Elements of the Differential and Integral Calculus

Elements of the Differential and Integral Calculus  (1911)  by William Anthony Granville

ELEMENTS OF THE DIFFERENTIAL
AND INTEGRAL CALCULUS

(REVISED EDITION)

BY

WILLIAM ANTHONY GRANVILLE, PH.D.. LL.D.

FORMERLY PRESIDENT OF PENNSYLYANIA COLLEGE

WITH THE EDITORIAL COOPERATION OF

PERCEY F. SMITH, PH.D.

PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL
YALE UNIVERSITY

---

Table of Contents

PREFACE

Differential Calculus

CHAPTER I
COLLECTION OF FORMULAS

1. Formulas from Algebra, Trigonometry, and Analytic Geometry
2. Greek alphabet
3. Rules for signs in the four quadrants
4. Natural values of the trigonometric functions
5. Tables of logarithms

CHAPTER II
VARIABLES AND FUNCTIONS

6. Variables and constants
7. Interval of a variable
8. Continuous variation
9. Functions
10. Independent and dependent variables
11. Notation of functions
12. Values of the independent variable for which a function is defined

CHAPTER III
THEORY OF LIMITS

13. Limit of a variable
14. Division by zero excluded
15. Infinitesimals
16. The concept of infinity ($\infty$)
17. Limiting value of a function
18. Continuous and discontinuous functions
19. Continuity and discontinuity of functions illustrated by their graphs
20. Fundamental theorems on limits
21. Special limiting values
22. The limit of $\tfrac{\sin\ x}{x}$ as $x \dot= 0$
23. The number $e$
24. Expressions assuming the form $\tfrac{\infty}{\infty}$

CHAPTER IV
DIFFERENTIATION

25. Introduction
26. Increments
27. Comparison of increments
28. Derivative of a function of one variable
29. Symbols for derivatives
30. Differentiable functions
31. General rule for differentiation
32. Applications of the derivative to Geometry

CHAPTER V
RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORM

33. Importance of General Rule
34. Differentiation of a constant
35. Differentiation of a variable with respect to itself
36. Differentiation of a sum
37. Differentiation of the product of a constant and a function
38. Differentiation of the product of two functions
39. Differentiation of the product of any finite number of functions
40. Differentiation of a function with a constant exponent
41. Differentiation of a quotient
42. Differentiation of a function of a function
43. Differentiation of inverse functions
44. Differentiation of a logarithm
45. Differentiation of the simple exponential function
46. Differentiation of the general exponential function
47. Logarithmic differentiation
48. Differentiation of $\sin\ v$
49. Differentiation of $\cos\ v$
50. Differentiation of $\tan\ v$
51. Differentiation of $\cot\ v$
52. Differentiation of $\sec\ v$
53. Differentiation of $\csc\ v$
54. Differentiation of $\operatorname{vers}\ v$
55. Differentiation of $\arcsin\ v$
56. Differentiation of $\arccos\ v$
57. Differentiation of $\arctan\ v$
58. Differentiation of $\arccot\ v$
59. Differentiation of $\arcsec\ v$
60. Differentiation of $\arccsc\ v$
61. Differentiation of $\operatorname{arcvers}\ v$
62. Implicit functions
63. Differentiation of implicit functions

CHAPTER VI
SIMPLE APPLICATIONS OF THE DERIVATIVE

64. Direction of a curve
65. Equations of tangent and normal, lengths of subtangent and subnormal
66. Rectangular coördinates
67. Parametric equations of a curve
68. Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point
69. Lengths of polar subtangent and polar subnormal
70. Solution of equations having multiple roots
71. Applications of the derivative in mechanics. Velocity
72. Component velocities
73. Acceleration
74. Component accelerations

CHAPTER VII
SUCCESSIVE DIFFERENTIATION

74. Definition of successive derivatives
75. Notation
76. The nth derivative
77. Leibnitz's formula for the nth derivative of a product
78. Successive differentiation of implicit functions

CHAPTER VIII
MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING

79. Introduction
80. Increasing and decreasing functions
81. Tests for determining when a function is increasing and when decreasing
82. Maximum and minimum values of a function
83. First method for examining a function for maximum and minimum values
84. Second method for examining a function for maximum and minimum values
85. Definition of points of inflection and rule for finding points of inflection
86. Curve tracing

CHAPTER IX
DIFFERENTIALS

87. Introduction
88. Definitions
89. Infinitesimals
90. Derivative of the arc in rectangular coördinates
91. Derivative of the arc in polar coördinates
92. Formulas for finding the differentials of functions
93. Successive differentials

CHAPTER X
RATES

94. The derivative considered as the ratio of two rates

CHAPTER XI
CHANGE OF VARIABLE

95. Interchange of dependent and independent variables
96. Change of the dependent variable
97. Change of the independent variable
98. Simultaneous change of both independent and dependent variables

CHAPTER XII
CURVATURE. RADIUS OF CURVATURE

99. Curvature
100. Curvature of a circle
101. Curvature at a point
102. Formulas for curvature
103. Radius of curvature
104. Circle of curvature

