George A. (George Abbott) Osborne.

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AN

ELI TAEY TREATISE

ON THE

DIFFERENTIAL AND INTEGRAL
CALCULUS,

WITH EXAMPLES AND APPLICATIONS.

BY

GEORGE A. OSBORNE, S.B.,

Professor of Mathematics in the Massachusetts
Institute of Technology.

>**:<

D. C. HEATH & CO., PUBLISHERS

BOSTON NEW YORK CHICAGO
1903

fi

By GEORGE A. OSBORNE

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QA

PREFACE.

This work, intended as a text-book for colleges and scien-
tific schools, is based on the method of limits, as the most
rigorous and most intelligible form of presenting the first
principles of the subject. The method of limits has also the
important advantage of being a familiar method; for it is
now so generally introduced in the study of the more ele-
mentary branches of mathematics, that the student may be
assumed to be fully conversant with it on beginning the
Differential Calculus.

The rules or formulae for differentiation in Chapter III.
differ in one respect from those in similar text-books, in being
expressed in terms of u instead of x, u being any function
of x. They are thus directly applicable to all expressions,
without the aid of the usual theorem concerning a function of
a function.

After acquiring the processes of differentiation, the student
in Chapter V. is introduced to the differential notation, as a
convenient abbreviation of the corresponding expressions by
differential coefficients. This notation has manifest advan-
tages in the study of the Integral Calculus and in its
applications.

In Chapter IX. and subsequent pages I have introduced for

Partial Differentiation the notation — , which has recently

ox
come into such general use.

iv PREFACE.

The chapters on Maxima and Minima have been placed
after the applications to curves, as the consideration of that
subject is much simplified by representing the function by
the ordinate of a curve. Maxima and Minima may be taken,
if desired, with equal advantage immediately after Chapter
XIII.

In Chapter X., Integral Calculus, I have taken the problem
of finding the Moment of Inertia of a plane area, as a better
illustration of double integration than that of finding the
area itself. The student more readily comprehends the inde-
pendent variation of x and y in the double integral,

I I (?? + y*)d%dy, than in J I dxdy.

A few pages of Chapter XII., Integral Calculus, are devoted
to a description of the Hyperbolic Functions together with
their differentials, and a comparison is made with the cor-
responding Circular Functions.

G. A. OSBORNE.

Boston, 1895.

CONTENTS.

DIFFERENTIAL CALCULUS.

Chapter I.
arts. FUNCTIONS. PAGE8>

1-4. Definition and Classification of Functions 1, 2

5. Notation of Functions. Examples 3,4

Chapter II.

DIFFERENTIAL COEFFICIENT.

6, 7. Limit. Increment 5

8-10. Differential Coefficient. Examples 6-9

Chapter III.

DIFFERENTIATION.

11-13. Differentiation of Algebraic Functions. Examples 10-21

14-16. Differentiation of Logarithmic and Exponential Functions.

Examples 21-27

17, 18. Differentiation of Trigonometric Functions. Examples. . 27-32
19, 20. Differentiation of Inverse Trigonometric Functions. Ex-
amples 32-37

21,22. Differentiation of Inverse Function and Function of a

Function. Examples 37-40

Chapter IV.
SUCCESSIVE DIFFERENTIATION.

23, 24. Definition and Notation 41

25. The nth Differential Coefficient. Examples 42-45

26. Leibnitz's Theorem. Examples 45-47

v

VI CONTENTS.

Chapter V.
Arts. DIFFERENTIALS. PagK8

27. Differentials as related to Differential Coefficients 48, 49

28. Differentiation by Differentials 49

29. Successive Differentials. Examples 50, 51

Chapter VI.
IMPLICIT FUNCTIONS.

30. Differentiation of Implicit Functions. Examples 62-54

Chapter VII.

EXPANSION OF FUNCTIONS.

32-36. Maclaurin's Theorem. Examples 65-60

37-41. Taylor's Theorem. Examples 60-63

42-45. Rigorous Proof of Taylor's Theorem 64, 65

46-49. Remainder in Taylor's and Maclaurin's Theorems 66-68

Chapter VIII.
INDETERMINATE FORMS.

50, 51. Limiting Value of a Fraction 69

62, 53. Evaluation of §. Examples 70-73

64-57. Evaluation of §, oo, oo — co. Examples 73-76

68. Evaluation of Exponential Forms. Examples 76-78

Chapter IX.
PARTIAL DIFFERENTIATION.

