
,. tan X ,. cos x ^
lim = lim — — = 1.
Z^t/2 sec X x->ir/2 Ctn X
Likewise, if / (x) -» as 4>{x) becomes infinite, their product is of the
form X 00 , and it can be put into either of the preceding forms.
Thus, as X -> 0, log x becomes — 00 ; so that
lo&r X 1 /x
lim (x log x) = lim ^rr— = lini — Ti-i = ^^ ("" ^) = 0.
Other indetei*minate forms are 00 — 00 , 1*, OO, ooo. All these can be
made to depend on the forms already considered. For let a, /3, 7, 5, €,
be variables simultaneously approaching, respectively, 00, 00, 1, 0, 0.
Then a— P, 7«, 5«, a« take, respectively, the preceding four indeter-
minate forms. But
lim(a-g)=lim ^/^-j/^ ,
which is of the form 0/0; while the logarithms of the others,
log 7a = a log 7, log 6* = e log 5, log a« == e log a,
are each of the form X 00 .
Example 1. Thus, when x-* ir/2, (sin x)*"* takes the form 1*. But
(sin x)^° ' = f[log sin x]/otn x
which approaches the same limit as e"*^'/*^*', as x-* v/2, and this
limit is evidently e^ = 1.
Example 2. Similarly, when x becomes infinite, (l/x)^/<2a>+i) takes
the form 0°. It may be written in the form,
which approaches the same limit as e~^/2«, that is, the limit is gO =' 1, as
x->oo.
Example 3. As an example of the last form, 00 0, take (l/x)« as
X -> 0. This becomes
0-^~X I08 X
and approaches e^ = 1, as x -» 0.
Indeterminate forms in two variables cannot be evaluated, imless
one knows a law connecting the variables as they approach their lim-
its, which practically reduces the problem to a problem in one letter.
264
THE CALCULUS
[XVI, § 151
EXERCISES
Evaluate each of the following indeterminate forms, where (r) |a
means the limit of 4>(x) as x approaches a. Draw a graph in each case.
1. ^
2.
8
00
log ctn X
log cos X
secx
r/2
• log(ir/2~x)
4. g
7.
8.
9.
log X
log 2
«
00
«/2 Vi
10. a;2 ctn x |o
5.
6.
oo
log cos X
sin2 X
log sin^ g
log tan3 X
11. a;2iogx3|o.
18. x^*
14. (l+x)V«|o.
16. (H-n/x)*U.
16. (tanx)«<»*|x/2.
17. (sinac)"*"**|o.
12. (tan X— sec x) L/2. 18. (tanx ^ ) •
\ 7r/2-X/\^/2
Find the value of each of the following improper integrals, using
Table V, F, when necessary after integrating by parts:
00
19. f a
Jo
.00
X e~* dx.
20. f x2 e-x dx.
Jo
21
Jo
2 6"^ dx.
162. Infinite Series. An infinite series is an indicated sum
of an unending sequence of terms:
(1) Oo + ai + «2 + • • • + On + • • • ;
this has no meaning whatever until we make a definition, for
it is impossible to add all these terms. Let us take the simi
of the first n terms:
Sn = «0 + fll + ^2 + • • • + Oji-l,
which is perfectly finite; if the limit of s„ exists as n becomes
infinite, that limit is called the sum of the series (1) :
(2) S = lim Sn =ao + ai H h fln H
n-»oo
If lim Sn = S exists, the series is called convergent; if S
n— »oo
does not exist, the series is called divergent; if the series
XVI, §152] SERIES 265
formed by taking the numerical (or absolute) values of the
terms of (1) converges, then (1) is called absolutely con-
vergent. Infinite series which converge absolutely are most
convenient in actual practice, for extreme precaution is neces-
sary in dealing with other series. (See § 154. See also
Goursat-Hedrick, Mathematical Analysis, Vol. I, Chap. VIII.)
Example 1, The series 1 + r -^ r^ + •••+r» + -"is called a geo-
metric series; the number r is called the common ratio. A geometric
series converges absolutely for any value of r numericaUy less than 1; for
1 r"
6'„ = 1 + r + r2 H h r^-i = -z -z ,
1 — r 1 — r
hence
lim
n— wo
1
1-r
= lim
n-*oo
1-r
= 0, if lr|<l,
since r»» decreases below any number we might name as n becomes in-
finite. It follows that the sum S of the infinite series is
/S = limsn = :j -, if|r|<l;
and it is easy to see that the series still converges if r is negative, when
it is replaced by its numerical value \r\.
Example 2. Any series a© + Oi + Oa H h «n + • • • of positive num-
bers can be compared with the geometric series of Ex. 1. Let
<rn = «0 + Ol + O2 + • • • + On-i;
then it is evident that a-n increases with n. Comparing with the geomet-
ric series oo (1 + r + r^ + • • • + r» +•••)» it is clear that if
where Sn = 1 + r + • • • + r*»~^. Hence an approaches a limit if Sn does,
i.e. if < r < 1. It follows that the given series converges if a value of
r < 1 can be found for which On < aor^, that is, for which an -^ On-i < r
< 1. There are, however, some convergent series for which this test can-
not be applied satisfactorily. It may be applied in testing any series for
absolute convergence; or in testing any series of positive terms. For
example, consider the series
111 1
266
THE CALCULUS
[XVI, § 152
here a„ = 1/n!, a„-i = l/(n — 1)!, and therefore On/on-i = (n — l)!/n!
= 1/n, Hence On/On-iK 1/2 when w > 2,
1
^-^^ + r! + ^! +
+
>1 +
(l+^+l
+
+ 2«-2 j
(n-i;i
= 1 +«»-!,
where 8»-i = 1 + r + • • • + r»-2^ r = 1/2. It follows that the given
series converges and that its sum is less than 1+2 — 3. [Compare
§ 154, p. 269; it results that « < 3. Compare Ex. 2, p. 268.]
153. Taylor Series. General Convergence Test Series which
resemble the geometric series except for the insertion of con-
stant coeflScients of the powers of r,
(1)
A + Br + 07^ + 07^ +
arise through application of Taylor's Theorem [D *], § 147,
p. 254; such series are called Taylor series or power series.
The properties of a Taylor series are, like those of a geometric
series, comparatively simple. Comparing (1) with [D *],
we see that r takes the place of (x— a), while A,B,C,D, • • •
have the values:
If we consider the sum of n such terms:
^(n-1)!^^ ^^ '
we see by [D *], that
fix) = Sn + En, where \En\ ^ M^ '-^^ , M^ > \r\x)\;
n I
or
Sn=f{x)-En.
XVI, §153] SERIES 267
It follows that if En approaches zero as n becomes infinite, the
infinite Taylor Series
V (a) f" id)
+ ...+-'-^(x-a)«+...
converges, and its sum is S = lim Sn =f(x).*
This is certainly true, for example, whenever \f^^\x) \ re-
mains, for all values of n, less than some constant C, however
large, for all values of x between x = a and x = b. For in
that case
Um |£n I < lim C • l^Zl^ = c liml^^^ =
for all values of (x — a).t When | f^^\x) \ grows larger and
larger without a bound as n becomes infinite, we may still
often make | En \ approach zero by making (x — a) numeri-
cally small.
Example 1. Derive an infinite Taylor series in powers of x for the
function / (x) = sin x.
Since f(x) = sin x, we have /' (x) = cos x, /" (x) = — sin x, and, in
general, f^^^{x) — ± sin x, or ± cos x; hence
\/n(x)\<l,\un\En\<lim^.=0;
therefore the infinite series [D**] for a = is
sinx = + a; + 0-^x3 + o + ^,x5+...;
this series certainly converges and its sum is sin x for all values of x,
since lim \En\ = 0.
* This result is forecasted in § 147.
fThis results from the fact that n eventually exceeds (x—a) niuneri-
cally; afterwards an increase in n diminishes the value of En more and more
rapidly as n grows.
268 THE CALCULUS [XVI, § 153
Example 2. Derive an infinite series for e* in powers of (x — 2).
Since / (x) = «f*. we have / (x) =«*,--•, /(">(x) = c«; hence / (2) = «?,
f (2) = c», • • •, /(•> (2) = c», and |/t»>(x) | < & where 6 is the largest
value of X we shall consider. Then the series
e. = e« + e«(x-2) + ^(x-2)«+...+^(x-2)«+...
= 6?[l + (x-2)+^(x-2)« + ...+i-,(x-2)«+...]
conveiiges and its sum is e^, for all values of x less than 6; for
lim|^«| < lim ^'''";^'" =0.
Since 6 is any number we please, the series is convergent and ite sum is
«^ for all values of x.
EXERCISES
Derive the following series, and show, when possible, that theyxx)n-
verge for the indicated values of x.
1. oosx = l-x2/2! + x4/4!-x«/6!+..; (allx).
2. ©• = l+x + xV2!+x3/3!+---; (aUx).
8. «-' = I-x+xV2!-x3/3!+---; (aUx).
4. tanx=x + x3/3 + 2x5/15 + 17xV315+.--; (|x|<«'/4).
5. log(l+x)=x-xV2+x3/3-x4/4+...; (|x|<l).
6. sinhx = (^-c-')/2=x+x3/3! + x5/5!+---; (aUx).
7. coshx-(c« + 6-»)/2 = l+xV2! + x4/4!+ .-•; (aUx).
8. tanh x = sinhx/coshx = x - x3/3 + 2 x6/15 - 17 xV315 H ;
(aUx).
9. Show that the series of Ex. 6 can be obtained from those of |
Exs. 2 and 3 if the terms are combined separately. i
10. Show that the series of Ex. 3 results from the series of Ex. 2 if x
is replaced by — x.
11. Obtain the series for sin x in powers of (x — x/4).
I
12. Obtain the series for e» in terms of powers of (x — 1).
13. Obtain the series for log x in powers of (x — 1). Compare it
with the series of Ex. 5.
XVI, §154] SERIES 269
14. Obtain the series for log (1 — x) in powers of x, directly; also by
replacing a; by — a; in Ex. 5.
16. Using the fact that log [(1 +a;)/(l - a;)] = log (1 +a;) - log
(1 ~ x)j obtain the series for log [(1 +a;)/(l — x)] by combining the
separate terms of the two series of Ex. 14 and of Ex. 5. This series is
actually used for computing logarithms.
16. Show that the terms of the Maclaurin series for (a + x)^ in
powers of x are precisely those of the usual binomial theorem.
17. Show that the series for e*»+« in powers of a; is the same as the
series for e* all multipUed by e».
18. Show that the series for 10* is the same as the series for «» with
X replaced by x/Mj where M = 2.30- • •.
164. Precautions about Infinite Series. There are several
popular misconceptions concerning infinite series which yield
to very commonplace arguments.
(a) Infinite series are never used in compiUation. Contrary
to a popular belief infinite series are never used in computa-
tion. What is actually done is to use a few terms (that is,
a polynomial) for actual computation; one may or may
not consider how much error is made in doing this, with an
obvious effect on the trustworthiness of the result.
Thus we may write
x3 . a;6 a;2*+i
but in practical computation, we decide to use a few terms, say sin a; = a;
— a;3/3! -\-(jfi/5l The error in doing this can be estimated by § 147,
p. 253. It is 1^7 1 <\x'^/7\\. For reasonably small values of x [say
I x\ <14° <l/4 (radians)], | J^7 1 is exceedingly small.
Many of the more useful series are so rapid in their convergence that
it is really quite safe to use them without estimating the error made; but
if one proceeds without any idea of how much the error amounts to,
one usually computes more terms than necessary. Thus if it were re-
quired to calculate sin 14° to eight decimal places, most persons would
suppose it necessary to use quite a few terms of the preceding series, if
they had not estimated E7.
270 THE CALCULUS [XVI, § 154
9
(6) No faith can be placed in the fact that the terms are becomr
ing smaller. The instinctive feeling that if the terms become
quite small, one can reasonably stop and suppose the error
small, is unfortunately not justified.*
Thus the series
has terms which become small rather rapidly; one instinctively feels
that if about one hundred terms were computed, the rest would not
affect the result very much, because the next term is .001 and the suc-
ceeding ones are still smaller. This expectation is violently wrong.
As a matter of fact this series diverges; we can pass any conceivable
amount by continuing the term-adding process. For
A -f A > 2 • A = ^,
^ + ^ + '- +A>^'h =A,
A ■+- lie + • • • -f liiT > 8 • xiiy = A»
and so on; groups of terms which total more than 1/20 continue to ap-
pear forever; twenty such groups would total over 1; 200 such groups
would total over 10; and so on. The preceding series is therefore very
deceptive; practically it is useless for computation, though it might
appear quite promising to one who still trusted the instinctive- feeUng
mentioned above.
(c) If the terms are alternately positive and negative,, and if
the terms are numerically decreasing with zero as their limits
the instinctive feeling just mentioned in (b) is actually correct:
the series a© — ai + a2 — as + • • • converges if a„ approaches
zero; the error made in stopping with a„ is less than a„+i.t
For, the sum Sn = Oo — ai + • • • d= On-i evidently alternates
m
between an increase and a decrease as n increases, and this
* This fallacious instinctive feeling is doubtless actually used^ and it is
responsible for more errors than any other single fallacy. The example
here mentioned is certainly neither an unusual nor an artificial example.
f One must, however, make quite sure that the terms actually approach
zero, not merely that they become rather small ; the addition of .0000001 to
each term would often have no appreciable effect on the appearance of the
first few terms, but it would make any convergent series diverge.
XVI, §154] SERIES 271
alternate swinging forward and then backward dies out as n
increases, since a„ is precisely the amount of the nth swing.
On each swing s„ passes a point S which it again repasses
on the return swing; and its distance from that point is
never more than the next swing, — never more than a„-|-i.
Since a„ approaches zero, s» approaches S, as n becomes
infmite.
Thus the series for sin x is particularly easy to use in calculation: the
error made in using x — x^/3\ in place of sin x is certainly less than
ix^/6l The test of § 147 shows, in fact, that the error {Esl <M6|a;V5!|,
where Ms = 1.
The similar series for e*:
is not quite so convenient, since the swings are all in one direction for
positive values of x; certainly the error in stopping with any term is
greater than the first term omitted. The error can be estimated by § 147,
p. 253; thus E5 (for a; >0) is less than Msx^/5ly where Ms is the maximum
of f^{x) =0* between x = and a; = x, t.e. e*; hence ^6<e*x5/5!.
Note that e» > 1 for a; > 0.
Another means of convincing oneself that the preceding series con-
verges for a; < 1 is by comparison with a geometric series with a ratio
a;/2, as in Example 2, p. 265. But this method would require the com-
putation of a vast number of terms, to make sure that the error is small.
(d) A consistently small error in the values of a function may
make an enormmis error in the values of its derivative.
Thus the function y - x^ .00001 sin (100000 x) is very well ap-
proximated b^ the single term y = x, — in fact the graphs drawn ac-
curately on any ordinary scale will not show the slightest trace of dif-
ference between the two curves. Yet the slope of 2/ = x is always 1,
while the slope of y =x — .00001 sin (100000 x) oscillates between
and 2 with extreme rapidity. Draw the curves, and find dy/dx for
the given function.
One advantage in Taylor series and Taylor approximating
polynomials is the known fact — proved in advanced texts —
272 THE CALCULUS fXVI, § 154
that differentiation as well as integration is quite reliable on any
valid Taylor approximation*
Thus an attempt to expand the function y = x — .00001 sin (lOOOOOx)
in Taylor form gives
r 1000002 , , 1000004 _ i
y =a;— I a; ^ — a^ -\ ^ — a^ ~ •••!»
which would never be mistaken f or y = a; by any one; the series indeed
converges and represents y for every value of x, but a very hasty exam-
ination is sufficient to show that an enormous number of terms would
have to be talcen to get a reasonable approximation, and no one would
try to get the derivative by differentiating a single term.
If the relation expressed by the given equation was obtained by ex-
periment, however, no reliance can be placed in a formal differentiation,
even though Taylor approximations are used, for minute experimental
errors may cause large errors in the derivative. Attention is called to
the fact that the preceding example is not an unnatural one, — pre-
cisely such rapid minute variations as it contains occur very frequently
in nature.
EXERCISES
1. Show that the series obtained by long division for 1 -5- (1 + x)
is the same as that given by Taylor^s Series.
2. Obtain the series for log (1 -f- x) (see Ex. 5, § 153), by inte-
grating the terms of the series found in Ex. 1 separately.
3. Find the first four terms of the series for sin"^ x in powers of x
directly; then also by integration of the separate terms of the series for
i/vr^=^.
4. Proceed as in Ex. 3 for the functions tan~i x and 1/(1 -|- x^).
6. Show that the series for cos x in powers of re is obtained by dif-
ferentiating separately the terms of the series for sin x,
6. Show that repeated differentiation or integration of the separato
terms of the series for e* always results in the same series as the original
one.
7. From the series for tan~ia; compute x by using the identity
x/4 = 4 tan-i (1/5) - tan-i (1/239).
* See, e.g., Goursat-Hedrick, Mathematical AncUysis, Vol. I, p. 380.
rjn, §154] SERIES 273
2 /•*
9. The "error integral" is P (re) = — ^ / e~^ dx. Express P (x) as a
8. The Gudermannian of x is gd (x) — 2 tan~* e» ~ x/2; expand in
powers of a;, calculate gd(.l) = 6° 43', and gd (.7) = 37° 11'.
series in powers of x; calculate P(.l) = .1126, P(l) = .8427, P (2) =
.9953+.
10. Show that ^^^^dx =» LSI". 18. Show tha,tfdt/VT^^ = .608+.
11. Show that J^^^dx = 1.78+ 14. Show thatyd«/VT^= 1.31 1+.
12. Show that /*' sin ^/*xdx = .930^.
CHAPTER XVII
PARTIAL DERIVATIVES— APPLICATIONS
166. Partial Derivatives. If one quantity depends upon two
or more other quantities, its rate of change with respect to one
of them, while all the rest remain fixed, is called a partial
derivative.*
If z = f (x, y) is a function of x and i/, then, for a constant
value of i/, 1/ = fc, z is a function of x alone: z = / (x. A;) ; the
derivative of this function of x alone is called the partid
derivative of z with respect to x, and is denoted by any one of
the symbols
dz ^ df(x, y) ^ . .^^._ df{x,k)
Aa;— ►O ^X
A precisely similar formula defines the partial derivative
of z with respect to y which is denoted by dz/dy.
In* general, if w is a function of any number of variables x,
y, z, ' - ' , and if one calculates the first derivative of u with
respect to each of these variables, supposing all the others to
be fixed, the results are called the first partial derivatives of
u with respect to x, i/, z, • • • , respectively, and are denoted by
the symbols
du/dXj du/dy, du/dZj • • • .
* This notion is perhaps more prevalent in the world at large than the
notion of a derivative of a function of one variable, because quantities in
nature usually depend upon a great many influences. The notion of par-
tixd derivative is what is expressed in the ordinary phrases "the rate at
which a quantity changes, everything else being supposed equal," or
"... other things being the same."
274
XVII, § 156] PARTIAL DERIVATIVES— APPLICATIONS 275
Example 1. Given 2 = x^ + 1/2, to find bzjbx and bzjdy.
To find dg/dx, think of y as constant: y — h\ then
ds ^ a(x2+i/2) ^ rf(x2+A;2) =2x- - =2t/
dx dx dx ' dy ^'
Example 2. Given 2 = x^ sin(x +y^)j to find dsi/^x. and dz/dy.
dz ^ a|x2sin(x+y^)l _ r <^(x2sin(xH-fc2)|
dx dx
r dlx^am(x + mn
L dx Jy-i
^ 2 X sin (x -f 2/^) -f a?^ cos (x -f y^)-
dz ^ a(x2 8in(x4-y^)} ^ r d{k^8m(k + t^)n
dy dy L dy Ja;-*
=2 x2 y cos (x -f y^)-
166. Higher Partial Derivatives. Successive differentiation
is carried out as in the case of ©rdinary differentiation.
There are evidently four ways of getting a second partial
derivative: differentiating twice with respect to x; once with
respect to x, and then once with respect to y] once with
respect to y, and then once with respect to x; twice with re-
spect to y. These four second derivatives are denoted, re-
spectively, by the symbols
dxKdxJ dx^ J^^^>y)y ey\dxJ dydx •'^^^'^z;,
dy \dy/ " dy^ " "^^ ^^' ^^ ' dx \dyJ ' dxdy^ ^'' ^^' ^^ •
There is no new difficulty in carrying out these operations;
in fact the situation is simpler than one might suppose, for
it turns out that the two cross derivatives fxy and fyx are
always equal; the order of differentiation is immaterial,*
A similar notation is used for still higher derivatives:
^^ " dx3 "■ dx \dxy ' ^^"^ "■ d2/dx2 - dy \dx^)''
etc., and the order of differentiation is immaterial.
♦At least if the derivatives are themselves continuous. See Goursat-
Hedrick, Mathematical Analysis, Vol. I, p. 13.
276 THE CALCULUS [XVII, § 156
The ordef of a partial derivative is the total number of successive
differentiations performed to obtain it. The partial derivatives of the
first and second orders are very frequently represented by the letters
V> Qy r, «, t:
^ dx'^ dy'^ dx^'^ dxdy dydx' dy^'
Example 1. Given z — x^ sin (x + y^), show that fxy = fyx-
Continuing Example 2, § 155, we find:
^=1^(1) =|;[2=^«^ (- + «'*)+-' «»(- + »*)]
= 4 xy cos (x +y^) — 2x^y sin (x + y*).
dx dy dx \dy/ dxt v • if / j
^4xy C08 (x-{- y^) — 2 x^j/ sin {x + y^),
EXERCISES
Find the first and second partial derivatives, ds/dx, ds/dy^ d^s/dx^,
d2 z/dx dyy d2 g/dy QXy and d^ zjdy'^ for each of the following functions. In
each case verify the fact that d^z/dx dy = d'^z/dy dx,
1. z=x^ — 2/2. 4. 3 = e<w+»y. 7. 3 = (x + 2/) e^i''.
2. z^x^y-\- xy^. 6. z = tan-i iyjx), 8. s = (x^/ — 2 2/2)5/3.
8. s = sin (x2 + 2/2). 6. 3 = e*sin2/. 9. s = log (x2 + 2/2)i/2.
10. The volume of a right circular cylinder is v = nr2 A. Find the rate
of change of the volume with respect to r when h is constant, and ex-
press it as a partial derivative. Find dv/dA, and express its meaning.
11. The pressure p, the volume v, and temperature ^ of a gas are
connected by the relation yo — kO, where is measured from the ab-
solute zero, — 273° C. Assuming constant, find dp/dv and express
its meaning. If the volume is constant, express the rate of change of
pressure with respect to the temperature as a derivative, and find its
value.
12. Find the rate of change of the volume of a cone with respect to its
height, if the radius, is constant; and the rate of change of the volume
with respect to the radius, if the height is a constant.
18. Show that the functions in Exercises 1, 5, 6, and 9 satisfy the
equation ^z/dx^ -f ^z/dy^ = 0.
XVII, § 157] PARTIAL DERIVATIVES— APPLICATIONS 277
14. A point moves parallel to the x-axis. What are the rates of
change of its polar coordinates with respect to xl
16. Show that the rate of change of the total surface of a right
circular cylinder with respect to its altitude is dA/dh = 2 ir r; and that
its rate of change with respect to its radius is dA/dr = 2irk-{-4irr.
16. Calculate the rate of change of the hypotenuse of a right triangle
relative to a side, the other side being fixed; relative to an angle, the
opposite side being fixed.
17. Two sides and the included angle of a parallelogram are a, b, C,
respectively. Find the rate of change of the area with respect to each
of them, the other two being fixed; the same for the diagonal opposite
toC.
18. In a steady electric current C = V -^ /2, where C, V, R, denote
the current, the voltage (electric pressure), and the resistance, respec-
tively. Find d C/d V and d C/d R, and express the meaning of each of
them.
157. Geometric InterpretatioiL The first partial deriva-
tives of a function of two independent variables
can be interpreted geometrically in a simple manner. This
equation represents a
surface, which may
be plotted by erect-
ing at each point of
the xt/-plane a per-
pendicular of length
f(x, y); the upper
ends * of these per-
pendiculars are the
points of the surface.
Let ABCD be a
portion of this sur-
/"p<h,k,o)
Fig. 77.
* If 2 is negative, of course the lower end is the one to take.
278 THE CALCULUS • [XVII, § 157
face lying above an area abed of the x2/-plane. If x varies
while y remains fixed, say equal to fc, there is traced on the
surface the curve HK, the section of the surface by the plane
y = k. The slope of this curve is dz/dx.
Similarly, dz/dy is the slope of the curve cut from the
surface by a plane x = A.
168. Total Derivative. If in addition to the function
z= f{x,y)y a relation between x and y, say y = 4>{x), is given,
z reduces by simple svbstiiution to a function of one variable:
z=f{x, 2/), y = (t>{x) gives z =/(x, 0(x)).
Now any change Ax in x forces a change Ay iny; hence y
cannot remain constant (unless, indeed, (x) = const.).
Hence the change Az in the value of z is due both to the direct
change Ax in a: and also to the forced change Ay in y. We
shall call
Az = the total change in 2 = / (x + Ax, y + Ay) — / (x, j/),
Ax2 = the partial change due to Ax directly
= /(x + Ax, I/) -f(x,y),
AyZ = the partial change forced by the forced change Ay
= Az- A^z, = /(x + Ax, 2/ + A?/) -/(x + Ax, y).
It follows that
(1) ^ = lim— j^lim (/(^+^.vH-Ay)-/(..y)\>
= l^ ^ajg +AyZ { ^^m^ f f(x-{-/!iX,y)-f{x,y)
Aa>-^ Ax < AI^->0\ ^
.y)Ai/\l.
Aaj-K)
/ (X -\-Ax, V + Ay) -/ {x +Aa;
+ AV
XVII, 1 158] PARTIAL DERIVATIVES— APPLICATIONS 279
whence, if the partial derivatives exist and are continuous,*
dz ,. ^^xz ^vz^y\
or, multiplying both sides by dx(=Ax),
dz dzdy
di'^dydx'
(3)
dz , dz , . J dy ,
where dy = <t> (x) dx. Since <^(x) is
any function whatever, dy is really
perfectly arbitrary. Hence (3) holds
for any arbitrary values of dx and dy
whatever, where dz = {dz/dx) dx is
defined by (2); dz is called the
total differential of z.
These quantities are all repre-
sented in the figure geometrically: ^^^^^^^
thus A2 = A^ + AyZ is represented st=cd^av
by the geometric equation TQ = 'J^'XCTJ^I^q'^R^
SR + MQ. It should be noticed az=QoQ-PoP=tq='SR+mq
that dz is the height of the plane '^"^ ^:^«-«+a»
drawn tangent to the surface at P, since dz/dx and a^/dy
are the slopes of the sections of the. surface hy y = yp and
X = xp, respectively. [See also § 163.]
If the curve PqQo in the xi/-plane is given in parameter
form, X = 4>(f)y V = ^(^)» ^^ "^^^ divide both sides of (3)
by dl and write
(4)
dz _dzdx .dzdy
di " dx dt dy dt'
since dx^di = dx/dt, dy -^ dt = dy/dt.
*For a more detaUed proof using the law of the mean, see Goursat-Hed-
rick, Mathematical Analysis, I, pp. 38-42.
280 THE CALCULUS [XVII, § 159
159. Elementary Use. In elementary cases, many of which
have been dealt with successfully before § 158, the use of the
formulas (2), (3), and (4) of § 158 is quite self-evident.
Example 1. The area of a cylindrical cup with no top is
(1) A=2irM + irr2,
where h is the height, and r is the radius of the base. If the volume of
the cup, Tf^ hf ia known in advance, say xr^ h = 10 (cubic inches), we
actually do know a relation between h and r:
o. »-^,
whence
(3) A=2irr^ + irr2 = -+irra
from which dA/dr can be found. We did precisely the same work in
Ex. 26, p. 57. In fact even then we might have used (1) instead of (3),
and we might have written
(4) ^ = 2 irr - +2 tA + 2 irr, or dA = 2 irr dh + (2 irh-\-2 irr) dr,
where dh/dr is to be found from (2).
This is precisely what formula (2), § 158, does for us; for
(5) ^=2irA + 2^r, ^-2^r,
^ =(2irA+2irr) + (2irr)^, or dA^ {2 Th+2 irr) dr +2 xr dL
We used just such equations as (4) to get the critical values in finding
extremes for dA/dr = at a critical point. We may now use (2), § 158,
to find dA/dr] and the work is considerably shortened in some cases.
Example 2. The derivative dy/dx can be found from (2),* § 158, if
we know that z is constant.
Thus in § 24, p. 39, we had the equation
(1) a;2 + 2/2=:i,
and we wrote:
whence we found
XVII, § 160] PARTIAL DERIVATIVES— APPLICATIONS 281
This work may be thought of as follows:
Let 2 = x2 + ^; then
dx dx dx'^dydx '^^ dx*
but s = 1 by (1) above; hence dz/dx = 0, and
2x+2j,f^=0,or^ = -5.
dx dx y
Thus the use of the formulas of § 158 is essentially not at
ail new; the preceding exercises and the work we have done
in §§ 24, 29, etc., really employ the same principle. But
the same facts appear in a new light by means of § 158;
and the new formulas are a real assistance in many examples.
160. Small Errors. Partial Differentials. Another appUcation
closely allied to the work of § 145, p. 248, is found in the esti-
mation of smaU errors.
Example 1. The angle A of a right triangle ABC (C = 90°) may
be computed by the formula
tan A = Tf OT A = tan~i _^
where a, 6, c are the sides opposite A, B, C, If an error is made in
measuring a or 6, the computed value of A is of course false. We may
estimate the error in A caused by an error in measuring a, supposing
temporarily that b is correct, by § 145; this gives approximately
1
dA b b
1+52
where d is used in place of d of i 145, since A really depends on & also,
and we have simply supposed b constant temporarily. Likewise the
error in A caused by an error in 6 is approximately,
_ ^
62
282 THE CALCULUS [XVII, § 160
If errors are possible in both measurements, the total error in A is,
approximately, the sum of these two partial errors:
The methods of § 146, p. 251, give a means ot finding how nearly
correct these estimates of AoA, ^A, and A A are; in practice, such
values as those just found serve as a guide, since it is usually desired
only to give a general idea of the amounts of such errors.
This method is perfectly general. The differences in the
value of a function z = f{x, y) of two variables, x and i/, which
are caused by differences in the value of x alone, or of y alone,
are denoted by A^a:, Ay2, respectively. The total difference
in z caused by a change in both x and y is
A2 =/(a: + Ax, 2/ + At/)-/(a:, i/)
= [/(x + Ax, 2/ + Ai/) -/(x + Ax, y)] + [/(x + Ax, y)
-fix,y)]
as in § 158. The differences A^^ and AyZ are, approximately,*
whence, approximately,
Az = A^ + Ajs = £Az + ~Ay.
♦ More precisely, these errors are
A,^ = I • Ax + E',. A^ = %\_^^ A„ + E",.
where | E'2 \ and | E^'t \ are less than the maximum Mt of the_yalues of all of
the second derivatives of z near (x, y) multiplied by A?, or Ay*, respectively
(see §146). And since dz/dj/ is itself supposed to be continuous, we may
write
dz ^ , dz . , „
Az^^^Ax+^Ay-\-E2,
where | ^2 1 is less than Af2( | Ax | + 1 ^2/ 1 )*• [Law of the Mean. Compare
5146.]
XVII, § 160] PARTIAL DERIVATIVES— APPLICATIONS 283
The products (dz/dx) dx and (dz/dy) dy are often called
the partial differentials of z, and are denoted by
dz dz
b^ = -r- dXj dyZ = -r- dy, whence dz = djfi + d^z.
ox ay
We have therefore, approximately,
Az = d^ + dyZy
within an amount which can be estimated as in § 146 and in
the preceding footnote.
Similar formulas give an estimate of the values of the
changes in a function u = f{x, y, z) of the variables x^y^z;
we have, approximately,
A ^^ A A ^^ A A ^^ A
A.u=-Ax, AyU^^Ay, A,ii=-A.,
Au = AxU + AyU + A^u=— Ax + —Ay + — Az,
within an amount which can be estimated as in the preceding
footnote. The generalization to the case of more than three
variables is obvious.
EXERCISES
Find dy/dx in each of the following implicit equations by method of
Ex. 2, § 159:
1. a;2 + 4 2/2 = 1. 3. x^ + y^''Sxy = 0.
2. 7x2-9^2 = 36. 4. 2/2(2a-a;) =a^.
5. If Ay By C denote the angles, and a, 6, c the sides opposite them,
respectively, in a plane triangle, and if a, il, B are known by measure-
ments, 6 = a sin B/sin A, Show that the error in the computed value
of b due to an error da in measuring a is, approximately,
dab = sin B CSC A da.
Likewise show that
djj) = — a sin 5 CSC A ctn Ad Ay and dsb = a cos B cac Ad B]
and the maximum total error is, approximately, \db\ ^ \dab\4- \dAb\
+ I dsbl. Note that dA and dB are to be expressed in radian measure.
284 THE CALCULUS [XVII, § 160
6. The measured parts of a triangle and their probable errors are
a = 100 ± .01 ft., A = 100° ± 1', B = 40° ± 1'.
Show that the partial errors in the side h are
dab = ± .007 ft., dAh = ± .003 ft., a^ = ± .023 ft.
If these should all combine with like signs, the maximum total error
would be db^ ± .033 ft.
7. If a = 100 ft., B = 30°, A = 110°, and each is subject to an
error of 1%, find the per cent of error in 6.
8. Find the partial and total errors in angle B, when
a = 100 ± .01 ft., 6 = 159 ± .01 ft., A = 30° ± 1'.
9. The radius of the base and the altitude of a right circular cone
being measured to 1%, what is the possible per cent of error in the
volume? Ana. 3%.
10. The formula for index of refraction is m = sin I'/sin r, i being the
angle of incidence and r the angle of refraction. If i = 50° and r = 40°,
each subject to an error of 1%, what is m, and what its actual and its
percentage error?
11. Water is flowing through a pipe of length L ft., and diameter
D ft., imder a head of H ft. The flow, in cubic feet per minute, is
Q = 2356 V jr, j-30/) * H L = 1000, D = 2, and ^ = 100, determine
the change in Q due to an increase of 1% in H; in L; in D, Compare
the partial differentials with the partial increments.
161. Envelopes. The straight line
(1) y = kx — k^y
where fc is a constant to which various values may be assigned,
has a different position for each value of k. All the straight
lines which (1) represents may be tangents to some one curve.
If they are, the point P*, (x, y) at which (1) is tangent
to the curve, evidently depends on the value of k:
(2) x = 0(fc), y = 4^{k);
XVII, § 161] PARTIAL DERIVATIVES— APPLICATIONS 285
these equations may be considered to be the parameter
equations of the required
curve. The motive is to
find the functions <l>{k) and
^(fc) if possible.
Since Pt lies on (1) and
on (2), we may substitute
from (2) in (1) to obtain:
(3) ^(fc) = k<t>{k) - P,
which must hold for all
values of fc. Moreover,
since (1) is tangent to (2) at
Ptj the values of dy/dx
found from (1) and from (2)
must coincide :
(4)
^ J from (1) ^ J from (2)
Fig. 79.
or fc0'(fc) = ^'(fc).
To find 0(A;) and ^(fc) from the two equations (3) and (4), it
is evident that it is expedient to dififerentiate both sides of (3)
with respect to fc:
(3*) ^'(fc) = A;0'(fc) + 0(fc) - 2 fc;
this equation reduces by means of (4) to the form
(5) = + 0(fc) - 2 fc, or 0(fc) = 2 fc,
and then (3) gives
(6) ^(fc) = fc(2fc)-fc2==fc2.
Hence the parameter equations (2) of the desired curve are
(7) x = 2fc, y = k\
and the equation in usual form results by elimination of fc:
(8)
X
286 THE CALCULUS [XVII, § 161
It is easy to show that the tangents to (8) are precisely the
straight lines (1).
The preceding method is perfectly general. Given any set of curves
(1)' F (X, y, k)^0,
where k may have various values, a curve to which they are all tangent
is called their envelope; its equations may be written
(2)' X = (fc), y = iA {k)\
whence by substitution in (1)',
(3)' F[4> (kh ^ (k), k] = 0,
for all values of k. Differentiating (3)' with respect to k,
* dF{x,y,k) __dFdx dFdy , dF ^
^ ^ dk dx dk '^ dydk'^ dk
Moreover, since (1)' is tangent to (2)',
dF . dF _dyl _dy-[
^ ^ dx ' dy dxj ^om (1)' dxj ^om (2)'
_dy , dx,
dk ' dk'
whence (3*)' reduces to the form
(^)' af = «'
and then (3)' and (5)' may be solved as simultaneous equations to find
(A;) and ^ {k) as in the preceding example.
The envelope may be found speedily by simply writing
down the equations (1)' and (5)', and then eliminating k
between them It is recommended very strongly that this
should not be done until the student is familiar with the
direct solution as shown in the preceding example.
162. Envelope of Nonnals. Evolute. If y = /(x) is a given
curve and if yt and m* respectively denote the ordinate and
slope when x = fc, the equation of any normal may be
written
(1) 2/ - y* = - — (a^ - fc),
or F{x, y, k) = yrriu - ytnit + x - fc =
XVII, § 162] PARTIAL DERIVATIVES— APPLICATIONS 287
Hence by (5'), § 161, we have, for the envelope of the system
of lines (1) when k is regarded as a variable parameter,
dF
(2)
dk
= y • 6* - y* • fe* - w* • w* - 1 = 0.
(Remember that in forming dF/dk, x and y are regarded
as constant, and only k,
w»> Vk, are regarded as
variable. We have used
bi to stand for dmjdk,)
Solving (2) for y we
have
(3) y = y. + ^^-
This value of y in (1)
gives
Fig. 80.
(4)
X = fc — m*
l+m\
Equations (3) and (4) are the parametric equations of the
envelope of the system of normals (1), and are precisely
equations (1) of § 98, with only a change of notation. Hence
the envelope of the systems of normals to a given curve is the
evolute of that curve.
EXERCISES
Find the envelopes of each of the following families of curves:
1. y =Skx'- k^. Ans, y^ = 4 sfi.
2. y = 4 fcr — A;*. Ana, y^ = 27x^.
3. y^ =^kx — k^, Ans. y = ± ix.
4. y = kx ± Vl 4- k^» Ans, a;^ + ^ = 1.
5. y^ ='J^x — 2k, Ans, xy^ = — 1.
288 THE CALCULUS [XVII, § 162
6. (x - A;)2 + 2r^ = 2 k. Ans. y^ ^2x + l.
7. 4x2 + (y- A;)2 = 1 - ^2. jins. y^+8x^ = 2.
