ELEMENTS OF CALCULUS
HARPER'S MATHEMATICS SERIES
Charles A. Hutchinson, Editor
ELEMENTS
OF
CALCULUS
SECOND EDITION
Th u rman S. Peterson, Ph.D.
PROFESSOR OF MATHEMATICS
PORTLAND STATE COLLEGE
HARPER & BROTHERS
PUBLISHERS
NEW YORK
ELEMENTS OF CALCULUS, Second Edition
Copyright 1950 by Harper & Brothers
Copyright © 1960 by Thurman S. Peterson
Printed in the United States of America
All rights in this book are reserved.
No parts of the book may be used or reproduced
in any manner whatsoever without written permission except in the case of brief quotations
embodied in critical articles and reviews. For
information address Harper & Brothers,
49 East 33rd Street, New York 16, N.Y.
Library of Congress catalog card number: 59-13918
Contents
xi
Preface
Introduction
Reference formulas and curves
I
Variables, Functions, and Limits
13
1. Introduction. 2. Rate of Change. 3. The Concept of Area.
4. Constants and Variables. 5. Functions. 6. Limit of a
Function. 7. Continuity. 8. Infinity. 9. Limit of a Sequence.
2 Differentiation and Applications
33
10. Increments. 11. Derivative. 12. Derivatives of Powers
of x. 13. Slope of a Curve. 14. Velocity and Acceleration.
15. Maxima and Minima; Critical Points. 16. Higher Derivatives. 17. Points of Inflection; Concavity. 18. Applications
of Maxima and Minima. 19. Differentials. 20. Approximations and Errors.
3 Integration of Powers
63
21. Antidifferentiation. 22. Integration of Powers. 23. Constant of Integration. 24. Differential of Area. 25. Area as an
Integral. 26. Calculation of Areas. 27. Area as a Limit.
28. Definite Integrals. 29. Fundamental Theorem. 30. Plane
Areas in Rectangular Coordinates. 31. Volumes of Solids of
Revolution.
4 Applications of Integration
87
32. Moment of Mass; Centroids. 33. Centroid of a Plane Area.
34. Centroid of a Solid of Revolution. 35. Moment of Inertia;
Radius of Gyration. 36. Moment of Inertia of an Area.
37. Moment of Inertia of a Solid of Revolution. 38. Fluid
Pressure.
39. Work.
5 Differentiation of Algebraic Functions
40. Introduction. 41. Formulas for Differentiation. 42. Formulas for Differentiation (Continued). 43. Differentiation of
Implicit Functions. 44. Summary of Formulas.
V
108
Contents
vi
6 Differentiation of Transcendental Functions
121
45. Transcendental Functions. 46. Properties of Trigonometric Functions. 47. Limit of sin 0/0. 48. Derivatives of
Trigonometric Functions. 49. Properties of Inverse Trigonometric Functions. 50. Derivatives of Inverse Trigonometric
Functions. 51. Exponential and Logarithmic Functions.
52. Derivatives of Logarithmic Functions. 53. Derivatives of
Exponential Functions. 54. Summary and Applications.
7 Parametric Equations, Curvature, and Roots
55. Parametric Representations. 56. Derivatives in Parametric
Form. 57. Differential of Arc Length. 58. Curvature.
59. Circle of Curvature. 60. Center of Curvature. 61. Evolutes. 62. Newton's Method.
146
8 Differentiation with Respect to Time
160
63. Time-Rates. 64. Curvilinear Motion. 65. Tangential and
Normal Components of Acceleration. 66. Angular Velocity
and Acceleration.
9 Polar Coordinates
67. Polar Coordinates.
173
68. Locus of a Polar Equation.
69. Intersection of Polar Curves. 70. Angle Between the
Radius Vector and Tangent.
71. Differential of Arc.
72. Curvature. 73. Radial and Transverse Components of
Velocity and Acceleration.
10 Indeterminate Forms
74. Limits. 75. Rolle's
185
76. Law of the Mean.
77. Cauchy's Theorem. 78. The Indeterminate Form 0/0.
79. The Indeterminate Form oc/oo. 80. The Indeterminate
Form 0 oo. 81. The Indeterminate Form co - oo. 82. The
Theorem.