59,60. Partial Differential Coefficients of First Order. Exam-
ples 79, 80

61-63. Partial Differential Coefficients of Higher Orders. Exam-
ples 80-82

64,65. Total Differential of Functions of Several Variables.

Examples 82-84

66. Condition for an Exact Differential. Examples 85

67. Differentiation of Implicit Functions 86

08, 69. Taylor's Theorem for Several Variables 87, 88

CONTENTS. VU

Chapter X.

CHANGE OF VARIABLES IN DIFFERENTIAL

COEFFICIENTS.
Arts. Pages.

70. Changing from x to y 89

71, 72. Changing from y to z 90

73. Changing from x to z. Examples 90-92

Chapter XI.

REPRESENTATION OF VARIOUS CURVES.

74-85. Rectangular Co-ordinates 93-98

86-93. Polar Co-ordinates 98-102

Chapter XII.

DIRECTION OF CURVE. TANGENT AND NORMAL.
ASYMPTOTES.

91-97. Direction of Curve. Subtangent and Subnormal.

Examples 103-108

98,98£. Differential Coefficient of the Arc 108,109

99. Equation of the Tangent and Normal. Examples . . . 109-112

100-106. Asymptotes. Examples 112-116

Chapter XIII.

DIRECTION OF CURVATURE. POINTS OF INFLEXION.

107-109. Direction of Curvature 117

110. Points of Inflexion. Examples 118,119

Chapter XIV.

CURVATURE. CIRCLE OF CURVATURE. EVOLUTE
AND INVOLUTE.

111-113. Definition of Curvature ; Uniform and Variable 120,121

114,115. Radius of Curvature. Examples 121-124

116. Centre of Curvature 124, 126

117-121. Evolute and Involute. Examples 125-128

Vlll CONTENTS.

Chapter XV.

ORDER OF CONTACT. OSCULATING CIRCLE. „
Arts. Pages.

122, 123. Consecutive Common Points 129, 130

124,125. Osculating Curves 130,131

126-128. Analytical Conditions for Contact 131-133

129,130. Osculating Circle. Examples 133-136

Chapter XVI.

ENVELOPES.

131-133. Series of Curves. Definition of Envelope 137, 138

134-136. Equation of Envelope 138-140

137. Evolute, the Envelope of Normals. Examples 140-144

Chapter XVII.

SINGULAR POINTS OF CURVES.

138-141. Multiple Points 145-148

142, 143. Points of Osculation. Cusps 149, 150

144. Conjugate Points. Examples 150-152

Chapter XVIII.

MAXIMA AND MINIMA OF FUNCTIONS OF ONE
INDEPENDENT VARIABLE.

145-149. Definition. Conditions for Maxima and Minima de-
rived from Curves 153-157

150, 151. Conditions for Maxima and Minima by Taylor's

Theorem. Examples 157-162

Problems in Maxima and Minima 162-164

Chapter XIX.

MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL
INDEPENDENT VARIABLES.

152-155. Definition. Conditions for Maxima and Minima by

Taylor's Theorem. Examples 165-171

CONTENTS.

INTEGRAL CALCULUS.

Chapter I.
ART8 elementary forms of integration. p AQE8

1, 2. Definition of Integration. Elementary Principles 173, 174

3. Fundamental Integrals 175, 176

4-7. Derivation and Application of Fundamental Formulae.

Examples 176-187

Chapter II.
INTEGRATION OF RATIONAL FRACTIONS.

8, 9. Preliminary Operation. Factors of Denominator 188, 189

10. Case I. Examples 189-191

11. Case II. Examples 191,192

12. Case III. Examples 192-195

13. Case IV. Examples 195-198

Chapter III.

INTEGRATION BY RATIONALIZATION.

14-16. Fractional Powers of x and of a + bx. Examples 199-201

17. Fractional Powers of a + b x 2 . Exampl es 201, 202

18,19. Expressions containing V±x a +ax+6. Examples... 202-204

20. Integration by Substitution. Examples 204, 205

Chapter IV.

INTEGRATION BY PARTS. INTEGRATION BY SUCCESSIVE
REDUCTION.

21. Integration by Parts. Examples 206-208

22-24. Formulae of Reduction. Examples 208-214

Chapter V.

TRIGONOMETRIC INTEGRALS.