8. X cos ^ + y sin ^ = 10. Arw. x^+y^ = 100.
9. Show that the envelope of a family of circles through the origin
with their centers on the parabola y^ ^ 2 x ia y^(x -\- 1) -^ 3fi — 0,
10. Show that the envelope of the family of straight lines ax -\-by
= 1 where a + 6 = a6, is the parabola x^ + y^ = 1.
11. Show that the envelope of the family of parabolas represented by
the equation y = x tan a — mx^ sec^ aiay == 1/(4 m) — mx^.
[Note. If m = g/(2 vq^), the given equation represents the path of a
projectile fired from the origin with initial speed vq at an angle of eleva-
tion a.]
12. The lemni scate (x ^ -j- 2/2)2 = ^2( 3.2 — y 2) m^y ^^ written in the
form X = a cos t Vcos 2 1] y = Sismt Vcos 2 1, Show that the evolute
is (x2/3 + 2/2/3)2 (a;2/3 _ ^2/3) = 4 ^2/9.
CHAPTER XVIII
CURVED SURFACES — CURVES IN SPACE
163. Tangent Plane to a Surface. Let Pq be the point
(xo 2/0, Zq) on the surface z = f(x,y). Let Po^i be the tangent
line at Pq to the curve cut
from the surface by the
plane y = yo and P0T2 the
tangent line to the curve
cut from the surface by
the plane x = Xq. The
plane containing these
two lines is the tangent
plane to the surface at Po.
Since this plane goes
through Po, its equation
can be thrown into the form
(1) 2 - 2o = A{x - Xo)+B(y- yo).
If we set y = yo we find the equation of Po^i in the form:
(2) z — Zo = A{x — Xo),
But, from § 28, p. 49, the equation of Po^i may be written in
the form :
Fig. 81.
(3)
Hence
(4)
= t1^ "
likewise B — -;r\ '
^2/ Jo
289
290 THE CALCULUS [XVIII, § 163
Thus the equation of the tangent plane is
or, what is the same thing,
(6) 2 - 2, = f^(x - X,) + |]^(y - j/o).
It is important to notice the great similarity between this
equation and the equation
(^> '^ = S]o'^ + |].'^«'
of § 158. Indeed (7) expresses the fact that if dx, dy are
measured parallel to the x and y axes from the point of tan-
gency (xo, 2/0, ^o), dz represents the height of the tangent
plane above (xo, 2/0, ^o). Equation (7) furnishes a good
means of remembering (6).
164. Extremes on a Surface. If a function z — f{x^y)]&
represented geometrically by a surface, it is evident that
the extreme values of z are represented by the points on the
surface which are the highest, or the lowest, points in their
neighborhood:
(1) / (xo, 2/0) > / (iCo + A, 2/0 + fc), if / (xo, 2/0) is a maximum^
(2) / (xo, 2/0) <f{xo + h,yo + k),iif (xq, yo) is a minimum^
for all values of h and k for which h^ + k^ is not zero and is not
too large.
It is evident directly from the geometry of the figure that
the tangent plane at such a point is horizontal.
This results also, however, from the fact that the section
of the surface by the plane x = xo must have an extreme at
(^0, 2/0); hence [df/dy]oy which is the slope of this section at
{xoyo)y must be zero; likewise [df/dx]o, the slope of the
XVIII, § 164] CURVED SURFACES 291
section through (xo, 2/0) by the plane y = yo, must be zero.
Hence equation (5), § 163, reduces to z — Zo = which is a
horizontal plane.
A point at which the tangent plane is horizontal is called
a critical point on the surface. The following cases may
present themselves.
(1) The surface may cut through its tangent plane; then
there is no extreme at (xo, 2/0).
This is what happens at a point on a surface of the saddleback type
shown by a hyperbolic paraboloid at the origin; a homeher example is
the depression between the knuckles of a clenched fist.
(2) The surface may just touch its tangent plane along a
whole line, but not pierce through; then there is what is often
called a weak extreme at (xo, yo); that is, z = f{x, y) has the
same value along a whole line that it has at (xo, 2/0), but
otherwise / (x, y) is less than [or greater than] / {xq, i/o).
This is what happens on the top of a surface which has a rim, such as
the upper edge of a water glass, or the highest point of an anchor ring
lying on its side. Most objects intended to stand on a table are provided
with a rim on which to sit; they touch the table all along this rim, but
do not pierce through the table.
(3) The surface may touch its tangent plane only at the point
(^0, yo); then z = f{x, y) is an extreme at (xo, 2/0): a mini-
mum, if the surface is wholly above the tangent plane near
(^0, 2/0) ; a maximum, if the surface is wholly below.
. The shape of the clenched fist gives many good illustrations of this
type also. Examples of formal algebraic character occur below.
Example 1. For the elliptic paraboloid s = x^ + 2/^ the tangent
plane at (xq, 2/0, 3d) is
2 — 30 = 2 xo (x — xo) + 2 2/0 (2/ — 2/0), ,
which is horizontal if 2 xq = 2 2/0 = 0; this gives Xq = yo = 3b = 0,
hence (x = 0, y = 0) is the only critical point.
292 THE CALCULUS [XVIII, § 164
At (x = 0, y = 0), ar has the value 0; for any other values of x and j/,
s ( = x2 + y^) is surely positive. It follows that ;? is a minimum at
a; = 0, y = 0.
Example 2. In experiments with a pulley block the weight «? to be
lifted and the pull p necessary to lift it Were found in three trials to
be (in pounds) (pj = 5, t^^i = 20), (p2 = 9, «^ = 50), {jpz = 15, 103 =
90). Assuming that p = au> + /3, find the values of a. and /3 which
make the sum S of the squares of the errors least. (Compare Ex. 37,
p. 58.)
Computing p by the formula aw + /3, the three values are p'l =
20 a + ^, p'2 = 50 a + /3, p'a = 90 a + /3. Hence the sum of the
squares of the errors is
S = (p'l - pi)2 + (p'2 - P2)2 + (p'3- Pzy
= (20 a + /3 - 5)2 + (50 a + /3 - 9)2 + (9Q a + /3 - 15)2.
In order that a5 be a minimum, we must have
^ = 2-[20 (20a + ^ - 5) + 50 (50a + /3- 9) +90 (90a + /3 - 15)] = 0.
da
^ = 2 [(20a + /3- 5) + (50 a + /3 - 9) + (90a + ^ - 15)] = 0.
op
that is, after reduction,
1100 + 16 ^ - 190 = 0, a = m = .143,
160 a + 3 /3 - 29 = 0, /3 = \%<^ = 2.03.
If the usual graph of the values of p and w is drawn, it will be seen
that p = aw -{- represents these values very well for a = .143,
/3 = 2.03 and it is evident from the geometry of the figure that these
values render S a minimum, S = .0545; for any considerable increase
in either a or j8 very evidently makes S increase. Since this is the
only critical point, it surely corresponds to a minimum, for the function
S has no singularities.
This conclusion can also be reached by thinking oi S aa represented
by the heights of a surface over an a/3 plane, and considering the sectien
of that surface by the tangent plane at the point just found as in Ex. 3
below; but in this problem the preceding argument is simpler.
It is customary to assume that the values of a and /3 which make S a
minimum are the best compromise, or the " most probable values'* ;
hence the most probable formula for p is p = .143 w + 2.03.
The work based on more than three trials is quite similar; the only
change being that S has n terms instead of 3 if n trials are made.
The plane
^-300
XVIII, § 164] CURVED SURFACES 293
Example 3. Find the most economical dimensions for a rectangular
bin with an open top which is to hold 500 cu. ft. of grain.
Let X, y, h represent the width, length, and height of the bin, respec-
tively. Then the volume is xyh; hence xyh = 500; and the total area z
of the sides and bottom is
, . , o 1 . o I. 1 1000 , 1000
(a) z = xy + 2hy-\-2hx=xy-] 1 •
X y
If this area (which represents the amount of material used) is to be a
minimum, we must have
,,, dz 1000 ^ dz 1000 ^
Substituting from the first of these the value y = lOOO/x^ in the
second, we find
(c) X — r^r^ = 0, whence a; = 0, or a; = 10. *'
lo^
The value x = is obviously not worthy of
any consideration; but the value x = 10 gives
y = lOOO/x^ = 10 and h = 500/{xy) = 5. ^ i^ *
The value of z when x = 10, 2/ = 10 is 300. If ' ^^^' ^'
the equation (a) is represented graphically by a surface, the values of z
being drawn vertical, the section of the surface by the plane z — 300
is represented by the equation
(d) xy + ^509 + 122? = 300, or x^y^ - 300 xy + 1000 (x + 2/) = 0.
X y
This equation is of course satisfied by x = 10, 2/ = 10. If we attempt
to plot the curve near (10, 10), — for example, if we set y = 10 + A;
and try to solve for x in the resulting equation:
(10 + A;)2x2 - (300 A; + 2000) X + 1000(10 + A;) = 0,
the usual rule for imaginary roots of any quadratic ax^ + 6x + c =
shows that
62 - 4 oc = - 1000 A;2 [4 A; + 30] <
for all values of k greater than — 7.5. Hence it is impossible to find
any other point on the curve near (10, 10). It follows that the hori-
zontal tangent plane z = 300 cuts the surface in a single point; hence
the surface lies entirely on one side of that tangent plane. Trial of
any one convenient pair of values of x and y near (10, 10) shows that z
is greater near (10, 10) than at (10, 10) ; hence the area 2 is a minimum
when X = 10, y = 10, which gives A = 5.
294
THE CALCULUS
[XVIII, § 165
165. Final Tests. Final tests to determine whether a func-
tion f{x, y) has a maximum or a minimum or neither, are
somewhat diiSScult to obtain in reliable form. Comparatively
simple and natural examples are known which escape all set
rules of an elementary nature.* (See Example 1 below.)
One elementary fact is often useful: if the surface has a
maximum at (xo, 2/0), every vertical section through (xo, t/o) has a
maximum there. Thus any critical point (xo, 2/0) may be dis-
carded if the section by the plane x = Xo has no extreme at
that point, or if it has the opposite sort of extreme to the
section made by y = 2/0.
Example 1. The surface a = (y — x^) (j/ — 2 x^) has critical points
where
dx
6 X2/ + 8 a;3 = 0, ^ = 2 y - 3 a;2 = 0;
that is, the only critical point is (x = 0, 2/ =0). The tangent plane at
that point is a = 0. This tangent plane cuts the surface where
(y- x^) (2/ - 2 x2) = 0;
that is, along the two parabolas
y = x^^y — 2x^. Atx ^ 0,y =
1, the value of 2 is + 1 ; hence s is
positive for points (x, y) inside
the parabola y = 2x^. Atx = 1,
y = Oy the value of z is + 2;
hence 2 is positive for all points
{Xy y) outside the parabola y =
a;2. At the point x = 1, 2/ = 1.5,
the value of a is — .25; hence a is
negative between the two pa-
rabolas. It is evident, therefore,
that a has. no extreme at x == 0,
Fig. 83. 2^ = 0.
A qualitative model of this extremely interesting surface can be
made quickly by molding putty or plaster of paris in elevations in the
* For a detailed discussion, see Goursat-Hedrick, Maihematiccd Analysis^
Vol. I, p. 119.
XVIII, § 165] CURVED SURFACES 295
unshaded regions indicated above, with a deprescion in the shaded
portion.
Another interesting fact is that every vertical section of this surface
through (0, 0) has a minimum at (0, 0); this fact shows that the rule
about vertical sections stated above cannot be reversed. Moreover,
this surface ehides every other known elementary test except that used
above.
EXERCISES
Find the equation of the tangent plane to each of the following
surfaces at the point specified:
1. = x2 + 9 2/2, (2, 1, 13). Ans. z = 4:X + I8y - IS.
2. = 2x2-42/2,(3,2,2). Ans. z =- 12 x - 16 y - 2.
3. z = xyj (2, — 3, — 6). Ans. Sx — 2y + z = Q.
4. 2 = (x + y)2, (1, 1, 4)* . Ans. 4x + 4y — = 4.
5. = 2 X2/2 + 2/3, (2, 0, 0) Ans. = 0.
6-10. The straight hne perpendicular to the tangent plane at its
point of tangency is called the normal to the surface.
Find the normal to each of the surfaces in Exs. 1-5, at the point
specified.
11. At what angle does the plane x + 2y — z -{- S = cut the
paraboloid x2 + 2/2 = 4 at the point (6, 8, 25)?
12. Find the angle between the surfaces of Exs. 1 and 2 at the point
(A/i3, 1, 22).
Find the angle between each pair of surfaces in Exs. 1-5, at some one
of their points of intersection, if they intersect.
13. Find the tangent plane to the sphere x2 + 2/2 + 02 = 25 at the
point (3, 4, 0); at (2, 4, VS).
14. At what angles does the line x = 2 2/ = 3 cut the paraboloid
2/ = x2 + ;^?
15. Find a point at which the tangent plane to the surface 1 is
horizontal.
Draw the contour lines of the surface near that point and show
whether the point is a minimum or a maximum or neither.
296 THE CALCULUS [XVIII, § 165
16. Proceed as in Ex. 15 for each of the surfaces of Exs. 2-5, and
verify the following facts:
(2.) Horizontal tangent plane at (0, 0); no extreme.
(3.) Horizontal tangent plane at (0, 0); no extreme.
(4.) Horizontal tangent plane at every point on the liye x + y = 0;
weak minimum at each point.
(5.) Horizontal tangent plane at every point where j/ = 0; no
extreme at any point.
Find the extremes, if any, on each of the following surfaces:
17. 2 =: x2 + 4 2/2 - 4 a;. (Minimum at (2, 0, - 4).)
18. a=a:3 - 3 X - 2/2. (See Tables, Fig. Ii.)
19. = a:3 - 3 a; + 2/2 (x - 4). (See Tables, Fig. I2.)
20. s = [(x - a)2 + 2/2] [(x + a)2 + 2/2]. (Similar to Tables, Fig. I7.)
21. 2 =x3 — 6 a; — ^2 (Draw auxiliary curve as for Fig. Ii.)
22. a = x' — 4 2/2 + xy^. (Draw auxiliary curve as for Fig. I2.)
23. 2 = X* + 2/^ — 3 xy. (Draw by rotating x2/-plane through ir/4.)
24. Redetermine the values of a and fi in Example 2, § 164, if the
additional information (p = 23, lo = 135) is given.
26. Find the values of u and v for which the expression
(aiu + biv — ci)2 + (a2U + b2V — 02)2 + (asu +bzv — ci)^
becomes a minimum. (Compare Ex. 24.)
26. Show that the most economical rectangular covered box is
cubical.
27. Show that the rectangular parallelopiped of greatest volume that
can be inscribed in a sphere is a cube.
[Hint. The equation of the sphere is x2 + y^ + 2^ = 1 ; one corner o f
the parallelopiped is at (x, y, 2); then F= 8 xyz, where a = Vl — x^ — y^.]
28. Show that the greatest rectangular parallelopiped which can be
inscribed in an ellipsoid x2/a2 -f- 2/2/62 -j- ^/c^ == 1 has a volume V =
Sahc/(SV3).
29. The points (2, 4), (6, 7), (10, 9) do not lie on a straight line.
Under the assumptions of § 164, show that the best compromise for a
straight Une which is experimentally determined by these values is
24 2/ = 15X + 70.
XVIII, § 166] CURVED SURFACES 297
30. The linear extension E (in inches) of a copper wire stretched by
a load W (in pounds) was found by experiment (Gibson) to be {W == 10,
E = .06), {W = 30, E = .17), {W = m, E = .32). Find values of a
and /3 in the formula E = aW -{- fi under the assumptions of § 164.
31. The readings of a standard gas meter S and that of a meter T
being tested were found to be (T = 4300, S = 500), (T = 4390, S =
600), (T = 4475, S = 700). Find the most probable values in the
equation T = aS + fi and explain the meaning of a and of /3.
32. The temperatures 6° C. at a depth d in feet below the surface of
the ground in a mine were found to be d = 100 ft., 6 = 15°.7, d = 200
ft., = 16°.5, c? = 300 ft., = 17°.4. Find an expression for the tem-
perature at any depth.
33. The points (10, 3.1), (3.3, l.Q), (1.25, .7) lie very nearly on a
curve of the form a/x + ^/y = 1. Use the reciprocals of the given
values to find the most probable values of a and /3.
34. The sizes of boiler flues and pressures under whiqh they collapsed
were found by Clark to he (d = 30, p = 76), {d = 40, p = 45), (d = 50,
p = 30). These values satisfy very nearly an equation of the form
p = jfc . (in or log p = n log d + log A;, where d is the diameter in inches,
and p is the pressure in pounds per square inch. Using the logarithms
of the given numbers, find the most probable values for n and log k,
166. Tangent Planes. Implicit Fonns. If the equation of
a surface is given in implicit form, F(Xy y, z) = 0, taking the
total differential we find:
But, by virtue of F (x, y, z) = 0, any one of the variables, say
2, is a function of the other two; hence
(2) (fe = |dx + g%.
Putting this in the total differential above and rearranging:
298 THE CALCULUS [XVIII, § 166
But dx and dy are independent arbitrary increments of x and
of y; and since the equation is to hold for all their possible
pairs of values, the coefficients of dx and dy must vanish
separately. This gives
dz dF/dx dz dF/dy
^^^ dx dF/dz ' dy dF/dz
substituting these values in the equation of the tangent plane,
and clearing of fractions, we obtain
(^) S o(^ - ^^ + f ] 0^^ - ^o) + So^^ - ^) = «'
the equation of the tangent plane at (xq, yoy zq) to the surface
F (x, y, z)=0,
167. Line Normal to a Surface. The direction cosines of
the tangent plane to a surface whose equation is given in the
explicit form z = f(x, y) are proportional (§ 163) to
(1) ' dz/dx]o, dz/dy]^^ and — 1.
Hence the equations of the normal at (xq, yo, Zq) are
(<y\ x — xq ^ y — yo ^ z — zp
^^^ dz/dx]o dz/dy]o -1
The direction cosines of a surface whose equation is given
in the implicit form F (x, y, 2) = are proportional to
(3) dF/dx]o, dF/dyh, dF/dz]o,
*
so that the equations of the normal to this surface are
x — xq y-yo z-zo
(4)
dF/dx]o dF/dy]o dF/dz]Q
168. Parametric Forms of Equations. A surface S may
also be represented by expressing the coordinates of any point
on it in terms of two auxiliary variables or parameters:
[S] x==f(u, v), y = <t>{u, v), z = \f^(u, v).
XVIII, § 168] CURVED SURFACES 299
If we eliminate u and v between these equations, we obtain
the equation of the surface in the form F (x, y, z) = 0,
Similarly a curve C may be represented by giving x, y, z in
terms of a single auxiliary variable or parameter t:
[C] x=f{t),y = 4>{t),z = Ht),
The elimination of t from each of two pairs of these equations
gives the equations of two surfaces on each of which the
curve lies. In particular, taking t =x gives the curve as the
intersection of the projecting cylinders:
[P] y = <t>{x), z = yp{x).
If, in the parametric equations of a surface, one parameter (say u) is
kept fixed while the other varies, a space-curve is described which lies
on the surface. Now if u varies, this curve varies as a whole and de-
scribes the surface. The curve on which u keeps the value k is called
the curve u = k. Similarly, keeping v fixed while u varies gives a curve
V = k\ The intersection of a curve u = k with a curve v = k' gives
one or more points {k, k') on the surface. The numbers A;, k' are called
the curvilinear coordinates of points on the surface.
Simple examples of such coordinates are the ordinary rectangular
coordinate system and the polar coordinate system in a plane. Thus
(2, 3) means the point at the intersection of the hnes a; = 2, y = 3 of
the plane; in polar coordinates, (5, 30°) means the point at the intersec-
tion of the circle r = 5 with the line 6 ■— 30°.
Example 1. The equations of the plane x-\-y -{-z = 1 may be
written, in the parametric form:
x = w, y — Vy z = l — u — v.
Let the student draw a figure from these equations by inserting ar-
bitrary values of u and v and finding associated values of x, y^ z. An-
other set of parameter equations which represent the same plane is
x=u-\-v, y = u^Vf 3 = — 2w+l.
Thus several different sets of parameter equations may represent the
same surface.
In the first form, put u = k. Then, as v varies, we obtain the straight
line
X = k, y = Vf z - l — k^Vj
300 THE CALCULUS [XVIII, § 168
which lies in the given plane. As k varies this Hne varies; its different
positions map out the entire plane. Likewise, t; = A;' is a Hne varying
with k' and describing the plane. The intersection of two of these
lines, one from each system, is point (k, k') of the plane.
Example 2. The sphere x^ -\- y*^ -\- z^ = c? may be represented by
the equations:
2=asin^cos0, y=a sin ^ sin d, z^acos^.
Here the parameters ^ and are respectively the co-latitude and the
longitude. Thus = A; is a parallel of latitude; d = A;' is a meridian;
and their intersection {k, k') is a point of co-latitude k and longitude k\
[If a is allowed to vary, the equationi^^of this example define polar
coordinates in space; but 90° — <f>m often used in place of <f>.]
I
Example 3. The equations
x=acoatf y=a am tf z=Uy
represent a space curve, namely a helix drawn on a cylinder of radius
a with its axis along the 2-axis. The total rise of the curve during
each revolution is 2 irb.
If a is replaced by a variable parameter w, the helix varies with m,
and describes the surface
x^ucoat, y=su am t, 2=&^.
which is called a helicoid. The blade of a propeller screw is a piece
of such a surface.
169. Tangent Planes and Normals. Parameter Fonns.
When a surface is given by means of parametric equations,
(1) X = / (u, v), y = <f> (w, v), 2 = ^ (w, v)y
the equation of the tangent plane is found as follows. Elimination of u
and V would give the equation in the implicit form F (x, y, z) = 0. If
the parametric values of Xj j/, z are substituted in this equation, the
resultiag equation is identically true, since it must hold for all values of
the independent parameters w, v; hence
that is
^^ dx du dy du dz du ' dx dv^ dy dv dz dv
XVIII, § 169J CURVED SURFACES
301
Solving these, we find:
(4)
dF dF dF
dx ' dy ' dz
dy dz
du du
dz dx
du du
dx dy
du du
dy dz
dv dv
•
dz dx
dv dv
•
dx dy
dv dv
hence the equation of the tangent plane is
(x — Xq)
du
dz
du
dy
dz
dv
dv
+ (2/ - Vo)
dz
dx
du
du
dz
dv
dx
dv
+ (2 - 3o)
dx
du
9y
du
dx
dv
dy
dv
while the equations of the normal are
x — xq __ y — yp _
z
So
dy dz
dz dx
dx dy
du du
du du
du du
dy dz
dz dx
dx dy
dv dv
dv dv
dv dv
-0;
EXERCISES
1. Determine the tangent plane and the normal to the ellipsoid
a;2 + 4 2/2 + 2^ = 36 at the point (4, 2, 2), first by solving for s, by the
methods of § 163; then, without solving for z, by the methods of
166- 167.
Determine the tangent planes and the normals to each of the fol-
lowing surfaces, at the points specified:
2. x^ + y^+z^ =a^ at (xq, ^o. ^o)-
3. a;2 - 4 2/2 + 22 = 36 at (6, 1, 2).
4. a;2 - 4 2/2 - 9 32 = 36 at (7, 1, 1).
6. a;2+2/2-z2=0at (3,4,5).
6. a;3 + a;22/ - 2 22 = at (1, 1, - 1).
7. 22 = e«+v at (0, 2, c).
8. Find the angle between the tangent planes to the eUipsoid
4 a;2 + 9 2/2 + 36 22 = 36 at the points (2, 1, 20) and (- 1, - 1, 21).
9. At what angle does the 2-axis cut the surface 22 =s e*-*?
302
THE CALCULUS
[XVIII, § 169
10. Obtain the equation of the tangent plane to the helicoid
X = u cos Vj y = usin Vf 2 = v,
at the point w = 1, v = ir/4.
11. Talcing the equations of a sphere in terms of the latitude and
longitude (Example 2, § 168), find the equation of its tangent plane and
the equations of the normal at a point where 6 = 4> ^ 45°; at a point
where ^ = 60% = 30**.
12. EUminate u and v from the equations x = u + Vf y — u — Vy
z — uoyijo obtain an equation in x, y, and z. Find the equation of the
tangent plane at a point where t* = 3, t; = 2, by the methods of § 166;
then by the methods of § 169 directly from the given equations.
13. Write the equation of the tangent plane to the surface used in
Ex. 7 at any point (xq, yo, zq). At what point is the tangent plane
horizontal? Is z an extreme at that point?
Proceed as in Ex. 7 for each of the following surfaces:
14. X =r cos dy y = r Bin 6, z == r, a,t r = 2, 6 = ir/4.
^. uv -{-1 u — V uv— 1 ^ _ ^
16. X = — j — , y = — — , z = — i — , at w = 2, » = — 1.
u -{-V u -{-V u -\-v '
16. a; = -"3w + 2v, y —2u — v, s = e**+f, at (uqj Vq),
17. X =2 cos 6 cos 0, 2/ = 3 cos 6 sin 0, a = sin ^, at ^ = = t/4.
18. The surfaces z = x^ — 4y^ and z = Qx intersect in a curve,
whose equations are the two given equations. Find the tangent line
to this curve at the point (8, 2, 48) by first finding the tangent planes to
each of the surfaces at that point; the line of intersection of these planes
is the required line.
19. Find the tangent line to the curve defined by the two equations
16x2-3^2 =4 sand 9x2+32/2-;^ = 20 at (1, 2, 1).
170. Area of a Curved Surface. Let /S be a portion of a curved
surface and R its projection on the X2/-plane. In R take an element
AxAy and on it erect a prism cutting an element AS out of S. At any
point of AS, draw a tangent plane. The prism cuts from thi? an ele-
ment A A. The smaller Ax Ay (and therefore AS) becomes, the more
nearly will the ratio aA/aS approach unity, since we assume that the
limit of this ratio is 1.
XVllI, § 170] CURVED SURFACES
303
Suppose now that the area i2 is all divided up into elements AxAy and
that on each a prism is erected.
The area S will thus be divided up
into elements AS and there will be'
cut from the tangent plane at a
point of each an element AA. One
thus gets
(1) S = Km y. AA,
Ax— ^0'
Ay-»0
But if 7 is the acute angle that
the normal to any AA makes with
the 2-axis, we have
Fig. 84.
(2)
hence
AA = sec 7 Aa; Ay;
(3) iS = Um y Ail = lim T* (sec y AxAy) = f faec y dx dy.
Ax-»0
Ay— K)
Ar-»0
Ay-»0
Of course sec 7 is a variable to be expressed in terms of x and y from the
equation of the surface. The limits of integration to be inserted are
the same as if the area of R were to be found by means of the integral
ffdx dy.
If the surface doubles back on itself, so that the projecting prisms cut
it more than once, it will usually be best to calculate each piece sep-
arately.
When the equation of the surface is given in the form 2 =» / (a;, y)y
the direction cosines of the normal are given by
dz dz ^
cos a : cos j8 : cos 7 = ^ : -:r : — 1.
dx dy
Taking cos 7 positive, that is 7 acute, we may write
(4)
and
sec 7
S)"+ (S)"+ ■■
» - //VlMif^"'*-
The determination of sec 7, when the surface is given in the form
F(Xy y, z) = 0, is performed by straightforward transformations similar
to those used in §§ 167-169; they are left to the student.
304 THE CALCULUS [XVIII, § 170
EXERCISES
1. Calculate the area of a spiiere by the preceding method.
2. A square hole is cut centrally through a sphere. How much of
the spherical surface is removed?
3. A cylinder intersects a sphere so that an element of the cylinder
coincides with a diameter of the sphere. If the diameter of the cylinder
equals the radius of the sphere, what part of the spherical surface hes
within the cyHnder?
4. How much of the surface s =xy lies within the cylinder
a;2 + 2/2 = 1?
6. How much of the conical surface ^ = a;^ + y^ lies above a
square in the a^-plane. whose center is the origin?
6. Show that if the region 22 of § 170 be referred to ordinary polar
coordinates, AA = r sec 7 Ar A6, approximately. (See § 92, p. 149.)
7. Using the result of Ex. 6, show that S —ffr sec 7 dr dd.
8. Show that, for a surface of revolution formed by revolving a
curve whose equation is s = / (x) about the z-axis,
sec 7 = Vl 4- [rf/(r)/dr]2, where r = Vx^ + y^.
9. By means of Exs. 7, 8, show that the area of the surface of revo-
lution mentioned in Ex. 8 is
-XT' nR¥J— J:'^FOT*.
where a is the value of r at the end of the arc of the generating curve.
10. Compute the area of a sphere by the method of Ex. 9.
11. Find the area of the portion of the paraboloid of revolution
formed by revolving the curve z^ = 2 mx about the x axis, from x =
to a; = A;.
12. Show that the area of the surface of an ellipsoid of revolution
is 2 7r6 [6 + (<»A) sin~ie], where a and h are the semiaxes and e the
eccentricity, of the generating ellipse.
13. Show that the area generated by revolving one arch of a cycloid
about its base is 64 -n-a^/S.
14. Show that the area of the surface generated by revolving the
curve ^2/3 -|- 2/2/3 = a^/z about one of the axes is 12 iral^/5.
XVIII, § 172] CURVED SURFACES
305
171. Tangent to a Space Curve. Let the equation of the curve
be given in parametric form a; == / (0, 2/. = (0, 2 = \A (0- Let Pq = (xq,
2/0, 2^) be the point on the curve where t = ^. Let Q be a neighboring
point on the curve where t = to ■{• At
The direction cosines of the secant PqQ are proportional to Ax/At,
Ay /At, As/At; hence its equations are
(1)
x
xo _ y — yo ^ g— 2o
Ax/At Ay I At Az/At
As Ai -^ 0, these become
20
^ ^ dx/dt]Q dy/dt]Q dz/dth'
the equations^f the tangent at
the point Pq.
If the curve is given as the
intersection of two projecting
cylinders y = fix), z = <l>(x),
we may join to these the third
Fia.
equation x = x, thus conceiving of x, y, and 3 as all expressed in terms
of x. The equations of the tangent then become
x — xo _ y — yo _ z— zq
1
(3)
dy/dx]o dz/dxlo
If the curve is given as the intersection of two surfaces, / (x, y, z) = 0,
F (x, y, z) =fi, and if we think of x, y, zaa depending upon a parameter
t, we find
^ ^^^,^dy ^dz ^
dt dxdt '^ dydt '^ dzdt *
and
dF ^ d£^,dF^dydF^dz^^
dt dx dt'^ dy dt'^ dz dt
From these .equations we obtain dx/dt : dy/dt : dz/dt, and we may write
the equations of the tangent at Pq in the form:
X— Xo
2/ -2/0
z— Zo
df df
dy dz
df df
dz dx
dx dy
dFdF
dy dz
dFdF
dz dx
dF dF
dx dy
172. Length of a Space Curve. The length of the chord joining
two points t and t +A< of the curve
(1) a;=/(O,2/ = 0(O,s = ^(O,
306 THE CALCULUS [XVIII, § 172
is Ac = VAX* + Aj/* + Aar*, or^
r9^ A^ |ax2 , Aj/a , As*.,
(2) ^ = \a?+A^+Z?^'-
Defining the length of a curve between two points as the limit of the
sum of the inscribed chords, we find for that length:
EXERCISES
1. At what angle does a straight line joining the earth's South pole
with a point in 40® North latitude cut the 40th parallel?
2. At what angle does the helix x = 2 cos ^, 2/ = 2 siu ^, a = 0, cut
the sphere a;^ + y^ _j- g2 — 9?
3. Find the angle of intersection of the ellipse and parabola that are
cut from the cone s^ = x^ +y^hy the planes 2z = 1 — x and a = 1 -f a;
respectively.
4. Show that the curves of intersection of the three surfaces
2 = 2/, a;2 = 2/2 4. 22^ a;2 + 2/2 _j. 32 = 1^
cut each other mutually at right angles.
6. Show the same for the curves of intersection of the surfaces
4x2 + 9^2 +3632 = 36, 3a;2 + 62/2 - 6^2 = 6, 10x2 - 15y^ - 6^2 =30.
6. Calculate the length of the curve x = tyy = fi, z = 2 ^/2, from
^ = to < = 1.
7. Find the length of the helix x = a cos ^, 2^ = a sin ^, a = bd, from
5 = ^0 to ^ = ^1. What is the length of one turn?
8. Find the length of the curve a; = sin s, y = cos a, from (1, 0, t/2)
to (0, - 1, ir).
GENERAL REVIEW EXERCISES
[The exercises marked with an asterisk are of more than usual
difficulty. Some of them contain new concepts of value for which
it is hoped that time may be found. 'Hiose of the greatest theoretical
value are marked f.
Attention is called to the reviews of doul^le and 'riple integration.]
1. Given u =^ xy, x = r cos 6, y = r ain dj find du/dr and du/dS,
first by actually expressing u in terms of r and 6; then directly from the
given equations.
Xyill, § 172] CURVED SURFACES 307
2. Proceed as in Ex. 1 for the function u — tan~^ (y/x).
3. Given u = t^^^, x = r cos d, y — r sin dy find du/dz and du/dy
first by expressing u in terms of x and y; then directly from the given
equations.
[Hint. In the second part, it is convenient here to solve the last
two equations for r and 6 in terms of x and y. But see Ex. 4.]
4.* lix = r cos 6 and y = r sin 0, show by differentiation that
- = l = -cos9-r8ine-, and - =0 = -sm » + r cos 9 -■
Solve these equations for &r/dx and dO/dx, and show that du/dx may be
found in Ex. 3 by means of the equation
dx^^ dx de dx'
6.* If, in general, tt is a function of the two variables (r, ^), show
that the last equation in Ex. 4 hold^ true. Find a similar equation for
du/dy, and evaluate du/dy in Ex. 3 by means of it.
6.*t If w is a function of any two variables p and q, and if p and q
are given in terms of x and y by two equations x — f(p, q)jy = 4>{Pj q),
obtain du/dx and du/dy by a process analogous to that of Exs. 4, 5.
Proceed as in Ex. 3, by the methods of Exs. 4, 5, in each of the fol-
lowing cases:
7. w = r^ __ cQg2 g 8. w = refi*. 9. u — d log r.
10. Find the volume of that portion of a sphere of radius 4 ft. which
is bounded by two parallel planes at distances 2 ft. and 3 ft., respectively,
from the center, on the same side of the center.
11. Determine the position of the center of mass of the solid de-
scribed in Ex. 10.
12. What is the nature of the field of integration in the integral
/ fix,y)dydx7
•^ X
Show that the same integral may be written in the form
I f(x,y)dydx + 1 I f {x, y) dx dy,
"^ -^ a/V2 *^
13. Find the volume cut from the sphere x^ + ^ + a^ = a^ by the
cylinder a^ -\']^ — ax =0.
308 THE CALCULUS [XVIII, § 172
14. Find the volume cut from the sphere a;^ + i/S -f s* = a^ by the
cone (x — a)2 + 2/2 — a^ — Q^
16. Show that the surface of a zone of a sphere depends only upon
the radius of the sphere and the height 6 — a of the zone, where the
bounding planes are z = a and z =b.
16. Find the area of that part of the surface kH = xy within the
cylinder a^ + 2/* = A;*.
17. Find the center of gravity of the portion of the surface described
in Ex. 16, when A; = 1.
18. Find the moment of inertia about its edge, of a wedge whose
cross section, perpendicular to the edge, is a sector of a circle of radius
1 and angle 30*^, if the length of the edge is 1, and the density is 1.
19. The thrust due to water flowing against an element of a surface
is proportional to the area of the element and to the square of the com-
ponent of the speed perpendicular to the element. Show that the
total thrust on a cone whose axis lies in the direction of the flow is
A;7rrV/(r2 + A2)i.
20. Calculate the total thrust due to water flowing against a seg-
ment of a paraboloid of revolution whose axis lies in the direction of
the flow. (See Ex. 19.)
21. Show that the thrust due to water flowing against a sphere is
2 kin^i^/S. Compare with the thrust due to the flow normally against
a diametral plane of this sphere.
22. Find the critical points, if any exist, for the surface z = x^ +2y2
— 4 rr — 4 y + 10. Is the value of z an extreme at that point? Draw
the contour lines near the point.
23. Determine the greatest rectangular parallelopiped which can be
inscribed in a sphere of radius a.
24. The volume of CO2 dissolved in a given amount of water at tem-
perature ^ is ( ^ 5 10 15,
(v 1.80 1.45 1.18 1.00.
Determine the most probable relation of the form v = a -{-bO.
26. Determine the most probable relation of the form S = a +hF^
from the data: \P 550 650 750 850,
S 26 35 52 70.
I
1
I
XVIII, § 172] CURVED SURFACES 309
26. Determine the most probable relation of the form y = (m^ from
the data: \x 1 2 3 4,
\y .74 .27 .16 .04.
27. The barometric pressure P (inches) at height H (thousands feet)
P 30 28 26 24 22 20 18 16,
H 1.8 3.8 5.9 8.1 10.5 13.2 16.0.
Determine the most probable values of the constants in each of the
assumed relations: (a) ^ = a + 6P; (b) H = a-hbP+cP^; (c) H =
a + 6 log P or P = Ae^^. Which is the best approximation?
28. t If the observed values of one quantity yaxe mi, rM2, m3, corre-
sponding to values h, h, fe of a quantity x on which y depends, and if
2/ = ox + 6, show that the sum
S = {ah + 6 - mi)2 + (ofe + 6 ~ mz)^ _|- (afe _|- 6 - ;^)2
is least when
h {all -{-b — mi) + fc (0/2 + 6 — W2) + 13 {ah + 6 — ma) = 0,
{all + 6 — mi) + {al2 -f & — ^2) + (a/3 +b — ms) = 0;
that is, when
a ' J^h^ -^b ' J^h - J^mih =Oanda. J^h -{- S b -J^mi =0,
or
Sj^mih — J^i'^^ 'Jj^ J^h^' ^wi — ^mi^i • J^h
where ^ indicates the sum of such terms as that which follows it.
[Theory of Least Squares.]
29. Show that the equation of the tangent plane to 2 s = x^ -\- y^ &t
{xoj yo)mz-\'3o ^xxo-\- yyo.
30. Determine the tangent plane and normal line to the hyperboloid
a;2 _ 4 2^ + 9 ;^ == 36 at the point (2, 1, 2).
31. Study the surface xyz = 1. Show that the volume included be-
tween any tangent plane and the coordinate planes is constant.
32. Study the surface 2 = {x^ -\- y^) {x^ -]- y^ — 1). Determine the
extremes.
33. At what angle does a Une through the origin and equally in-
clined to the positive axes cut the surface 2 s = x^ + y^?