Indeterminate Forms 0°, oo°, 1°°.
I
I
Curve Tracing
196
83. Graphs of Curves in Rectangular Coordinates. 84. Oblique Asymptotes Determined by Inspection. 85. Asymptotes
to an Algebraic Curve. 86. Singular Points of Algebraic
Curves.
87. Summary of Curve Tracing.
12 Integration Formulas
88. Introduction. 89. Formulas of Integration. 90. Integration of Powers. 91. Integration of Exponential Functions.
207
Contents
vii
92. Integration of Trigonometric Functions. 93. Transformations of Trigonometric Integrals. 94. Integrals Giving Inverse
Trigonometric Functions. 95. Additional Formulas of Integration. 96. Improper Integrals.
13 Integration Procedures
227
97. Introduction. 98. Integration by Parts. 99. Algebraic
Substitutions. 100. Trigonometric Substitutions. 101. Integration of Rational Fractions. 102. Miscellaneous Substitutions. 103. Use of Integration Tables. 104. Approximate
Integration; Trapezoidal Rule. 105. Simpson's Rule.
14 Additional Applications of Integration
259
106. Area in Polar Coordinates. 107. Length of a Plane
Curve.
108. Centroid and Moment of Inertia of Arc.
109. Area of a Surface of Revolution. 110. Volumes of Solids
with Known Cross Sections. 111. Average Value.
15 Infinite Series
276
112. Sequences and Series. 113. Convergent and Divergent
Series. 114. Theorems on Convergence. 115. The Integral
Test. 116. Comparison Tests. 117. Ratio Test. 118. Alternating Series. 119. Absolute and Conditional Convergence.
120. Power Series.
16 Expansion of Functions
295
121. Introduction. 122. Maclaurin's Series. 123. Algebraic
Operations with Power Series. 124. Differentiation and Integration of Power Series. 125. Approximation Formulas
Derived from Power Series. 126. Taylor's Series. 127. Taylor's Theorem.
17 Hyperbolic Functions
313
128. Definitions of the Hyperbolic Functions. 129. Identities
Involving Hyperbolic Functions. 130. Derivatives and Integrals of Hyperbolic Functions. 131. The Inverse Hyperbolic
Functions. 132. Derivatives of the Inverse Hyperbolic Functions. 133. Integrals Leading to Inverse Hyperbolic Functions.
134. Relations Between Trigonometric and Hyperbolic Functions. 135. Geometric Interpretation of Hyperbolic Functions.
18 Solid Analytic Geometry
136. Rectangular Coordinates. 137. Distance Between Two
Points.
138. Point on the Line Joining Two Points.
325
viii
Contents
139. Direction of a Line. 140. Angle Between Two Lines.
141. Locus of a Point in Space. 142. Equation of a Plane.
143. Normal Equation of a Plane. 144. Planes Determined by
Three Conditions. 145. Equations of a Line. 146. Symmetric
Equations of a Line. 147. Equation of a Surface. 148. Quadric Surfaces.
19 Partial Differentiation
353
Functions of Two or More Variables ; Continuity.
150. Partial Derivatives. 151. Geometric Interpretation of
Partial Derivatives. 152. Partial Derivatives of Higher Order.
153. Increment and Total Differential of a Function.
154. Approximations and Errors. 155. Total Derivatives.
156. Chain Rule for Partial Derivatives. 157. Differentiation
of Implicit Functions. 158. Tangent Line and Normal Plane
to a Curve. 159. Normal Line and Tangent Plane to a Surface.
160. Maxima and Minima. 161. Differentiation of a Definite
Integral. 162. Taylor's Series for Functions of Two Variables.
163. Sufficient Condition for a Maximum or Minimum.
149.
20 Multiple Integrals
387
165. Iterated Integrals. 166. Iterated
Integrals in Rectangular Coordinates. 167. Plane Areas by
Double Integration. 168. Centroid and Moment of Inertia of
a Plane Area. 169. Iterated Integrals in Polar Coordinates.
170. Plane Areas in Polar Coordinates. 171. Volumes by
Double Integration. 172. Volumes in Cylindrical Coordinates.
173. Areas of Curved Surfaces.
174. Triple Integrals.
164. Double Integrals.
175. Iterated Integrals.
Rectangular Coordinates.