25-27. Integration of tan n xdx; of seCscefc; of tan m xsec n xdcc.

Examples 215-218

28, 29. Integration of sin m x cos n xdx. Examples 219-222

X CONTENTS.

Arts. Pages.
30. Trigonometric, transformed into Algebraic, Integrals.

Examples 222-224

31, 32. Trigonometric Formulae of Reduction. Examples 224-226

33-35. Integration of — and — ;

a + b sin x a+b cos x

of e ax siii7ixdx and e 8 * cos nxdx.

Examples 226-229

Chapter VI.
INTEGRALS FOR REFERENCE.

36. Integrals containing Va 2 — a? ; Vx z ±a* ; ± ax 2 + bx + c. 230-235

Chapter VII.

INTEGRATION AS A SUMMATION. DEFINITE INTEGRALS.

37-40. Integration, the Summation of an Infinite Series 236-^240

41-43. Definition of Definite Integral. Examples 240-244

Chapter VIII.

APPLICATION OF INTEGRATION TO PLANE CURVES.
APPLICATION TO CERTAIN VOLUMES.

44-47. Areas of Curves. Examples 245-249

48, 49. Lengths of Curves. Examples 249-252

50,51. Surfaces of Revolution. Examples 252-255

52. Other Volumes. Examples 255-257

Chapter IX.

SUCCESSIVE INTEGRATION.

53-56. Double and Triple Integrals. Examples 258-260

Chapter X.

DOUBLE INTEGRATION APPLIED TO PLANE AREAS
AND MOMENT OF INERTIA.

57-60. Double Integration. Rectangular Co-ordinates. Ex-
amples 261-264

61-03. Double Integration. Polar Co-ordinates. Examples . . 264-266

CONTENTS. XI

Chapter XI.

Arts surface and volume of any solid. Paok8

64, 65. Area of any Surface. Examples 267-270

66, 67. Volume of any Solid. Examples 270-273

Chapter XII.

HYPERBOLIC FUNCTIONS. CYCLOID, EPICYCLOID, AND
HYPOCYCLOID. INTRINSIC EQUATION OF A CURVE.

69-71. Definitions of Hyperbolic, and Inverse Hyperbolic,

Functions 274-276

72, 73. Differentiation of Hyperbolic Functions. Inverse Hyper-
bolic Functions as Integrals 276, 277

74, 75. Hyperbolic Functions and the Hyperbola. Exercises . . 278-280

76-82. Equation and Properties of the Cycloid 280-284

83-89. Equations and Properties of the Epicycloid, and Hypo-
cycloid 284-288

90-93. Intrinsic Equation of a Curve and of its Evolute.

Examples 289-292

DIFFERENTIAL CALCULUS.

CHAPTER I.

FUNCTIONS.

1. Definition of a Function. When the value of one variable
quantity so depends upon that of another, that any change
in the latter produces a corresponding change in the former,
the former is said to be a function of the latter.

For example, the area of a square is a function of its side ;
the volume of a sphere is a function of its radius ; the sine,
cosine, and tangent are functions of the angle ; the expressions

x>, logO^ + l), Vx{x + 1),
are functions of x.

A quantity may be a function of two or more variables.
For example, the area of a rectangle is a function of two
adjacent sides ; either side of a right triangle is a function of
the two other sides; the volume of a rectangular parallelo-
piped is a function of its three dimensions.

The expressions

rf + xy + tf-, log(ar J + 2 / 2 ), of*,

are functions of x and y.
The expressions

xy + yz+zx, \p+^, log (x 2 + y - z),

^ z

are functions of x, y, and z.

2. Dependent and Independent Variables. If y is a function
of x, as in the equations

y^x 2 , y = ta,n4x, y = e x ,

2 DIFFERENTIAL CALCULUS.

x is called the independent variable, and y the dependent
variable.

It is evident that whenever y is a function of x, x may be
also regarded as a function of y, and the positions of dependent
and independent variables reversed. Thus from the preceding
equations, _

aj=Vy, £c = £tan -1 2/, x = log e y.

In equations involving more than two variables, as

z + x — y = 0, w -f- wz -f zx + y = 0,

one must be regarded as the dependent variable, and the
others as independent variables.

3. Explicit and Implicit Functions. When one quantity is
expressed directly in terms of another, the former is said to
be an explicit function of the latter.

For example, y is an explicit function of x in the equations,

y = x 2 -\-2x, y= -y/x 2 + 1.