310 THE CALCULUS [XVIII, § 172
34. Determine the tangent line and the nonnal plane at the point
(1, 3/8, 5/8) on the curve of intersection of the surfaces x + y + a = 2
and r^ + 4 ^ - 4 22 = 0.
35. Detennine the tangent line and the normal plane to the curve
X — 2 cos ty y —28inty z = fi a,t t == ir/2 and at < = ir.
36. Find the length of one turn of the conical spiral x ^^ t cos (a log 0,
y = / sin (a log t), z — htj starting from t = t.
37. Determine the length of the curve x — a cos $ cos ^ y = a cos
$ am <t>t ^ ~ <i sin 0, from = ^i to ^ = ^, where 6 is given in terms
of 4> by the equation 5 = A; log ctn (x/4 — <f>/2). (Loxodrome on the
sphere.)
38,*t Show that the surfaces /(a;, y, z) = and ^ (x, y, z) =0 cut
each other at right angles '^ fx<f*x •\- fy<l>y + f»4>s ==0.
39.* Show that the surfaces
xV(«2+X)+2/V(^+X)+22/(c2+X) = 1, a>6>c>0,
are always (t) ellipsoids if X > — c^, (ii) hyperboloids of one sheet if
— 6^ < X < — c2, (m) hyperboloids of two sheets if — a^ < X < — 6^.
(CONFOCAL QUADRICS.)
Show also thaji these surfaces cut each other mutually at right angles.
If X = r cos 6 cos 4>, y — r cos 9 sin ^, z = r sin B (polar coordinates),
find du/dfy du/ddy and du/d<t> for each of the following functions:
40. w = x2 + 2/2 + z^, 41. w = x2 + y2 _ 22. 42. u = ae»+».
43. Compute du/dXy du/dyy and dw/dz if w = r? (sin^ e + sin^ ^),
where r, d, are defined as in Exs. 40-42.
44.t Show that the centroid (x, y) of a plane area in polar coordinates
(p, e) is
J J p2 COS e dp de J J p2 sin ^ dp dS
X = , y = ,
jjp dp de J J p dp de
where the integrals are extended over the given area.
CHAPTER XIX
DIFFERENTIAL EQUATIONS
PART I. ORDINARY PIFFERENTIAL EQUATIONS OF
THE FIRST ORDER
173. Definitions. An equation involving derivatives or
differentials is called a differential equation. An ordinary
differential equation is one involving only total derivatives.
A partial differential equation is one involving partial
derivatives.
The order of a differential equation is the order of the
highest derivative present in it.
The degree of a differential equation is the exponent of
the highest power of the highest derivative, the equation
having been made rational and integral in the derivatives
which occur in it.
Examples.
(« g-«
(First order, first degree.)
<« l+S-*-
(Second order, first degree.)
<» [' + ©7 ■
-»(S)'
(Second order, second degree.)
'« S+S-'
(Second order, first degree.)
Such equations constantly arise in the applications of
mathematics to the physical sciences. Many simple examples
have already been treated in the text.
311
312 THE CALCULUS pOX, § 174
174 Elimination of Constants. Differential equations also
arise in the elimination of arbitrary constants from an
equation.
Example 1. Thus, if A and B are arbitrary constants, then equation
y = Ax + B represents a straight line in the plane, and by a proper
choice of A and B represents any line one pleases in the plane except a
vertical line. One differentiation gives m = dy/dx = A, which repre-
sents all hues of slope A. A second differentiation gives
(1) flexion = 6 = d^y/dx^ = 0,
which represents all non-vertical lines in the plane, since all these and
on other curves have a flexion identically zero.
Example 2. Any circle whose radius is a given constant r is repre-
sented by the equation
(2) {X - A)2 + (2/ - B)2 = r2,
from which A and B may be eliminated as in the preceding example.
Differentiating once,
(3) x-A + iy- B)y' = 0,
where y' = dy/dx. Differentiating again,
(4) 1 + 2/'2 + (2/ - B)y'' = 0,
where y" = d!^y/dx^. Solving (3) and (4) for a; — A and y-rB and sub-
stituting these values into (2), so as to eliminate A and B, we find
(5) (1 + 2/'2)3 = r22/"2.
This sayB that every one of these circles, regardless of the position of its
center, has the curvature 1/r, — a statement which absolutely charac-
terizes these circles.
In general, if
(6) Six, y, cu C2, • • • , c„) =
is an equation involving x, y, and n independent arbitrary
constants Ci, C2, • • •, c^, n differentiations in succession with
regard to x give
these equations, together with (6), form a system of w + 1
XIX, § 175] DIFFERENTIAL EQUATIONS
313
equations from which the constants Ci, C2, • • • , c» may be
eliminated. The result is a differential equation of the nth
order free from arbitrary, constants, and of the form
(8) 0(^,2/,2/',y", •••,2/^"^)=.0.
Equation (6) is called the primitive or the general solution
of (8). The term general solution is used because it can be
shown that all possible solutions of an ordinary differential
equation of the nth order can be produced from any solution
that involves n independent arbitrary constants, with the ex-
ception of certain so-called '* singular solutions" not derivable
from the one general solution (6).
Thus, to solve an ordinary differential equation of the nth
order is understood to mean to find a relation between the
variables and n arbitrary constants. These latter are called
the constants of integration.
If, in the general solution, particular values are assigned to
the constants of integration, a particular solution of the dif-
ferential equation is obtained.
175. Integral Curves. An ordinary differential equation
of the first order,
(1) <t>(x> y> y') = 0, or I/' = / (x, 2/),
where y' = dy/dx, has a general
solution involving one arbitrary
constant c:
(2) F{x,y,c) = Q,
This represents a singly infinite
set or family of curves, there being
in general one curve for each value
of c. Any curve of the family can
be singled out by assigning to the
proper value. Fig. 86.
314 THE CALCULUS [XIX, § 175
The differential equation determines these curves by-
assigning, for each pair of values of x and y, that is, at each
point of the plane, a value of the slope y' [ = /(^, 2/)] of the
particular curve going through that point. Thus the curves
are outlined by the directions of their tangents in much the
way that iron filings sprinkled over a glass plate arrange
themselves in what seem to the* eye to be curves when a
magnet is placed beneath the glass. Straws on water in
motion create the same optical illusion.
A differential equation of the second order:
<t>(x, y, y\ 2/") =0, or y" = /(x, y, y'),
has a general solution involving two arbitrary constants,
F(x, y, ci, cz) = 0.
This represents a doubly infinite or two-parameter family of
curves; for each cqnstant, independently of the other, can
have any value whatever. The extension of these concepts
to equations of higher order is obvious.
The curves which constitute the solutions are called the
integral curves of the differential equation.
EXERCISES
Find the differential equations whose general solutions are the follow-
ing, the c's denoting arbitrary constants:
1. a;2 + y2 = q2 jIyis, X -i-yy' =0.
2. a;2 — 1/2 = ex. Ans, x^ +'y^ = 2 xyy\
3. y — C&' —i (sin a; + cos x). Ans, y' = y -\-8mx,
4. y = ex -\-€^, Ans. y = y'x + y^.
6. y = cx+f(c). Ans. y = y'x-\-f(y'),
6. y - Cie^' + 026^. Ans. i/" — 5 2/' + 6 y = 0.
7. y = cie«* + Ci^ Ans. y" -- (a + h)y'+ aby = 0.
8. xy = o-{- (^x. ' Ans. x^y^ = y'x + y.
9' y = (ci+ x)€^ + c^. Ans. I/" - 4 1/' + 3 1/ = 2 e3*.
XIX, § 175] DIFFERENTIAL EQUATIONS 315
10. 2/ = cie* + o^ + C3e3*. Am, y'" — 6 2/" + 11 2/' — 6 y = 0.
11. r ^ c sin Q, An&. r cos = r' sin d,
12. r = e^. Ans. r log r = r'd.
13. Assuming the differential equation found in Ex. 1, indicate the
values of y\— — x/y) at a large number of points (x, y) by short
straight-line segments through each point in the correct direction.
Continue doing this at points distributed over the plane until a set of
curves is outlined. Are these curves given in Ex. 1?
14. Proceed as in Ex. 13 for the equation y' — yfx. Do you recog-
nize the set of curves? Can you jirove that your guess is correct?
16. Draw a figure to illustrate the meaning of y' == x^. Find y.
Generalize the problem to the case y' = }{x).
16. Find that curve of the set given in Ex. 1 which passes through
(1,2). Find its slope (value of y') at that point. Do these three values
of (x, y^ y') satisfy the differential equation given as the answer in No. 1?
17. Proceed as in Ex. 16 for the equation of Ex. 2.
18. Proceed as in Ex. 16 for the first equation of Ex. 15.
19. Find the differential equation of all circles having their centers
at the origin.
20. Find the differential equation of all parabolas with given latus
rectum and axes coincident with the a;-axis.
21. Find the differential equation of all parabolas with axes falling
in the x-axis.
22. Find the differential equation of a system of confocal ellipses.
23. Find the differential equation of a system of confocal hyperbolas.
24. Find the differential equation of the curves in which the sub-
tangent equals the abscissa of the point of contact of the tangent.
26. A point is moving at each instant in a direction whose slope
equals the abscissa of the point. Find the differential equation of all
the possible paths.
26. Write the differential equation of linear motion with constant
acceleration; of Unear motion whose acceleration varies as the square of
the displacement. The same for angular motion of rotation.
27. A bullet is fired from a gun. Write the differential equations
which govern its motion, air resistance being neglected. How must
these equations be modified, if air resistance is assumed proportional to
velocity?
316 THE CALCULUS [XIX, § 176
176. General Statement We shall now consider methods for
solving dififerential equations. Since the most common proper-
ties of curves involve slope and curvature, and since in the
theory of motion we deal constantly with speed and ac-
celeration, the differential equations of the first and second
orders are of prime importance.
Ordinary differential equations of the first order and first
degree have the form
(1) M + N^ = 0,oTMdx + Ndy = 0,
where M and N are functions of x and y.
No general method is known for solving all such differential
equations in terms of elementary functions. We proceed to
give some standard methods of solution in special cases.
177. Type I. Separation of Variables. It may happen that
M involves x only, and N involves y only. The variables are
then said to be separated and the primitive is f oimd by direct
integration:
CMdx+fNdy^C,
C being an arbitrary constant.
Example 1. A particle is falling through air such that the resistance
is proportional to the speed. If the particle starts from rest, what is
its speed at any time?
Since acceleration is dv/dt, and since this is equal to g diminished by
a term proportional to v, we have
dv
Separating variables:
g — av
Integrating:
a
or fif — av = e-a(«+*).
Separating: ^ j^ dy + -g— rr dx ^0.
XIX, § 178] DIFFERENTIAL EQUATIONS 317
Since the particle starts from rest we have v *« when t = 0, Sub-
stituting these values in the last equation we have g = e~**;
hence g — av = e-<«-at = ge~^t,
or V = (g/a){l — e-^).
Example 2. Given (x^ + I) (y + 1) dy +xy^dx = 0. Determine
the relation between x and y.
Integrating: \o%y — l/y + log Vx^ + 1 = A;.
Let ^ = — log Cf rearrange and combine terms; the result Is
log(cyVa;2 + l) = l/y
or cyVs^~+l =eV».
178. Typell. Homogeneous Equations. When M and N
are homogeneous * in a: and y and of the same degree, the
equation is said to be homogeneous. If we write the equation
in the form
dy _ _M^
and make the substitution
dy , xdv
we obtain a new equation in which the variables can be
separated.
Example 1.
(1) {xy -\-y^)dx + {xy — x^) dy = 0,
or
(2) dy ^ xy-\-y^
dx x^ — xy
Substituting as above :
* Polynomials are homogeneous in x and y when each term is of the
same degree. In general, / (x, y) is homogeneous if / (Ax, ky) = k^fix^ y) for
some one value of n and for all values of k.
318 THE CALCULUS [XLX, § 178
(3). t; + x^=^^3^-j-^,
dv 2v^
or aJ^- =
dx 1-v'
separating variables, ^ dt; = —
Integrating: — 2I; "" 2 ^^^ ""^ ^^^ ^ "^ ^*
Replacing t; by y/x,
or logxy = 2 c;
henoe
(4) xy = e-*/»-2«,
or xy = ke-'^y,
where A; = e-^.
Check: Differentiating both sides of (4) with respect to x, we find
(5) ydx+xdy^ke^/y^- ^^'^'"'^^ ']^,
dividing the two sides of (5) by the corresponding sides of (4) respectively
x.,N r J 1 J 1 ydx — xdy
(6) \ydx-{-xdy\-^xy^ -^ -^ ^;
show that (6) agrees with (1).
EXERCISES
Solve the following exercises by separating the variables:
1. x dy -{- y dx =^ 0. Ans. xy = c,
2. xVm^dx - yVT+x^dy = 0. Ans. Vl+x^ = VT+¥+ c.
3. sin ^ c?r + r cos d dd — 0. Ans. r sin ^ = c.
4. xVl +ydx= yVl -\-x dy.
Solve the following homogeneous equations
6. (x-\-y)dx + {X'-' y) dy =0. Ans. x^ +2xy — f^ = c.
6. (x2 + 2/2) cte = 2 xy dy. Ans. x^ — y^=cx.
7. (3 x2 - 2/2) dy = 2 xy dx. Ans. x^ — y^^cifi.
8. (x2 -\-2xy — 2/2) ctx = (x^ — 2xy — y^) dy.
Arw. a;2 + 2/2 = c(x + y).
XIX, § 178] DIFFERENTIAL EQUATIONS 319
The following Exs. 9- 18 axe intended partially for practice in recog-
nizing types:
9. Vl — y2 dx + Vl — x^ dy = 0. Ans. sin-i x + sin"! y = c.
10. 3fidx-\- (3 x^y + 2 y^) dy = 0. ilrw. x^-\-2y^ ^ cVx^ + y^.
11. dy + ysmxdx = sin x cte. 12. r d^ = tan d dr.
13. (2/ — 1) cte = (a; + 1) dt/. 14. ydx + ix — y) dy = 0.
16. x{l +u^)dx = y(l +a;2) dy. 16. (9 x^+y^)dx=2 xy dy,
17, ^ -^ X = c. 18. -^ 4- 2^ = x.
dx dx
19. In Ex. 1 above, draw a figure to represent the direction of the
integral curves at various points. Hence solve the equation geomet-
rically.
20. A point moves so that the angle between the a;-axis and the di-
rection of the motion is always double the vectorial angle. Determine
the possible paths. . xy , ^ n
^ ^ Am, 0,9 -cx;c>0.
21. Proceed as in Ex. 20 for a point moving so that its radius vector
always makes equal angles with the direction of the motion and the
X-axis. Ans, r = c sin d.
22. The speed of a moving point varies jointly as the displacement
and the sine of the time. Determine the displacement in terms of the
time. Ans, asce*"*®®"'.
23. Find the value of y if its logarithmic derivative with respect to
a; is a;2.
24. Determine the curve whose subnormal is constant and which
passes through the point (2, 5).
26. Determine the curve whose subtangent at any point {x, y) is
(1 + x)y and which passes through (0, 3).
26. Determine the curve passing through (5, 4) such that the length
of the normal at any point (§ 30) equals the distance of the point from
the origin.
320 THE CALCULUS [XIX, § 178
27. When a wheel is driven by a belt the tension at P and the angle
are connected by the equation dT/d$ = kT,
B ^^
^2
k being a known constant.
If Ti = 200 lb..
*-.. \
>
V k = 0.1,
y
1 and ACB = 60°,
/ what is T2?
If Ti = 500 lb.,
Ti = 550 lb..
^1
Fig. 87.
and ACB = 90°,
what is A;?
28. In a chemical reaction A is the quantity of active matter originally
present, q the quantity of product at time t; these are related through
the equation dq/dt = k (A — q).
Express g as a function of it.
29. In a bimolecular chemical reaction the original amounts of ac-
tive substances are A and B; the product q formed in time i is to be
determined from the equation
dq/dt ^k{A-'q)(B-' q).
Express q in terms of t. Consider the special case A =^ B
30. The differential equation of the adiabatic expansion of a gas
\a kp dv -\- V dp ^ Q. Show that p = cv^. Find c if A; = .001, and
V = 100 when p = 10.
31. The rectilinear motion of a particle under the action of a central
force which varies as the inverse square of the distance from a fixed
point is V dv/dt = A;^/^, where v is the speed and t the time. Express
V in terms of t
32. Solve Helmholtz's equation for the strength of an electric cur-
rent, C =E/R- {L/R) idC/dt)y E, L and R being constants. If
C = when t = 0, show that C = (E/R) (1 - 6-«/^).
179. Type in. Linear Equations. This name is applied
to equations of the form
(1) l+^^=«'
XIX, § 179] DIFFERENTIAL EQUATIONS 321
where P and Q do not involve y, but may contain x. Its solu-
tion can be obtained by first finding a particular solution of
the reduced equation,
(10 1 + ^^ = 0'
where ^ is a new quantity introduced for convenience in what
follows; and where Q is replaced by zero. In (1') the vari-
ables can be- separated (see § 179), and we get
as a particular solution, the constant C of integration being
given the particular value 0.
If we make the substitution
(2) 'y = V'y,
where t; is a function of x to be determined, the equation (1)
becomes
The first term vanishes by (1') leaving
Hence
y^ = Q, or dv = ^dx = [Qe^^'^dx.
and
(3) y = vy = e'^''^ \f[Qe^''^ dx + c j •
This equation expresses the solution of any linear equation.
It should not be used as a formula; rather, the substitution
(2) should be made in each example.
322 THE CALCULUS [XIX, § 179
' Example. Given
(1) ^ + 3x^y=3fi,
the reduced equation in the new letter y == y/v is
(10 ^ "*■ ^ ^^^ "" ^' whence y = er^.
Hence the substitution y = v -y becomes
^==«-x»^-
and (1) takes the form
(2) y = V e-*', whence -^ = ^~*';^ "" ^ vx^e^f
[
e-^'^ - 3 wrSe-^l + 3 x^ [w^T = afi.
This reduces, as we foresaw in general above, to the form
.dv , dv - ,
e-*' 3- = a;6, or -r- = a:^ e«'
aa; ax
whence v =^J*x^^'dx + c = J [^^ ^' ~ e*'] + c,
or, returning by (2) to y:
(3) 2/ = ve^' = i [aj3 - 1] + ce^'.
Check. DiflPerentiating both sides,
(4) ^--x^-3x^ce-^;
ehminating c by multiplying (3) by 3 x^ and adding to (4),
The result (3) may also be obtained by direct substitution from (1).
Sufficient practice in the direct solution, as in the preceding example,
is strongly advised.
180. Equations Reducible to Linear Equations. Certain
forms of equations may be reduced to linear equations by a
proper change of variable. No general rule can be given,
and the proper substitution is usually to be formed by
trial.
dv
Example. sec^ 2/ ;£ + 3 a^ tan y =3fi.
Letting tan y = z, we have
dx
which is linear in z and has the same form as the example solved in
§ 179.
XIX, § 180] DIFFERENTIAL EQUATIONS 323
The equation
^ + Py = Qy^y (n a constant.)
called the. extended linear equation is always reducible to
the linear type by putting y^~^ = z.
Example. Given
Thus -(1/2)2/3^+1=0:2/3
and ^-2^^-2x.
ax X
Here P = - ?, fpdx = - 2 log x, eJ^^^ = x-^;
so that z = 3^ If dx-)rc\ = — 2 a;2 log a; + cx^ = y-2,
and finally x^y^ (c — 2 log x) = 1. Check this result.
EXERCISES
Solve the following equations and check each answer.
1. ^ "■ ^ ~ ^/^- ^' 5^ + y cos a; « sin 2 a;.
2. ^+3x22/=3»5. 4. x^£+y^\ogx.
6.
I+I-^-
7.
S-2r. = r2^.
6.
g+. = x^.
8.
.2/^2-^=^-
9.
C082x|+y =
tans.
10.
r^^ = (l+r2)sm^.
11.
12.
In-.-.-.
13. dy — ydx^mix dx. 14. sec ^ dr + (r — 1) dd = 0.
324 THE CALCULUS [XIX, § 180
16. (a;2 -\-l) dy ^ (xy -\-k) dx, 16. xdy -\-ydx = xy^ log x dx.
17. The equation of a variable electric current is
where L and R are constants of the circuit, i is the current, and e the
electromotive force of the circuit. Calculate i in terms of /, 1**, if e is
constant; 2°, if e = eo sin (at.
Ans. 2** i = ,- ^Q sin (ut - 0) + ce-»</^, = arc tan (« L/i?).
181. Other Methods. Non-linear Equations. A variety of
other methods are given in treatises on Differential Equa-
tions; some of these are indicated among the exercises which
follow; Noteworthy among these are the possibility of
making advantageous substitutions; and — what amounts
to a special type of substitution — the possibility of writing
the given equation in the fofrm of a total differential, dz = 0,
where 2 is a known function of x and y which leads to the
general solution z = constant (see Exs. 5-12, below).
Equations not linear in y' may often be solved. If the
given equation can be solved for y\ several values of t/' may
be found, each of which constitutes a differential equation:
the general solution of the given equation means the totality
of all of the solutions of all of these new equations.
EXERCISES
Solve the following equations, using the indicated substitutions:
1. y^dy + (y^+x)dx = 0, (Put v = 2/3.)
2. 8dt — td8='2s(t'-s)dt. (Put s = tv.)
3. xdy — ydx = (x^ — y^) dy. (Put y — vx,)
4. u^v^ iudv -{-V du) = (v -i-iP) dv. (Put w = x, v = y,)
6. Solve the equation (3 x^ + y) dx -\- (x -\- 3 y^) dy = 0.
[Hint. If we put z = x^ -\- xy -\- y^, this equation reduces to cfo = 0;
for dz = (dz/dx) dx + (dz/dy) dy. But dz = gives z = const., hence
/
XIX, § 181] DIFFERENTIAL EQUATIONS 325
35^ + a?t/ + 2/3 = c is the general solution. Such an equation as that
given in this example is called an exact differential equation,]
6. Solve the equation xdy — ydx =0.
[Hint. This equation can be solved by previous methods; but it is
easier to divide both sides by x^ and notice that the resulting equation
is d (y/x) = 0; hence the general solution is y/x == c. A factor which
renders an equation exact (l/x^ in this example) is called an integrating
factor.]
m
7. Solve the equation {x^ + 2 xj^) dx-+ (2 a^ y -h 2/2) ^^ = q.
lHint. Put z = x3/3 + a;22/2 + 2/3/3.] '
8. Solve the equation {s -\-t sin s) ds + (< — cos «) dt = 0.
[Hint. Arrange: s cfo + [< sin s cfe — cos a dt] -{-t dt = 0) integrate
this, knowing that the bracketed term is — d (/ cos s).]
9. Solve the equation xdy—{y — x)dx = (),
[Hint. Arrange: [xdy — y dx] -\- x dx =0; divide by x^t and com-
pare Ex. 6.]
10. Show that [/ (x) + 2 xy^ dx -\-[2 x^y + 4^ iy)] dy = can always
be solved by analogy to Ex. 7.
11. Show that [/ (x) -\- y] dx — X dy can always be solved by analogy
to Ex. 9. Solve (x^ +y) dx — xdy = 0. Ans, x — y/x — c,
12. Solve the equation (r — tan 6) dd + (r sec ^ + tan d) dr = 0.
[Hint. Multiply both sides by the integrating factor cos d; — sin 6
dd -i- r dr + d (r sin 6) =0; integrate term by term.]
13. When a family of curves crosses those of another family every-
where at right angles, the curves of either family are called the orthog-
onal trajectories of those of the other family.
Find the orthogonal trajectories of the family of circles
a;2 _j_ 2^2 = j&^
[Hint. If the differential equation of the first family be dy/dx =
/ (^> y)t then the differential equation of the orthogonal trajectories is
dx/dy = — /(x, y), for any point of intersection (x, y) the slope of the
curve of one system is the negative reciprocal of the slope of the curve
of the other.
In this example the differential equation of the given family is x dx +
ydy^O. It is evident that the differential equation of the orthogonal
326 THE CALCULUS pOX, § 181
family is obtained by replacing dy and dxhy — dx and dy, respectively;
hence the desired equation is xdy — ydx = 0, whence the curves are
y '^cx, i.e. the family of all straight Unes through the origin.]
14. Find the orthogonal trajectories of the exponential curves.
y = e» + A;.
[Hint. The differential equation is dy/dx = e». The orthogonal
family is defined by the equation dy/dx = — «"*, whence the trajec-
tories are 2/ = «~* + c. Draw the figure.]
Determine the orthogonal trajectories of the following families, and
draw diagrams in illustratioh of each:
15. x + y ^k, 18. 0^ ^ ^ - 2 log a; -h c.
16. xy==k. 19. 2a;2 4-2/2=^^2,
17. y^=4k(x+k). 20. x^+y^:=kx.
PART II. ORDINARY DIFFERENTIAL EQUATIONS
OF THE SECOND ORDER
182. Special lypes. We first .consider some very special
forms of equations of the second order that are most fre-
quently used in the application of mathematics to physics,
namely:
[I] ^^=: ±ft^y [fc = constant.]
[II] ^^^2 + B^ + Cy ^0 [A,B,C, constants.]
[III] il§ + B^ + Cy = F(x). [A, B, C, constants.]
These are all special forms of the general equation of the
second order (x, y, dy/dx^ cP y/dx^) = 0.
[IV] We shall consider other special forms also, some of
which include the above; namely, the cases that arise when
one or more of the quantities x, 1/, dy/dx, are absent from the
equation. (See § 186, p. 334.)
XIX, § 183] DIFFERENTIAL EQUATIONS 327
183. Type I. This type of equation arises in problems on
motion in which the tangential acceleration cP s/dfi is propor-
tional to the distance passed over:
(1) S = ±fc^«'
a form which is equivalent to [I], written in the letters s and
t If we multiply both sides of this equation by the speed
V = ds/dt and then integrate with respect to t, we obtain
but we know that
and
hence (2) becomes*
Case 1. If the sign before Jlt^ is +, (3) becomes
(4) ^ = Jt^ kVW+^u
whence i > ^ == \ kdt + C2,
(5) log(s + Vs2 + Ci)=fc« + C2;
or, solving for s,
(6) s = AeM + Be-^.
♦ This is often called the energy integral, for if we multiply through by
the mass m, the expression mv*/2 on the left is precisely the kinetic energy
of the body.
328
THE CALCULUS
[XIX, § 183
where 2 A = e^* and 2 J5 = — Ci c"^' are. two new arbitrary
constants.
By means of the hyperbolic functions sinh w = (e" — €"^)/2
and cosh w(e" + e"")/2 this result may also be written in
the form
(7) s = a sinh (kt) + b cosh (kt),
where b + a = 2A and 6 — a = 2 J5.
Case 2. If the sign before fe* is — , Ci must be negative
also, or else v is imaginary; hence we set Ci = — a^ and write
(42)
or
V = TT = fc Va2 — s2
whence
sin-i ( y = fc< + Cs;
(52)
or solving for s:
(62) s = fl sin (kt + C2) =A sin kt + B cos kt,
where A = a cos C2 and J5 = a sin C2 are two new arbitrary
constants.
Equation (62) is the characteristic equation of simple har-
monic motion; the amplitude of the motion is a, the period is
2 7r/fc, and the phase is — C2/k,
The differential equation (1) was first found in § 88, p. 155.
We now see that the general simple harmonic motion (62) is
the only possible motion in which the tangential acceleration
is a negative constant times the distance from a fixed point;
i.e. it is the only possible type of natural vibration xmder the
assumptions of § 76, p. 125.
XIX, § 184] DIFFERENTIAL EQUATIONS 329
EXERCISES
Solve each of the following equations:
. (P8 _ ^ <P8
6. Find the curves for which the flexion {^y/dx^) is proportional to
the ordinate iy).
6. Determine the motion described by the equation of Ex. 1 if the
speed t; ( = ds/dt) and the distance traversed s are both zero when < = 0.
7. Proceed as in Ex. 6 for Ex. 3, and explain your result.
8. Write the solution of Ex. 1 in terms of sinh t and cosh t. Deter-
mine the arbitrary constants by the conditions of Ex. 6, and show that
the final answer agrees precisely with that of Ex. 6.
9. Determine the motion described by the equation of Ex. 3 if
t; = 2 and « = 10 when < = 0; if t; = and 5=5 when < = 0.
18i. Type n. Homogeneous Linear Equations of the Second
Order with Constant Coefficients. The form of this equation is
(1) *^^'%*c,-o.
where A, B, C are constants.
The type just considered is a special case of this one. Fol-
lowing the indications of the results we obtained in § 183, it
is natural to ask whether there are solutions of any one of the
types we found in the special case.
Trial of e**. If we substitute y = e^'m (1) we obtain the
equation:
(2) [Ak^ + Bk + C]^ = 0.
The factor e** is never zero; hence k miLst satisfy the quad-
ratic equation
(1*) Ak^ + Bk + C = 0,
330 THE CALCULUS [XIX, § 184
which is called the auxiliary equation to (1). If the roots of
(1*) are real and distinct, i.e. if
(3) D=J52-4AC>0,
then ffiese roots ki and h ore possible values for k, and the gen-
eral solution of (1) is
(4) y = Cie*i' + C te^,
since a trial is sufficient to convince one that the sum of two
solutions of (1) is also a solution of (1) ; and that a constant
times a solution is also a solution.
Trial of y = e'^* -v. If (3) is not satisfied, the substitution
(5) y = e** • V
changes (1) to the form
(6) A^, + [2KA+J5]^ + [Aic2 + J5K + C]t; = 0,
which becomes quite simple if we determine k so that the
term in dv/dx is zero:
(7) 2 ic A + 5 = 0, whence ic = - B/2A)
then (6) takes the form
(8) X5 1-7-2 — ^ = - K^Vy
dx" 4 A
where K = V4 AC - By {2 A) = V- D/2 A is reoZ if
(9) D=52-4AC^0,
which is the case we could not solve before.
If Z>< 0, the solutions of (8) are
(10) t; = Ci sin (Kx) + C2 cos (Kx),
by (62), § 183, p. 328; hence the solutions of (1) are
(11) y = e'^ .v = e^' [Ci sin (Kx) + C» cos (Kx)],
where k = - B/{2 A) and K = V^/{2 A) ; these values of
XIX, § 184] DIFFERENTIAL EQUATIONS
331
K and K are most readily found by solving (1 *) for k, since
the solutions of (1*) are k=(-B ± VD)/(2 A)=k±KV^.
If Z> = 0, X = 0, and the solutions of (8) are
(12) v^CiX + C2;
hence the solutions of (1) are
(13) y=e^.v = e^' [Cix + CJ,
where k= —5/(2 A) is the solution of (1 *); since when
D = 0, (1*) has only one root k= - B/(2 A),
It follows that the solutions of (1) are surely of one of the
three forms (4), (11), (13), according as D = B^ — 4 AC is
+, — , or 0; that is, according as the roots of the auxiliary
equation (1*) are real and distinct j imaginary, or equal; in
r&um^:
D = B»-4AC
Character of
Roots of (1*)
Values of Roots
OF (1*)
Solution of (1)
+
Real, unequal
A^i, k2
(4)
—
Imaginary
K±KV-1
(11)
Equal
K
(13)
We illustrate by some examples put, for convenience of
comparison, in tabular form.
Examples
1
•
2
s
Equation (1)
32/"-42/'+2/=0
32/"-42/'+t2/=0
32/"-42/'+22/ =
Auxiliary equa-
tion (1*)
3A;2-4A;+I=0
3A;2-4A;+J=0
3A;2-4A;+2=0
Roots of (1*)
1, 1/3
2/3, 2/3
i(2±V-2)
Solution of (1)
y=Cie*+C2e*/3
y=e^^Hci+(^)
y = e2*/8 (Cl COS
\/2 , . V2 ,
g a:-fc2sm ^ x)
332 THE CALCULUS [XIX, § 184
EXERCISES
1. 2/"-42/' + 32/ = 0. ' 9. y"-92/' + 14y = 0.
2. 2/" + 32/' + 2y = 0. 10. 22/"-32/'+2/=0.
5. 52/"-42/' + 2/=0. 11. 6y"-13y'+62/=0.
4. 9y" + 12|/'+4y =0. 12. 2/"-3y' = 0.
6. 2/"-22/'+2/=0. 13. y"-4y = 0.
6. y" + y' + 2/ = 0. 14. y" + 9y = 0.
7. y"-22/' + 3y = 0. 16. y" + %' = 0.
8. 3 2/" + 5 2^' + 2 2^=0. 16. y" ± A;2/ = 0.
17. If a particle is acted on by a force that varies as the distance and
by a resistance proportional to its speed, the differential equation of its
motion is
dPx/dfi + b dx/dt + ex = 0,
where c > if the force attracts, and c < if the force repels. Solve the
equation in each case.
18. If in Ex. 17, 6 = c = 1, and the particle starts from rest at a
distance 1, determine its distance and speed at any time t. Is the
motion oscillatory? If so, what is the period? Solve when the initial
speed is vo.
19. If in Ex. 17, 6 = 1 and c = — 1, discuss the motion as in Ex. 18.
185. Type m. Non-homogeneous Equations. This type is
of the form:
where A,BjC, are constants, and F (x) is a function of x only.
We proceed to show that this form can be solved in a manner
exactly analogous to § 179, p. 320. First write down the
reduced equation in the new letter v:
XIX, § 185] DIFFERENTIAL EQUATIONS 333
and solve (1*) by the method of § 184. Let v = <t>{x) be any
one particular solution of (1*) (the simpler, the better, except
that v = is excluded). Then the substitution
(2) y = viu
transforms (1) into
(3) {Av" + Bv' + Cv) u + (2Av' + Bv)^ + Av^ = F (x);
but, since v satisfies (1*), the first term of (3) is zero; and
if we now set du/dx = w temporarily, this equation can be
written as the linear equation:
(^) -dx + i Av [^^^ir>
which is precisely of the form solved in § 179. Comparing
(4) with (1), § 179, we have
(5) p.2jy + B. FM.
^ ^ Av Av
Having found t/; by § 179, we have
M = I wdx + Ci, y = uv = v\ I wdx + c» L
which is the required solution of (1).
Example 1. Given the equation
we write the reduced equation
this is easily solyed by the method of § 184; the simplest particular solu-
tion is t; = er». Substituting t; = e-* in the general work above, we find
P = T-^ — = 1 and Q = — p^ = e» sm x;
Av Av
334 THE CALCULUS [XIX, § 186
heDce eS^^ = e*, and
tt? = e-* fe^ mixdx -{-Ci = ■» e* (2 sin x - cos x) + Cier*,
w (to + C2 = Y^j e* (sin a; — 3 cos x) — Cie~* + C2t
y = u*v — — (sin a; — 3 cos x) — Cie~^ + C2e~*.
EXERCISES
1. y" — 3 y' + 2 y = cos X.
il?w. y — ^ (cos a; — 3 sin a;) + cie* + cae^.
2. y"-42/' + 2y = a;.
^ns. y = i (aJ + 2) + CleC»^-^2)x 4. o^&-<2)x^
S. y" + 3j/' +2 2/ = e». ilns. 2/ = e*/6 — ci^-^* +' cae-^.
4. y" — 2y' ■}- y — X, Ans. y = a: + 2 +e»(ci +caa;).
5. 2/" + y = sin X. Ans. 2/ = — J a; cos a; + Ci sin a; + C2 cos a;.
6. y^' — y' — 2y = sin x.
Ans. y — ^ (cos a; — 3 sin x) + cie"* + cae^.
7. y" + 4 y = x^ + cos X.
Ans. 2/ = i (2 a;2 — 1) + J cos x + Ci cos 2 x + ca sin 2 x.
8. 2/" - 2 ^' = 62x ^ 1, ^^, 2/ = ia; (e2» - 1) + ci + cae^*.
9. ^" — 4 2/' + 3 2/ = 2 c3*. Ans. y = xe^ + cie* + c^.
10. If a particle moves under the action of a periodic force through a
medium resisting as the speed, the equation of motion is
(Ps/dfi + Ads/dt = B sin C «.
Express s and the speed in terms of ^. If A = B = C = 1, what is
the distance passed over and the speed after 5 seconds, the particle
starting from rest?
186. Type IV. One of the quantities x, y^ y' absent.
Type IV ai <l>(y") = 0. Solve for y", to obtain a solution,
say y" = a. Then integrate twice. The general solution
for each value of y" is of the form y = J ax^ + CiX + c^.
XIX, § 186] DIFFERENTIAL EQUATIONS 335
In problems of motiorij this type is equivalent to the statement that
4> (jj) = 0, where 7 J, = cPs/dfi = dv/dL Hence jj, may have any one of
the several constant values which satisfy <t> (J^) = 0; but if j^, = kf
8 = kfi/2 + ci< + C2 (see Ex. 24, p. 64).
Type TVbi y missing, ^(x, y', y") = 0. The substitution
m = y' = d\f/dx, dm/dx = d^y/dx^ = y", reduces the given
equation to an equation of the first order in m, Xj dm/dx.
Solving, if possible, one gets a relation of the form
/ (m, X, c) = 0.
This is again an equation of the first order in x and y, and
may be integrated by methods given in Part I, §§ 177-181.
The interpretation in motion problems is particularly vivid and
beautiful. Thus v = ds/dt and jj, = d/o/dt = ^s/df\ hence finy equa-
tion in jj,i Vf t, wUh 8 ahserUf is a differential equation of the first order
in V, Solving this, we get an equation in v and t; since v =^ ds/dt, this
new equation is of the first order in s and t
Example 1. l+x+a^^=0.
Setting dy/dx = m, l+x+x^ -r- = 0.
1 4-x
Separating variables, — dm = — g" ^*
Integrating, — m = 1- log a; + ci.
X
Integrating again, y = Ipg a; — a; log a; + (1 — ci) a; — C2.
Interpret this as a problem in motion, with s and i in place of y and x,
andjr = dv/dt = cPs/dfi.
Example 2. In a certain motion the space passed over s, the speed
», and the acceleration jr are connected with the time by the relation
1 + v^ — jr = 0; find s in terms of t
Placing JT = dv/dtj the equation
1+^-1=0
is of the first order. The variables can be separated, and the integral is
tan-i t; = ^ -f- ci or » = tan (t + ci),
336 THE CALCULUS [XIK, § 186
which is itself a differential equation of the ftrst order if we replace v by
ds/dL Integrating this new equation:
Jds =y tan (t + Ci) d^ + C2, or 5 = — log cos (t + a) + 02.
In such a motion problem we usually know the values of v and s for
some value of ^. If » = and « = 10 when ^ = 0, for example, ci must be
zero (or else a multiple of w) and Ci must be 10; henc6
s = — log cos t + 10.