176.
Iterated Triple Integrals in
177. Volumes by Triple Integra-
tion. 178. Center of Gravity and Moment of Inertia of a
Solid. 179. Triple Integrals in Cylindrical Coordinates.
180. Triple Integrals in Spherical Coordinates.
21
Differential Equations
182. Solutions of Differential Equations.
183. Differential Equations of First Order and First Degree.
184. Exact Differential Equations. 185. Linear Equations
of the First Order. 186. Equations Reducible to Linear
Equations. 187. Second Order Equations Reducible to First
Order. 188. Applications of First Order Differential Equa181. Definitions.
tions.
189.
Linear Differential Equations of Order it.
425
Contents
190.
ix
Homogeneous Equations with Constant Coefficients.
191. Non-homogeneous Equations with Constant Coefficients.
192. Applications of Linear Differential Equations.
22 Vector Analysis
459
194. Addition of Vectors. 195. Scalar
Multiplication of Vectors. 196. Vector Multiplication of
Vectors. 197. Scalar Triple Product. 198. Vector Triple
Product. 199. Derivative of a Vector. 200. The Gradient.
193. Introduction.
201. The Divergence. 202. The Curl or Rotation. 203. Summary of Vector Differentiation.
204. Line Integrals.
Integrals.
206. Divergence
205. Surface
Theorem.
207. Stokes's Theorem. 208. Summary of Integration.
Table of Integrals
495
Numerical Tables
503
II. Natural Logarithms.
III. Exponential and Hyperbolic Functions. IV. Trigonometric Functions. V. Common Logarithms.
I. Powers, Roots, Reciprocals.
Index
13
Preface
This text is designed to serve as an introductory course in calculus
for students who have some familiarity with the basic principles of
algebra, trigonometry, and plane analytic geometry. The book provides sufficient material for classes that meet each day for one academic
year. The arrangement of the topics, however, facilitates the organization of shorter courses.
Purpose. The basic theory of calculus is developed in this book
with a view to giving the student both a sound understanding of the
fundamental concepts of calculus and a thorough appreciation of its
many applied uses.
Considerable stress has been placed on the logical
structure of the theory in order that students might gain some active
experience in making original mathematical developments. To serve
this purpose all definitions, theorems, and general procedures are
presented in as much detail as the demands of clarity and relative
simplicity permit.
Special features. The fundamental principles of calculus are
clearly stated in precise mathematical terms. Numerous illustrations
and illustrative examples are presented to clarify both the theoretical
and the applied aspects of the subject.
For the convenience of the many students of calculus who are
primarily interested in engineering and other applied fields, the concept of integration and its applications are introduced in the early
part of the book. Also, to assist in the computational aspects of
calculus, the formulas and curves of more elementary mathematics
are given in an Introduction, five numerical tables are given on pp.
503 if., and the answers to all odd-numbered problems are included
in the text. A pamphlet containing the answers to the even-numbered
problems is available to instructors.
Revised features. In order to make the text as adaptable as
possible to modern needs, the basic material on limits, functions, and
continuity has been thoroughly revised with a view to emphasizing
modern mathematical techniques. Extensive additions on the
theoretical structure of calculus have been made, both in illustrations
and exercises, throughout the book.
On the applied side, considerable material has been added to illustrate the many applications of differential equations. Another
important addition supplementing the applied work is a chapter on
xi
xii
Preface
the calculus of vectors. This chapter covers completely the algebra,
differentiation, and integration of vectors, including discussions of
directional derivatives, line integrals, and surface integrals.
In addition to the preceding changes, a completely new chapter on
solid analytic geometry has been included in order to augment the
increased emphasis which has been placed on the applications of
calculus to a space of three dimensions.
Acknowledgments. The author wishes to take this opportunity
to express his appreciation to his many friends and colleagues who have
so graciously criticized and assisted in the revision of the text. In
particular the author would like to thank the many users of the
original edition who have contributed many worth-while suggestions
for the revision.
T. S. P.
Portland, Oregon
May, 1960
ELEMENTS OF CALCULUS
Introduction
Mathematical Formulas
The following list of mathematical formulas and graphs is given to
summarize briefly the computational aspects of more elementary mathematics and to serve as a convenient reference in future work.