When the relation between y and x is given by an equation
containing these quantities, but not solved with reference to y,
y is said to be an implicit function of x, as in the equations,

2xy + y 2 = x 2 + 1, y + logy = x.

Sometimes, as in the first of these equations, we can solve
the equation with reference to y, and thus change the function
from implicit to explicit. Thus we find from this equation,

y = — x±V2x 2 + l.

4. Algebraic and Transcendental Functions. An algebraic
function is one that involves only the operations of addition,
subtraction, multiplication, division, involution and evolution
with constant exponents. All other functions are called tran-
scendental functions, including logarithmic, exponential, trigo-
nometric, and inverse trigonometric, functions.

/

FUNCTIONS. 3

5. Notation of Functions. The symbols F(x), f(x), (x),
ij/(x), and the like, are used to denote functions of x. Thus
instead of " y is a function of a;," we may write

y=f(x) or y = (x).

A functional symbol occurring more than once in the same
problem or discussion is understood to denote the same func-
tion or operation, although applied to different quantities.
Thus, if

/(*)=*■ + 6, . . (1)

then f(y) = 2/ 2 + 5, /(a) = a 2 + 5,

/(a + l) = (a + l) 2 + 5 = a 2 + 2a + 6,

/(2) = 2 2 + 5 = 9, /(1) = 6.

In all these expressions /( ) denotes the same operation as
defined by (1) ; that is, the operation of squaring the quantity
-and adding 5 to the result.

The following examples will further illustrate the notation
of functions.

EXAMPLES.

1. If f(x) = 2s 3 -ar J -7z + 6, show that
/(3) = 30, /(2) = 4, /(0) = 6, /(1) = 0,
/(-2) = 0, /(f) = 0, f(x - 2) = 2a?- 13^ + 21*,
f{x + ft) = 2x? + (6 ft - 1) x 2 + (6ft 2 - 2ft - 7) a + 2ft 3

2. Given /,(#) = 2y 4 -f-^l, f(y)=7y'-6y-hl', show that

/,(1)=/,(1), A(t)-»Ai /i(-2)=/ 2 (-2),
/i(0)=/ 2 (0).

3. If /(o) = ^Lz^, show that

^ v ' a + 1

/(«)-/(&) =
!+/(«)/(&) l + a6

4 DIFFERENTIAL CALCULUS.

4. If \$(ro) = (ro + l)m(m-l)(ro-2), show that
(1) = \ x + )~"ft( g) , as ft approaches zero,
da; ft

The differential coefficient of a function may then be defined

DIFFERENTIAL COEFFICIENT. 9

as the limiting value of the ratio of the increment of the function
to the increment of the variable, as these increments approach
zero. That is, the differential coefficient of the function (»)
with respect to x, is

the limit of +(* + *> -»(*) ,
h

as h is indefinitely diminished.

The differential coefficient is sometimes called the derivative.

Note. — In Art. 94 will be found a geometrical illustration
of the differential coefficient.

EXAMPLES.

Following the process of Art. 9, derive the following dif-
ferential coefficients :

>/ 1. y = 3x*-2x. ^ = 6x-2.

u dx

y 2. y = x 4 +5. Ic = 4a;3 -

3. y = (x-l)(2x + 3). §p = 4z + l.

V4 - ^-_!

y ~~ x' dx~ x 2 '

a dy 2 a

Xm x — a dy 2a

7. y = x J .

v 8. 2/ = V« 2 -2.
2

-9. y =

dx

'(* +

a)*

dy_
dx

3 a?*

: 2 *

dy

X

dx

Vx 2 -

-2

dy

1

10. y = x\$.

dy l

d» 3 a*

CHAPTER III.

DIFFERENTIATION.

U. The process of finding the differential coefficient of a
given function is called differentiation. The examples in the
preceding chapter are introduced to illustrate the meaning
of the differential coefficient, but this elementary method of
differentiation is too tedious for general use.

Differentiation is more readily performed by the application
of certain general rules, which may be expressed by formulae.
In these formulae u and v will denote variable quantities, func-
tions of x ; and c and n, constant quantities.