Examples. 1 +a;f^ +0^2^ = = 1 +a;m +
dm
This can be written dm/dx + m/x = — l/x^,
which is linear in m and a;, the solution being
m = — -loga; + --
x X
The second integration gives
y - — i [log a;]2 +ci log x + C2.
Interpret this as a motion problem, and determine ci and C2 to make
y = 10 and m = 3 when a; = 1.
Type IVc: X missing. <t>(y, y\ y") = 0. The substitution
m = y' gives
,/_^ „n^dy' _dy' dy_dm
and the transformed equation is an equation of the first *
order in y and m. We solve this and then restore y' in place
of m, whereupon we have left to solve another equation
(in X and y) of the first order. »
This is precisely the way in which we solved Type I, § 183,
Type I being only an important special case of Tj^e IV.
Example 1. If the acceleration jj, is given in terms of the distance
passed over (compare § 188), we have
d^s ^ . . dv ^ . .
This is transformed by the relation
. _dv __ du^ds __ dv
^T- dt ~ dsdt ' ds^'
XIX, § 186] DIFFERENTIAL EQUATIONS 337
(which is itself a moat vaLuahle formula) into
do ^ . V
^» = «(8)
in which the variables can be separated; integration gives
which is called the energy integral (see footnote, p. 327).
The work cannot be carried further than this without knowing an
exact expression for ^ («). When <f> («) is given, we proceed as in § 183,
replacing v by ds/dt and integrating the new equation:
^ v^y («) (fe + 2 c
Unfortunately the indicated integrations are difficult in many cases;
often they can be performed by means of a table of integrals. One case
in which the Integrations are comparatively easy is that already done
in § 183.
EXERCISES '
1. y"2^'— 4 x2 = 0. Ans. y = ±l/3x^ + cix + eg.
2. 2/" = Vl + 2/^2. Ans, 2 2/ = cie* + e-^/ci + C2.
8. xy" + y' = x^. Ans. y — x^/g 4- ci log x + C2.
4. s cPs/dfi + ds/dt = 1. Atis. s^ = ^ + ci^ + C2.
9. ^ = rc* cos a;. 10.
11. ^ = «• - cos 2 a;. 12.
dx^
XKry
•
(jpy
dx^ "■
^.
dx^
x + Z
sin a;.
cPy
dx^
1 +
(1)
13/2
13. Show that Ex. 12 is equivalent to the problem, to find a curve
whose radius of curvature is unity.
14. The flexion {(^/dx^) of a beam rigidly embedded at one end, and
loaded at the other end, which is unsupported, mki} — x)y where A; is a
constant and I is the length of the beam. Find ^, and determine the
constants of integration from the fact that y — Q and dy/dx = at the
embedded end, where a; = 0.
338 THE CALCULUS [XIX, § 186
15. Find the form of a uniformly loaded beam of length /, embedded
at one end only, if the flexion is proportional to P — 2 te + a^, where
a; = at the embedded end.
16. Find the form of a uniformly loaded beam of length I, freely
supported at both ends, if the flexion is proportional to P — 4 a:^ in each
half, where x is measured horizontally from the center of the beam.
PART III. GENERALIZATIONS
187. Ordinaxy Equations of Higher Order. An equation
whose order is greater than two is called an equation of higher
order; the reason for this is the comparative rarity in
applications of equations above the second order. We shall
state briefly the generalizations to equations of higher order,
however, since they do occur in a few problems, and since
it is interesting to know that pratically (he same rules apply
in certain types for higher orders as those we foimd for
order two.
188. Linear Homogeneous Type. The work of § 184 can be
generalized to any linear homogeneous equation with constant
coefficients:
Thus if we set y =? e**, as in § 184, we find
(1*) fc" + aik^-' + . . . + On-ifc + a„ = 0,
again called the auxiliary equation. Corresponding to any
real root ki there is therefore a solution e*** ; if all the roots are
real and distinct^ the general solution of (1) is
(2) y = Cie*i^ + Ca 6*2^ + • • • + C„6*«^
where fci, fe, • • • , A;„ are the roots of (1). Curiously enough,
the chief difficulty is not in any operation of the Calculus;
rather it is in solving the algebraic equation (1*).
1
XIX, § 188] DIFFERENTIAL EQUATIONS 339
It is easy to show by extensions of the methods of § 184
that any pair of imaginary roots of (1*), k = kiLK V— 1 cor-
responds to a solution of the form^
(3) y = e*^^ [C sin {Kx) + C" cos {Kx)],
which then takes the place of two of the terms of (2).
Finally, if a root k = k of (1*) occurs more than once, i.e. if
the left-hand side of (1*) has a factor (fc — kY, the cor-
responding solviion obtained as above should be multiplied by
the polynomial
(4) Bo + BiX + B2:t'+'"+ B^,^^-\
where p is the order of multiplicity of the root {i.e. the expo-
nent of (fc — k)^, and where the B^s are arbitrary constants
which replace those lost from (2) by the condensation of
several terms into one.
The proof is most easily effected by making the substitution y = e^'u,
whereupon the transformed differential equation contains no derivative
below d^u/dx^; hence u = the polynomial (4) is a solution of the new
equation, and y = e** times the polynomial (4) is a solution of (1). This
work may be carried out by the student in any example below in which
(1*) has multiple roots. t
t This fact is often made plausible by the use of the equations
e«V-i = cos u + V— 1 sin u, e-"V-i = cos u — V— 1 sin u;
these equations can be derived formally by using the Taylor series for e*,
cos w, sin w, with » = wV— 1, but they remain only plausible until after a
study of the theory of imaginary numbers. The solutions e* =*= K"^-^ are
indicated formally by (2) ; hence it is plausible that (3) is correct.
A more direct process which avoids any uncertainty concerning imagi-
naries is almost as easy. For the substitution j/ = e**«w (see §184) gives a
new equation in u and x which, together with its auxiliary, has coefficients
of the form (d^A( k)/d k^) -r-nl, where A (k) represents the left-hand side
of (1*). Now irV — 1 is a solution of the new auxiliary by development of
A(:k) in powers of (k—K); hence u = sin (Kx) and u =cos (Kx) are solu-
tions of the new differential equation, as a comparison of coefficients dem-
onstrates. This process constitutes a rigorous proof of (3).
t To avoid using i mag inary powers of e, if that is desired, substitute
2/ = c** [cos (Kx ) -h V — 1 sin (Kx)]u, when the multiple root is imaginary,
k = K + KV^^l.
340
THE CALCULUS
[XIX, § 188
These extensions of § 184 should be verified by the student by a direct
check in each exercise.
Example
1
%
s
(1)
2/'" - 2/' =
2/*^H-62/'"+12y"+82/'-0
y'" + 82/=0
(1*)
A;3-A; =
fc4 + 6A:3 + 12A;2+8A; =
A;3 + 8 =
k-=
0, 1, - 1
0,-2,-2,-2
- 2, 1 ± V3 V- 1
y
Ci+cje^+cje"*
c\ + e"^ (C2 + Cjo: + c^pt^)
cie-2*4-e*(c2CosV3a;
+ Cg sin y/Zx)
189. Non-homogeneous Type. The non-homogeneous type
(1)
d^y
d^ y
(ir»''"^'dx"-^ +
+ an-i^ + an^F(x)
cannot be solved in general by an extension of § 185. But in
the majority of cases*which actually arise in practice,* a suffi-
cient method consists in differentiating both sides of (1) re-
peatedly until an elimination of the rigfA^-hand sides becomes
possible. The new equation will be of higher order still:
(2)
d^y I aS«\ I
"T Am-i -IT 4" Am — 0,
but its right-hand side is zero. Solve this equation by § 188
and then substitute the result in (1) for trial; of course there
will be too many arbitrary constants; the superfluous ones
are determined by comparison of coefficients, as in the
examples below.
Example 1. y'" + j/' = sin x.
Differentiating both sides twice and adding the result to the given
equation:
y^+22/'" + j/' = 0.
* For more general methods, see any work on Differential Equations;
eg, Forsyth, Differential Equations,
XIX, § 189] DIFFERENTIAL EQUATIONS 341
The auxiliary equation k^ -\- 2 k^ + k = has the roots A; = 0, A; = ±
V— 1 (twice). Hence we first write as a trial solution yt the solution
of the new equation: yi = ci + fe + cax) cos a: + (c4 + c^x) sin x; sub-
stituting this in the given equation, we find — 2 ca cos x — 2 c^sinx =
gin Xf whence ca = and c^ = — 1/2; substituting these values in the
trial solution yt gives the general solution of the given equation:
y = ci + C2 cos a; + (C4 — x/2) sin x,
I
EXERCISES
1. 2/'" — 3 y" ^ 0. Arw. y = ci + cax + cge*".
2. y'" — 2/" - 4 y' + 4 y = 0. Arw. y = cie» + C2e^ + cae-^*.
8. ^^ — 16 y = 0. Atw. y = cie^ + C2e~2» + cs cos 2 a;+C4 sin 2 x,
4. y*^ — 6 y" + 9 = 0. Arw. ^ = e* V3(ci + cax) + e-*V3(c3 + C4x).
6. y^+62/'"+92/' = 0.
Ans. y = ci + (c2 + cax) cos Vs x + (c4 + csx) sin VSx.
6. y^- 162/'" + 64 2/ = 0, A;=2, 2, -1 ±iV3, -l±iV3.
ilns. y = 6^(ci + cax) + e"* [(cs + C4a;) cos VSx + (cs + cea;) sin V3 x.
7. 2/"-5 2/'+4 2/ = e2«. ^^. 2/ = cie»«- (l/2)e2» +p2e4*.
8. 32/" + 42/' + 2/ = sina;. 10. 2/'" - 2/" - 4 2/' + 4 2/ = e».
9. 2/'"- 32/" + 22/' = a;. 11. 2/*^ - 5 2/" + 4 2/ = 62«.
12. Solve the equation 2/'" + 2/' = by first setting ^' = p.
Solve the following equations by setting y' = p or else y" = q.
13. 3 2/'" -4 2/" + 2^' =0. 16. 2/'" + 3 2/"+2 2/' = e».
14. 2/'" + y" + 2/' = 0. 17. 2/*^-2/" = 0.
16. 2/'" + 2/' = sin X. 18. 2^^ + 2/" = e».
The following equations, though not linear, may be solved by first
setting 2/' = p or 2/" = g or ^"' = r.
19. 2/' = 2/" + VI + 2/"2. 21. l+x+xV = 0.
20. 2^" + 2/"'a; = (2/")2 a^. 22. 3:2/*^ + 2/'" = a;^.
23. Solve the equation x^y" +xy' — y = logx,
[Hint. Put x = «•; then
dy ^dy dz^ _\dy , ^ ^ d_ fldy\ dz _ J:/^_^Y
dx~ dzdx " X dz' dx^ ds \a; ds/ <ia; a^ \d2^ ds/ '
so that the transformed equation is
— I — y = 2, whence 2/ = cie* + 026"* — 3 = CiX + C2X~^ — log a;.]
342 THE CALCULUS . [XIX, § 189
Solve the equations,
24. x^y" — xy' — Zy^O, 25. rty" — y' = log x,
26. (a; + l)2 2/"-4(x + l)2/' + 6y = x, (x + 1 =».
27. (a + 6x)V' + (a + 6a;)y' - 2/ = log (a + 6a;), (« + &«=«•).
28. a;V"-6j/=l+a;.
190. Systems of Differential Equations. Let us finally con-
sider systems of two equations, and let us suppose the equa-
tions to be linear in the derivatives, that is, to involve only
the first powers of these derivatives.
191. Linear System of the First Order. Let the equations be
(1) 2/' = ax 4- &2/ + C2 + d,
(2) z' = aix + biy + CiZ + di,
where the coefficients are constant. We wish to determine
y and z as functions of x.
Differentiating (1) with respect to x gives
(3) 2/" = a + 6i/' + c2';
then the elimination of z and 2' between the three equations
(1;, (2), (3), gives a differential equation of the second order
in y, which should be solved for y.
192. dx/P = dy/Q = dz/R. Here P, Q, and R are functions
of X, y, z. Let X, /i> v be any multipHers, either constants or
functions of x, y, z. Then, by the laws of algebra,
. V •dx _dy _dz _ \dx + fidy + vdz
^^^ P~ Q^ R~ \P + ^Q+vR '
Suppose that we can select from these ratios (or from these
together with others obtainable from them by giving suitable
values to X, /i> »') two equal ratios free from 2, i.e. containing
only x and y. Such an equation is an ordinary differential
equation of the first order in x andt/. Solving it, we obtain
(2) fix, 2/, ci) = 0.
XIX, § 192] DIFFERENTIAL EQUATIONS
343
Suppose that a second pair of ratios can be found, free from
another of the variables, say y. The result is an equation of
the first order in x and z. Let its solution be
(3) F{x,z,C2) = 0.
Then (2) and (3) form the complete solution of the system.
Conversely, differentiating (2) and (3) with respect to x,
eliminating ci and Ca, and solving for dx:dy: dz, we find a
system like (1). In selecting the second pair of ratios, the
result (2) of the first integration may be utilized to eliminate
the variable whose absence is desired.
Example 1.
dx/x^ = dy/xy = dz/^.
The first two ratios give dx/x = dy/yy whence y = c\x. Putting this
value of y in dy/xy = dz/:? gives dy/{ciy^) = dz/^, so that
c\y z
or, a = Ci2/ + ciCzyz = x + c^z. Hence
the solutions are given by the two
equations y = ciXy s = a; + o^z.
Interpreted geometrically, the solu-
tions represent a family of planes and
a family of hyperboloids. These are
the integral surfaces of the differ-
ential equation. Each plane cuts
each hyperboloid in a space curve,
forming a doubly infinite system of
curves, the integral curves of the
differential equation. The system may be written dx:dy:dz =a^:xy:y^.
But the direction cosines of the tangent to a space curve are proportional
to dxj dy, dz. Thus the given equations define at each point a direction
whose cosines are proportional to x^, xy^ y^. Our solution is a system of
curves having at each point the proper direction. What curve of the
above system goes through (4, 2, 3)? What are the angles which the
tangent to the curve at this point malces with the coordinate axes?
I'
Pig. 88.
344 THE CALCULUS [XLX, § 192
— , ^ dx dy d%
Example 2. = — ^- =
y — z z — X X — y
Let X =5 /* = y = 1. Then each of the above fractions equal
dx -\-dy -\-dz
m ■■■■■!■ •
But since the given ratios are in general finite, this gives
dx + dy + dz = Oy whence x +y + z = ci.
Again, let X = a;, a* = j/, y = 2. This gives
xdx -\-ydy -\-zdz = 0, whence x^ + y2 _j_ ^2 = ^j.
Thus the integral surfaces are planes and spheres, and the integral
curves are the circles in which they intersect.
In this example the multipHers X, /k, v have been chosen so as to get
exact differentials.
Ti o dx dy dz
Example 3. = — f— = — •
x—y x+y z
The first two ratios are free from z and give
arc tan {y/x) = log [c\x^/^x^ + 2/^].
Using the multipliers \ — Xj /x = y, v = 0, and equating the ratio thus
obtained to the last of the given ratios, we find '
xdx+ydy dz , ^ i a ^ji
^_^y2 = 7> whence x^+y^= ^z^.
EXERCISES
1. x dxly^ = y dy/x^ = dz/z, Ans, a?* — ^ = ci; a? = c^ (x? + ^).
2. dx/x = dy/y = — dz/z. Ans. yz = a; y = CiX-
3. dx/yz = dy/xz = d&/(x -\- y).
Ana. ^ -2{x + y) +ci;3i^ ^^ =02.
4. dx/(y + z) = dy/(x + z) =dz (x + y),
Ans. {x — y) = ci (x — s)
{x — yY{x + y-\-z)^ a.
5. dx/{x^ + ^) = dy/(2 xy) = dz/{xz + yz).
Ans. 2 y = ci (a;2 — y^); x + y =c^.
ft ^ — 2a?y . dz^ _. 2xz
dx" 3^^y^ — s^' dx s? — y^ — ^
Ans. y = CiZ = 02 (a^ + ^ + s?).
XIX, § 192] DIFFERENTIAL EQUATIONS 345
„ dy _ 2 — 3a; . dz _ 2x — y
dx~'3y-2z' dx^Sy-2z
Ans, x + 2y + 3s = ci;a? +1^ + ^=^0i.
8. dx == — ky dt] dy == kx dt,
Ans, X =^ A COS kt-\-B gin kt; y = A sin kt" B cos kt,
9. dx/dt =» 3 x — y; dy/dt =^x-\-y,
Ans, x=^(A+Bt)&^''yy = (A''B+Bt)&^K
10. Determine the curves in which the direction cosines of the tan-
gent are respectively proportional to the coordinates of the point of
contact; to the squares of those coordinates.
11. A particle moves in a plane so that the sum of the exial compo-
nents of the speed always equals the sum of the coordinates of the
particle, while the difference of the components is a constant A;. De-
termine the possible paths. Ans, x -\-y = ci<^\x — y =kt-\-Ci,
12. If the particle in Exercise 11 is at (1, 1) when i = 0, where is it
;rlien 2 = 5? Approximately how far has it traveled?
INDEX TO TABLES
Beferences to pages of the Tahles in Italic numerals.
References to pages of the body of the book in Roman numerals.
PA6S8
Table I. Signs and Abbreviations IS
Table IL Standard Formulas S-IS
Table III. Standard Curves 19-S4
Table IV. Standard Integrals . • . . • . • S6-60
Table V. Numerical Tables . , • .• # • • 61-60
m
Greek Alphabet
Lettsbs Nambs
LsTTKBB Nambs
Lettebs Names
Lettbkg
\ NAMB9
Aa
Alpha
H 17 Eta
N I' Nu
Tr
Tau
BiS
Beta
e Theta
S^ Xi
T V
Upsilon
r7
Gamma
It lota
Omicron
$0
Phi
A8
Delta
E jc Kappa
Hit Pi
Xx
Chi
Ee
Epsilon
A X Lambda
Pp Rho
yfrrj,
Psi
zi-
Zeta
M/i Mu
Z <r f Sigma
w
Omega
TABLES
[Boman page numbers refer to the body of the text ; italic page nambers refer to these
Tables.}
TABLE I
SIGNS AND ABBREVIATIONS
1. Elementary signs assumed known without explanation .
+ ; db; T; — ; =; ay.h=^a'h = ab\ a-i-b = a/b = a : 6 = - ?
cfi; a3; a»; a-*=l/a»; a^^^Va; aPf^^VaP; aO = l; (); [];
a', a", ..., a(») (accents) ; ai, 02, ..., «« (subscripts).
^ , not equal to. >, greater than or equal to.
>, greater than. -^^ less than or equal to.
<, less than. n\ (or |n) , factorial n = n(»— 1) •••3.2.1.
q.p., approximately. | a |, absolute or numerical value of a.
3. Signs peculiar to The Calculus and its Applications :
(a) Given a plane curve y =f(x) in rectangular coordinates («, y) ;
m = slope = dy/dx =f(x) = y' = first derivative ; see p. 19.
[Also occasionally D^y, f^, y, p, by some writers.]
a = angle between positive a>-axis and curve = tan-i m.
Ay, A2y, ..., A*»y, first, second, -.., n*^ differences (or increments) of y,
dy =f(x) . Ax, (Py =zff(x) • A?, •••, d^y =/(«)(a;) • Ax**, first, second,
— , n*^ differentials of y.
rr = relative rate of increase, or logarithmic derivative ; see p. 114 ;
=/(«) -5-/(«) = (dy/dx) -J- y = d (log y)/dx = r^ -5- 100.
Tp = percentage rate of increase = 100 • rr.
b = flexion = d^/dx^ z=zff(x) = y" = second derivative ; see p. 62.
d^/dx:^ =/(«)(x) = y<'») = n^ derivative.
K= curvature =zl •%- B; .B = radius of curvature = 1 -i- JST; p. 140.
1
2 SIGNS AND ABBREVIATIONS [1, 3
Sf(x) dx = indefinite integral otf(x); see p. 83.
Cf(x)dx = f'~ /(x)da; = definite integral of /(«); see p. 87.
= arc between x = a and x = b.
]b -ix = b
= A = area between y = 0, y = f(x), « = «, x = 6 ; seep. 90*
a -lz = a
(6) (^it?€n a cwroe p =/(^) in polar coordinates (p, ^) ;
^ = Z (radius vector and curve) = ctn-i [(dp/d0) -5- p]
= ctn-i[d(logp)/d^].
= Z (circle about and curve) = tan-i [(dp/dB) -;- p]
= tan-i[d(logp)/d^].
= -4 = area between p =f(^6), 6 = a, = ^; see p. 160.
a Jtfssa
(c) For problems in plane motion :
s = distance. r, = horizontal speed = projection of v on Ox.
t = time. Vy = vertical speed = projection of v on Oy.
m = mass. ^*, = horizontal acceleration = proj. of Jon Ox,
v = speed. jy = vertical acceleration = proj. of J on Oy.
V = velocity (vector), jy = normal ace. = proj. of J on the normal.
J = ace. (vector). jj. = tangential ace. = proj. of j on the tangent
= angle (of rotation), a = angular acceleration.
(V = angular speed. g = acceleration due to gravity.
(d) Problems in space; functions z =f(x, y, •••) of several variables •
Previous notations are generalized when possible v^ithout ambiguity,
exceptions are ^^^j^^^f^, ^^g^/^^^^.
r = Z'^zjdx^ = /,, ; « = ^zjhx dy =f,y=^fy,', t = d^z/dy^ = fyy.
[The notation (dz/dx)^ used by some vnriters for dz/dx is ambiguous.]
4. Other letters commonly used with special meanings:
IT = ratio of circumference to diameter of circle = 3.14159- ••.
e = base of Napierian (or hyperbolic) logarithms = 2.71828-..
ilf = logio e = modulus of Napierian to common logarithms = 0.434-*-
^ = " sum of such term as " ; thus : ^ '~^a^ = ai + ag* + ••• ««*•
(«» i^j 7)» — direction angles of a line in space.
(Z, w, n), — direction cosines ; I = cos a, etc.
5. H.'M. — simple harmonic motion.
e or e, — eccentricity of a conic ; also phase angle of a S. H. M.
II, A] EXPONENTS AND LOGARITHMS 3
a, — amplitude of a S. H. M.
(a, 6), — semiaxes of a conic ; (a, 6, c), semiaxes of a conicoid.
A = difference (of two values of a quantity).
p = density ; also radius vector, radius of curvature, radius of gyration.
6. Trigonometric, logarithmic^ hyperbolic^ and other transcendental
functions : See Tables, II, A ; II, F, 3 ; II, G ; II, H ; and consult Index.
6. Inverse function notations :
If y =/(«), then f-^iy) =x; f-^ denotes an inverse function, [This
notation is ambiguous ; confusion with {/(«)}"^ = 1 h- /(«).]
sin-i X or arc sin x, — inverse of sin sc, or anti-sine of x, or arc sine a?,
or angle whose sine is x. [Other inverse trigonometric functions, and
hyperbolic functions, follow the same notations. See Tables, II, G, 18 ;
H,7.]
TABLE II
STANDARD FORMULAS
A. Exponents and I^ogarithms.
(The letters B, 6, etc. indicate base; L, I, *- indicate logarithm; N, n,
..• indicate number; base arbitrary when not stated. See § 69, p. 99.)
Laws of Exponknts Rules or Logarithms
(1) Nz=B^;in particular (1)' L = logjiVT, i.e. N = B'°»*^; andr
1=»50; B^Bt; \/B= B^K logl = 0; loggB^i; logB(l/5)=-l.
(2) BL.B = B^+K (2)' log (A^ . n) = log N + log n.
(3) B^-^Bi = B^-K (3) ' log (N^n)= log N - log »u
(4) (Bt)» = B^L^ (4)' log (iV») = n log JST.
(5) N=B^,B = b^,N= 6*^. (5) ' logj, iV^ = logi, B . logs N,
B^6, &=10 gives A;=0.4842945=jr=logio6; \og^^N=-M' XogtN,
-ff=10, 6=« gives A=2.302585=l+Jf=logel0; logei^=(l-!-Jlf) logio^^.
&=iV gives L=l/*, l=log65. loga&; «.^., log, 10=1 -i-logio«.
L=^x gives 10«=«**-*^; e*=10**
N^x gives log,o x^M > log, x ; log, 85= (1-f-if ) logjo x.
(6) y = cx» gives r = nw + A;, w = logio x, t? = logi«,y. A; = logio c.
(7) y = ce«* gives v^mx-\-k, t? = logio y, wi =« logio e = aJlf,
* = logio c.
STANDARD FORMULAS
[II. B
B. Factori.
(1) a2 - &2 = (flf _ 5)^^ ^ 5), (2) (a ± 6)« = a* ± 2a6 + 6».
(3) a»— 6» = (a — 6)(a»-i + a*-^ 6 + a»-«62 ^ ... ^ 5»-i).
(4) a2»+i + 62»+i = (a + 6)(a2» - a2»-i6 + ... + fes*).
See also Tables, IV, Nos. 16, 20, 21, 49, 60.
(5) Polynomials : if /(a) = 0, f(x) has a factor as — a; in general ;
f(x) -J- (x — a) gives remainder /(a) .
(6) (a ± 6)» = a» ± -a-ift + ILilLzil a»-262 + ... + (±l)»6»
See II, B, 1, p. 7. ^ ^ * ^
C. Solation of Bquations.
(I)ax2 + 6x + c = 0, roots: x = — ^ ± :^^^Ei«£ = - A ± >^,
^ ^ ' 2a 2a 2a 2a'
where
D = b^'- 4ac'j roots of (1) are
real
coincident
. imaginary j
when D
>0
=
<0.
z.;
(2) a;» + pia:»-i ^ pax"-! + •• + p«-ix + P» = 0. Roots : Xi, Xa, — ,
then ^Xi = — 1)1, 2ja;t«,- =P2, 2jx»xyX* = — ps, etc.
(3) /(x)— 0(x)=O: roots given by intersections of y rr/Cx), y=0(x).
(Logarithmic chart often useful. ) Find roots approximately ; redraw
figure on larger scale near intersection. (Generalized Horner Process.)
(4) Simultaneous Equations : /(x, y) = 0, 0(x, y) = : roots (x,
are points of intersection ; redraw on larger scale as in (3).
(5) Linear Equations :
(a) 2 equations in 2 unknowns : ^ . ^ [•
I a2X + 62!/ = C2 J
Solutions : x =
y =
cibi
C262
aiCi
a^c^
a\b\
a^b^
aibi
azb^
= (C162 — C261) -i- (ai6a — aa&i))
= (aiCa — aaCi) -5- (0162 — fl2&i)»
11, D]
FORMULAS OF ALGEBRA
(1) n equations in n unknowns
2f ••• f ft.
Qi bi "• pi ••• A^i
02 O2 • • • P2 '•• Jco
ciiXi + biX2 + ••• + kiXn =P« ; i = 1,
Solutions : o^ =
where
(^n^n'-'Pn '••^n
■J- 7} J ^^°™° of ^'8 replaces colnmn of 1
\ coefficients of <bj* )
2>=
di 61 ••• Jfei
G2 &2 *•• ^2
On &«••• A^n
6a ••• X^
61 ••• ki
63 ••• ^8
63 • • • ^8
= ai
• •
• •
-a2
• •
• •
• •
bn"'K
• •
bn ••• A;„
+ -+(-l)»-ia.
61 •••
A:i
02 ••'
.A:2
•
•
•
•
•
•
6»-i
*.-i
[Coefficient of a,- skips tth row of D, The last formula is a general
definition of a determinant.]
D. Applications of Algebra.
1. Interest. (P = principal ; p = rate per cent ; r =p-i- 100 ; n =
number of years ; An = amount after n years.)
(a) Simple interest : An = P(^l + nr).
(6) Yearly compound interest: ^ = P- (1 + r)«.
(c) Semiannually compounded : ^, = P ( 1 + r/2)2*.
(d) Compounded once each wth part of year : -4, = P(l + r/m)*^.
(e) Continuously compounded i An = P lim (1 + r/m)*^ = Pe*^.
2. udranm^ies. Depreciation. (/= yearly income (or depreciation or
X)ayment or charge) ; n = number of years annuity, or depreciation, runs.)
(a) Present worth Pof yearly annuity I:
P= 7[(1 + r)n - l]-^[r(l + r)«].
(6) Annuity J purchasable by present amount Pj or, yearly deprecia*
tion I of plant of value P :
1= P[r(l + r)»»]H-[(l + r)»-l].
(c) Final value ^^ of n yearly payments :
^ = I(l + r)[(l + r)«-l]+r.
6 STANDARD FORMULAS [II, D
3. Permutations Pn,n <^^^ Combinations Cn.ri of n things r at & time,
without repetitions :
(a) Pn,r = w(«— 1) •• (n — r+ 1)= «! -f-(n — r)!
(6) C«,, = P„.,-^rI=[n(«-l)... (n-r + l)]-rl
4. Chance and Probability.
(a) Chance of an event = (number of favorable cases) -?- (total number
of trials) <1.
Chance of succeBsive (independent) events — product of separate chances <1.
Chance of at least one of several (independent) events — sum of separate chances.
(6) Probable value v of an observed quantity :
V = [ymi\-^n= arithmetic mean of n measurements wi, m^^ •••,
w„ ; probable error in « = ± .6745 \ ( ^ (^ — w*»)^) -?- w (» — 1).
(If the observations are unequally reliable, count each one a number of times, p^, which
represents its estimated reliability ; Jt),- = " weight " of m^).
(c) Probable value of k in formula v = kx:
A; = 2 XiVi -5- 2***' ^^^^ ^ measurements (a;i, «i), (iCg, W2), •••, (x«, «»);
probable error in A; = ± .6746 \/ 5) (*^» - v»)2 -r- (n — 1) ^a^^^ g^e Exs.
37, p. 58 ; 28, p. 309.
(d) Probable values of A;, Z, m, •••, in formula t? = A;aj + Zy + m« + —
are solutions of the equations :
k 2)^*^ + ^ S ^♦^/i + wi 2^ ^'^^ "•■ **' " S^*"^<
k ^XiZi + I 2J y»2f» + m ^Zi^ + ... = V^f^v^
See also Exs. 37-42, p. 58, Example 2, p. 292, and Exs. 24-31, p. 308.
{Bules for Least Squares, See also Observational Errors^ No. Ill, J.)
II, E] SERIES 7
E. Series.
1. Binomial Theorem : Expansion of (a + 6)"
(a) n a positive integer : (a + 6)» = a'»+ 2*^^ C'„,ra"^6'' ;
[Cn,r: see No. II, D, 8, p. 6, and also II, B, 6, p. A.]
(6) n fractional or negative, | a | > | 6 | :
(a -|-6)«=a» + —a^-^b + ^i^=iiia«-262+ ... + Cn,ra«-*'6'+ - (forever)
II iB I
(c) Special cases :
= (1 ± «)-! — 1 T » + a^ T «»'+ «* T ••• ; ( I » I < 1). (Geometric progression.)
l±x
2 2«.2I 2»-81
2. Arithmetic series : a + (a -f d) + (a + 2 d) + ... + (o + (w — l)d) ;
last term = 2 = a +(» — l)d; sum = s = n(a + Z)/2.
3. Geometric series : a •\- ar -\- at^ + ar^ + •••.
(a) n terms : Z = ar^-^ ; « = ^ " ^ = a — ^ •
^ ^ ' r-1 r-1
(6) infinite series, | r | < 1 : « = a/ (I — r).
4. 1 + 2+3 + 4+ ••• +(n-l)+n = n(n+ l)/2.
5. 2 + 4 + 6 + 8 + ... +(2n-2) + 2n = n(7i + l).
6. 1 + 3 + 5 + 7+ ... +(2?i-3) + (2n-l) = n2.
7. 12 + 22 + 32+ ... +(w-l)2+n2 = n(n + l)(2n+l)-^31
8. 18 + 28+38+ ... +(n-l)« + n8 = [n(» + l)/2]2.
9. 1 + 1/1 1 + 1/2! + 1/31+ ... =lim(l + iy = 6 = 2.71828.-..
10. e» = l+x/l! + a;2/2i + ic«/3l... ; (all x); a* = e*i<««.
11. log.(l ±a;) = ±a-a;2/2±a:8/3-x*/4±ic5/6 ; (-l<a;< + 1).
12. log, [(l+x)/(l-a;)]=2[x + a;V3 + x^/6+-];(-l <«< + !).
[Computition of log JV^: set W-(l + x)/(i - x); then oj = (iV- l)/(iV^+ 1); use II, A, 6'.]
8 STANDARD FORMULAS [n,E
13. sin a; = a;/l ! - xV3 ! + x^/s i - x^ll I + — ; (all x).
14. co8a; = l-xV2I + x*/41-a^/6!+ ...;(aU«)-
15. tan a; = a; + ie»/3 + 2x6/15 + 17x7315+ ... ; (|a;|<ir/2).
General term : 22» (2»" - l)52n-l -i-(2»)! ; see iJ;», Tbftfo*, V, N, p. m,
16. ctnx=l/x-x/3-x8/45-V252,_i(2x)a»-i^(2n)!;
(0<|x|<ir).
17. secx = 1 + xV2 ! + 6 x*/4 ! +2^ [-B2,.x2»/(2 n)l] ; ( | x |< ir/2).
18. C8CX=l/x+x/31 + 2j[2(22«+i-l)B2n+ix2»+V(2n+2)!];
(0<|x|<ir).
19. 8in-ix=ir/2-C08-ix=x+xV(2.3) + l. 3x5/(2.4. 5)+.. .;(|x|<l).
20. tan-ix = ir/2-ctn-ix = x-xV3 + x6/5 — xV7 + ...; (|x|<l).
21. (e« H- 6-»)/2 = cosh x = 1 + xV2 1 + x*/4 1 + x^/e !+..•; (all x).
22. (e* - 6-*)/2 = sinh x = x + x8/3 1 + xV5 1 + xV? I + ... ; (all x).
23. e-«* = l-x2 + x*/21-a^/3! + x8/4I ; (allx).
24. /(x)=/(a)+/(a)(x-a)+/"(a)(x-a)V21 + ...
+/(— i)(a)(x - a)^y{n - 1) ! +^,.
Taylor's Theorem ; Remainder En' \Kn\< [Max. | f^^^x) I ] I (a - a)» | -»- n 1 ;
^n =/*»la+i;(a5-a)](aj-a)n+n ! ; En =(1 -i>)«-l/^»)[a +/>(» - a)] (a? - a)Vn ! ;
IPl<l.
Set a - 0: /(x) -/(O) +/'(0)aj +/"(0)a5V2 ! + ••• +/(«-l>(0)a»»-V(n-l)! +i?i»;
[JToc/owrtn].
Set » - r + A, a - r : /(r + A) -/(r) + A/'(r) + A«/"(r)/2 I + ... + ^«.
25. fix + h,y + k) -/(fl5, y) + [hMx, y) + k/y(x, y)]
+ [A?/,, + 2 A*/,„ + ifc«/„„l + 2 1 + ... + i;;
I JT,, I < M{ |A| + |£|)"-s-ft!, if— maxim am of absolute values of all n^ deriyatiTes.
26. If fix) — afl/2 + a^ cos + a^ cos 2 + a^ cos 8 + ...
+ &i sin 05 + &j sin 2 » + &8 s**! 8 CD + ... ; (—«-<»< + «-).
On = - 1 /(») cos «» rfoj ; &n = - I /(») sin «a> <to. Fourier Theorem.
F. Qeometrlc Magnitudes. Mensuration.
I = length (or perimeter) ; A = area ; V= volume.
11, Fl
MENSURATION
9
Dimensions ob Equations
FOBMITXAS
2. Trapezoid.
3. Circle.
Z, OEB = a/2 ;
/_ OBT^ 90° ;
ZFBT^a\
/, FBO = 90" - a
^ =Z,FTB\
4. Ellipse. «<1.
g + ||-1, (origin at 0).
or P = ; — ^ — :;t
'^ 1 — ecostf
(pole at F) ;
Foci, #, -F" ; Center ; 0.
Sides: a^h^c. Angles: A,B,C.
Altitude from ^ on a » A^.
« - (a + 6 + c)/2 ;
ir = ^ + ^+^=180<»;
= («-«) tan (J/2) ;
c = 6 cos J + a cos ^ ;
<fi =^ a* + 1* - 2 ab cos C.
h = height. &i, &j = bases.
■ ■ — ■ ■ ■ ■ ■ ■ I —
r «> radias ; d = diameter ;
a = COB at center
arc CB , ,, s
= (radians)
r
180 arc CB .,
B= (degrees) ;
n T
V2 = CEB, «^ = 2 a = 2>0^;
sin a =^B -i- r = 1 -i- esc a ;
cos a = 0F-¥- r = 1 -5- sec a ;
tan a = 7!S -5- r = 1 -i- ctn a ;
vers a^FC-i-r = 1 — cos a ;
ex sec a = CT-i- r = sec a— 1.
= a + i» + c = 2«;
^akJ2^hhiJ2^chJ2
=»(l/2)a&sin C, etc.
= ra =%/« («-a)(«-&)(«~c);
sin J sin B sin (7
tan
B-C
b-c , A
b + c 2
:4=A(6i + &a)/2.
I = 2 irr = ird = 2 A/r ;
^ = wr« = jr<«V4 =- 1 r/2 ;
arc (TiS :^r»a^ (a in radians)
= irra/180, (a in degrees);
Chord DB =» 2 r sin a
=-2r8in(.^/2);
Sector ODCB = ^ nr*, (a in
degrees) ;
Triangle DOB = r* sin a cos a
= (l/2)r«sin2a-,
Segment Di^^C- r« [ira/180
- (sin 2 a)/2].
tan (a/2) ^BF^EF^ si n a/Cl + cos a ) ;
sin (a/2) = 5F-4- iF^ =V (1 - cos tt)/2 ;
cos (a/2) ^EF-h-EB = V(l + cos a)/2.
a, &, semiaxes ; /*, r', radii,
c = Va2 - Ifi ;
« s da =- Va2"3^/a,
(eccentricity);
p « yt/c = a(l - eV« ;
a «=■ tan~i { t^ ) = eccentric
angle ;
05 = a cos a, y = & sin a ;
25 . , 25.
a,»«cos^,y = 68ln — ,
CD «e a sin <^, y => & cos ^ \
^ = ir/2 - a.
r + r' = const. =» 2 a. ^ — waft ;
aft .05
«^P-^a
T~'"'a'
=» rt J Vl — e»co8»a tf a :
where cos a = aj/a.
Arc-ffP= I \— i ~dx
JO ^ a* — <t«
= a r Vl-.e«sin»«^<?«^
Jo
(^ s ir/2— a ; sin <^ « oj/a.
STANDARD FORMULAS
[II, If
BTbolft.
r,r''. ndil;
r-- 1- -MBit, -2a;
fi-BMtorOFP-^1oe(2 + «)
f
\Y '
e
FukboU. «-l.