GEOMETRY
In the following formulas, r denotes radius, h altitude, l slant height,
b base, B area of base, 0 central angle expressed in radians.
1.
Triangle. Area =
2. Rectangle. Area = bh. Diagonal = 1/b2 + h2.
3. Trapezoid. Area = 2h(bl + b2).
4. Circle. Arc = rO. Circumference = 27rr. Area = nrr2.
5. Circular sector. Area = 2r20.
6. Circular segment. Area = 2r2(0 - sin 0).
7. Rectangular parallelepiped. If a, b, c are the sides
Diagonal = \/a2 + b2 + c2.
Volume = abc.
8. Prism. Volume = Bh.
2bh.
9.
10.
Pyramid. Volume = 3Bh.
Right circular cylinder. Lateral surface = 27rrh. Volume =
7rr2h.
I.
12.
13.
14.
15.
Right circular cone. Lateral surface = rrrl. Volume = 37rr2h.
Sphere. Surface = 47rr2. Volume = 43L?Tr3.
Spherical segment. Volume = 37rh2(3r - h).
Frustum of a pyramid. Volume = 3h(BI + B2 + 1/BiB2).
Frustum of a right circular cone.
Lateral surface = 7rl(r1 + r2). Volume = 37rh(r2 + r2 + r1r2).
16.
Prismatoid. Volume = 6h(B1 + 4B. + B2)-
17.
Quadratic formula. If axe + bx + c = 0, a 0 0, then
ALGEBRA
x=
- b ± 02 - 4ac
2a
1
Introduction
2
18.
Properties of logarithms.
(a) log (11IN) = log M + log N,
(c) log M n = it log ill,
(e)
(b) log (M/N) = log M - log N,
(d) log /M = (1/n) log Al,
(f) logo 1 = 0.
logb b = 1,
19. Factorial numbers. n! = 1.2.3 . . . (n - 1) n. 0! = 1.
20. Binomial expansion.
(a + b)n = an + nClan-lb + C2an-2b2 _}.... .
+
nCran-rbr + ... + bn,
where
n!
n(n - 1) ...,
....
nCr =
r! (n - r)!'
1-2
Arithmetic progression. If a denotes the first term, d the
nCl
21.
n
= 1'
nC2 =
common difference, n the number of terms, l the last term, and
S the sum, then
22.
S = 2n(a + l),
S = ?n[2a + (n - 1)d].
l = a + (n - 1) d,
Geometric progression. If a denotes the first term, r the
common ratio, n the number of terms, l the last term, and S the
sum, then
n
23.
S= 1- r
l = am-1,
S= a 1- r
Infinite geometric progression. S = a/(1 - r), if r2 < 1.
24.
Radians and degrees. 360° = 27r radians = 1 revolution.
TRIGONOMETRY
1 radian = 57.2957
25.
.
degrees.
1 degree = 0.0174532... radian.
Values of trigonometric functions for certain angles.
Angle in
Degrees
sin
cos
tan
0
1
0
00
30°
cot
sec
csc
Angle in
Radians
0
1
2
2
4
2
2
2
1
1
0
0
-1
0
0
1
0
1
271
Mathematical Formulas-Trigonometry
3
Fundamental identities.
26.
sec x = 1 /cos x,
cot x = 1 /tan x,
cot x = cos x/sin x,
tan x = sin x/cos x,
1 + cot2 x = csc2 X.
1 + tang x = sect x,
sine x + cost x = 1,
csc x = 1 /sin x,
Reduction relations.
27.
tanCo-
Cosine
Tangent
- sin x
cos x
- tan x
- cot x
sec x
- csc x
90° - x
cos x
sin x
cot x
tan x
csc x
sec x
90° + x
cos x
- sin x
- cot x
- tan x
- csc x
sec x
-x
sin x
- cos x
- tan x
- cot x
- sec x
csc x
180° + x
- sin x
- cos x
tan x
cot x
- sec x
- csc x
270° - x
- cos x
- sin x
cot x
tan x
- csc x
- sec x
270° + x
- cos x
sin x
- cot x
- tan x
csc x
- sec x
360° - x
-sin x
cos x
-tan x
-cot x
sec x
-csc x
Angle
Sine
_x
1800
28.
gent
Secant Cosecant
Formulas for the sum and difference of two angles.
sin (x ± y) = sin x cosy ± cosx sin y,
cos (x ± y) = cos x cosy T- sin x sin y,
29.
tan (x ± y) = tan x ± tan y
1 + tan x tan y
Double-angle formulas.
sin 2x = 2 sin x cos x,
cos 2x = cost x - sine x,
tan 2x =
30.