It is frequently convenient to write the differential co-
efficient of a quantity

■ — u, instead of — •
dx dx

Thus the differential coefficient of (u + v) is more con-
veniently written

A( M + V ) rather than d ( u + v ) .
dx K " dx

12. Formulas for Differentiation of Algebraic Functions,

T ^_ i
dx

II. — = 0.
dx

md f , \ du , dv
dx dx dx

tit d , \ du , dv

DIFFEBENTIATION. 11

V.

d , x du

■ — . leu) = c •

dx dx

VI.

du dv
V u

d A A dx dx

dx\v)~ v 2

VII.

— (u n ) = nu n "* —
dx dx

These formulae express the following general rules of dif-
ferentiation :

I. TJie differential coefficient of a variable with respect to itself
is unity.

II. The differential coefficient of a constant is zero.

III. The differential coefficient of the sum of two variables is
the sum of their differential coefficients.

IV. The differential coefficient of the product of two variables
is the sum of the products of each variable by the differential
coefficient of the other.

V. The differential coefficient of the product of a constant and
a variable is the product of the constant and the differential co-
efficient of the variable.

VI. The differential coefficient of a fraction is the differential
coefficient of the numerator multiplied by the denominator minus
the differential coefficient of the denominator multiplied by the
numerator, this difference being divided by the square of the
denominator.

VII. The differential coefficient of any power of a variable is
the product of the exponent, the power with exponent diminished
by 1, and the differential coefficient of the variable.

13. Derivation of Formulas.

Proof of I. This follows immediately from the definition of

a differential coefficient. For since — = 1, its limit — - = 1.

A# dx

Proof of II. A constant is a quantity whose value does

not vary. Hence

12 DIFFERENTIAL CALCULUS.

Ac = and — = 0;

therefore its limit — = 0.

dx

Proof of III. Let y = u+v, and suppose that when x is
changed into x + h, y, u, and v become y' } u\ and v' j then

2/'= w'-j-'y';

therefore y f — y = w'— u + v'— v;

that is, Ay = Aw + Aw.

Divide by Aa;; then

A^__ Aw Av

Ax Ax Ax

Now suppose Ax to diminish and approach zero, and we
have, for the limits of these fractions,

dy _du dv
dx dx dx

If in this we substitute for y, u+v, we have

d , , v du . dv

— (u + v) = 1

dx dx dx

It is evident that the same proof would apply to any
number of variables connected by plus or minus signs. We
should then have

d , , , v du , dv , dw ,

dx dx dx dx

Proof of IV. Let y = uv\ then
y f =u'v',
and y'—y = uV~wv = (m'-«)v'+«(v'-v);

that is, Ay = v'Aw -f wAv.

DIFFERENTIATION. 13

Divide by Aa;; then

Aw ,Au . Av
Aa; Ax Aa;

Now suppose Aa? to approach zero, and, noticing that the
limit of v' is v, we have

dy du . dv
dx dx dx

, i , • d , ^ du , dv

that is, — (uv) = v \-u —

dx dx dx

This formula may be extended to the product of three or
more variables. Thus we have

d , x d , x d/\, dm

— (uvw) = — (uv • w) = w — (uv) + uv —
dx dx dx dx

(du . dv\ ,
dx dx)

dw

uv —

dx

du . dv , dw

= vw \-uw h uv •

dx dx dx

Similarly, for the product of four functions, we have

d , s. du . dv , dw , dz

— {uvwz) = vwz h wzu f- zuv h uvw

dx dx dx dx dx

A similar relation holds for the product of any number of

variables.

dc
Proof of V. This is a special case of IV., — being zero.

But we may derive it independently thus :

y = cw,

y'z=cu',

y'—y = c(u f

-w),

Ay = cAu,

Ay __ Au

Ax Ax

14 DIFFERENTIAL CALCULUS.

dy du d , s. du

— = c — , or ■ — (cu) = c —
dx dx dx dx

Proof of VI. Let y = -,

then 2/'=^;

,, £ , u' u u'v — uv' (u'—u)v — u(v ( ~-V)
therefore y—y = -r = -, = — ; — -J

V V

that is, Ay =

V V vv vv

vAu — uAv

v'v

Au Av

v u —

Ay Ax Ax

Ax v'v

Now suppose Ax to diminish towards zero, and, noticing
that the limit of v' is v, we have

Since
therefore
By IV,

therefore

dy

du

v

dx

dv
dx

dx

V 2

VI. from IV.

thu!

u

y = ->

V

yv = u.

dx dx

du
dx'

dy

v-£ =

dx

du v
dx ^

i dv #
i dx*

dy _

du

dx

dv

u —

dx