V P— -
'f M
iAr^lstusreemm.
OF-plt-LNH.
j,i-2p«, (origin at 0);
ArMOiVPJf-lVa^l/W;
(Bee raW«, p. W, No. 4« {»).
v|t '
7. Prlem.
i-htlght.
T-B.h.
8
Pn
«mOid(SlM
^.|o»8rl»s8(i™);
jr-nilddleaecdoD;
r-uppprb.ae; ft- height.
(SMiiii«roS(«, iv,G,p. *?.)
9. Pyramid (.ny
A-t.tat,t\af»\
ft-hrtght.
r-.i.»A
10. Biglltcaroulap
OyUnder.
ft-liefglit;«-bi^<arM).
II, F]
MENSURATION
11
Dimensions os Equatiokb
FOBICITLAB
,
11. Bight Circular r » rftdias of base ;
s = \/r* + A» ;
Cone. See Fig. 19, p. 76.
tan a=r/h ;
cos a=h/8 ; sin a=r/8.
A = height ; B — base ;
8 = slant height ;
a = half vertex angle.
-4 (curved )= nr \/r* -|- A« =- irr« ;
^(total)*=irr (« + r);
F==ir»-iA/8 = J?A/8.
r = radius lower base ;
R « radius upper base ;
h = height; «= slant height.
12. Frustum of
Cone.
^= lower base (area);
7*= upper base.
fi = V(y,;-r)« + A«;
-4. (curved) = ir» (^ + r) ;
F = irA {R» + Rr + f*)/S.
13. Sphere.
(a) Entire Sphere.
r = radius ; d = diameter ;
(7= great circle (area).
^ = 4 irr» - ire«« - 4 C;
F = 4irr»/8 = ird«/6
«^.r/8 = 4 0/8.
(6) Spherical Seg-
ment.
Other notations as
above.
a — radius of base of seg-
ment;
h Bs height of segment.
a« = A(2r-A);
/4 = 2 irrA = w (a« -|- A«) ;
F= irA(8a« + A«)/6
= irA« (8 r - A)/3.
(c) Spherical Zone.
h = height of zone ;
a^h^ radii of bases.
^ = 2 irrA ;
F= irA (8 a« -1- 3 62 + A«)/6.
id) Spherical Lune.
a = angle of lune (degrees'^.
^ = irr«a/90.
(«) Spherical Tri-
angle.
Sides a, ^, y.
Angles ^, J5, C.
k = V[8in (« — a) sin (8 -
E^A + B + C-lSa^;
S=iA + B+C)/2\
« = (a + ^ + y)/2,
-i8)sln(«-y)1/sln«;
sin ^ _ sin 5 _ sin <?..
sin a sin p sin y '
cos a -■ cos jS cos y
-f-sinjS.siny cos.<l;
^=: \/ - cos S/lco& (aS - ^) cos {S - B) cos (-S - C)]-
-H sin jS sin C cos a ;
tan(^/2)=*/8in(«-a);
tan (a/2) - JTcos {S- A).
14. Mlipsoid.
Semiazes, a, &, c.
««/a« + y»/2>* + e»/c» = 1.
r>='4nabc/8.
16. Paraboloid of
Bevolution.
r » radius of base;
A = height.
F-irr«A/2 = ir;>A«.
16. Anchor Bang.
r«= radius, generating cir-
cle;
R = mean radius of ring.
^-4ir»ftr;
r^2ir*Rt*.
v^aJ>+y»± Vr^ - e« = i?.
[See also Standard Applications of Integration, Tables, IV, H, p. 4S.^
12
STANDARD FORMULAS
[n,G
Q. Trigonometric Relations. For Trigonometric Mensuration For- /
mulas, see II, F, 1, 3, 13 e, p. 9,
1. Definitions. See also II, F, 3, p. 9,
sin A = y/r ; cos A = x/r ; tan A = y/z ;
CSC A = r/y ; 8ecA = r/x ; ctn ^ = r/y ;
vers A = l — cos A ; exsec -4 = sec -4 — 1.
2. Special Values^ Signs, etc, for sine, cosine, and tangent.
Angle
0«»
±0
1
±0
80«»
46'»
60*
90-
1
±0
±00
ISO*
±0
-1
±0
270"
-1
±0
±00
860° ±^
orO'*±A
90'*±A
ISO* ± A
270* ±^
sin
1/2
>/2/2
V3/2
±sin^
+ cos^
qpslnul
— cos^
cos
Vs/2
V2/2
1/2
+ cos^
=F8inJl
— cos J.
±sinJl
tan
Vs/8
1
V3
±t&nA
TotnA
d:tan^
Tctn Jl
[± and ± 00 indicate that the function changes sign.]
3. CSC A = 1/sin A; sec ^ = 1/cos A ; tan A = 1/ctn A.
4. a;2+y2=,4; cos2^+sin2^=l; l+tan2^=sec2^;ctn2^+l=csc2A
5. sin (A ±B)= sin AcosB ± cos A sin 5.
6. cos (A± B)= cos -4 cos .B T sin ^ sin B.
7. tan(Jl±5) = [tan^± tanJ?]-T-[l T tanJl tanS].
8. sin 2 ^ = 2 sin A cos A; sin a = 2 sin (a/2) cos (a/2).
9. cos 2 ^ = cos2 A — 8in2 ^ = 1 - 2 sin^^ = 2 cos2 Jl — 1 ;
cos a = cos2 (a/2) — sin2 (a/2) ; see also II, F, 3, p. 9.
10. sin 3 -4 = 3 sin ^ — 4 sin^ A. 11. cos 3 ^ = 4 cos^ ^ — 3 cos A
12. tan 2 ^ = 2 tan ^ ^ [1 - tan2 A]. [See also II, F, 3, p. P].
13. 2sin^cos-B = sin(^ + 5) + sin(^-5);
sin a ± sin /3 = 2 sin [(a ± /3)/2] cos [(a ::f /3)/2].
14. 2 cos -4 cos 5 = cos (^ - 5) H- cos (A + -B) ;
cos a + cos /3 = 2 cos [(a + /3)/2] cos [(a - /3)/2].
16. 2 sin ^ sin B = cos (A — B)— cos (^ + B) ;
cos a - cos /3 = - 2 sin [(a + /3)/2] sin [(a - /8)/2].
11, H]
TRIGONOMETRY
13
16. sin2 ^ - sin2 B = cos2 J5 - cos? ul - gin (^ + B) sin (A-B),
17. cos2 A - sin2 B = cos^ B — sin^ ^ = cos (^ + ^) cos (^ - J5).
18. Definitions of Inverse Trigonometric Functions:
(a) y = sin-i x = arc sin a; = angle whose sine is x, if a; = sin j^ ;
usually y is selected in 1st or 4tli quadrant].
(6) y = co8"i X = arc cos x, if x = cos y ; [take y in 1st or 2d quadrant],
(c) y = tan-i x = arc tan x, if x = tan y ; [take y in 1st or 4th quadrant]
19. sin-i X = t/2 — cos-i x = cos-i VI -x^ = tan-i [x/ Vl - x2]
= csc-i (1/x) = sec-i[l/ Vn=^ = ctn-i[ vT3^/x].
20. cos-i X = ir/2 - sin-i x = sin-i Vl - x^ = tan-i [Vl-xV«]
= sec-i (1/x) = csc-i [1/ >/ri^] = ctn-i [x/ VH^^].
21. tan-i X = ir/2 - ctn-ix = ctn-i (1/x) = sin-i [x/Vl + 2c'^]
= cos-i [l/Vl + x2] = sec- VI + x2 = csc-i [vT+^/x],
22. Special values, correct quadrants, etc., for inverse functions.
Value
+
—
n/2
1
ir/2
»/4
-1
1/2
ir/6
n/S
0.46
V2/2
V8/2
V8/8
>1
-Jk
Bln-lflj
UtQ
4th Q
-ir/2
ir/4
,r/8
0.62
- sin-l (+ k)
cos -1 05
UtQ
2dQ
w
V4
ir/6
0.96
IT — 0O8-l(+A)
tan-lflj
1
UtQ
4th Q
-ir/4
•
0.62
0.71 .
ir/6
>ir/4
-tan-l(+*)
H. H]rpeTbolic Functions.
1. Definitions. (See figures III, E, J2, pp. 22, SO ; and V, C, p. S4,
sinh X = («* — c-*)/2 ; cosh x = (c* + e-*)/2 ;
tanh X = sinh x/cosh x = (e» — e-*)/(e» + e"*) ;
etnh X = 1/tanh x ; sech x = 1/cosh x ; csch x = 1/sinh x,
= Gudermannian of x = ^dx = tan-i (sinhx) ; tan = sinh x.
= tan-i [(e« - c-»)/2] = 2 tan-ie« — ir/2
2. cosh2 X — sinh2 x = 1. 3. 1 — tanh^ x = sech^ ».
4. 1 — ctnh^x = csch2x.
5. sinh (x±y) = sinh x cosh y ^ cosh x sinh y.
14 STANDARD FORMULAS [II, H
6. cosh (x±y)= cosh x cosh ^ i: sinh a; sinh y.
7. y = sinh-^a; = arg sinh x = inverse hyperbolic sine, if as = sinh y
[Similar inverse forms corresponding to cosh x, tanh x, etc.]
8. sinh-ix = cosh-i Vx^ + 1 = csch i (1/a;) = log (x + Vx^ + 1).
9. cosh-i X = sinh-i Vx^ — 1 = sech-^ (1/x) = log (x + Vx^ — 1).
10. tanh-ix = ctnh-i (1/x) = (1/2) log [(l + x)/(l - x)].
11. It 4> = gd Xy sinh x = tan <p, cosh x = ctn 0, tanh x = sin 0.
I. a. Plane Analytic Oeometry
[(», y) or (a, 6) denote a point ; (ajj, y^) and (a-a, y,) two points ; etc.]
1. Distance I = P1P2 = V(X2 - Xi)2 -f {y^ - yi)'^ = Vax^ + Ay^
2. Projection of P1P2 on Ox = Ax = X2 — xi = Z cos «, where
a = Z(Ox, P1P2).
3. Projection of PiP2 on Oy = Ay = ^2 — 2/i = ^ sin a,
4. Slope of P1P2 = tan « = (^2 — yO/(^ — asi) = Ay/ Ax.
5. Division point of P1P2 in ratio r : (xi + r Ax, yi + r Ay).
6. Equation Ax + By -{-C = 0: straight line,
(a) y = r?»x + b : slope, w ; y-intercept, b.
(&) y—yo = m(x — xo): slope, m ; passes through (xo, yo).
(c) (y-yi)/(y2-yi)=(«—«i)/(x2-xi): passes through (xi,yi), (X2,y2).
(d) X cosa + ycos/3 =p: distance to origin, p; a^Z{Ox, n) ;
|3 = Z(Oyy n); n = normal through origin. •
[General equation Ax + By + C—0 reduces to this on division by \^A* + B*.}
7. Angle between lines of slopes wi, W2=tan-i [(wi — m2)/(l+mim2)].
[Parallel, if mi = 7712 ; perpendicular, if l+?»iW2=0, i.e. if wi = — l/m2.]
8. Transformation x = x' -^ h, y = y' -{• k. [Translation to (h, *).]
9. Transformation x = ex', y = ky'. [Increase of scale in ratio c on
X-axis ; in ratio k on y-axis. ]
10. Transformation, x = x' cos $ — y' sin 0, y = x' sin ^ + y' cos $,
[Rotation of axes through angle $.']
11. Transformation to polar coordinates (p, B): x =p cos 6, y = p sin $.
Reverse transformation : p = Vx^ + y^, $ = tan-i (y/x).
II, J] ANALYTIC GEOMETRY 15
12. Circle : (x— ay + (y — by = r^; center, (a, b) ; radius, r ; or
(« — «) = *• cos dy (y — 6) = r sin d. (d variable.)
13. Parabola : y^ — 2px : vertex at origin ; latus rectum 2 p.
14. Ellipse : x'^/a^ + yV^ = 1 • ce^iter at origin ; semiaxes, a, b,
(See n, F, 4, p. 9.)
15. Hyperbola : x^/a'^ — y^/b^ = 1 ; center at origin ; semiaxes, a, b ;
asymptotes, x/a ± y/b = 0. See II, F, 5, p. 10.
(a) If a = 6, x2 — 2^2 _ q2 . retaugular hyperbola.
(b) xy = k, rectangular hyperbola ; asymptotes : the axes.
(c) y=(ax-{-b)/{cx-{-d), rectangular hyperbola ; asymptotes: x=z^d/c,
y = a/c,
16. Parabolic Curves: y = ao + a\X + a2X^ + ••• H- an«*
[Graph of polynomial ; see also Figs. A, B, pp. 19 ^ 20,']
17. Lagrange Interpolation Formula. Qiven y = /(x)^ the poly-
nomial approximation of degree n — 1 [parabolic curv'^e through n points,
(«i, yi), (X2,y2)y •-, (a?„, 2/J] is
y = r(Qc) = ViP^ix) + y.^p^{x) + ... + VnPni^),
where the polynomials pi(x), i?2(^)» •••» Pn(jx) are
n,(a;) = -X*-"" ^i)(^_":.^2V- (a; - y<-i)(a; - a^f+i) ••- (a; - gn)
(X< - Xi){Xi -X2)'" {Xi - Xi-\){Xi - Xi+{) -"{Xi-Xn)
[Numerator skips (« — aj^) ; denominator skips (x< — Xi). Proof by
direct check.]
[For a variety of other curves, see Tables^ III, pp. 19-34. ]
I. b. Solid Analytic Geometry.
[(x, y, z) denotes a point ; (xi, y^ Zi) and (X2, Vi, 22) two points, etc.
O denotes the origin, (0, 0, 0).]
1. Distance : P1P2 = V(X2 - Xi)2 + (2/2 - 2/1)2 ^ (^^ _ zi)2.
2. Distance from the origin : OP = Vx^ + y2 ^ 22.
3. Direction cosines of a line L : cos a, cos/3, cos 7, if a, /3, 7 denote
the angles L makes with the x, y, z axes, respectively ; and we have
cos^a + cos2j8 + COS27 = 1.
4. Direction cosines proportional to given numbers :
If a: b : c = cos a : cos/3 : cos 7, .and B^ = a'^ -\- b^ -\- c^, then
Qosa = a/JB, cos/3 = &/JB, cos 7 = c/i2.
16
STANDARD FORMULAS
[11, 1
6. Angle B between lines L and U with direction cosines (Z, m, n) and
(r, m', n') :
cos^ = IV + mm' + nn'.
JAnes parallel if W + mm' -\- nn' = 1, or if i = i', m = m', n = n'.
• Lines perpendicular if W + mm' = nn' = 0.
6. Direction cosines of a plane P = direction cosines of any line
perpendicular to P.
7. Equation of a plane P :
te + my + nz = p, or x cos a + y cos /3 + « C0S7 = p,
where (Z, m, n) are the direction cosines of P, and p is the length of the
perpendicular from to P.
8. General equation of a plane : Ax + J5y + C« + D = 0.
If 122 = ^2 4. ^ ^. C2, i = cosa = A/R, m = B/B, n= C/E, p^-B/B,
9. Plane with intercepts a, 6, c, on the axes :
x/a + 2//6 + z/c = 1.
10. Plane determined by {xu Vi, ^i)? i^, 1/2, 2^2), (acj, ys» zi):
X y z 1
»i yi 2i 1
X2 2/2 ^2 1
353 2/3 23 1
11. Angle $ between two planes (I, m, n ; p), (I, m, n ; p);
cos $ = 11' -{- mm' + nn'.
12. Angle ^ between planes Ax -\- By + C« -f D = and
A'x + 2?'y + C'« + D' = ;
cos e = ^^'H-P^^+ CC 2J2 = ^2 4. ^ 4. (72 R/2 = ^'2 + J5/2 + 0'«
BB' . .' T T , ,
Planes paraHeZ if W + mm' + nn' = 1, or if ^ = A', B= B', C = C
Planes petpendicuiar if A A' + 55' + CC = 0.
13. Distance d from point (xi, yi, «i) to plane (i, m, n ; p):
d = 1x1 + fny\ + W2i — p.
14. Distance d from (xi, yi, 2i) to -4x + 5y + Cz + D = :
^^Aci + 5y^+C2i + i) 222 = ^2 + ^2+02.
15. Direction cosines (2, m, n) of a line determined by two planes
^ + 5y + Cz + D = 0, A'x-\- B'y + C'z + D' = ;
=
limine
B
C
•
c
A
•
A
B
B'
•
c
A'
•
A'
&
see 4.
n,Ji
ANALYTIC GEOMETRY
17
16. Line through (xu Vu 2i) and (0521 y2i 22) •
x — xi __ y — yi _ z^Zi ^
X2 - xi 2/2 - 2/1 22 — Zl
17. Line through (xo, yoi 2©) in direction (i, m, n):
18. Line through (xo, yo» 20) perpendicular to plane Ax-{- By -\-Cz^O:
x — xq __ y — vo _, z — zq
A B C '
19. Plane through (Xi, 2/1, Zi) perpendicular to line of fonnula 17 :
A(x-Xi) + B(^y^ yi) + C (2 - 2i) = 0.
20. Sphere of center (a, 6, c) and radius r :
(X - ay + (y- 6)2 + (2 — c)2 = r«.
21. Cones with vertex at :
352 t/2 22
a* 6* c2
Imaginary J if all signs are alike ; otherwise real, and sections parallel to
one of the reference planes are ellipses.
22. Ellipsoids and hyperboloids with centers at O (see Tables III
Signs on left :
All + : ellipsoid
One — : hyperboloid of one sheet
Two —
All -
23. Paraboloids on z-axis with vertices at (see Tables III N4, 5) :
Signs on left :
< Alike : elliptic paraboloid
Different : hyperbolic paraboloid
24.. Contour lines on curved surface F(x, y, z)=0:
Sections by z = a are F(x, y, a) = 0.
25. Curves in space :
(a) Intersection of two surfaces : Fi(x, y, 2)= 0, 1^2 (x, y, 2) = 0.
(6) Solve for y and z : y =/(x), z = 0(x).
(c) Parameter forms : x =/(0» y = 0(O» 2 = ^(Q.
26. General cylinder with elements parallel to z-axis : /(x, y) = 0.
X* . 2/2 . 2*
hyperboloid of two sheets
surface imaginary
. x2 . 2/2
o2 62
18 STANDARD FORMULAS [11, J
J. Diflerential Formulas.
1.1/= /(x) : dy =/'(x) dc, f'(x) = dy -^ dx = dy/dx.
2. r(x, 2/) = : F^dx + Fydy = 0, or dy = _[F, ^ F^jcte.
3. x=f(t), y = i>(t):
{a) dx = f (t) dt, dy = <p'(t)dt, dy/dx = <pf{t) -i-f^Ct).
(6) d:hf/dx;^ = dldy/dxydx = dl<p' -^fydx = [0"/' -/"0'] -5- (/')«.
(c) d8y/d:i^3 = j[^2^/da;-^]/tZx = d[(0"/' -/"0') -^ (/')^]M -/'•
4. Transformation x = /(f) : y = 0(ic) becomes y = 0(/(O) = ^ CO*
(a) dy/dx becomes dy/df -^/'(O »' C^^® ^ C^)]-
(6) <r^/eix2 becomes [(c?V<^«^) '/'(O - (<^yM)/"(0] -*■ [/'(O]';
[see 3 (6)].
5. Transformation ic=f(tjU), y = ^(tfU): y = F(x) becomes
w = *(0.
(a) dy/dx becomes ^^^ or rM+?^ . ^"1 ^ fV+^t . ^l^
d« dt Idt du dtj Idt du dtj
(6) d?y/dx'^ becomes d[dy/dx]/dt -f- dx/dt ; [compute as in 6 (a)].
6. Polar Transformation x = p cos ^, y = p sin ^ .
dx = cos ^ dp — p sin ^ d^ ; d^/ = sin ddp + p cos ^ d^,
d^x = cos 0d^p — 2 sin ddpdd — p cos ^ df^,
(?V = sin ^d^p + 2 cos ^ dp d^ — p sin ^d^-i.
. 7. z = F*(i€,y): dz = F^dx-\-Fj,dy=pdx + qdy; [see I, 3(d),
p. £],
8. Transformation x =f(u, v), y = (p(u, v) : z = F(x, y) = *(u, »).
5w 5x Sm 5y du ' 5v 5x 5v dy dv '
dx du dv dy du dv
lA, B, C, D found by solving 8 (a) for dz/dx and dz/dy.^
(c)^ = Af^\=Af^j?£+5?£^
^ "^ dx^ dx\dxj dx\ du dvl
= A^(a^-^ + B^^] + B ^ (a^-^+ B^\.
du \ du dvj dv\ du dv)
[Similar expressions for d'^z/dy^ and higher derivatives.]
STANDARD CURVES
A Gnrves y — x" all pass through (1 1) - positive powers also through
(0 0) nega, e powers asj mp otic to the y sjcie Spec al cases 1
20 STANDARD CURVES ' [1
are ■tralgbt litieB ; h = 2, 1/2 are ordinary paiabolaa ; n = — 1
ordinary hyperbola ; n = 3/2, 2/3 are Boml-oabical parabolas.
B. Logaritbinlo Paper ; Carves y = x", y = kx'*. Logarithmic
paper is need cliiefly in experimentai deUrmtnation of the eonstanU k and
n ; and for graphical tables. In Fig. B, k=l except where given.
[See |lS9,p.334, »1>iitb; *]90 TiniMin-HB»n, ffjrfrouHo T^iblet; Trmtirine, Xa-
gtn6eri' Handboi^k; D'Ocagne, Somoeraphit.\ Tbellnsy — »— Igiiee tbe rtcipnteak
r/ 7in7itf'era by ilircct readlngB.
Ill, Dl
ELEMENTARY FUNCTIONS
C. Trlgouometrlo Fnnctiona. The invDrHe tiigonometric innc-
tions are given by reading y fitst.
2.303 logi«x. The values of the expo-
= e" are given bj reading y first. See E.
i
'P"-' \"'-\
1 ■ U_--- ^---T~1
-'■ Li'.'B
rrDjius
i.
--■li-
:::r|.::
-
'
- '- - ■ ^
^ljr"_
i
//'■'
: '"'r—A
•^ ■ -:-
i'i'ii' "^■■■M. ,i,-i..-h-riJ.-4lii^i|
22
STANDARD CURVES
[III, I
B. BzponentSal and Hyperbolic Functlona. Tbe catenaiy
(hyperbolic cosine) [?/ = cosh i = (e* + e— )/2] and the bjrperbolic
•Ine [y =sinha5= (e*— e-')/2] are ahown in their relation to the ei-
pODBiitial curve* i; =e; y - e"'. Notice that both hyperbolic cnrrea
are SHymptotic to y = e'/2.
' is the standard damping ci
ni, PI
HARMONIC CURVES
F. Haimonlc Curvea. The general type of Bimpls haimonlo Oiuv*
is y = a sin (^kx + 1):
».,.
«■!■ (!« + ()
■IBZ
O..
■■■a«
(i/2)«iBCe»-
..J
Binplltuds
■ a
1
1
1
1/2
S>/*
2ir
w
Jr/a
-,/k
•
«/2
o-i
A eompoand taEumonlo cnrvs is lonned by superpoaiag aimple hur
monies : In Fig. i\, j = Bin 2 a; + (1/2) sin (Bx- 1.2) ia drawn.
A
= ..n
21
,(Cr-
1J2)
r
//
K^
...
^H'
n{Or-
"'
/
\
/
\
f
r^
\
V'
'
V
A.
^
A
/
\\
\
y.
y
v/
^
1
V.
^
^
A
Id theorids of rlbnUons, gonnd, elKtrleltf.
24
STANDARD CURVES
The Bimplciit type of damped vlbtatloiia U
Bhowa y — e-''*sin 3 r The general form iey—c
damped simp e Tib a odb may be auperposed o
damped v bratioua.
[Ill, P
< = «-" sin kx : Fig, Fj
-"sin (to + Such
. otber damped or im-
O. The RoulettM.
A roulette is the path of any point rigidly connected with a moving
curve which rolls without slipping on another (fixed) curve.
TbeCTcMd
Figure Gi shows the ordinary cycloid, a roulette formed by a point
P on the rim of a wheel of radius a, which rolls on a straight line OK
See also Fig. 30, p. 144. The equations are
jfherefl = ^ifCP
Ill, GJ
ROULETTES
25
Figure G2 shows the curves traced by a point on a spoke of the
wheel of Fig. H, or the spoke produced. These are called trochoids ;
their equations are
x = ad—h sin^,
y =z a — h cos 0^
Fig. G»
Xhe Tieclioids
where h is the distance PC, If & > a, the curve is called an epitrochoidi
if 5 < a, a hypotrochoid.
Figure G3 shows the epicycloid ;
x = (a + 6) cos ^ — 5 cos
[^']
y]= (a + 5) sin <? - 6 sin f^Ll^ $],
e
Fig. Gj
formed by a point on the circumference of a circle of radius b rolling on
the exterior of a circle of radius a.
26
STANDARD CURVES
[III.G
y}^
^
f c
r
V ^
y
v^
Fio. G4
Figure Gg shows the hypocycloid :
« = (a - 6) cos^ + 6 cos ["^^^^ ^"j,
y = (a _ 6) sin ^ - 6 sin T^i^ <?1,
formed by a point on the circumference
of a circle of radius 6 rolling on the
interior of a circle of radius a.
Hypocycloid
Figure G4 shows the special epicycloid, a = 6,
x — 2a cos $ — a cos 2 0,
y = 2 a sin ^ — a sin 2 ^,
which is called the cardloid ; its equation in
polar coordinates (p, <p) with pole at (^ is
p = 2a(l — COS0).
Fig. G4
Figure Ge shows the special hypocycloid, a = 4 6,
x = a cos^ 0,
1
y = a sin8 0,
or a;2^ + y2/8 = aa/«,
Fig. G«
which is called the four-cusped hypocycloid, 01
astroid.
H. The Tractrlz. This
curve is the path of a particle
P drawn by a cord PQ of fixed
length a attached to a point
Q which moves along the
X-axis from to ± 00. Its
equation is
V
^
Vp
The Tractrix.
^
^ a
V
X
V
X-:;__
k
Q
X
FiQ. H
x = alog^ + ^^'^y'^v^^^-3jr2.
y
in, I]
CUBICS— CONTOUR LINES
27
I. Cubic and Quartic Curves.
Figure Ii shows the contour lines of the surface z = ofi — ^x — y'^ cut
out by the planes z = k^ for &=— 6, —4, —2, 0, 2, 4; that is, the
cubic curves x^'~Zx — y'^ = k,
•
The surfiMse has a maximum at (»« — 1, ^ — 0; the point (e = 1, ^ « is also a critical
point, but the surface cuts through its tangent plane there, along the curve ik = — 2 ;
y« = a5» - 8 a; + 2. ,
These curves are drawn by means of the auxiliary curve ^'=3'*— 3», itself a type of cubic
curve ; then y =y/q — k. is readily computed.
Fig. Ii
Fig. t
Figure I2 shows the contour lines of the surface 2J=a8— 3 a;+2^« («— 4)
for « = A; = — 6, — 4, — 2, 0, 2, 4 ; that is, the cubic curves
y2=(ic«-3x-A:)/(4 - x).
The Burfl&ce has a maximum at (- 1, 0). At (1, 0) the horizontal tangent plane « — -2
cuts the surface in the strophoid y«= (a5« - 8cb -f 2)/(4 - oj) whose equation with the
rew origin C>' is y» = a?'* (3 + aj') /(3 _ a;'). The line tp « 4 is an asymptote for each of the
curves.
28
STANDARD CUftVES
[m,i
Figure I3 shows anothei cubic : the cissoid,
famous for its use in the ancient problem oi
the ** duplication of the cube/' Its equation is
y2 =
ofi
2a — a; *
or p = 2atan(^ sin^.
It can be drawn by using an anxiliaiy carve as above ;
or bj means of its geometric definition : OP — QB^ when
Oy and AB are vertical tangents to the circle OQA,
Figure I4 shows the conchoid of Nicomedes, used by the ancients in
the problem of trisection of an angle. Its equation is
= -a?a +
-^ — ) , or p = a sec ^ :t 5.
X — a/
FlQ. I4
Condhohl
Follmn
of
Descartes
Fig. h
Figure Ig shows the cubic «* + j^ — 3 axy = 0, called the Folium
of Descartes ; see Example, p. 45.
The ^
Witch ^^
2a ^'•«-..^,,^
*^"i«+4a«
*-
^
X
Fio. Is
Figure le shows the witch of Agnesi : y = S aV(x2 + 4 a*) ; see Ezs.
I, 36, p. 131 J 62, p. 162 ; and see III, J, below.
QUARTICS— CONTOUR LINES
Figure It shows the CaBmlnlan ovals, defiaed geometrically by the
equation FF • FFi = i* ; or by the qnartic equation,
o)« + j/'][Ca; + a)'4-y'] = H
where a = OF (= I in Fig. I;). The special oval Ifi = o? Is called the
leminlKato, (i^ + y*)= = 2 a" (2= - y') or ^^ = 2 «« cos 2 S.
which has minima at (2 = ±a, y = 0), and a critical point with r
STANDARD CURVES
J. Error or Probability Cmvaa.
Figure Ji is the so-called oorve ot error, or probability ci
where ft is i
of precision. See Tables.
IV, H, 148, p. SO ; and V,
y = seeta x = 2/(e' + e-)
In some instances thiscurve, or the witch (Fig. I«), maybeusedinplac
ot Fig. J]. Any of
these curves, on a
proper scale, give
good approzimatioiiB
to the probable dis-
tribution of any ac-
cldeotal data which
t«nd to group tLem-
selves about a mean. Fiu. Ji
K. PolyTtoinial Appro^mationa.
Figure K], shows the first Taylor polynomial approxiniatloiui to tha
CuncUoii y = sin X. (See § 147, p. 2
APPROXIMATION CURVES
Figure K3 shows the Slmpaon-LagreuiBe approzlmatlotU) : (1) bj
i broken line ; (2) b; an ordinarj parabola ; (3) by a cubic, wbicb
However degenerates
into a parabola in this
example. (Lagrange
iDtcrpolation For-
mula, Ta6(M, p. 15.)
'^''SS^SSi
s
11V1.1™ point, ^ uk.n - ffil TTiTrfm-f fflTf fFf^hH ^
jloher together thu ■■ H4+i-|-rT I |-|4+|-T I H"|- H+|-H->+ i M-f: :44-|-hH+
Fio. Ki
3*- 1.2I3» - .405ir', O^bSit.
.it/3.w,-Sir/3,B,-i.
L. Trlgonomettlo ApprozlniBtiaiia.
(1)
.4
/
^
\\
C
,
E'ou
ler
APP
lex
Q«tl
//
A
A
1
1
JU
...
^
«<0
r
\
j
I
/('
).
'/2
\
<TJ
^
<
'1
-
f^
%
T
}
\
li
Ap^
™
Hii
mat
Z"
\
uy
//(»
1 "
J, =2. tor
D
^S.
•■H
./;
>-'-""i ■»•"*•"*
i-i
1 1 1 1 1 1 1
Fio. L
Fignre L shows the approximation to the two detached llne-eegmenls
r = -x/2, (~x<ar<0),y = ir/2, (0<i<») by means ot an eipres-
flon of the form a, + ai sin le + og siu 2 x + — + a, sin nx. See TI, E, 28.
STANDAKD CUKVES
\
ao- /60*
i^
/l90"
^S^
' \
\
t
Arehlmei
MtnBjrtraK
\>-
'^
/ ,.
off ^
>/'\
/
^^
,^
\
m, N] SPIRALS — QUADfilC SURFACES
U. SplralB.
Tignies M: and Mg represent the Archimedean Spiral p = aS, and the
Hyparbollo Spiral p$ = a, respectivety.
Jl. Quadtic StuiaoeB.
These are standard figures of the nsnal equatioDS.
HTporiralatd of ons Sheat
34
STANDARD INTEGRALS
[in,N
Hyperboloid of two Sheets
i=0O
>h
V7
if
I
M
V
V
/
.*Vx'
Fig. Nt
Elliptic Fftraboloid
TlO. ^«
z
a
Hjrperbolic Paraboloid
TABLE IV
STANDARD INTEGRALS
Index ;
A. Fundamental General Formulas, p. S5,
B. Integrand — Rational Algebraic, p. 36
C. Integrand Irrational, p. S9.
(a) Linear radical r = Vaag -h 6, p . S9,
(6) Quadratic radical V± x^ ^ ^2^ p, jp,
D. Binomial Differentials — Reduction Formulas, o. ^1.
is. Integrand Transcendental, p. 41.
(a) Trigonometric, p. 41,
(6) Trigonometric — Algebraic, p. 44-
(c) Jnverse Trigonometric, p. 4^-
(d) Exponential and Logarithmic, p. 45.
F. Important Definite Integrals, p. ^6.
G. Approximation Formulas, p. ^7.
H. Standard Applications, p. 48.
A. Fundamental General Formulas.
•■ • It ^— = -r— , then w = t; + constant. [Fondanaental Theorem*.
2. If I t« dX = I, then -^ = w. [General Check.!
•^ da?
3. \cudx = c\udoc.
4. \[u + v'\dx=(udx^-(vdx.
6. \UdV = UV—lvdU, {Parts. J
8. r f /(u) dwl = f /[^(a5)] ^^ cfaj. :SubgtltotIon.J
35
36 . STANDARD INTEGRALS [IV, B
B. Integrand — Rational Algebraic.
7. \x^dx = — , n=7t-l, see 8.
•^ n+l
Notes, (a) /(Any Polynomial) das, — use 8, 4, 7.
(b) /(Product of Two Polynomials) (to, -~ expand, then use SfA,t»
(c) /ctto — 00, by 8, 7.
8. f ^ = loge 05 = (logio a;)aog. 10) = (2.302685) logio x.
Notes, (a) /(l/a5'")(fa5, — use 7 withn — - wi if m=^l ; useSif m — 1,
(6) / [(Any Polynomial)/*"*] dx^ — use short division, then 7 and &•
9. f-^=larctaii^ = ltan-i^ = lcte-i^
1 cr
= — — Ctn-l— [+ const.].
a a
L f_J^= _L log ^I^ = JL log «^=^[+ const.].
Note. All rational functions are integrated by reductions to 7, 8, 9. The redoe*
tions are performed by 8, 4, 6. No. 10 and all that follow are results of this process.
11. ((ax + b)^dx = 1 (<^^ + ^)"'^\ n^-1. (See No. 12.) [From 7.1
J a n + 1
12. f — ^ — = llog(ax + b), [From 8.1
J (oic + 6) a
Notes, (a) { "^ dx, — use long division, then 7 and 12.
fX + b
^ j^ |. Any Polynomial ^^^ _ ^^^ j^^^ division, then 7 and 12.
J ax-\-b
13. rr-^7T- = -7 .x7^ vx i >(^=?fcl)' [From 11.1
JCox + fe)"* a (m-l)(ax+6)'"-^ -^ / l j
14. fy-^^r-r = -^r— TT + l^S (aa + 6)1 • [From 11, 12.]
Notes, (a) f "*" dx, — combine A times No. 14 and B times No. 18, m = 2.
' (aa5 + b)*
^j^ J. (Any Polynomial) ^^^j^^ i^ng division, then 7 and 14 (a); oP use 16
J (ow + &)■
IV, B] RATIONAL ALGEBRAIC 37
16. r f-^C*, ax + b) dxl = " f -^ ("— » A ^u, [From 6.]
Notes, (a) BestaUment : put t* for ace + 6, 2LZL_ for 05,-^ for dx.
a a
(5) lWiax + mdx,-nsel6. ^«,.1[-1 + -1^]^^^.
(^j f (Any Polynomial) ^ _ use 16 ; then 8 (6).
^^j J. (Any Polynomial) ^^^^ __ ^^^ jg ^^^^^ m <8, ; but see 12 (ft), 14 (ft).
J (005 + ft)»»
(e) Jaj»(aaj + ft)"»<to, — use 16 if >n > «; use 7 (6) if wi<n ; see also 61-64.
16. ? =— i — r-« ^— 1
(aa? + ft) (ca? + d) ad — be Laoc + 6 ex + dj
NoTBS. (a) f ^^ , — use 16, then 12. Special cases, — see 10 and 16 (6>
-»(aa5 + b)(fix + d)
(M r ^« ^-ff- ^^Id*. (Special case of 16 (o).)
Ma.(aa, + 6) ftJ Laj f/a--i-ftJ *^
(c) r Ax + B <faj-. use 16, then long division, 12.
^^^ f (Any Polynomial) ^^ ^^^^ jg^ ^^^ 1^^^ division, 7, 12.
-» (aa5 + ft)(ca5 + tf)
(«) If a£{ - ftc — 0, 18 can be used.
17. l(Fix,ax + l»dxl ^=liF(-^, -^).^^.
L^ J*^^^ -^ \w — a w — a/ (tt — a)*
NoTss. (a^ Beatatsment:
Put t* for ^1^+^; -^^for«; -^1*- for a« + ft ; -^^. forcto.
flj t* - a «- a (t* - a)> ^
(jx r dx 1_ |. (u-a)"*+n-2 ^^ ^ ^^^ ^^^ g ^
•' «B«(aa5 + 6)"» jm+n-lJ i*"»
/ -v f dx ^ « — g log t» .,v f dx ^ «« — 4 am- 2a«logt» .
^^^ J «B*(aa5 + 6) " ft« * ^ ^ •'a5«(aaj + ft)'" 2ft«
18. f— ^_ = JLtan-i«\/^,if a>0, 6>0. [See 9.]
J ax^ + b y/ab ^ ft
= 1 log^^"^^^,ifg>0, 6<0. [See 10.]
N«™- («)J;i^,-usel8(2ndpart);6 — c. (ft)J-^ — 1^^.
38 STANDARD INTEGRALS [IV, B
19- f-^l^ = :^ log («a;2 + 6).
J ax^ + b 2 a
NOTB8. (a) J , — use loDg division, then 18.
(ft) fil^iL^ da), — use 18, 19.
. , . r(-A-ny Polynomial) ,
(c) J aa^ + ft ^05, — use long division, then 18, 19.
20. 1 = 1 r m'^ _ mx — n n ^
^^""- ^^^ hmx + nnaa^ + b) ^'"^ -^*« ^0. then 12, 18, 19.
(&) ^<g«-l--Bia;-K7 _A 1 . -» ^ , /^ ^ft^»\ 1
(tmc + w)(oa5« + 6)^amaj + n maa5» + 6\ a m) (mx+n)(cM*+b) '
, . - Any Polynomial _ . ,. . . , , ^
^^^ J (n»fl; + 7i)(q<p« + fr) '^'*'' ~ "*® ^^"^ division, then 20 6, 12. 18, 20 a.