2 tan x
1 - tang x
Half -angle formulas.
sinx2 = ±
x
tan 2 =
/1 - cosx
2
+J l - cosx
1 + cosx
x
cos 2
/1 + cos x
2
1 - cosx
sin x
sin x
1 + cos x
Introduction
4
31.
Sum formulas.
sin x + sin y = 2 sin 2(x + y)cos1(x - y),
sin x - sin y = 2 cos 2(x + y)sin 2(x - y),
cos x + cosy = 2 cos (x + y) cos (x - y),
?
2
cos x - cos y = - 2 sin 1 (x + y) sin (x - y).
32.
z
Product formulas.
sin x sin y = cos (x - y) - 2 cos (x + y),
sin x cos y = sin (x - y) + 2 sin (x + y),
cos x cos y = 2 cos (x - y) + 2 cos (x + Y).
Inverse formulas. When a > 0,
Cot-' (- a) = IT - Tan-1 (1/a),
Sin-' (-a) = -Sin-' a,
Cos-' (-a) = it - Cos-' a, Sec-' (-a) = Cos-' (1 /a) - 7r,
Csc-' (- a) = Sin-' (1/a) - IT,
Tan-' (-a) = - Tan-' a,
Sin-' a =
1 - a2,
Cos-' a = Sin-'1/l - a2.
When a > 0, b > 0,
Sin-' a - Sin-' b = Sin-'(a-N/1 - b2 - b\/1 - a2),
Tan-' a - Tan-' b = Tan-' (a - b)/(1 + ab),
33.
Cos-'
1Cos-'[ab - V(1 - a2)(1- b2)],
Cos-' a + Cos-' b = it/2,
if a2 + b2 > 1,
a2 + b2 = 1,
IT - Cos-'[-x/(l - a2)(1 - b2) - ab], a2 + b2 < 1,
if ab < 1,
Tan-'(a + b)/(1 - ab),
ab = 1,
Tan-' a + Tan-' b = it/2,
ab > 1.
7T- Tan-'(a + b)/(ab - 1),
34.
Formulas for any triangles. Sides, a, b, c; opposite angles,
A, B, C ; s = 2 (a + b + c) ; radius of circumcircle, R ; radius of
incircle, r.
a = b = c
= 2R,
Law of sines.
sin C
sin B
sin A
Law of cosines. a2 = b2 + c2 - 2bc cos A,
(s - a)(s - b)(s - c),
r
s
Area = lab sin C
_ a2 sin B sin C
2 sin (B + C)
= Vs(s - a) (s - b) (s - c).
Formulas-Plane Analytic Geometry
5
PLANE ANALYTIC GEOMETRY
35.
For two points Pi(xi,yi) and P2(x2,y2).
Distance PiP2. d = 1/(x1 - X2)2 + (yi - y2) 2.
Slope of P1P2. m = Y1 - Y2
xi - X2
Mid-point of P1P2. x = 2(xi + X2), y = 2(yi + y2).
36. Angle between two lines with slopes m1 and M2M1 - M2
1 + m1m2
tan
For parallel lines mi = M2; for perpendicular lines m1
37.
-1/m2.
Equations of straight lines.
Point-slope form. y - yi = m(x - xi).
Slope-intercept form. y = mx + b.
= Y2 - yl
Two-point form. Y - Y1
X - x1 x2 - xi
Intercept form.
X
+ b = 1.
Horizontal line. y = b.
Vertical line. x = a.
38.
Distance from P(xi,yi) to the line Ax + By + C = 0.
d=
Axi+Byi+C
+1/A2+B2
40.
Relations between rectangular and polar coordinates.
r = ± 1/x2 + y2,
x = r cos 0,
0 = tan-'(y/x).
y = r sin 0.