21. a«2 + 6a; + c = araJ + -^T-^i=-l«^.
L 2aJ 4a
(c) J ^^^^^^Tg^ ^'^» — ^<>"8^ division, then 7, 21, 21 &, 18, and 19.
/.,v r Any Polynomial ,
K"') J — Any Cubic — ' ~ ^°^ division, then find one real &ctor of cubic, then use
81, 21 ft. [If the cubic has a double factor, set u = that ikctor, then use 17 c]
22 f ^^ 1 1
J (0x2 + 6)2 2aaic2 + 6'
23 f ^^ — ^ ■ 1 r da; ., ,j>
* J (ax2 + 6)2 2 6(ax2 + 6)'^26Jaa;2 + 6' ^''®'' ^^•
^' L-f^ = r^ f-l > tl^«^ 7 or 8.
25. f ^ =_- i g_ ■ 2m-8 r (fo;
J (ax^+b)^ 2 6(w-l) (ax2 + 6)«»-i "^ 2(m - 1)6 J {ax^+b)'^-i
Notes, (a) Use 25 repeatedly to reach 23 and thence 18.
(6) Final forms in partial firaction reduction are of types 12, 24, 26 (by use of 21)
IV. C] IRRATIONAL ALGEBRAIC 39
C. (a) Integrand Irrational : involving r — Vox + h.
26. r {Fix, y/ax + h) dx\ ^ ' =z i F (^"-^ ^ r]"^ dr. *
27. i Vaac + 6cte= \t— dT = -—i*^ r = Vax + 6.
•7 J a 3a
28. J.V5^T5d. = AJ(r*-5^)dr = ^[5-|].
29. f ^ =gfdr = gr.
30. r ^_ = f-M?L;use9orlO.
-^xy/ax+b -^r^-^
31. f ^^- = 2 a f /^ ^ ; use 23.
Note, ^/ax + 6 = (ooj + b)/Vax + ^ ; (Vaa5 + 6)« - (flO) + 6) V</a + b.
(b) Integrand Irrational : involving V±x^ ± a'^.
32. f-=_^_=:zz = arc8in-=sin-i^=-cos-i-+ [const.]-
33. f _ -<^ap ^ ^ logCa? + >/i2±a2) = sinhi^ [+ const.] for +,
or cosh~i - [+ const.] for — .
a
34. f ^^ = sin-i f ?^l^^ = - cos-i ^-^I^-^ [ + const.]
= vers-^ (x/a) + const.
f — ^ = Isec-i^ = -cos-ig =- Icac-ig r+ const.].
•7 '*'. •v/'Ka _ /,2 a a a jc a a
35
36 f ^^^ =-v^g^«. 38. fa;V^^^dx=::^(>/S2:=^2)8.
• J \/a2-x2 -^ 3
37. f /^^^ =Vx2 4-qa. 39. fa Va2 + a^da; = J {y/x^ + a2)^
Notes, (a) 82 and 88 Airnlsh the basis for all which follow.
{V) 86, 87, 88, 89 follow from xdx^di9^ + const.) /2.
40 STANDARD INTEGRALS [IV, C
40. f_^^=-?V523:T3 + «?8in-i?.
41. f ^^ =.llogp + ^^^^^^1.
•^a;Va2±«2 a L a; J
42. f ^^ =--I,Va2^xg.
43. (a) I Va2-a;2(fa=?Va«-««+ — 8in-i^.
•^ 2 2 a
(6) (^^^^i^dx = V5n:F2 - a log « +^^:^.
^ X X
•^ x2 X a
44 f g^^ =^Vg«4- a«T^logrx + Va^rfca«).
•^ Vx2 ± a2 2 2
46. f dx ^^V^2^a2^
46. (a) f Vx2 ± a2dx = ? Vx? ± a^ ± ^ log (x + Vx2 ± o^).
(6) r Vx2 j:aa ^ ^ Vx2 ± a2 ± a^ C — ^ then 35 or 41.
^ « J a; Vx2 ± a2
(c) fvgZgdx = -^'^^^^Vf-^^^,theD32or38.
^ x2 X ^ Vx2 ± a2
. f /^ = ? 48. f
•^ f Vo2 - x2^» a2 Va2 - x2 -^
dx _ ±«
( Vo2 - x2)» a2 Va2 — x2 •^(Vx2ia2)» a« Vx» db a*
NoTKS. 7V*t0ronofn0^ic Subsiitutiona. If the desired form is not found in 82-^, try
79. Then use Nos. 65-79, see 79. {h) See also D 51-54, below.
49. V±(ax2 + 6x + c) = Va V±w2 ± A?, where
u = x + Aandfc2 = &i=l«£.
2a 4a2
IV, E] REDUCTION FORMULAS 41
50 -x l^^ "^ ^ = , ax + b y/(ax + b)icx-{-d) ^
^cx + d"" V(aa; + 6)(ca; + d) .cx + d
NoTBS. (a) Integrals containing V{ax + b)/{cx + d) : use 50, then 49, then 82-48.
(5) Sabstttntion of u B'\/(ax + b)/(cx + d) is successftd without 50,
D. Integrals of Binomial Differentials — Reduction Formulas.
Symbols : u = ax* + b; a, 6, p, m, n, any numbers for which no de.
nominator in the formula vanishes.
51. (x^(ax^-\-h)Pdx= [7^-^Hp-{-npb xx^u^-^dx].
J m-\- np -\-l J
= ^ [— ic«+iwi'+i 4- (m + TO + np + 1) (x^uP-^^ doc],
bn(p + 1) J
53. (af^{ax:^ + b)Pdx
= = [a:«+iwP+i — a(m -{■ n + np + 1) \ x'^-^^uP doc].
{m-{-\)b J
54. (x^{ax^ + h)P dx
a(m + wp + 1) J
Notes, (a) These reduotion form.ula8 useflil when p, m, or n are firaotiotial ;
hence applications to Irrational Integrands.
(6) Bepeated application may reduce to one of 82-48.
(o) Do not apply if p, m, n, are all integral, unless n ^2 andp large. Note 11, 15, 17-2£^
Ea. Integrand Transcendental : Trigonometric Functions.
55. Isin X dx = — cos a?.
56. ( sin2 xdx=i—\ cos a; sin x + J » = — J sin 2 a; + J a;.
Note. / sin* fepcfoj, — set kx =» m, and use 56. Likewise in 55-78.
67. Uwzdx = - «'""-' '^oo^^ + a^il f gjp^a x(fa...
Note. If » is odd, put sin« aj = 1 — cos« x and use 62.
42 STANDARD INTEGRALS [IV, g
58. Ccos X dx = sin x*
6Q. ( coB^xdx = i sin xcos x + i ^ = i sin 2 x + |x.
60. fco8»ic<te = 52«n£«nf + «nlfco8»-Sxd«.
J n n J
NoTB. If n is odd, put co8*(0 >■ 1 — 8in*fl9 and use 63.
61. Isin xcosxdx=— i cos 2x = i sin^ a; [ .+ const.].
62. f8ina;cos»a:dx=-^^?^,n:?fc-l.
J n + 1
63. rsin»xcosa;dx = ^^^^^ — -,71=^—1.
J n+1
64. f sin" X cos- X cte = sin-+i x cos >»-i x ^ m-^1 T .^, ^ ^^^^.^ ^ ^^
J m + n TO + n./
— sin*-* X C0S"»+1X , W — 1 r„:„n-2« «A.ei»/^y7^
= 1 I sin*^^ X cos"* xox.
w + n TO+ n J
NoTK. If n is an odd Integer, set sin* a; « 1 — cos> ce and use 62. If m is odd, use 63.
i»e C ' f \ / \j cosr(m4-n)xl cosfCm — n)x]
66. i sin (mx) cos (nx) dx = ^^ — -^^ — -^ ;r^ r-^»
J ^ ' ^ ^ 2(w + n) 2(m - n)
wi T^ db ««
66. rsin(mx)sin(nx)dx = ^^"i:(^"^)^] -- '^^"J/'^+^^^^^
J ^ ^ ^ ; 2(m-n) 2(w + n)
67. rcos(mx)cos(nx)dx= »^"C(^^ ^>^3 + "^°£<"* + ")^3 , m:^±»».
68. I tanx(Zx=— logcosx. 69. ftan2xdx = tanx — x.
70. ('tan»xdx = ^^"**~^ ^ - f Un*-2xdx.
J n — 1 J
71. fctnxdx = logsinx. 72. J ctn«x(ix =— ctnx— SB.
73. rctn»xdx=-^i?^^^-fctnn-2xdx,
*; n— 1 ./
74» Jsecxifo =: Iqgtanf | + ?Uj = log(8ecx + tanx)[+con8t.].
IV, E]
TRIGONOMETRIC
43
75. Tcsc xdx =\og tan | = — log (esc x + ctn x) [ + const. ].
76.1
sec^xdx = tanx.
■ I
77. I csc^ xdx = — ctn x.
78
. ( sec"*
xcac*^xdx
J si
dx
sin* X cos"* 05
(See also 64.)
dx
= — = — sec"*^i X csc«-^ X + ^"^^ — = f sec"*-2 x esc* x
wi — 1 w — 1 J
= = — sec"*-! X cse**-i x -|- ^"^^ ~" rsec"» x csc**^ xdx
n— 1 n— 1 J
NoTxs. (a) In 64 and 78 and many others, m and n may have negative valaes.
(&) To reduce J[sin*CB/co8"*CB]<?(D take m negative in 64.
(o) To reduce /[cos"»0/sin** x] cte take n negative in 64.
79. Substitutions:
0)
(2)
(8)
(4)
(ft)
u —
since
costs
tana?
sec CD
(0
tan —
2
du
cos (0 (f {0
— Ainwdoi
sec* £D ef (0
sec CD tan (to
-8ec«-(fflJ
811100
«
Vl - U«
u
Vl + w«
Vw« - 1
u
2tf
1 +M«
COS 00
>/l -««
-u
Vi + ws
1_
14
1 -!<»
l + tf*
tanas
7*
V7^
u*
Vi-
M«
u
t»
Vwa-
-1
2u
1 -M«
S0
sin-* tt
cos-i f*
tan"*!*
sec-*t*
2 tan-* 1*
cto
(fu
Vi -i*«
dfM
Vi - t*«
du
1+u*
du
t*Vtt«- 1
2<gu
Replace ctn x, sec x, esc x by 1/tan x, 1/cos x, 1/sin x, respectively.
Notes, (a) J /'(sin 0) cos cb ef «, — use 79, (1).
(b) / ^(cos aj) sin (to, — use 79, (2).
(0) / ^(tan x) sec« 85 efoj, — use 79, (8) .
id) Inspection of this table shows deHrahle aubatituUona from trigonometric to
algebraic, and conversely. Thus, if only tan (0, sin* x, cos* x appear, use 79, (8).
44 STANDARD INTEGRALS [TV, E
80. f ^ = ^ gin-ifc + asinx jf „*> j^
..^ a + 6 sin X Va' — 6* a + 6 ain a;
= _i_ log &>V6^^^ + atan(V2)^ .^^,^ ^
y/l^-a^ 5 + V62 _ ^a + o tan (a;/2)
81. f ^— = i_ tan-i rJ^EItaDq, a^>6^;
J a + & cos a: v^2 _ 52 L ^a + 6 2 J
V62 - a2 V6 + a -V6-atan (a;/2) '
82. f . ^ = 1 logtan^ilg. « = sin-i ^ ■
y a sm X + 6 cos X VaM-~6^ ^ Vo^ + 6*
Notes, (a) J — . . .^ , — use 79, (1), = ^ log (» + ^ sin »)•
* a + 2>siii(D ^ a + sinoi b ^ b •' a+6 sin aj'
then use 82 a, 80.
(c) Many others similar to (a) and (6) ; e.g, /[sin cp/(a + & cos a)] rfa>, — use 79, (2).
{d) r . , , — -^z T— and like forms, — nse 79, (8) ; see 79, note d.
^ a* sin* x + b* cos« x » v / »
(«) As last resort, ase 79, (5), for any rational trigonometric integral.
Hb' Integrand Transcendental : Trigonometxic-Algebraio.
83. f x« sin ic daj = — a?» cos a; + wi f a:«-i cos x dx,
84. j a^ cos 05 (to = a?» sin a; — TO j a:"»-i sin x dx.
Notes, (a) JccsintB^to*? — (BcoscD + Jcos^Jcto, — use 58.
(6) /a?* sin xdx^ — repeat 88 to reach 68.
(e) / (Any Polynomial) sin x dx, — split up and use 88.
(d) For cos 85 repeat (o), (6), (c).
Q- fsin X dx — sin X ,1 f cos x ^^ ^ _j. 1
J X~ (to — 1) X*»-l TO — 1 •/ X™-1
gg rcoaxdx^ C08X l_Cmidx,m=^l.
J X" (m—1) x"-! m — 1 J x"*-!
IV. E] TRANSCENDENTAL 45
87. C^Mdx= f fl - ^ + ^ "I dx ; see II, B, 13, p. 8,
88. (^^^dx = f rl - £. + ^ "1 dx ; see II, B, 14, p. 8.
J X J La5 2 1 4 1 J
NoTS. Other trigonometrie-algebraic combinations, use 5 ; or 79 followed by 89-84.
Be> Integrand Transcendental : Inverse Trlgonometrio.
89. Isin-i xdx=:x sin-i x + Vl^^. [From 6.]
90. Tcos-i xdx = x cos-i x — Vl — x^,
91. jtan-i xdx = x tan-i x — J log (1 + x*).
92. (xn 8in-i xdx = ?:!li^ll^ - JL f ^!ii^ , then 53 or 64, 32, 36.
J n + 1 n-\-lJ VrZ^
93. rx»co8-ixdx = ^'*^'^^«''^ + ^- f^::li^,then63or64,32,36.
J n + 1 w+1 -^ Vl— x^
94. Cxntaa-ixdx = ^^'^^"'^ L_ f?!!li^, then 19 (c).
J n + 1 n + l-Jl+a;* ^'^
NoTKS. (a) Replace ctn-ioj by ^ - tan-ijp ; or by tan"* (l/oj) and Bubstltate l/oj — 1»,
(fj) Replace sec'ia by cos~i(l/a5), c8C~iqj by 8ln~i(l/aj) and substltate \/x = u.
(c) /(Any Polynomial) sin~icpr7aj, split up and use 92. (Similarly for cos'ijo, etc.)
id) i/(^) 8in-»fl8<ffl8, - use (5) with u — sln"»a). (Similarly for co8~ia and tan-i<n.)
(«) Other Inverse Trigonometric Integrands, use 79 or 5*
Hd' Integrand Transcendental : Exponential and Logarithmic
96. (a^dx = -^ = -f^ logio e = rf^ 0.4848.
J lege a logio<^ logio^
96. (€^dx = e».
Notes, (a) / «*»rfa5 -= «»* -^ *. (5) Notice a* - 6(lo».«)» - e**, * - log, a.
97. fx^e** (Zx = - x»e*» - - f x*»-ic*' dx.
^ k kJ
NoTKS. (a) J «■€*'(/» = a?e**/* - e**/*«. (b) ^x'^e^dx, —repeat 97 to reach 97 Ca)
(c) /(Any Polynomial) e^'dx, split up and use 97.
46
STANDARD INTEGRALS
[IV, E
98. (—dx = — + ^ (- — dx (repeat to reach 99).
J x"» (m — 1) a;"»~i w — 1 ^ a;"*"^
99. C^dx =("[- + 1 + — 4- — + •••! <^tt, u = kx; see ra6Ze«, V, H
J X J Lu 2131 J
Bmnxdx = e^
k sin nx— n cos fix
100. fe**8i_. - ,., ^
J *2 + n2
102. I logxdx = a; logx — jc.
103. f(logx)-^ = il2£^i:±\ «=jt_l.
J X n + 1
104. C^= ffMM^ ti = logx; see 99 and ro6Zc«, V, H.
105. f X" log X dx = x"+i ri?l^ 1 1 .
J Lw + 1 (n + l)2j
106. f e** logx(?x = - e** logx - i f — dx, see 99.
J k kJ X
F. Some Important Definite Integrals.
•«cfa5 1
108. P-
JD a'
a?»» f»— 1
* dx H
, if m > 1 (otherwise non-existent).
2 + 62x2 2ab
109. f *a5»»6-»cfa5 = r(n + 1) = n I if n is integral. See V, F, p. 56.
NoTjBS. (a) In general, r (n + 1 ) = « • r (n), as for n I, If »> 0.
(6) r(2) = r(i) = i, r(i/2) = v^. r(« + i)-n(»).
110. fx-(l-x)ndx = i:(!?L±I)iX!L±i}.
J» r(m + n + 2)
■X'
111. i sinnx-Hinmxdx
/•n
cosna!;co8nia;cfiz;:=0, if 9f»^n,
if m and n are integral.
112. J sin2 nx dx=\ em^nxdx = ir/2 ; n integral, see 56, 59.
tV, G] DEFINITE INTEGRALS 47
3. C^e-'^dx^l/k. 114. P[(8in nx)/aj] da;= ir/2.
5. f *6-** sin nx dx = n/(k^ + w^) , if A; > 0.
6. Pe-** cos mx dx = k/(Jc^ + wi^) , if A; > 0.
7. f *e-*'x»d« = ^^^t^^ = -^ , if n is integral. See 109.
Jo A;'*+^ A;*+i
,8. Pe-*»'Vx = v^/(2*).
. i e-*^* cos mx (Zx = -— , if A; > 0.
J.) 2 A5
20. r_l^?_=r--^^- = -^. 121. C(\o^xydx = {-\Yn\
Jo efct+g-te Jo cosiiA-x 2 k Jo ^ '^ ^ ^ ^
;. r ''^^iog sin xdx= \ ^log cos xdx = — ^ log 2.
Jo Jo 2
23. i 8in2«+i xdx= \ cos2»*+i xdx=i „ ^ „ — — - ( n, positive
J. J» 3.6.7...(2»+l)\j^^^^^
24. f "''sin''- a;<te = ("'^os^xdx = ^I'^/'J^l'^^ H (»- P<»i«™
^ Jo 2.4.6...2» 2 j^j^^^^
Q. Approzimation Formulas.
• ) f (^') ^x = / (c) (6 — a), a < c < 6, [Law of the Mean.]
if
25
26
28
. (\f(x)dx = /W+/(^) (6 - a). [Trapezoid Rule— precise
for a straight line.]
27. I f(x)dx. lExtended Trapezoid Eule.']
[/(a)/2+/(a+Ax)+/(a+2Ax)+ ••• 4-/[a4-(w-l)Ax]+/(&)/2]Ax.
. Cfi^)dx=iM±im±±3m±mi^i,^a).
[Prlamold Rule ; or second Simpson- Lagrange ^.p^roximaXion ; precise
/(x) is any quadratic or cubic ; see § 124, p. 202.]
.29. £f(x)dx = ^ If (a) + 4/(a + Ax) + 2/(a + 2 Ax)
+ 4/(a + 3 Ax)H-2/(a 4-4 Ax)+ - +/(&)].
[Blmpson^B Rule ; or extended prismoid rule. Ex. 13, p. 207.]
48 STANDARD* INTEGRALS [IV, G
130. Cf(z)dx
[A third Simpson-Lagrange Approximation. Extend as in 129.]
131. f /(») (to
[A fourth Simpson-Lagrange Approximation ; see Lagrange interpola-
tion formula, n, I, 17, p. i5.]
H. Standard AppllcationB of Integratdon.
132. Areas of Plane Figures : id A.
ia\ Strips AA parallel to 2^-axls : dA — y dx.
(2>) Strips ^A parallel to avaxls : dA — x dy, I
(c) Rectangles ^A =^^ix^y:dA = dxdy, A = Jfdxdy,
(d) Parameter form of equation : A =(1/2) / {x dy. — y dx).
(e) Polar sectors bounded by radii : dA = (p'/2) d9.
(/) Polar rectangles 6.A '^ p^p^9: dA^pdpd9\ A "UpdpdB,
133. Lengths of Plane Curves : i ds,
(a) Equation in form y^f(x): d% = ^/l +[/'(a5)l«cte.
(6) Equation in form x = 4>(y) : ds = Vl +l</>'(y)P<^y.
(c) Parameter equations : ds —y/d^ + <fy*.
(tf) Polar equation : tf« =■ Vd^"+p«^^.
134. Volumes of Solids : (dV.
(a) Frustum (area of cross section A): dV'* Adh\ V^jAdh where h is the variable
height perpendicular to the cross section A,
(b) Solid of revolution about aj-axis : dV^ny*dx.
(c) Solid of revolution about y-axis : <i 7 =» iraj* dy.
(d) Bectangular coordinate divisions :dV=dxdyd0\
V'= J^dzdydx= ^sdydx== S{Jzdy}dx — jAd».
(e) Polar coordinate divisions ; dV^p^sinBdpd^ dB,
rsr,H] APPLICATIONS 49
135. Area of a Surface : \ f sec + dx dp,
where ^ is the angle between the element ds of the surface and its pro-
jection dxdy,
(a) Surface of Bevolation about avaxis : A— j 2 try da.
(6) Sorfoce of Bevolution about y-axis : A—l^rrxds.
136. Length of twisted arcs : i ds.
(a) Bectaugalar Coordinates : ds =« y/d«^ + dy* + rf««.
(&) Explicit Equations y =/ (a?) , » « {x) : rf« = v^l + [/' (»)]* + [*' (a)]*.
(c) a-/(0, y - 0(0, « = '/'W : <?« = v^[/'(0?+[<^'(0]«+['/''(OP.
(c?) Polar Coordinates : d« = Vrf p« + pSci^* + p* cos* citf».
137. Mass of a body : M = \dM = fp cf F,
where p is the density (mass per unit volume).
(a) If p is constant : Jtf"™ p / d F; see 184.
(jb) On any curve : c? F= rf«, if p = mass per unit length.
(c) On any surface (or plane) : dV= dA^ if p » mass per unit area.
138. Average value of a variable quantity q : A. V* of q* :
(a) throughout a solid : q =»/(», y, «) ; A.V.otq.==^q dV-i- Jd V.
(5) on an area ^ : A. V. of q. — ^qdA-i- J dA.
(c)onanarc«: A.V»otq.^ jqds-^ ^ds.
139. Center of Mass, (x, y, i) : sc= i x dM -i- i dM,
with similar formulas for y and z. See dM, 137.
{a) for a volume : <?if ■= p c? F.
(6) for an area : dM^ p cf ul.
(c) for an arc : dM» p da.
139.* Theorems of Papi)«« or Guldin :
(a) Surface generated by an arc of a plane curve revolved about an
axis in its plane = length of arc x length of path of center of mass of arc.
(6) Volume generated by revolving a closed plane, contour about an
axis in its plane = area of contour x length of path of its center of mass.
140. Moment of IneHia : I = f r2 dM. (See 137, 139.)
{a) For plane figures, Tx + Iy^Io, where /«, ly, 1q are taken about the te-azis, the
y-axis, the origin, respectively.
(b) For space figures, /« + /y + /«= Tq.
(c) /jp = /- -f (05 — fiB)«Jf, where /^ is taken about a line || to the aj-axis.
50 STANDARD INTEGRALS [IV, H
141. Badim of Gyration : k^^I-h M= f r8 dM -s- (dM.
[In 140 and 141, r may be the distance flroro some fixed point, or line, or plane.]
142. Liquid pressure : P= iph dA,
where P is the total pressure, dA is the elementary strip parallel to the
surface ; h is the depth below the surface ; and p is the weight per unit
volume of the liquid.
143. Center of liquid pressure : h= i h^dA ± KhdA.
144. Work of a variable force : W =\f cos + ds,
where / is the numerical magnitude of the force, ds is the element of the
arc of the path, and yp is the angle between / and ds.
145. Attraction exerted by a solid : F=: k f ^^^ ^
where k \a the attraction between two unit masses at unit distance, m is
the attracted particle, dM is an element of the attracting body ; r is the
distance from m to dM.
Components Fg, Fy. F, of F along Ox, Oy, Oz are :
^,.t™J?2i±l^, F,-kmf-!!^^, j.._fcm;22i^,
where a, ^, y are the direction angles of a line joining m to dJf.
146. Work in an expanding gas : W = \p dv»
147. Distance «, speed r, tangential acceleration jr*
Jr = I V cZ< = i I isdtX dt.
[Similar forms for angular speed and acceleration.]
148. Errors of observation :
y (f CD, where y ia the
4,.w„^„— ~-^„^, ^. w- .— ^ — . *~*
(&) The usual formula y - (A/Vw) «-*"** gives: P- (A/Vw) J «-**** cfoj, where h
is the so-called measure of precision.
(c) Probability of an error between x^— a and a5-«+a; P(a)— I y dx.
(d) Probable error = (0.477) /A =- value of a for which P(a) — 1/2.
(fl) Mean error ^ \ «yd<r+ I y tf » =- 1 / ( An/w )
Jo Jo
V. NUMERICAL TABLES
A. TRIGONOMETRIC FUNCTIONS
[Characteristics nf Logarithms omitted — determine by the usual rule from the value]
Radians
De-
Sine
Tangbnt
Cotangent
Cosine
grees
Value logio
Value logio
Value
logio
Value
logio
0000
0°
.0000
— CO
.0000 -00
00
00
1.0000
0000
90°
1.5708
.0175
1°
.0176
2419
.0175 2419
57.290
7581
.9998
9999
89°
1.5533
.0349
2°
.0349
5428
.0349 5431
28.636
4569
.9994
9997
88°
1.5359
.0524
3°
.0523
7188
.0524 7194
19.081
2806
.9986
9994
87°
1.5184
.0698
4°
.0698
8436
.0699 8446
14.301
1564
.9976
9989
86°
1.5010
.0873
5°
.0872
9403
.0876 9420
11.430
0580
.9962
9983
85°
1.4835
.1047
6°
.1046
0192
.1051 0216
9.5144
9784
.9945
9976
84°
1.4661
.1222
7°
.1219
0869
.1228 0891
8.1443
9109
.9926
9968
"83°
1.4486
.1396
.8°
.1392
1436
.1405 1478
7.1154
8522
.i)903
9958
82°
1.4312
.1571
9°
.1564
1943
.1684 1997
6.3138
8003
.9877
9946
81°
1.4137
.1746
10°
.1736
2397
.1763 2463
5.6713
7537
.9848
9934
80°
1.3963
.1920
11°
.1908
2806
.1944 2887
5.1446
7113
.9816
9919
79°
1.3788
.2094
12°
.2079
3179
.2126 3276
4.7046
6726
.9781
9904
78°
1.3614
.2269
13°
.2250
3521
.2309 3634
4.3315
61366
.9744
9887
77°
1.3439
.2443
14°
.2419
3837
.2493 3968
4.0108
6032
.9703
9869
76°
1.3266
.2618
15°
.2688
4130
.2679 4281
3.7321
5719
.9659
9849
75°
1.3090
.2793
16°
.2756
4403
.2867 4576
3.4874
5425
.9613
9828
74°
1.2916
.2967
17°
.2924
4669
.3057 4853
3.2709
5147
.9563
9806
73°
1.2741
.3142
18°
.3090
4900
.3249 5118
3.0777
4882
.9511
9782
72°
1.25(K5
.3316
19°
.3256
6126
.3443 5370
2.9042
4630
.9455
9757
71°
1.2392
.3491
20°
.3420
6341
.3640 5611
2.7475
4380
.9397
9730
70°
1.2217
.3665
21°
.3584
5643
.3839 5842
2.6051
4158
.9336
9702
69°
1.2043
.3840
22°
.3746
5736
.4040 6064
2.4751
3936
.9272
9672
68°
1.1868
.4014
23°
.3907
5919
.4245 6279
2.3559
3721
.9205
9640
67°
1.1694
.4189
24°
.4067
6093
.4452 6486
2.2460
3514
.9135
9607
66°
1.1519
.4363
25°
.4226
6269
.4663 6687
2.1445
3313
.9063
9573
65°
1.1345
.4538
26°
.4384
6418
.4877 6882
2.0503
3118
.8988
9537
64°
1.1170
.4712
27°
.4540
6570
.5096 7072
1.9626
2f)28
.8910
9499
63°
1.0996
.4887
28°
.46i*5
6716
.5317 7257
1.8807
2743
.8829
9459
62°
1.0821
.6061
29°
.4848
6856
.6543 7438
1.8040
2562
.8746
9418
61°
1.0647
.5236
30°
.5000
6990
.5774 7614
1.7321
2386
.8660
9375
60°
1.0472
.5411
31°
.5150
7118
.6009 7788
1.6643
2212
.8572
9331
69°
1.0297
.5585
32°
.5299
7242
.6249 7958
1.6003
2042
.8480
928i
58°
1.0123
.5760
33°
.5446
7361
.6494 8125
1.5399
1875
.8387
9236
67°
.9948
.5934
34°
.5592
7476
.6745 8290
1.4826
1710
.8290
9186
66°
.9774
.6109
a5°
.6736
7586
.7002 8452
1.4281
1548
.8192
9134
55°
.9599
.6283
36°
.5878
7692
.7265 8613
1.3764
1387
.80^)0
9080
54°
.9425
.6468
37°
.6018
7796
.7536 8771
1.3270
1229
.7986
9023
53°
.9250
.6632
38°
.6157
7893
.7813 8928
1.2799
1072
.7880
8965
52°
.9076
.6807
39°
.6293
7989
.8098 9084
1.2349
0916
.7771
8905
51°
.8901
.6981
40°
.6428
8081
.8391 9238
1.1918
0762
.7660
8843
50°
.8727
.7166
41°
.6561
8169
.8693 9392
1.1504
0608
.7547
8778
49°
.8552
.7330
42°
.6691
8265
.9004 9544
1.1106
0456
.7431
8711
48°
.8378
.7606
43°
.6820
8338
.9326 9697
1.0724
0303
.7314
8641
47°
.8203
.7679
44°
.6947
8418
.9667 9848
1.0356
0152
.7193
8569
46°
.8029
.7864
45°
.7071
8495
1.0000 0000
1.0000
0000
.7071
8495
45°
.7854
Value
logio
Value lopjo
Value
loPio
Value
logic
De-
Radians
COSINR
Cotangent
Tangent
Sine |
grees
51
52
NUMERICAL TABLES
[V, B
B. COMMON LOGARITHMS
N
1
2
3
4
5
6
7
8
9
I>
10
0000
0O43
0086
0128
0170
0212
0253
0294
0334
0374
42
11
0414
0463
0492
0631
0569
0607
0645
0682
0719
0765
38
12
0792
0828
0864
0899
0934
0969
1004
1038
1072
1106
35
13
1139
1173
1206
1239
1271
1303
1336
1367
1399
1430
32
14
1461
1492
1523
1553
1584
1614
1644
1673
1703
1732
30
15
1761
1790
1818
1847
1876
1903
1931
1959
1987
2014
28
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
26
17
2304
2330
2365
2380
2405
2430
2466
2480
2504
2529
25
18
2563
2577
2601
2625
2648
2672
2696
2718
2742
2765
24
19
2788
2810
2833
2856
2878
2900
2923
2945
2967
2989
22
20
3010
3032
3064
3075
3096
3118
3139
3160
3181
3201
21
21
3222
3243
3263
3284
3304
3324
3345
3365
3385
3404
20
22
3424
3444
3464
3483
3602
3622
3541
3660
3579
3698
19
23
3617
3636
3655
3674
3692
3711
3729
3747
3766
3784
18
24
3802
3820
3838
3856
3874
3892
3909
3927
3945
3962
18
25
3979
3997
4014
4031
4048
4066
4082
4099
4116
4133
17
26
4150
4166
4183
4200
4216
4232
4249
4265
4281
4298
16
27
4314
4330
4346
4362
4378
4393
4409
4425
4440
4456
16
28
4472
4487
4502
4518
4533
4648
4664
4579
4694
4609
15
29
4624
4639
4654
4669
4683
4698
4713
4728
4742
4757
15
30
4771
4786
4800
4814
4829
4843
4867
4871
4886
4900
14
31
4914
4928
4942
4955
4969
4983
4997
5011
6024
5038
14
32
5051
5065
5079
5092
5105
6119
6132
5145
5169
5172
13
33
6185
5198
5211
6224
5237
5250
6263
5276
5289
5302
13
34
6316
5328
5340
5363
5366
6378
6391
6403
5416
5428
13
35
5441
5453
5465
5478
5490
6502
5514
5527
6539
5651
12
36
5663
5575
5687
5699
5611
5623
6635
6647
5658
5670
12
37
6682
5694
5705
5717
5729
6740
5762
5763
6775
5786
12
38
6798
5809
5821
5832
5843
6866
5866
5877
5888
5899
11
39
6911
5922
5a33
5944
5965
5966
5977
5988
5999
6010
11
40
6021
6031
6042
6053
6064
60Z5
6085
6096
6107
6117
11
41
6128
6138
6149
6160
6170
6180
6191
6201
6212
6222
10
42
6232
6243
6263
6263
6274
6284
6294
6304
6314
6325
10
43
6335
6345
6365
6365
6375
6385
6395
6406
6415
6425
10
44
6435
6444
6454
6464
6474
6484
6493
6603
6613
6622
10
45
6632
6542
6551
6561
6571
6680
6590
6599
6609
■6618
10
46
6628
6637
6646
6666
6665
6675
6684
6693
6702
6712
9
47
6721
6730
6739
6749
6758
6767
6776
6785
6794
6803
9
48
6812
6821
6830
6839
6848
6867
6866
6875
6884
6893
9
49
6902
6911
6920
6928
6937
6946
6965
6964
6972
6981
9
50
6990
6998
7007
7016
7024
7033
7042
7050
7069
7067
9
51
7076
7084
7093
7101
7110
7118
7126
7135
7143
7152
8
52
7160
7168
7177
7185
7193
7202
7210
7218
7226
7235
8
53
7243
7251
7269
7267
7275
7284
7292
7300
7308
7316
8
54
7324
7332
7340
7348
7356
7364
7372
7380
•
7388
7396
8
V.B]
COMMON LOGARITHMS
53
N
1
2
3
4
5
6
7
8
9
D
8
55
7404 .
7412
7419
7427
7435
7443
7451
7459
7466
7474
56
7482
7490
7497
7505
7513
7520
7528
7536
7543
7551
8
67
7559
7566
7574
7582
7589
7597
7604
7612
7619
7627
8
58
7634
7642
7649
7657
76(>4
7672
7679
7686
7694
7701
7
59
7709
7716
7723
7731
7738
7745
7752
7760
7767
7774
7
60
7782
7789
7796
7803
7^10
7818
7825
7832
7839
7846
7
61
7863
7860
7868
7875
7882
7889
7896
7903
7910
7917
7
62
7924
7931
7938
7945
7952
7959
7966
7973
7980
7987
7
68
7993
8000
8007
8014
8021
8028
8a35
8041
8048
8055
7
64
8062
8069
8075
8082
8089
8096
8102
8109
8116
8122
7
65
8129
8136
8142
8149
8156
8162
8169
8176
8182
8189
7
66
8195
8202
8209
8215
8222
8228
8235
8241
8248
8254
7
67
8261
8267
8274
8280
8287
8293
8299
8306
8312
8319
6
68
8325
8331
8338
8344
8351
8357
8363
8370
8376
8382
6
69
8388
8395
8401
8407
8414
8420
8426
8432
8439
8445
6
70
8461
8457
8463
8470
8476
8482
8488
8494
8500
8506
6
71
8513
8519
8525
8531
8637
8543
8649
8555
8561
8567
6
72
8573
8579
8585
8591
8597
8603
8609
8615
8621
8627
6
78
8633
8639
8645
8651
8657
8663
8669
8675
8681
8686
6
74
8692
8698
8704
8710
8716
8722
8727
8733
•
8739
8745
6
75
8751
8756
8762
8768
8774
8779
8785
8791
8797
8802
6
76
8808
8814
8820
8825
8831
8837
8842
8848
8864
8859
6
77
8865
8871
8876
8882
8887
8893
8899
8904
8910
8915
6
78
8i*21
8927
8932
8938
8943
8949
8954
8960
8965
8971
6
79
8976
8982
8987
8993
8998
9004
9009
9016
9020
9025
5
80
<mi
9036
9042
9047
9053
9058
9063
9069
9074
9079
5
81
9085
9090
9096
9101
9106
9112
9117
9122
9128
9133
6
82
9138
9143
9149
9154
9159
9165
9170
9176
9180
9186
5
88
9191
9196
9201
9206
9212
9217
9222
9227
9232
9238
5
84
9243
9248
9253
9258
9263
9269
9274
9279
9284
9289
5
85
9294
9299
9304
9309
9315
9320
9325
9330
9335
9340
6
86
9345
9350
9355
9360
9365
9370
9375
9380
9386
9390
5
87
9395
9400
9405
9410
9415
9420
9425
9430
9436
9440
5
88
9445
9450
9455
9460
9465
9469
9474
9479
9484
9489
5
89
9494
9499
9504
9509
9513
9518
9523
9528
9533
9538
5
90
9542
9547
9552
9557
9562
9566
9571
9576
9581
9586
5
91
9590
9595
9600
9606
9609
9614
9619
9624
9628
9633
5
92
9638
9643
9647
9652
9657
9661
9666
9671
9675
9680
5
98
9685
9689
9694
9699
9703
9708
9713
9717
9722
9727
5
94
9731
9736
9741
9745
9750
9754
9759
9763
9768
9773
6
95
9777
9782
9786
9791
9795
9800
9805
9809
9814
9818
6
96
9823
9827
9832
9836
9841
9845
9850
9854
9859
9863
6
97
9868
9872
9877
9881
9886
9890
9894
9899
9903
9908
4
98
9912
9917
9921
9926
9930
9934
9939
9943
9948
9952
4
99
9956
9i)61
9965
9969
9974
9978
9983
9987
9991
9996
4
54
NUMERICAL TABLES
[V,C
C. EXPONENTIAI, AHD HYPERBOLIC FUNCTIONS
e*
e-
-z
slnhat;
eoshoo
(0
logeOJ
Value
logio
Value
logio
Value
lo&io
Value
loffio
0.0
— 00
1.000
0.000
1.000
0.000
0.000
— 00
1.000
0.1
-2.303
1.105
0.043
0.905
9.957
0.100
9.001
1.006
0.002
0.2
-1.610
1.221
0.087
0.819
9.913
0.201
9.304
1.020
0.009
0.3
-1.204
1.350
0.130
0.741
9.870
0.305
9.484
1.045
0.019
0.4
-0.916
1.492
0.174
0.670
9.826
0.411
9.614
1.081
0.034
0.5
-0.693
1.649
0.217
0.607
9.783
0.621
9.717
1.128
0.052
0.6
-0.511
1.822
0.261
0.549
9.739
0.637
9.804
1.185
0.074
0.7
-0.367
2.014
0304
0.497
9.696
0.769
9.880
1.255
0.099
0.8
-0.223
2.226
0.347
0.449
9.663
0.888
9.948
1.337
0.126
0.9
-0.106
2.460
0.391
0.407
<f.609
1.027
0.011
1.433
0.156
1.0
0.000
2.718
0.434
0368
9.666
1.175
0.070
1.543
0.188
1.1
0.095
3.004
0.478
0.333
9.622
1.336
0.126
1.669
0.222
1.2
0.182
3.320
0.521
0.301
9.479
1.609
0.179
1.811
0.258
1.3
0.262
3.669
0.566
0.273
9.435
1.698
0.230
1.971
0.295
1.4
0.336
4.055
0.608
0.247
9.392
1.904
0.280
2.151
o.:«3
1.6
0.406
4.482
0.651
0.223
9.349
2.129
0.328
2.an2
0.372
1.6
0.470
4.953
0.695
0.202
9.305
2.376
0.376
2.577
0.411
1.7
0.631
6.474
0.738
0.183
9.262
2.646
0.423
2.828
0.452
1.8
0.588
6.050
0.782
0.165
9.218
2.942
0.469
3.107
0.492
1.9
0.642
6.686
0.826
0.160
9.175
3.268
0.614
3.418
0.534
2.0
0.693
7.389
0.869
0.135
9.131
3.627
0.660
a762
0.576
2.1
0.742
8.166
0.912
0.122
9.088
4.022
0.604
4.144
0.617
2.2
0.788
9.025
0.955
0.111
9.046
4.467
0.649
4.668
0.660
2.3
0.833
9.974
0.999
0.100
9.001
4.937
0.690
5.037
0.702
2.4
0.875
11.02
1.023
0.091
8.958
6.466
0.738
6.657
0.745
2.6
0.916
12.18
1.086
0.082
8.914
6.060
0.782
6.132
0.788
2.6
0.956
13.46
1.129
0.074
8.871
6.695
0.826
6.769
0.831
2.7
0.993
14.88
1.173
0.067
8.827
7.406
0.870
7.473
0.874
2.8
1.030
16.44
1.216
0.061
8.784
8.192
0.913
8.253
0.917
2.9
1.005
18.17
1.259
0.055
8.741
9.060
0.957
9.115
0.960
3.0
1.099
20.09
1.303
0.050
8.697
10.018
1.001
10.068
1.003
3.6
1.253
33.12
1.620
0.03Q
8.480
16.643
1.219
16.673
1.219
4.0
1.386
54.60
1.737
0.018
8.263
27.290
1.436
27.308
1.436
4.5
1.504
90.02
1.964
0.011
8.046
46.003
1.653
45.014
1.653
6.0
1.609
148.4
2.171
0.007
7.829
74.203
1.870
74.210
1.870
6.0
1.792
403.4
2.606
0.002
7.394
201.7
2.305
201.7
2.305
7.0
1.946
1096.6
3.0i0
0.001
6.960
548.3
2.739
648.3
2.739
8.0
2.079
2981.0
3.474
0.000
6.526
1490.6
3.173
1490.5
3.173
9.0
2.197
8103.1
3.909
0.000
6.091
4051.6
3.608
4051.5
3.608
10.0
2.303
22026.