Equation of circle. Center (h,k), radius r.
41.
Equation of ellipse. Center (h,k), semimajor axis a, semiminor
39.
(x - h) 2 + (y
- k) 2 = r2.
axis b.
(x-h)2+ (y-k)2=
a2
42.
b2
1
or
(y-k)2+ (x-h)2=
a2
b2
Equation of hyperbola. Center (h,k), semitransverse axis a,
semiconjugate axis b.
(x-h)2
a2
(y-k)2=
b2
1
or
(y-k)2- (x-h)2=
a2
Equilateral hyperbola with center at origin xy = C.
b2
1.
Introduction
6
43.
Equation of parabola. Vertex (h,k), focal distance p.
(y-k)2=4p(x-h)
(x-h)2=4p(y-k).
or
SOLID ANALYTIC GEOMETRY
44.
For two points P,(xl,yl,zl) and P2(x2,y2,z2).
Distance P,P2. d = V (xl - x2) 2 + (yl - y2) 2 + (z1 - z2) 2.
Direction numbers of P1P2. [XI - X2, y1 - Y2, z1 - z2].
Mid-point of P,P2. x = 2 (xl + x2), y = 2 (yl + Y2), z = (zl + z2)
2
45.
Angle between two lines whose direction numbers are
[al, b1, c1] and [a2, b2, c2].
cos
ala2 + b,b2 + ClC2
= _
Perpendicular lines.
+b2+
2
C22
a,a2 + b,b2 + clc2 = 0-
Equation of a plane through P(xl,yl,z1) and perpendicular
to a line with direction numbers [A,B,C].
+B(y-y1) +C(z-z1) = 0.
A(x-x1)
47.
2
al
c1
b,
a2=b2=C2
Parallel lines.
46.
i+c1'Va2
a2j +b2
Equations of a line through P(xl,ylizl) with direction
numbers [a,b,c].
48.
x- x1
y - yl
z -z1
a
b
c
Distance from P(x1iyl,zl) to the plane Ax + By + Cz +
D = 0.
d-Ax,+By,+Cz,+D
\/A2+B2+C2
49.
50.
Relations between rectangular and cylindrical coordinates.
x=rcos0,
y=rsin0,
z=z;
r = ± -V x2 + y2,
6 = tan-'
z = z.
(y/x),
Relations between rectangular and spherical coordinates.
x = r sin 0 cos 0,
y = r sin 0 sin 0, z = r cos 0;
r = ± V x2 + y2 + z2, 0 = tan-1 (.y/x),
0 = tan-l(V/x2 + y2)/z.
Formulas-Curves
51.
Sine curve.
7
52.
Cosine curve.
y=sin x
53.
y = cos x
Tangent curve.
54.
Secant curve.
Y
X
3n
11
2
55.
Inverse sine curve.
56.
Inverse cosine curve.
Y
11 .
art
2
+n
Principal values
-1--+
-1
0
II
y=cos 1 x
X
/
/
X
Introduction
8
57.
Inverse tangent curve.
Inverse secant curve.
58.
Y
Y
IT
n
1
-- --------
z
-1
X
O
-----X
O
Principal values
V\-',
Principal values
-n+
-n
y=tan' x
59.
y = sec' x
Logarithmic curve.
60.
Exponential curve.
x
x
y=logbx, (b>1)
61.
Hyperbolic sine curve.
62.
Hyperbolic cosine curve.
X
X
y=cosh x
y=sinh x
Formulas-Curves
63.
Cubical parabola.
64.
9
Semicubical parabola.
x
x
y = ax3, (a > 0)
65.
y2=ax3, (a> 0)
Probability curve.
66.
Parabolic arc.
x
y = e-x 2
67.
1
I
1
x2+ y2=a2
Folium of Descartes.
68.
The cissoid of Diodes.
x
x3 + y3 -3axy=O
y4=x3/(2a - x)
Introduction
10
Astroid.
69.
70.
The witch of Agnesi.
x2y=4a2 (2a-y)
2
x3+y3-a32
2
71.
Strophoid.
72.
X s In
y
2
73.
Tractrix.
a+ o2_Y2
a2_ y2
0+X
x2o-x
Cardioid.
74.
Limacon.