4.343
0.000
6.667
11013.
4.041
11013.
4.041
log<, X. = (logio x)-^M ; M= .4342944819. logi© e*+» = logio e* -r logio ev.
Sinhx and coshaj approach e*/2 as x increases (see Fig. E, p. 2^). The
formula logjo (e'/2) = M -x — logio 2 represents logjo sinh x and logjo cosh x to
three decimal places when a; > 3.5 ; four places when a; > 5 ; to five places when
x>6] to eight places when x > 10.
V.E]
ELLIPTIC INTEGRALS
55
D. VALUES OF
dx
Vl-A:2siii2e '^o V(l-a52)(l-A;2ac2)
[Elliptic Integral of the First Kind.]
a? = sinO
Jb»
<^-5o
<f>^10P
*-16«
<l>=ZOP
<^»45o
<^»60o
«fr-75o
K
-ir/ao
=ir/18
=ir/12
= 7r/6
-ir/4
-ir/8
-6ir/12
«fr-90">
-ir/2
0.0
0.087
0.175
0.262
0.524
0.786
1.047
1.309
1.571
0.1
0.087
0.175
0.262
0.524
0.786
1.049
1.312
1.575
0.2
0.087
0.175
262
0.525
0.789
1.054
1.321
1.588
0.3
0.087
0.176
0.262
0.526
0.792
1.062
1.336
1.610
0.4
0.087
0.175
0.262
0.527
0.798
1.074
1.358
1.643
0.5
0.087
0.175
0.263
0.629
0.804
1.090
1.385
1.686
0.6
0.087
0.175
0.263
0.532
0.814
1.112
1.426
1.762
0.7
0.087
0.175
0.263
0.536
0.826
1.142
1.488
1.854
0.8
0.087
0.176
0.264
0.539
0.839
1.178
1.566
1.993
0.9
0.087
0.175
0.264
0.544
0.858
1.233
1.703
2.275
1.0
0.087
0.175
0.265
0.549
0.881
1.317
2.028
oo
£(*, ♦)
Jo
E. VALUES OF
hi sin^ e d^
Jo
[Elliptic Integral of the Second Kind.]
dx^
X
u
sine
sin^
h^
*=6«>
<^=10«
<^»16o
*=80«
<^-46«
«fr=60«»
*=75"»
JS
=ir/86
-ir/18
=ir/12
= ir/6
= ir/4
= ir/8
-6ir/12
^=90°
= ir/2
0.0
0087
0.176
0.262
0.524
0.786
1.047
1.309
1.571
0.1
0.087
0.176
0.262
0.623
0.785
1.046
1.306
1.566
0.2
0.087
0.174
0.262
0.523
0.782
1.041
1.297
1.554
0.3
0.087
0.174
0.262
0.521
0.779
1.033
1.283
1.533
0.4
0.087
0.174
0.261
0.520
0.773
1.026
1.264
1.504
0.5
0.087
0.174
0.261
0.518
0.767
1.008
1.240
1.467
0.6
0.087
0.174
0.261
0.515
0.759
0.989
1.207
1.417
0.7
0.087
0.174
0.2(i0
0.512
0.748
0.965
1.163
1.351
0.8
0.087
0.174
0.260
0.509
0.737
0.940
1.117
1.278
0.9
0.087
0.174
0.259
0.505
0.723
0.907
1.053
1.173
1.0
0.087
0.174
0.259
0.500
0.707
0.866
0.966
1.000
56
NUMERICAL TABLES
F. VALUSS OF n (p) =T (p + 1) = (e-^xPdSD
Jo
p A PROPER FRACTION
[n fn) =B r (n + 1) a= n I, if n is a positive integer.]
[V.I
1>?»0.0
p-0.1
p-0.2
p-0.8
p-0.4
1>»0.5
p»0.6
p=0.7
p^O.S
p=M
0.9G2
r(p+i)=
1.000
0.951
0.918
0.897
0.887
0.886=v^/2
0.894
0.909
0.931
r (* + 1) = * r (*) , if ifc > ; hence T(k + 1) can be calculated at intervals of 0.1.
Minimum value of r(p + 1) is .88560 atp = .46163.
6. VALUES OF THE PROBABILITY INTEGRAL:
Vi
Jo
e-^^dXi
X
.0
.1
.2
.8
.4
.6
.6
.7
.8
.9
0.
1.
2.
.0000
.8427
.9953
.1125
.8802
.9970
.2227
.9103
.9981
.3286
.9340
.9989
.4284
.9523
.9993
.5205
.9661
.9996
.6039
.9763
.9998
.6778
.9838
.9999
.7421
.9891
.99911
.7969
.9928
1.0000
H. VALUES OF THE INTEGRAL C
[Note break at a; = 0.]
^^dx
<x> (C
n»l
n«2
ra=s8
n»4
n»6
n»6
n=7
n=8
n^i
X =*— n ♦
a;=— n/10
-.2194
-1.823
-.0489
— 1.223
-.0130
-.9057
-.0038
-.7024
-.0012
-.6598
-.0004
-.4544
-.0001
— .3738
— .0000
-.3106
-.0000
- .wa
a;=+n/10
x=4-n
— 1.623
1.895
- .8218
4.954
-.3027
9.934
+ .1048
19.63
.4542
40.18
.7699
85.99
1.065
191.5
1.347
440.4
1.623
1038
•Note
'« dx
-00
e*dx
=—00. Values on each side of a = can be used safely.
\ -^^ and f — dx reduce to the integral here tabulated ; see IV, 99, 104, p. 46.
Jo log* J z*
V.I]
RECIPROCALS SQUARES CUBES
57
Ii. RECIPROCALS OF NUMBERS FROM 1 TO 9.9
1
.0
.1
.2
.8
.4
.6
.6
.7
.8
.9
1.000
0.909
0.833
0.769
0.714
0.667
0.625
0.588
0.556
0.526
2
0.500
0.476
0.465
0.435
0.417
0.400
0.385
0.370
0.357
0.345
3
0.333
0.323
0.313
0.303
0.294
0.286
0.278
0.270
0.263
0.256
4
0.250
0.244
0.238
0.233
0.227
0.222
0.217
0.213
0.208
0.204
5
0.!»0
0.196
0.192
0.189
0.185
0.183
0179
0.175
o.m
0.169
6
0.167
0.164
0.161
0.159
0.156
0.154
0.152
0.149
0.147
0.145
7
0.143
0.141
0.139
0.137
0.136
0.133
0.132
0.130
0.128
0.127
8
0.125
0.123
0.122
0.120
0.119
0.118
0.116
0.116
0.114
0.112
9
0.111
0.110
0.109
0.108
0.106
0.105
0.104
0.103
0.102
0.101
I2. SQUARES OF NUMBERS FROM 10 TO 99
1
1
2
S
4
6
6
7
8
9
100
121
144
169
196
226
256
• 289
324
361
2
400
441
484
529
576
625
676
729
784
841
3
900
961
1024
1089
1156
1225
1296
1369
1444
1621
4
1600
1681
1764
1849
1936
2026
2116
2209
2304
2401
5
2500
2001
2704
2809
2916
9025
3136
3249
3364
3481
6
3600
3721
3844
3969
4096
4226
4356
4489
4624
4761
7
4900
5041
5184
5329
6476
5626
5776
6929
6084
6241
8
6400
6561
6724
6889
7056
7226
7396
7569
7744
7921
9
8100
8281
8464
8649
8836
9025
9216
9409
9604
9801
Is. CUBES OF NUMBERS FROM 1 TO 9.9
1
.0
.1
.2
.8
.4
.5
.6
.7
.8
.9
1.00
1.33
1.73
2.20
2.74
3.37
4.10
4.91
5.83
6.86
2
8.00
9.26
10.66
12.17
13.82
15.62
17.58
19.68
21.96
24.39
3
27.00
29.79
32.77
a5.94
39.30
42.87
46.66
50.65
64.87
66.32
4
(>4.0
68.9
74.1
79.5
86.2
91.1
97.3
103.8
110.6
117.6
5
125.0
132.7
140.6
148.9
157.5
166.4
175.6
185.2
195.1
205.4
6
216.0
227.0
238.3
250.0
262.1
274.6
287.5
300.8
314.4
328.5
7
343.0
357.9
373.2
389.0
405.2
421.9
439.0
456.6
474.6
493.0
8
512.0
631.4
551.4
671.8
592.7
614.1
636.1
658.6
681.5
705.0
r
729.0
753.6
778.7
804.4
830.6
857.4
884.7
912.7
941.2
970.3
58
NUMERICAL TABLES
[V,J
Ji. SQUASE ROOTS OF NUMBERS FROM 1 TO 9.9
.0
.1
.2
.8
.4
.5
.6
.7
.8
.»
0.000
0.316
0.447
0.548
0.632
0.707
0.775
0.837
0.894
0.949
1
1.000
1.049
1.095
1.140
1.183
1.225
1.265
1.304
1.342
1.378
2
1.414
1.449
1.483
1.517
1.549
1.581
1.612
1.643
1.673
1.703
8
1.732
1.761
1.789
1.817
1.844
1.871
1.897
1.924
1J949
1.975
4
2.000
2.025
2.049
2.074
2.098
2.121
2.145
2.168
2.191
2.214
^ 6
2.336
2.258
2.280
2.802
2.324
2.345
2.366
2.387
2.408
2.429
6
2.449
2.470
2.490
2.510
2.530
2.560
2.569
2.588
2.608
2.627
7
2.646
2.665
2.683
2.702
2.720
2.739
2.757
2.775
2.793
2.811
8
2.828
2.846
2.864
2.881
2.898
2.915
2.933
2.950
2.966
2.983
9
3.000
3.017
3.033
3.050
3.066
3.082
3.098
3.114
3.130
3.146
J2. SQUARE ROOTS OF NUMBERS FROM 10 TO 99
1
1
2
8
4
5
«
7
8 9
3.162
3.317
3.464
3.606
3.742
3.873
4.000
4.123
4.243 4.359
2
4.472
4.583
4.690
4.796
4.899
6.000
5.099
5.196
5.292 6.385
8
5.477
5.568
5.657
5.745
5.831
5.916
6.000
6.083
6.164 6.245
4
6.325
6.403
6.481
6.557
6.633
6.708
6.782
6.856
6.928 7.000
6
7.071
7.141
7.211
7.280
7.348
7.416
7.483
7.550
7.616 7.681
6
7.746
7.810
7.874
7.937
8.000
8.062
8.124
8.185
8.246 8.307
7
8.367
8.426
8.485
8.544
8.602
8.660
8.718
8.775
8.832 8.888
8
8.944
9.000
9.055
9.110
9.165
9.220
9.274
9.327
9.381 9.434
9
9.487
9.539
9.592
9.644
9.696
9.747
9.798
9.849
9.899 9.950
K. RADIANS TO DEGREES
1
2
8
4
5
6
7
8
9
Radians
Tenths
Hundredths
Thousandths
Ten-thousandths
57°17'44".8
114°35'29".6
171°53'14".4
229°10'59".2
286°28'44".0
343°46'28".8
401° 4' 13" .6
458°21'58".4
515°39'43".3
5°4;V46".5
11°27'33".0
17°11'19".4
22°55'05".9
28°38'52".4
34°22'38".9
40° 6'25".4
45°50'11".8
61°33'58".3
0°34'22".6
1° 8'45".3
1°43'07".9
2°17'30".6
2°51'53".2
3°26'15".9
4° 0'38".5
4°35'01".2
5° 9'23".8
0° 3'26".3
0° 6'52".5
0°10'18".8
0°13'45".l
0°17'11".3
0°20'37".6
0°24'03".9
0°27'30".l
0°30'56".4
0° 0'20".6
0° 0'41".3
0° 1'01".9
0° 1'22".5
0° l'43".l
0° 2'03".8
0° 2'24".4
0° 2'45".0
0° 3'05".6
V.M]
CONSTANTS
59
L. IMPORTANT CONSTANTS AND THEIR COIIMON
LOGARITHMS
^=» Number
Value of 2^
LoOio^A^
TT
3.14159265
0.49714987
1-i-ir
0.31830989
9.60286013
X2
9.86960440
0.99429975
V¥
1.77245385
0.24857494
e = Napierian Base
2.71828183
0.43429448
M=lOgio€
0.43429448
9.63778431
l-5-3f=logelO
2.30258509
0.36221569
180 -f-ir = degrees in 1 radian
67.2957795
1.75812263
ir -f. 180 = radians in 1°
0.01745329
8.24187737
TT ^ 10800 = radians in 1'
0.0002908882
6.46372612
V -*. 648000 = radians in 1"
0.000004848136811095
4 68567487
sin 1"
0.000004848136811076
4.68557487
tan 1"
0.000004848136811133
4.68657487
centimeters in 1 ft.
30.480
1.4840158
feet in 1 cm.
0.032808
8.6169842
inches in 1 m.
39.37
1.5951654
pounds in 1 kg.
2.20462
0.3433340
kilograms in 1 lb.
0.453693
9.6666660
g (average value)
32.16 ft./sec./sec.
1.5073
= 981 cm./ sec/sec.
2.9916690
weight of 1 cu. ft. of water
62.425 lb. (max. density)
1.7953686
weight of 1. cu. ft. of air
0.0807 lb. (at 32° F.)
8.907
cu. in. in 1 (U. S.) gallon
231
2.3636120
ft. lb. per sec. in 1 H. P.
560.
2.7403627
kg. m. per sec. in 1 H. P.
76.0404
1.8810445
watts in 1 H. P.
745.957
2.8727135
M. DEGREES TO RADIANS
1°
.01745
10°
.17463
100°
1.74533
6'
.00175
6"
.00003
2°
.03191
20°
.34907
110°
1.91986
r
.00204
7"
.00003
3°
.05236
30°
.52360
120°
2.09440
8'
.00233
8"
.00004
4°
.06981
40°
.69813
130°
2.26893
9'
.00262
9"
.00004
5°
.08727
60°
.87266
140°
2.44346
10'
.00291
10"
.00006
6°
.10472
60°
1.04720
160°
2.61799
20'
.00682
20"
.00010
7°
.12217
70°
1.22173
160°
2.79253
30'
.00873
30"
.00016
8°
.13963
80°
1.39626
170°
2.96706
40'
.01164
40"
.00019
9°
.16708
90°
1.57080
180°
3.14159
50'
.01464
60"
.00024
60
NUMERICAL TABLES
(V, N
N. SHORT CONVERSION TABLES AND OTHER DATA:
MULTIPLES, POWERS, ETC., FOR VARIOUS NUMBERS
n»l
n»2
n»c8
n=4 n=5 n=6
»=7
n=8
M=9
TT »n
3.1416
6.2832
9.4248
12.566 15.708 18.860
21.991
25.133
28.274
TT . 71^/4
.78540
3.1416
7.0686
12.666 19.635 28.274
38.485
60.265
63.617
V . n8/6
.62360
4.1888
14.137
33.610 65.450 113.10
179.59
268.08
381.70
ir-rw
3.1416
1.5708
1.0472
.78540 .62382 .52360
.44880
.39270
.34907
n-T-'f'
.31831
.63662
.95493
1.2732 1.5915 1.9099
2.2282
2.5465
2.8648
(ir/180) . n
.01745
.03491
.05236
.06981 .08727 .10472
.12217
.13963
.15708
(180/ir) . n
67.296
114.59
171.89
229.18 286.48 343.77
401.07
458.37
515.66
e'n
2.7183
5.4366
8.1548
10.873 13.591 16.310
19.028
21.746
24.465
Mn
.43429
.86859
,1.3028
1.7371 2.1714 2.6057
3.0400
3.4744
3.9087
(l-r3f).n
2.3026
4.6052
6.9078
9.2103 11.513 13.816
16.118
18.421
20.723
\-7-n
1.0000
.50000
.33333
.25000 .20000 .16667
.14286
.12500
.11111
n3
1.
4.
9.
16. 25. 36.
49.
64.
81.
n8
1.
8.
27.
84. . 125. 216.
343.
512.
729.
n*
1.
16.
81.
256. 625. 1296.
2401.
4096.
€561.
n6
1.
32.
243.
1024. 3125. 7776.
16807.
32768.
59049.
26.2*
64.
128.
256.
. 512. 1024. 2048.
4096.
8192.
16384.
3»
3.
9.
27.
81. 243. 729.
2187.
6561.
19683.
V7i
1.
1.4142
1.7321
2. 2.2361 2.4495
2.6458
2.8284
3.
3,-
1.
1.2599
1.4422
1.5874 1.7100 1.8171
1.9129
2.
2.0801
n!
1.
2.
6.
24. 120. 720.
5040.
40320.
862SS0.
l^n!
1.
0.5
.16667
.04167 .00833 .00139
.00020
.00002
.000003
^»*
l-r6
1.
1^30
5. 1-^42 61.
l-r30
1385.
6-7-66
cm. in 71 in.
2.5400
5.0800
7.6200
10.160 12.700 16.240
17.780
20.320
22.860
in. in n cm.
.39370
.78740
1.1811
1.5748 1.9685 2.3622
2.7559
3.1496
3.5438t
m. in n ft.
.30480
.60960
.91440
1.2192 1.6240 1.8288
2.1336
2.4384
2.7432
ft. in 71 m.
3.2808
6.5617
9.8425
13.123 16.404 19.685
22.966
26.247
29.527
km. in n mi.
1.6093
3.2187
4.8280
6.4374 8.0467 9.6661
11.265
12.875
14.484
mi. in n km.
0.6214
1.2427
1.8641
2.4855 3.1069 3.7282
4.3496
4.9710
5.5923
kg. in n lb.
.45359
.90719
1.3608
1.8144 2.2680 2.7216
3.1751
3.6287
4.0823
lb. in n kg.
2.2046
4.4092
6.6139
8.8185 11.023 13.228
15.432
17.637
19.842
1. in 71 qt.
.94636
1.8927
2.8391
3.7854 4.7318 6.6782
6.6245
7.6709
8.5172
qt. in 71 1.
1.0567
2.1134
3.1700
4.2267 6.2834 6.3401
7.3968
8.4634
9.5101
* Bn = nth Bernoulli number ; see II, E, 15-18, p. 8,
1 Exact legal values in U. S.
INDEX
[Numbers in roman type refer to pages of the body of the book; those in
italics refer to pages of the Tables.]
Absolute value, 14.
Acceleration, 60, 62; angular, 72;
component, 63; of a reaction, 78;
tangential, 60; total, 63.
Algebraic functions, 24, 41.
Amplitude of S. H. M., 126.
Analytic geometry, formulas, 16.
See also Curves.
Anchor ring, 11.
Annuity, S.
Approximate integration, 193, 47.
Approximation. See also Error, La-
grange, Prismoid, Simpson, Tay-
lor.
Approximations, formulas for, 4'^:
polynomial, 234, 255, SO; Simpson-
LaGrange, 31; Taylor, 30; trigono-
metric, 5, 31.
Area, polar coordinates, 149; of a
surface, 302 ; surface of revolution,
137, 200.
Areas, 90, 215, 4^.
Astroid, ^6.
Asymptotes, 188, 189.
Atmospheric pressiu*e. 111.
Attraction, 232, 50.
Average value, 224, 231, 49,
Bacterial growth. 111.
Beams, 71, 213.
Bernouilli numbers, 60,
Binomial differentials, 184, 4^.
Binomial theorem, 269, 7.
Cardioid, 136, 26.
Cassinian ovals, 29.
Catenary, 108, 137, 22.
Cavalieri's Theorem, 202.
Center of gravity, 224, 225, 226, 49.
Center of mass, 49. See also Center
of gravity.
Center of pressure, 231.
Centroid. See Center of gravity.
Chance, 6.
Circle, 9, 15.
Circular measure, 119. See also
Radian.
Cissoid, 229, 28.
Coefficient of expansion, 114.
Combinations, 6.
Compound interest law, 110.
Concavity, 65.
Conchoid, 28.
Cone, 11.
Confocal quadrics, 310.
Constants, 1; of integration, 313; 60.
Continuity. See Fimctions, con-
tinuous.
Contour lines, 27.
Conversion tables, 60.
Cooling, in fluid. 111.
Critical point, on a surface, 291.
Critical points, for extremes, 53.
Cubes, table of, 57.
Curvature, 139, 154; center of, 142;
radius of, 141.
•Curves, 19, see also Functions; cubic,
27; parabolic, 15, 19, see also
Polynomials; quartic, 27.
Curvilinear coordinates, 299.
Cycloid, 136, 143, 24.
Cylinder, 10; projecting, 299.
Cylindrical coordinates, 233.
Damping, of vibrations, 24»
Definite integrals, 46>
ai
62
INDEX
Depreciation, 6.
Derivative, 19; of a constant, 25;
of a function of a function, 31 ; of a
power, 25, 27, 34; of a product, 30;
of a quotient, 28; of a sum, 25;
logarithmic, see Logarithmic; par-
tial, 274, 18; total, 278.
Derivatives, notation for, 19; second,
61; of inverse trigonometric func-
tions, 128; of exponentials, 107; of
logarithms, 103; of trigonometric
functions, 120, 121.
Derived. curves, 68.
Determinant, 5.
Difference quotient, 6.
Differential, partial, 283; total, 279.
Differential coefficient, 19.
Differential equations, 82, 127, 311;
exact, 325; extended linear, 323;
higher order, 338; homogeneous,
317, 329, 338; linear, 320; linear,
constant coefficients, 338; non-
homogeneous, 332, 340; ordinary,
311; partial, 311; second order,
326; separable, 316; special types,
334; systems of, 342.
Differential formulas, 44, 132, 15.
See also Derivatives.
Differentials, 43; exact, 325; notation
for, 43; transformation of, 18.
Differentiation, 20; formulas for,
35, 44, 132, 15.
Direction cosines, 16.
Distribution of data, SO.
Electric current, 114, 277.
Elimination of constants, 312.
Ellipse, 9, 15.
Ellipsoid, 11, 14'
Elliptic functions, 55.
Elliptic intervals, 183.
Empirical curves, 234.
Energy integral, 337.
Envelopes, 284.
Epicycloid, ^5.
Epitrochoid, £4-
Equations, differential, see Differ-
ential; in parameter form, 32,
see also Parameter ; solution of, 4-
Error curve, SO,
Errors, of observation, 50,
Evolute, 142, 145, 286.
Explicit functions, 40.
Exponentials, 107, 22, see also
Logarithms; differentiations of,
107; table of, 54.
Exponents, S.
Extremes, 52, 257, 290; final tests
for, 54, 66, 294; weak, 291.
Factors, 4-
Falling bodies, 87, 208.
Family, of curves, 21.
Finite differences, 248. See cUso
Increments.
Flexion, 61.
Flow of water, 308.
Fluid pressure. See Water pressure.
Atmospheric pressure, etc.
Folium, 41, 28.
Force, work done by, 50.
Fourier's theorem, 8, SI.
Frustum, of a cone, 11; formula, 96;
of a solid, 95.
Functions, 1; continuous, 11, 15;
derived, 19; notation for, 1;
implicit, etc., see Implicit, etc.; of
functions, 31; algebraic, rational,
etc., see Algebraic functions, etc.;
classification of, 24.
Gamma function, 56.
Gases, expansion of, 38, 48, 78, 105,
110, 50.
Geometry, of space, 16.
Graphs, 2.
Gudermannian, 131, IS.
Guldin and Pappus, Theorem, 4^.
Gyration, radius of. See Radius.
Harmonic functions, 23. See also
Trigometrio.
Helicoid, 300.
Helix, 300.
Hooke's Law, 125.
Hyperbola, 10, 15.
Hyperbolic functions, 108, IS, 22, 54,
inverse, see Inverse.
Hyperbolic logarithm. See Loga-
rithms.
INDEX
63
Hyperboloid, 33.
Hypocycloid, S6.
Hypotrochoid, SS.
Implicit functions, 40.
Improper integrals, 190.
Increments, 4, 248; method of, 238;
second, 239, see also Finite
differences.
Indeterminate forms, 259, 262.
Inertia, moment of. See Moment.
Infinite series. See Series.
Infinitesimal, 14; principal part, 261.
Infinitesimals, higher order, 261.
Infinity, 16.
Inflexion, point of, 65.
Integral, as limit of sun, 192, 197;
fundamental theorem, 86; in-
definite, 83; notation for, 83.
Integral curves, 313, 343.
Integrals, definite, 87, 4^; double,
210; elliptic, 183, 9, 66; improper,
190; infinite limits, 188; infinite
integrand, 189; multiple, 208, 215;
table of, 36; triple, 208.
Integral surfaces, of a differential
equation, 343.
Integrand, 83.
Integraph, 243.
Integrating factor, 325.
Integration, 83; approximate, 193,
see also Approximation; by parts,
163, 36: by substitution, 158, 36,
43; formulas for, 156, 36; of a
sum, 84; of binomial differentials,
184, 4U of irrational functions,
129, 39; of linear radicals, 172, 39;
of polynomials, 84, 158; of quad-
ratic radicals, 173, 39; of rational
functions, 165, 36; of trigonometric
functions, 157, 172, 177-182, 41;
reduction formulas, 184, 4^^ 4^!
repeated, 208; successive, 209.
Interpolation, Lagrange's formula,
16. See also Lagrange.
Inverse functions, 3.
Inverse hyperbolic functions, 131,
14, 64.
Inverse trigonometric functions, 128.
Involute, 146.
Irrational functions, 25; differen-
tiation of, 34; integration of, 164,
172.
Isothermal expansion, 105.
Kinetic energy, 231.
Lagrange interpolation formula, 16^
47.
Law of the mean, 247, 47; extended,
253, see also Taylor's theorem.
Least squares, 58, 296, 309, 0.
Lemniscate, S9.
Length, 133, 4^; polar coordinates,
152; of a space curve, 305.
Limits, 14; arc to chord, 133; proper-
ties of, 15; sin 6 to 0, 119.
Liquid pressure, 50.
Logarithmic derivative, 115. See '
also Rates, relative.
Logarithmic plotting, 234, 20.
Logarithms, computation of, 7;
graph of, 21; hyperbolic, 102;
Napierian, 102, 64; natural, 102;
rules of operation, 99, 3; table of,
62.
Maclaurin's Theorem, 258. See also
Taylor's Theorem.
Mass, 49.
Mathematical symbols, 1-3.
Maximum, 6. See also Extremes.
Mean square ordinate, 231.
Mensuration, 9.
Minimum, 6. See also Extremes.
Modulus, of logarithms, 103.
Moment of inertia, 219, 221, 49;
polar coordinates, 220.
Motion. See Speed, Acceleration, etc.
Napierian base e, 102. See also
Logarithms.
Natural logarithms. See Logarithms.
Normal, 5, 49; length of, 50; to a
surface, 298, 300.
Notation, 1.
Numbers, e, M. See Logarithms.
Organic growth, law of. 111.
Orthogonal trajectories, 325.
64
INDEX
Pappus Theorem, 49,
Parabola, 10, See also Curves,
parabolic.
Paraboloid, 11, 34-,
Parameter forms, 32, 50, 134.
Partial derivative, 274, see dUo
Derivative; order of, 275.
Partial derivatives, geometric inter-
pretation, 277; transformation, 18,
Partial differential. See Differential.
Partial fractions, 165.
Pendulum, 127, 238, 250.
Percentage rate of increase, 112.
See also Rates.
Period, of S. H. M., 126.
Permutations, 6.
Phase, of S. H. M., 126.
Plane, equation of, 16,
Planimeter, 243.
Point of inflexion, 65.
Polar coordinates, 5, 147; plane area,
216; moment of inertia, 220; space,
300.
Polynomial, approximations. See
Approximations.
Polynomials, 24, see also Curves,
parabolic; differentiation of, 25;
roots of, 65; integration of, 84, 158.
Power cvirves, 19.
Power series. See Taylor series.
Primitive, of a differential equation,
313.
Prism, 10.
Prismoid, defined, 205.
Prismoid rule, 202, 10, 47.
Probability, 6, see also Least Squares,
Error curve, 30; integral, SO, 56,
Psrramid, 10,
Pythagorean formula, 134.
Quadric surfaces, 33; confocal, 310.
Quartic curves, ;?7.
Radian measure, table of, 61, 68,
Radium, dissipation of, 114.
Radius of curvature. See Curvature.
Radius of gjn-ation, 220, 50.
Rates, average, 18; instantaneous,
19; percentage, 112; related, 74;
relative, 112, 114; reversal of, 79,
see also Integrals; time, 10, 60,
Rational functions, 24; differenti-
ation of, 28; integration of, 165.
Reactions, rates of, 78, 114.
Reciprocals, table of, 57,
Reduction formulas, 179, 185, 41-
Relative rate of increase, 112, 114.
See also Rates and Logarithmic
derivative.
RoUe's Theorem, 247.
Roulettes, ^4,
Semi-logarithmic plotting, 236.
Series, alternating, 270; convergence
tests, 266; differentiation of, 272;
geometric, 265, 7; infinite, 7;
integration of, 272; precautions,
269; Taylor, 266.
Simple harmonic motion, 124, 328,
^3.
Simpson-Lagrange approximations,
31.
Simpson's rule, 207, 47,
Singular solution of a differential
equation, 313.
Slope, 4.
Solution of equations, 4-
Speed, 9, 60, 62, see also Motion;
component, 10; angular, 72; total,
42, 134; of a reaction, 78.
Sphere, 11.
Spherical coordinates, 232, 300.
Spirals, 3$.
Square roots, table of, 58,
Squares, table of, 57,
Strophoid, ^7.
Subnormal, 50.
Subtangent, 50.
Summation, approximate, 193 ; exact,
196.
Summation formula, 197.
Surfaces, quadric, 33.
Table of integrals, 156, 36,
Tables. See special titles.
Tangent, equation of, 5, 49; length
of, 50; to a space curve, 305.
Tangent plane, to the surface, 289, '
297, 300.
Taylor series, 266, 30,
INDEX
65
Taylor's Theorem, 263, 8, see also
Law of the Mean.
Time rates. See Rates.
Total derivative, 278.
Total differentials, 279.
Tractrix, e6.
Trajectories, orthogonal, 325.
Transcendental functions, 25.
Trapezoid rule, 47-
Trigonometric functions, table of,
12, 21,61.
Trigonometry, 9,
Trochoid, 24-
Variable, 1; dependent, 1; inde-
pendent, 1.
Velocity, 60. See also Speed.
Vibration, 125, 23; electric, 125.
Volume, of frustum, 95, 199; of solid
of revolution, 94.
Volumes, 94, 95, 210, 48.
Water pressure, 201, 231, SO,
Witch, 28.
Work, of a force, 60; on a gas, 105,
110.
ANSWERS TO EXERCISES
§ 6. Page 9
1. y = 2aj-3. 8. 8x — 2/ = ll. 6. 3x-y = 2. 7. 3x+y = 4.
§ 9. Page 12
8. 82 ; 114 ; 178 ; 50 + 32 T. 6. 2 ; 6 ; 40 ; 2 T. 7. 80 ; 16 ; - 80.
9. « = 5/2 ; 8 = 100 ; c = 0. 11. 1 ; 2 « ; Vl+TPT
§ 11. Pages 17-18
6. 9. 7. -5/12. 9. ^-1/2. 11. (a + b)/(c + d). ' 18. 1.
15. 0. 17. 3/5. 19. - 1/2. 23. 5/3. 26. 2. . 27. Va/p.
§ 14. Pages 22-23
1. 2x-4. 3. 3-3x2. 5. 8-4x3. 7. - l/(x - 1)2.
9. -2/x3. 11. 8/(x + 2)2. 13. 4x + 9y = 24. 16. Rises
when I X I > \/5~; falls when | x | < V5 ; slope zero when x = ± V6.
17. Rises when x > 2 ; falls when x < 2 ; slope zero when x = 2.
19. Hor. speed, dx/dt = 15 ; vert, speed, dy/dt='^ 32 1 + 15.
21. 8irr; 3 a2 ; 2 irrA/3.
§ 16. Pages 27-28
1. 12 x3. 8. 20 x4. 6. 50 (t* + 1). 7. 12 t(t^ - 1).
9. 18 t\l + «3). 11. 6 r (3 r + 2). 13. 2v- 1. 16. 11 rW + 12 vK
17. 13 ayi2 _ 9 aby^. 19. mfcx«-i + nAx"-i. 21. 3 n«3n-i — (n + 2)e»+i.
23. (-2, 0). 26. (1, 6). 27. Slopes at x = 0, 2, - 2, 4, - 4 resp.
are 0, 12, 36, 72, 120 ; slope is 9/2 at x = 3/2 and - 1/2 ; slppe is - 3/2
at X = 1/2.
§ 17. Pages 29-30
1. -l/x2. 8. 6/(x + 4)2. 6. -9/x4. 7. -28x-8-4x-3.
9. - («4 + 2 t)/(«3 _ 1)2. 11. 4 w/(u2 _ 1)2. 18. 16/3 ^ 1/^2.
16. 0. 17. - 6/2* - 6 2/(z2 + 1)2. 19. 2(1 - r^)/(r^ - r + 1)«.
21. 3(l-6y2 + 2 2/3)/(«3_i)2. 28. - 6 fT -{- S tr^,
26. -(r6+3i32)/(«4_i)2. 27. -x/(a» + 6)3.
29. (y2-2)/(y-l)4. 31. tan-i(-3).
1
2 ANSWERS
§ 20. Pages 33-34
1.6JB2-2. 8.4X + 3. 5. 12x2-5x4-2x. 7.4x(a;«-l).
9. 6x«(x«-3). 11. 24x(3a;S + 6)3. 18. 4(1 - 30(H-2«-3<?).
16. 3(3 «2 - l)(«3 _ t - 4)«. 17. 8(2 rs + 3 v)(v^ + 3 u* - 2).
19. -6 «V(«' + !)'• 21- (6a;2 - 3 x* - 18x)/(x2 + 2)1
28. - 8 8(2 82 - 3)-3. 25. 6(2 + 4 «2 - 5t))(3r - 5)(tj2 + 2)2.
27. 6^6+12 «H 12 «2+12. 29. 166-108X. 81. (2/5)[3(l-2x)-«+8].
88. 3/(2 «). 85. Horiz. tangents at X = 1/2, —4/3, -6/12.
§ 23. Pages 37-38
1. (4/3)xi 8. (4/3)x"*. 6. (3/4)x"^. 7. -6x-2-5x"**
9. (22/3)«* + 7 t^. 11. - 6x-4 - (l/2)x^ + (2/8)x"i.
13. ^ . 15. A±^. 17. -ll^ll
2V2 + 3X 2V2 + 3U
19. -^^-.- 81. lg''-l«^. SS.
8(l + ««)* 2V3a;-4
, ^ • 27. ''-^°' .
4^x + xVx 2u2Va2-u
2Vt2_3t
2x-l
2 VI -XH-X2
«-3«3
Vl -<2
81. 2(xj- VI + x2)2/ Vl + x2. _ Tangents: 88. 3x + 4y=5.
85. 2V2y - X = 2. 87. 2 V2y = 5x - 1. 89. tan-i(5/12).
41. dp/du=-1.41fcu-2.4i.
§ 25. Pages 41-42
1. - 2 y/x or - 2/x«. 8. (2 X — y)/x or 1 + 6/x2.
5. x/4i/orx/(2Vx2-36). 7. - x2/2/2 or - x2/(a» - x»)i
9. 3(1 - x2)/(2 y) or 3(1 - x2)/2V3x-x3.
11. (2 X - 2/2)/(2y + 2xy) or (2 + x)/2(l + x)i 18. - y/x.
15. l/(2y). 17. -Vy/x. 19. 4«;2y = x2.
21. « ; 27 y2 = 4(x - l)s. 23. (<2 - 1)/( 2Q ; x' + y2 = i.
27. dy/dx > when x < and v .v. 29. o = (1/2) V2 + 2«2.
§ 27. Pages 46-48
1. (2ax + 6)dx. 8. 3(2ax-|-6)(ax2+6x+c)2dx. 5. -a(axH-6)-2dx.
7. -(12t + l)d«. 9. «(a - 0K2 a - 5 t)d<. 11. (S - 2 t)dt/2VSt^.
18. (9/2)Vt«-3«(t2-l)dt. 15. -(l+r)diV(2i5 + i?*)*.
17. -(l+r)(2 + 3i?2 + a?8)d«/(t?3-.l)2. 19. 42/2(2 -.yS)-!^^.
ANSWERS 3
21. (l/2)(2H-y)(l + y)"W 28. 6nps«-i(a + 6««)P-ids.
25. - 6np«»-ids/(a + 6«»)p+i. 27. cte/[>/x(l - Vx)2].
29 y^ 81. (2^^~^^-^)<^ . 33. q(g^ +y^)dg
^ 1 — X * y + «^ * 2 y (1 — oaj) *
35 (3^2 + 20x4- y)dx, 37 y + 6x = ll. 48. Vt72.
2i/(a— 6x)
45. -(2 + ^) (1 + ^)2/^. 47. 9/(2 r). 49. 2(1 - <2)/(i + ^)2 ;
- 4 V(l + *2)2 ; 2/(1 + t2). 51. dc = (6 - tj)di)/(p - a/tj2 + 2 a6/tj3).
§ 31. Page 52
Tang. Norm. Subt. Subn. Tang. Norm.
1. 9x + y = 5. x-9y = 37. 4/9. 36. 4v^/9. 4V^.
8. 9x + 4y = 25. 9y-4x = 32. 16/9. 9. 4^97/9. V97.
5. x + 32/=:6. 3x-y = 8. 3. 1/3. VlO. VlO/S.
7. y = 3. X = 4. GO . 0. 00 . 3.
Tangents : 9. y + y^ = 2 AxcXq. 11. xxq + Wo = «*• 18. ft^xx^ ± a2j/yo
= a262. 15. axXo+ 6(xyo + Xq?/) + cyyo + ^i^ + «o) + e(y + Vo) +/ = 0.
35. Pages 56-59
1. 8. 5. 7. 9.
Max. x = 0. q=i—\, y = 0. n = l. t = 2/3.
Min. x = 4. q = 6, y =± V2. n = 3. t = 1.
11. 18. 15. 17. 19.
Max. 8 = 0. None. x = 4. (1 ± V6)/2. r = 2.
.Min. « = ±V572. x = -2. x = 0. (-l±\/6)/2. r=-2.
21. A square ; area 400. 88. (d: 7, d: 7). 25. Ht. = diam.
27. Width = 2 X depth. 29. Depth = V3 x breadth, 81. x/6+y/8=l.
88. Max. 3 ; Min. 1 : the variable x does not increase steadily when the
function D^ goes through its minimum or maximum, as the general
theory requires. 39. Compromise: 42^ a foot; average: 42.56^ a
foot. 45. Width = 1/2 x base of A. 47. Height = (2 v%/3) x radius
of sphere. 49. Rad. = V(6 + V5)/10 x radius of sphere. 51. 2 ab.
§ 40. Pages 64-65
1. 2x + 5, 2. 8. 4x-l,4. 5. 2x-6/2, 2.
7. 6(x2 + x-6),6(2xjfl). 9. 4x3- 6 x2+ lOx , I2x»-12x + 10.
11. 1/2 Vx + x/Vx2 + l, - l/(4\/x3)+ 1/V(x2 + 1)3.
18. (X + 2)2(5x2 + 4x - 3), 2(x + 2)(10x2+16x + l). 16. a, 0.
4
ANSWERS
17. 2 ox +6, 2 a. 19. [m(x — 6) +n(x — a)](a; - a)"»-i(aj — 6)»-i,
[m(m — l)(x — 6)2 + 2mn(x — a)(x — 6)+ n(Ti— l)(x— a)^]
X (X — a)'»-2(x - 6)'»-2.
27. m = 2(3x»-7x-13)(x + 3)(x-2)2
-llOx-10).
88. 85.
d/b. - 2 r3.
0. 6 r -».
6. 1.
d. - 2 t-K
V6« + (P.
0.
0.
0.
m
6:
V :
J-
Ji
0.
0.
-12M
25. m = -l/x2, 6 = 2/x3.
6 = 2(x-2)C15x»— 10x2
87.
-(1 + t)ytK
2(1 + 0V«'-
(1 + t)-K
-«-2.
Vt2 + 4
-2(1 + «)-'•
2r«^
2 V(l + «)-« + t-« .
8.
Max. X = — 1.
Min. X = 5.
Infl. X = 2.
§ 44. Page 70
5.
- 11/6.
5/6.
-1/2.
9.
7.
0.
±1.
11.
None.
None.
None.
Max. - B/(2 A), A<0; none, ^ > 0.
Min. None, ^ < ; - B/(2 A), A >0.
Infl. None.
28. 2/ = 3x2 + ox + 6; xV3 + 3x2/2 + ax+b; ox + 6.
27. 29.
Infl, None. x = 3 1/4.
Max. Defl. x = ± Z/2. x = 2(1 + V33)/16.
18.
None.
-2.
2^.
§ 46. Page 73
1. w = 3 <2/l000 degr./sec. ; a = Q t/lOOO degr./sec.«.
8. w=(-«3/4-l/32) rad./min. =[-«3/(8T)-l/(64ir)]rev./min.
= (- <3/240 - 1/1920) rad./sec.
a =— 3«2/4 rad./min.2 = - 3 tys v rev./min.2 =— t2/4800 rad./sec.2.
7. (ir/720)(«2 - ««/46) ft. /sec. ; (ir/860)(t - t2/30) ft./sec.2.
ANSWERS 6
§47. Page 76
1. 1026/(2304 r) ft./min. 3. Inversely as the cross-sections:
dhldi : dh'/dt = r'* : r^. 5. Inversely as areas of liquid surfaces ; their
dV dP 1
ratio varies as the area of remaining liquid; 5 =- r'cota;
dt dt 2
a = half -angle of cone, r =: radius of liquid surface in the funnel.
7. (fc/4) VS/e" cu. in./sec. 9. dy/d« = 4(4x - 1) ft./sec.;
12 ft. /sec. ; 44 ft./sec. 11. Vy = 3 x^u, = 30 as* ft. /sec. ; 1080 ft. /sec.
13. 8/V5 ft./sec. ; 4/\/5B ft. /sec. 15. 4 ft./sec. ; Nearly 312 ft./sec.
17. = 17** 41', 16° 18', 10° 22' ; max. h. = 62600 sin* 20° ft.
§ 50. Page 82
1. 2a5* + c. 3. x» + c. 6. -2x3 + c. 7. — xV6 + c. 9. 5xV2
+ 4x + c. 11. a = ty4 — 2 «2 4- 7 « + c. 13. y = axV2 + bx+c.
15. y = .002x8-. 001 x4+.003x» + c. 17. y = -x-V2 4-c.
19. »=-3/«-2/«* + c. 21. 2/=(4/3)x*-6xi + c.
§ 51. Tage 85
1. y = J(4 X* + 3 x)dx = 4 x'/S + 8 xV2 + c.
3. y = Jx-«dx=— X-V2 + c. 5. y = J(4x + 6)dx = 2x2 + 5x + c.
7. y = J9(Jx = 9x + c. 9. y= /(x* — x*)(ix =xV4 — xV6 + c.
11. y = J(x + Vx) dx = xV2 + 2x*/3 + c. 13. xV2 + 6x + c.
15. 3x*- 9x5/6 + c. 17. 16x + xV2- 2x3/3 + c.
19. 2x*-4x*/6 + c. 21. 2x^/7 -2xV^ + c.
23. x + 3x*/2 + 3x^/6 + c. 26. 4 tJ/7 + c. 27. 3u*/2-2Vu+c.
29. - l/t» - 1/(2 «*) + c. 31. 6 «*/8 + 6 «*/2 + c.
33. 3y"V/i3«6yJ/7 + c. 35. 12\/y+c. 37. 3u*/8 + c.
39. x*/2 + (2/6)x*+ c. 41. t»/3-tV6 + c. 48. ax*/2 + 6x»/3 + c.
45. ax*+i/(n+l)+6x"+V(« + 2) + c. 47. - t-i'Vl-^ + *^/2 + c.
49. (2/3)«*+(4/7)«* + c.
§ 54. Page 88
1. 3000 gal. ; 1600 gal. 3. s = tV^ + c ; 4 ; 76/4. 5. 46/(8 r) .
333/(200ir). 7. 16/6. 9. -940/3. 11. 22/3. 13. -1899.
15. 7 a/8. 17. 0. 19. 10a»/3. 21. 729/6. 23. -2. 25. 28/3;
27. .0002 29. 226/4. 31. s = flftV2 + 10 « ; 770. 33. lOflf; no.
6 ANSWERS
§ 55. Page 93
1. 1/3; 21. 8. 2/6; 62/6. 6. 2/3 ; - 18. 7. 2(2V2-l)/3 ;
2(5V6-2V2)/3. 9. 1/4; -226/4. 11. 6/6; (6/6)al- 18. 15/2.
16. 32. 17. 11/6. 19. 8/3. 21. 1/3.
§ 57. Page 97
1. 128 7r/7; 2ir/7. 8. 856ir/106; 167r/105.
5. For upper half of curve : For lower half of curve:
0to2; (14/3 + 4V3)t. (22/3 - 4 V3)ir.
- 1 to 1 ; 4 ir(l + 2 V2/3). 4 ir(l - 2 V2/3).
7. 296ir/81; tt (3x* - 6x2 - l)/3a;8]J. 9. 8ir ; ir(x* + 3)/3x]».
11. 4ir. 18. 778ir/6. [15. (2ir/21)(21 + 14V2 + 24v^8).
17. 32 ir; 48 tt; 2ir(62-a2). 19. 4irahyS; 4ira26/3. 21. 20ir/3.
28. 8V3/6; 2k/ VS. 26. Trkr^hy2i 2irkr^^h,
§ 60. Page 101
16. Iogio7/logioll= 0.812 17. 186.4± 19. 3.479 21. 2.862
§ 65. Page 106
1. i^. 8. _A^. 6. — — 7. — . 9. 2 + 21og«.
X l + 2x l+x X
11. ^ 18. LllMi. 15. Ml^gOf. n. . .713
2-6t + 3^ ^ St
19. 160.693 21. 2(e3-2)/3 + (e-2-l)/6. 28. 0.219
25. 2.302; log A:. 27. — X"*. 29. Max., none; Min., x = l; Infl.,
none. 81. Max., none; Min., x=±2; Infl., none. 88. 11.416
85. -0.396 87. 6.693 89. A:2 1ogc.
§ 67. Page 109
1. 3e3«. 8. e^i+^/(2\/rri). 5. e*(2x+x2). 7. (3 log 10)103«-m.
9. I. 11. -2x. 18. 2e«(6* + l). 15. (e^^ + e^^)/4v^.
19. 10.02 ; 2.36 ; sinh a.
21.
28.
25.
Max.
None.
None.
x = l.
Min.
x=-l.
None.
None.
Infl.
x=-2.
x = 0.
x = 2.
29. 0.632
81. 1.381
88.
(eio _ e-io)/8 - 5,
/2. 85. 26.762
ANSWERS • 7
§ 69. Page 113
1. 21 e3» ; 3. 8. (1 4- aJ)e» ; (1 + aj)/x. 6. 4 e^^+s ; 4.
7. (Jkoa; + fe6 + a)e*» ; (kax-\-kb + d)/(ax-\-b),
9. (3-4»-6x2)e-«'; (3 -4a; - 6aj2)/(3aj + 2).
11. About 963 sec. after ^ = 40°.
13. k = (1/5830) log (6/4) ; (1/1909) log 1.27 ; 26.7 in. ; 672 min. ; 1806
ft. ; 1266 m. 16. 60 log 2.
§ 71. Pago 117
1. - 2. 8. 3. 5. 10. 7. - 2 X + 3 kxK 9.-2 r^/(r^ + 1).
11. [3(1 - <2 + t4) (t2 + 1) log 10 - 2 « + 4 t3]/(l - t3 + t4).
13. [1 + log(l + X) ] (1 4- xy+'. 16. xV^=V2(i + log Vi).
17.6 + 22x4-18x2. 88. A;e*-«'/2. 36. &x». 87. fee**.
§ 74. Page 122
1. 4 cos 4. 3.-2 sec22 ^. 6. 4 x« cos x\ 7. — 3 sin 3 ^.
9. cosx— xsinx. 11. tanx. 18. cos x 4- 3 sin 2 x.
16. 2xsin(3-2x)-2(l 4- x^) cos (3 - 2 x).
17. e' cos (3 « - 1) [cos(3 t - 1) - 6 sin (3 « - 1)].
19. e«/io[(l/10)(co8i-4sin3 0-(sin«4.12cos3«)], 21. 1.
Maximum Minimum Points of Inflexion
23. X = 2 WTT 4- ir/2. 2 nir — ir. nir.
26. none. none. mr,
27. nir — 7r/4. nw 4- ir/4. nT/2.
29. 2 nir 4- ir/4. (2 n 4- l)ir 4- ir/4. nir.
81. nir - ir/12. (2 n 4- l)ir/2 - ir/12. (2 n 4- l)ir/4 - ir/12.
83, 86, 37. The functions differ at most by a constant.
89. 2. 41. -(l/2)cos2x4- fe. 48. sec t+k.
46. sin X — (3/2) cos 2 x 4- k, 47. (1/2) sin 2 x — x 4- fc.
Vg Vjf V Path.
63. — 6 sin 2 «. 6 cos 2 1 6 x'^ + y^ = 9.
56. cos t — sin L cos t. Vl — 2 sin t cos t 4- cosH. z^ — 2xy + y^=zl,
67. — 2 ir ft. /sec. ; ± ir V^ ft. /sec.
§ 76. Page 126
1. 2 cos 2 £, — 4 sin 2t. 8. cos £ — cos 2 £, — sin 1 4- 2 sin 2 1
6. 2cos2«4-0.9cos6«, — 4sin2 « — 6.4sin4^
7. ak cos (kt 4- e), — aA:2 sin (kt 4- e).
9. cP^/d«2 = - 20 ir2 sin 10 irt ; 0.2; 1/6; ±20ir2. 28. a; ak cos kt.
26. S. H. M. because cP0/dt^ = - k'^B,
8*
ANSWERS
1.
4x8
9.
15.
1
8.
§ 79. Page 130
-1 e -1
6.
^y/x^ - 1
(l + x«)tan-ix
11.
X'
l + 4x
4-2xtaii-i2v'x.
18.
VI — C2"*°*
17.
19.
e
— 2g
l + x*
-1
l + aj«
21 3(logtan-igy
(H- a;2) tan-i «
27. -ir/6.
28.
Vl-x*
25. x/3.
\/l-x2
Vl — ag/2Vx + sin-i Vx
(1 - «)*
29. (l/2)8in-i2x. 81. -tan-i(l-x).
88. 7r/6; ir/3. 85. ir/12 ; ir/6.
§ 84. Page 136
1. ViO; 2ViO; (6-a)\/iO.
8. Vl + m2 ; 2 Vl + w*^ ; (6 — a)VH-m2.
6. (2/3) (4 - \/2) ; (8/3)(2V2 - 1) ; (2/3)(6i - a*) V2.
7. 9. 11.
(to: V2(ft. 2(1 + 0^*- (1 + M)(i«.
s: 2\/2. 99. (6 - a) + (6-«- a-»)/3.
tJ ; v^. 2(1 + 0. 1 + 1-«.
§ 85. Page 139
1. 4irV6; 20irV6; 2irV6(6- a)(6+o-l).
8. 13irv^; 447r\/l0; irViO(6-a)(36 + 3a+4).
5. Sir; 4 IT. 7. (7r/2)(e2 + e-» + 6). 9. 263ir/64.
§87. Page 144
R
(1 + 4x2)
a
/3
i
8.
5.
(y» -f 4 Qg)
4a»
(1 4- cos^x)
— 4x8;
3x + 2a;
6x2+1
2>/x7a(2a — x).
4
sinx
7. coah^x;
« +
1 + cos* X .
ctnx
^ — siDhxcoshx;
2y.
1 + cos'x
cscx
ANSWERS
9
9.
11.
18.
15.
17.
19.
ib*afi + oV)*
2Va
3 a sin 2 9
OB 4- 2(x + y) ^; y + 2(x + y)-^/?.
(1 + <*)
i
2^3
0;
oco8d(l +2sin«^)
(l/6)(9 + 4t«)*|; 2-4«8/3;
0.
a sin ^(1 + 2 cos« ^).
2«8
3 1« - 1/2.
§ 91. Page 149
1. tan ^. 8. - (3/4) esc ^ — ctn 0. 5.(1 + cos« $)/(2 tan &).
7. e/2. 9. 1/2. 11. (1/3) tan 3d. 18. (1/3) tan(3d + 2ir/3).
15. -tan(d/2).. 17. (ecosd- l)/(esind). 19. -ctn(d/2).
§ 92. Page 151
1. ir8/6. 8. 3»V4. 5. 12v^T. 7. 3. 9. 626/2. 11. tt/S.
18. (V3-l)/2. 15. 4ir. 17. 4.
§ 93. Page 154
1. 5ir. 8. _V2(e»/2- 1). 5. tan 1 - tan (1/2)= 1.012+. 7. irV2.
9. (e* - e») V5.
§ 94. Page 155
1. py/2. 8. (a/^)(l +^)'*. 5. (1 + 8 cos* 3^)^/(10 + 8 cos* 3d).
7. (2/3) vTo^. 9. (62-a« + 2a/))*/(2 62-2a« + 3ap).
§ 98. Page 160
1. a; - xy2 + xys - x*/4 + c. 8. a^x + abx^ + 62x3/3 + c ; or,
(a+6x)«/(3 6)+c'. 5. (c^* - e-2»)/2 - 2 x + c.
7. laji-jx^ + fx^ + c. 45. .0686 47. 1/3. 49. log 3.
51. 5 V2/12. 58. irV2. 55. Sira*. 57. Areas : 2, 1, ir/2.
§ 99. Page 164
19. (2 +x + 2x« + x») tan-ix-2x-xV2 +c.
21. e*»(32x«-24x«+12x-163)/128+c. S5. 1. 27. ir/12+VS/2-l.
. (2-3e-»)/18. 81. (l+e-»)/2.- 88. 2-6/e.
10 ANSWERS
§ 105. Page 170
16. \ogi!L±^. 17. x + log^n?. 19. 1 .^c (x-2)Kx + 8)
27. llog^ + ltan-i?. 29. llog (^ + 1)^(^- 2),
86. log2-41og?-^. 87. - tan-i (cosx). 89. -log^ + ^.
41. log tan (x/2) . 43. log Izii! . 45. log (e« + e-*) . ~
1 -f- 6*
§ 107. Page 174
9. (6g- -4)(l +x)i/15. 11. 2tan-iV»=^. 18. Vx"=^ + sin-iVx.
tK Vx + 4 . , 1, Vx + 2 — 2 1
16. -t^-+_log_^__. „. A(4x-3)(l+x)»
19. -(3/10)(2 x+3)(l-x)* 21. (4/3)x*-.4xi + 4taii-ixi
28. cos-ix-2V(l-x)/(l+x). 26. -9-41og2.
27. 61og(3-Vsinx)-2VsEx. 29. (2/3) V2 + 3 tan x."
81. -slnx -4\/sinx-41og(l -VSnx).. 83. (l-fx2)^/3.
36. (l+x2)V6. 87. -.(flH^x2)-V6. 89. 2(a + 6x3)*/9 6.
Ai 2y/a + 6x» -« i V4x2 4- 1 — 1 , ,
41- :^ 49. logy^±^rJ: — L 61. Vx2-.l-.tan-iVx23T.
53. (x-Vi3:^)V2. 69. ( V^/2) log[4x + 1 + 2 V4x2 + 2x + 2].
61. -cos-i^ii. 63. J-iQg ^ + V3 - V x2 + 2x4-3
V2 V3 x-V'3-V22 + 2x + 3
2^ + 1./ . . „ . „, . 3
8
66- —J— VI + a; + x2 + - log (2x + 1 + 2 Vr+T+^).
«.« 1t^^ 2x+1^ 3
®^- 3^ "~r~-^~i6^^s(^+2* + 2i2); 12== vr+T+^.
69. log ^=_siir:ix
^ 1 + VI - X2 X
^, 6x2-8x — 3 . , 3x-8 /:; =
71. z sm-i X H z — VI — x2.
4 4
78. ?-^=^Vn=^ + ^^^sin-ix + x2cos-ix.
X 2
ANSWERS 11
§ 111. Page 183
cos X sin3 X . S . ^^^^ „. .
1. __ h-(x— cosxsinx).
4 8
8. (1/6) tan^x— (1/3) *an3x + tanx — X. 6. (l/5)tan6x.
tan2x . tan^x ^ ; 2ctn3x ctn^x
7. -^ + -j-. 9. -ctnx 3 ^.
11. — (1/2) cot2 X — log sin x. 13. — sin x — esc x.
16. (1/2) sec2 X + log tan x. 17. x cos a + sin a log sin x.
19. (l/3)sin3x-(l/6)sin5x. 21. -sinx-cscz. 28. sin4x(l+cos2x)/12.
25. (l/2)cos2x-logcosx. 27. -(l/2)(cosxcsc2x + logtanx).
29. — (l/l4)cos7x— (l/2)cosx.
§ 114. Page 185
■■A- '-W'-^-r.- •■ .^^.- '«'<—'
X* „" , (*-2)« - X , I, 1+x
7. -+3* + logJ^--^. 9. 2(l-x^)+4l°Sl^-
1 'i'X21 -X X .3,x — 1
^^- ^^^^l^-^ie'*^ 2- ^^- ^-2-(irzi) + ii^^5TT-
15. llog4=|. 17. :^log^±^^". 19. 2tan-xV?^l.
**'-8^x« + 2 4 ^x + 2+V2
_2 2x «^ 1 „
21. 23. . 26. 7^a+&x3-
oVa4- &x Vx- 1 ^ .
27. ^ 29. l(4x-7)(3x+7)*.
b(n -p) \/(a + 6x)»-p ^^
35. 12 f^-^ + ^-^+tan-i2/\ where 2/12 = X.
\9 7 6 o y
87. 4(3&x-4a)(a+ 6x)</(2162). 39. (4/405) (16 x +28) (3 x +7)*.
^ +1
2(x2+l)^2
41. (8/66)(4x2-3a)(a + x2)*. 43. ^T^^^Tn +otan-ix.
12(x2+3)2 ^36 V3 8(x2 + 2 X + 6) 16 2
*^- ^^ + r6«^«-^^ "• i(4x3-21)(7+4x3)i
68. 2x1:^1^ vi^3:T2+'^ log (x + VS^^r52).
o o
12 ANSWERS
55.
^ 57. 69. 4 8ina;-31ogtan(^ + |y
ay/a + bx^ Sa{a + lMfi)i ^ ^'
61. tan^-sec^. 68. _JL ipg 2 tan(^/2) + 5- V21,
V2r 2 tan {0/2) + 5 + V21
66. - aj - (1/3) ctn 8 x. . 67. 7/2 - 2 log 2.
. (3/4)log(6/3)-(2/V6)tan-iV273.
n. j.iog (8^V59)(6 + V21) ,3^^ 75. log (2 4- V3).
77. -log(3-V5). 79. 68.7 81. 0.833 88. 0.184
85. 4.037 87. 0.88 89. 2.274 91. 0.767 98. 0.940
95. 0.902 97. 0.254
§ 118. Page 195
1. 16/3. 8. 4/6. 5. - 1.307 7. 2. 9. ir/6. 11. ir/4.
18. V2/8. 15. 6.89 17. 0.263 19. 0.693 21. 0.746
88. 74/3. 25. 0.346 S7. 0.650
§ 120. Page 198
1. 12.4 8. 17.2 5. 2 7. 0.35 9. 67.6
11. 4; 1/2; (abs. val.). 18. 48. 15. 161. 17. 4600/ir.
19. 3600/3 rev., 4.7 sec, nearly.
§ 123. Page 202
I. iraV16. 8. rabh^/S. 5. 2V3aV3. 7. kw/2.
9. ir(e««-e-a»+4a)/4. 11. 64iraV3. 18. 32ira«/6. 15. 18,720 lb.
17. 12,480 lb. 19. 699 lb. nearly. 21. P* = c^P.
§ 124. Page 206
5. 1/2 9^. 7. 232/3. 9. 90. 15. 1.099 17. 7.912
19. 0.298
§ 127. Page 211
8. xV2 + ex + c'. 5. (4/16)(l - 0^ + ct + c'.
7. e^/QO - ^V12 + ce^ + c'0 + c". 9. e« + ex 4- C.
II. c +&U-U log u. 18. P/2 + sin e ; t^/Q — cos e -f 1.
15. Vl + t2; (</2) VTT^ + (1/2) log (t + VTTF).
17. 8. 19. (4/16)(6*-6* + 3^-32). 21. 32/3. 88. 1106/2.
ANSWERS 13
26. irr»/3. 27. 32 ir. 29. 8/9. 81. 6(4x2-Z2)/a
88. y = A:(3 te2 - iX^)/Q. 86. y = (3 aa;2 + 6x»)/6.
87. y = fc[loga/x)4-aj/Z-l].
§ 129. Page 218
1. 26/106. 8. 363/5. 6. (68 - 1)/16 +(5io-l)/30. 7. 20477/4.
9. 26/36. 11. 1/70. 18. 2aV3. 17. atx = 0.904 19. 1/12.
21. V3 + 2ir. 28. V5/3 + log (1 + V2). 26. 6V6/6. 27. ir/8;
. 3ir/2. 81. ir»/6. 88. 3ir/2. 86. 16 aVV6. 87. o^/O.
131. Page 223
1. 8/20. 8. 104V'2/35. 6. >/2(irV16 - 31/18) + 16/9.
7. 17/162. 9. ikay3 ; 2 ka'^/Z. 11. k7ra^/2. 18. kva^/^,
16. lch\b + Zb')/\2. 17. A:T(a2* - ai4)/2. 19. 5 ikay V3 ; side = 2 a.
21. ^kir/U, 28. 36fcir/16. 26. fc7rV20. 27. 19 A;ir/4.
29. kaK 81. 641 A;/756. 83. 846290 A;/189. 86. 21026 ik/10395.
§ 136. Page 227
1. 3/8. 8. 1/6. 6. 3/4. 7. 1/4. 9. 1/20. 11. 2/ir.
18. 2 a/3. 16. 3 VS. 17. (4a/3ir, 4 6/3ir).
19. 3 ^/4 from vertex. 21. 3 a/8 from center, on axis of revolution a.
28. Dist. fromcenter=2asina/(3a). 25. f-^, ^-IzL^lLniV
\e+l 4e — 4e-i/
27. (3/5,12/35). 29. (1,-1/2). 81. (1,0). 88. (-5/6,0).
86. [ (24 - 6 ir2)/ir», (2 ir* - 12)/ir2].
§ 136. Page 229
1. ir(aV2 + 62); a>6. 8. 3iraV4. 6. 1.558+. 7. ira62[ir/(2 a)
-(l/4)sin(2ir/a)]. 9. 4 7rr3/3. 11. jr^aK 18. 32iraV105.
16. ^ = log \/2 ; « = 0.6192 ; y = 0.2059 ; I = fe(x/4 - 23/48) .
17. ^ = 3ira2; x'=ira; y"=6a/6; I = (iraV3)(8ir2 + 5).
19. ^ = 7r(62- a2)/4 ; « = (a2 + a6 + 62)/2(6 + a) ; 1= 3 ir(6* - a*)/32.
21. ^ = 2c/3. 28. z=3/4. 26. x = ira/2 + 64 a/(46ir).
29. 4 7rr3(l - cos* a)/3 ; 2 a = angle of cone. 81. log tan (x/2 + ir/4).
88. 8 a. 86. (3/10) mass times square of radius.
87. (1/5) mass times sum of squares of other two semi-axes.
47. 2/x ; 1/2. 49. 2 r/ir. 61. 3025 irfc/27. 68. 382812.5 ir» ft. Ib./min.
61. IT.
14 ANSWERS
§ 140. Page 236
1. 2x = 3y-ll; 4x = y + ll. 9. u = 86.2P-w«. 11. ^i = .66 «-•».
18. d = 14.8t««. 19. y = e2«/8. 91. y = 10e-*. 98. y = 4e--2«.
§ 141. Page 240
1. /(x)= 0.6x2 -1.4 aj + 2.5 8. <f>(m) = m^ -^ m\ 5. P=e^'10-^^.
7. D = 10(»-JO/« 9. 0.012 X 10-7BI «.
11. Tungsten: C = .00000272 F»-«o ; IT = 10300 T-i**; B = 31.6 F«».
§ 145. Page 249 .
1. X = 1/2. 8. X = 1.88 11. About ± 40 ft. 18. About 420 ft.
15. T changes by about 1.1 9^, 1.4 96, .06 % respectively.
17. 0.520, 0.530, 0.541, 0.590 calories
§ 147. Page 256
1. tanx = x. 8. cosx = l— xV2.
The cubics are : 6. l+x + xV2 + xV6 ; | -E^l < .000004
7. X - xV2 + xV3 ; I ^1 < .0004
9. e-2[l - (X - 2) + (a; - 2)2/2 - (x - 2) V6] ;\E\< .0004
11. 1 + 2(X - ir/4)+ 2(X - 7r/4)2 +(8/3)(x - ir/4)8 ; | -E? |< .000007
16. X - xV6 + XV120. 17. X < 5^ 50'.
§ 148. Page 258
1. 8. 5. 7. 9.
Max. none. x = 1. none. (2 nir + ir/2)». 131°. 2 etc. none.
Min. X = 0. none. none. (2 rnr — ir/2)*. 291°.4 etc. x = 0.
§ 150. Page 262
1. 1. 8. 3. 5. -1. 7. 2/3. 9. log (a/6). 11. 1.
18. - 1/2. 15. 0. 17. 1. 19. log a - 1. 21. 0. . 28. 3.
25. 4. 27. 3. 29. 1. 81. 3. 83. x. 85. 2. 87. 3/2.
§ 151. Page 264
1. 0. 8.-00. 5. -1/2. 7. 0. 9. 0. 11. 0. 18. 1.
15. e». 17. 1. 19. 1. 21. 1.102
§ 153. Page 268
11. (V2/2)[l + (X - t/4)-(x - )r/4)V2 ! -(x - ir/4)V8 I + — ]•
18. (X - 1) - (x - 1)2/2 +(x-l)V3-.
ANSWERS • 15
§ 156. Page 276
• The answers for Ex's. 1-9 are in the order «,,, z^y, Zyy
1, 2, 0, - 2. 8. 2 cos (x2 + y2)- 4 x^ sin (aj2 + y2), ^4xymi(x^ + 2^),
2cos(x2 + 2,2)-4v2sin(x^ + 2^). 5. ^^^^ (J^,^ (^^,-
7. (6aj-h2y + 4a;» + 4a;2y)e«, 2(a; + y)(l +2aJ2^)e«,
(2aj + 6y + 4aJ2/2 + 4y3)e« ; u = «« + y*.
yg^ga -2xy x^-y^ . 11 «^, *.
(«2 + y2)2' (X2 + 2/2)2' (X2 4-^2)2 1, „
17. Area=jr; Z«= &sin C, Z^ = asin C, Kc^ahcosC; Diag. = D;
Da = 2(a — 6 cos 0), Dj, = 2(6 — a cos O), Do = 2 ah sin C.
§ 160. Page 283
1. — aj/(4y). 8. (y - x2)/(y2 — x). 7. Errors due to Aa, AB,
A4: 19fe; .30 9^; .84 9b. 9. 3%. 11. 20.6, 19.3, 101.6
§ 166. Page 295
7. (x-3)/12=(2-2/)/16 = 3-2. 9. {B=y, 2 = 4.
11. cos-i(2/\/39). 18. 3x + 42/ = 26; 2x + 4y+V62 = 26.
16. (0, 0, 0). 26. u and tj are solutions of the '* normal equations "
uZaxi + vSa6 = Sac ; uSa6 + rS66 = S6c ; where Soa = ai^ + au* + ^a*,
Sa6 = ai&i + 0262 + os&a, etc. 81. a = 7/8, /3 = 11690/3.
88. a = - 10.30, /3 = 6.48
§ 169. Page 301
I. 2x + 42/ + 2 = 18; (x - 4)/2 =(2/ - 2)/4 = 2 - 2.
8. 3x-22/+2 = 18; (x- 6)/3 =(1 - 2/)/2 = 2 - 2.
6. 3x + 42/-62 = 0; (x-3)/3=(y~ 4)/4 =(6 - 2)/6.
.7. ex + cy4-22 = 4c; x/e =(2/ - 2)/c =(2 - c)/2.
9. cos-i(±2/V6). _ 11. x+y-f-V2 2 = 2a, x = 2/ = 2/V2.
\/3 X4-3 y+2 2=4 a, x/ \/3 = y/^ = 2/2. 18. xxo - y^o - 2(2 + 2©) = ;
(0, 0, 0) ; no extreme. 16. x'^-\-i/^ = l -h z^\ -'X-\-Zy-\'Zz = l.
17. 9jB2 + 4y2 + 362i = 36; 9x + 6y + 18>/22 = 36.
19. 8x-3y-2 = l; 9x + 6y-2 = 20.
§ 170. Page 304
8. 2 a?(3 ir/4 - 2). 6. V2 a2 ; a = side of square.
II. (2 ir/3)( Vm(2 fe + m)« - w?).
16 ANSWERS
§ 172. Page 306
1. W. 8. coB-i (1/6). 7. (^1 - ^a) Vfl?TT2, 2 » Va^ + 6».
§ 172. Page 306 (Second List)
I. rsin.2^; f«co82d. 8. 2rc2«(co8d— sin^) ; 2fc««(cofltf + smtf).
7. 2 cos e(r* — sin« d)/r ; 2 sin ^(r* + cos* e)/r.
9. (^cos^— sinolog r)/r, (^ sin ^ + cos ^ log r)/r. 11. « = 286/116.
18. 4 a«(3 IT - 4)/9. 17. (0,0). 88. The inscribed cube.
25. iS' = -7.97 + .000107 P«. 87. (a) ff = 83.68 - 1.188 P.
(6) ff = 31 .64 - 0.966 P - 0.0087 P«. (c) ff = 40.86 - 10.73 log P.
88. sin-i (1/V3) and sin-i (1/3).
86. -x/2=(y-2)/0 = (4 2-ir«)/(4ir);
(x + 2)/0 = - y/2 = (2 - x2)/(2 r) ; 8 « - 4 jr2 + ir» = ;
y — irz +ir» = 0.
87. aVl + A;*(^— 0i). Value of y should be a cos sin 0.
§ 175. Page 314
17. aj2 - y* =- 3aj ; 6/4. 19. xdz + ydy = 0. 91. vy" + y'« = 0.
28. x* - y« — xj/(y' ~ l/y') = a« - 62. 26. dy = xdx,
*^- dt«~ ^' "Si"^' d^~ ^ *dt' dT* — dt
§ 178. Page 318
II. y = 1 4- ce^o". 18. y - 1 = c(x + 1). 16. 1 +y«= c(l + ai*).
17. 2 y = cx« + c'. 88. |/« = cC. 96. y = 3(1 + x).
97. Ti = 200 e»/» ; * = (2/ir) log 1.1
99. (B - 5')/(^ - g) = e('-^)(«+«). 81. «« = c - 2 *«/<•
§180. Page 323
1. y = (c + x)e^/«. 8. y=2sinx-2 + ce"**"*. 5. xj/a(2 + ex) = 1.
7. 2/r = 1 — ^ + cer^. 9. y = tan x — 1 + ce-**°». 11. 8 = c«-* — 1 + «.
18. y = ce» — (sinx + cosx)/2. 15. y = kx-\- c Vl + x».
§ 181. Page 324
1. 3y5 = l— 3x + cc-8». 8. (x + y)/(x - y) = ce*i'.
7. x» + 3 xV + y» = c. 9. xe*^* = c. 15. (a)x — y = c.
17. y* = 4 c(x + c). 19. y^ = kx.
ANSWERS 17
§ 183. Page 329
1. a = Ae^ + Be-*, 8. s = a sin (i + 6).
5. y = A&* + Ber** oxy — a sin Qci + 6).
7. S. H. M. with zero amplitude ; no motion.
9. a = \/l04, 6 = tan-i6 ; a = 6, 6 = ir/2.
§184. Page 332
1 = cie* + C26*». 8. y = Cie» + Cje-*'». 6. y = {c\Z + C8)e».
7. y = (ci sin ^2 « + C2 cos V2'x)e». 9. y = CiC** + Cje^..
11. y = cie*«/3 + C2e8»/2. . 18. y = cie^* + c^sr^, 15. y = ci + C2e-*».
17. % = c-*</2[cicVii=iSt/2 + Cjte~>/»«=^/«], c < 6V4 ;
X = e-w/«[ci sin V4 c — hHI'l + Cjcos V4c — 6««/2], c > ftV**
19. X = e-'/2^5-±2^c^^/2 + ^"^e-v^/'l ; No ;
L 10 10 J
^ ^ ^„ ri+A/6 + 2ro^^,,^V6^1>2.o^^^,n
L 2V6 2V6 J
§ 186. Page 337
5. 3 < = 2(V« - 2 c) V>/s + c + C. 7. y = e8»/4 + CiX + C2.
9. y = 6cosa5 + 4a5sinaB — «*cosx4 ca5 + c'.
11. yz=.^ —(1/16) C082 X + CiX* + C2X* + C3X + C4.
15. y = fcB«(x« - 4 te 4- 6 12)/12.
§189. Page 341
9. y = 3x/4 + x«/4 + Cie* + CjC«* + Cs-
11. y = cie» + cjc-* + csc-^ + (C4 + x/12)c2». 18. y = Cie» + Cje*/* + Cj.
15. y = — Jxsinx + cicosx + C2sinx + C8. 17. y = Cie»+C2e"*+C8X4-C4.
19. 2Vl + cc»4-log[(Vl + c€*- 1)/(V1 + ce* + 1)] + c'.
21. y = (x — x2/2) log X + CiX* + CaX + C3. 25. y = CiX* — x logx + C2.
27. y = (1 — 6)/62 - log (a + 6x)+ Ci(a + &x)«i + C2 (a + &x)"4, where
mi and m^ are the roots of m* + (6 — l)m — 6* = 0.
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