CHAPTER XIII
THEOREM OF MEAN VALUE. INDETERMINATE FORMS

105. Rolle's Theorem
106. The Theorem of Mean Value
107. The Extended Theorem of Mean Value
108. Maxima and minima treated analytically
109. Indeterminate forms
110. Evaluation of a function taking on an indeterminate form
111. Evaluation of the indeterminate form $\tfrac{0}{0}$
112. Evaluation of the indeterminate form $\tfrac{\infty}{\infty}$
113. Evaluation of the indeterminate form $0 \cdot \infty$
114. Evaluation of the indeterminate form $\infty - \infty$
115. Evaluation of the indeterminate forms $0^0, 1^\infty, \infty^0$

CHAPTER XIV
CIRCLE OF CURVATURE. CENTER OF CURVATURE

116. Circle of curvature. Center of curvature
117. Second method for finding center of curvature
118. Center of curvature the limiting position of the intersection of normals at neighboring points
119. Evolutes
120. Properties of the evolute
121. Involutes and their mechanical construction

CHAPTER XV
PARTIAL DIFFERENTIATION

122. Continuous functions of two or more independent variables
123. Partial derivatives
124. Partial derivatives interpreted geometrically
125. Total derivatives
126. Total differentials
127. Differentiation of implicit functions
128. Successive partial derivatives
129. Order of differentiation immaterial

CHAPTER XVI
ENVELOPES

130. Family of curves. Variable parameter
131. Envelope of a family of curves depending on one parameter
132. The evolute of a given curve considered as the envelope of its normals
133. Two parameters connected by one equation of condition

CHAPTER XVII
SERIES

134. Introduction
135. Infinite series
136. Existence of a limit
137. Fundamental test for convergence
138. Comparison test for convergence
139. Cauchy's ratio test for convergence
140. Alternating series
141. Absolute convergence
142. Power series

CHAPTER XVIII
EXPANSION OF FUNCTIONS

143. Introduction
144. Taylor's Theorem and Taylor's Series
145. Maclaurin's Theorem and Maclaurin's Series
146. Computation by series
147. Approximate formulas derived from series. Interpolation
148. Taylor's Theorem for functions of two or more variables
149. Maxima and minima of functions of two independent variables

CHAPTER XIX
ASYMPTOTES. SINGULAR POINTS

150. Rectilinear asymptotes
151. Asymptotes found by method of limiting intercepts
152. Method of determining asymptotes to algebraic curves
153. Asymptotes in polar coördinates
154. Singular points
155. Determination of the tangent to an algebraic curve at a given point by inspection
156. Nodes
157. Cusps
158. Conjugate or isolated points
159. Transcendental singularities

CHAPTER XX
APPLICATIONS TO GEOMETRY OF SPACE

160. Tangent line and normal plane to a skew curve whose equations are given in parametric form
161. Tangent plane to a surface
162. Normal line to a surface
163. Another form of the equations of the tangent line to a skew curve
164. Another form of the equation of the normal plane to a skew curve

CHAPTER XXI
CURVES FOR REFERENCE

Integral Calculus

CHAPTER XXII
INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS

165. Integration
166. Constant of integration. Indefinite integral
167. Rules for integrating standard elementary forms
168. Trigonometric differentials
169. Integration of expressions containing $\sqrt{a^2-x^2}$ or $\sqrt{x^2 \pm a^2}$ by a trigonometric substitution

CHAPTER XXIII
CONSTANT OF INTEGRATION

170. Determination of the constant of integration by means of initial conditions
171. Geometrical signification of the constant of integration
172. Physical signification of the constant of integration

CHAPTER XXIV
THE DEFINITE INTEGRAL

173. Differential of an area
174. The definite integral
175. Calculation of a definite integral
176. Calculation of areas
177. Geometrical representation of an integral
178. Mean value of $\Phi(x)$
179. Interchange of limits
180. Decomposition of the interval
181. The definite integral a function of its limits
182. Infinite limits
183. When $y = \Phi(x)$ is discontinuous

CHAPTER XXV
INTEGRATION OF RATIONAL FRACTIONS

184. Introduction
185. Case I
186. Case II
187. Case III
188. Case IV

CHAPTER XXVI
INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

189. Introduction
190. Differentials containing fractional powers of $x$ only
191. Differentials containing fractional powers of $a + bx$ only
192. Change in limits corresponding to change in variable
193. Differentials containing no radical except $\sqrt{a + bx + x^2}$
194. Differentials containing no radical except $\sqrt{a + bx - x^2}$
195. Binomial differentials
196. Conditions of integrability of binomial differentials
197. Transformation of trigonometric differentials
198. Miscellaneous substitutions

CHAPTER XXVII
INTEGRATION BY PARTS. REDUCTION FORMULAS

199. Formula for integration by parts
200. Reduction formulas for binomial differentials
201. Reduction formulas for trigonometric differentials
202. To find $\int e^{ax}\sin{nx}dx$ and $\int e^{ax} \cos{nx}dx$

CHAPTER XXVIII
INTEGRATION A PROCESS OF SUMMATION

203. Introduction
204. The fundamental theorem of Integral Calculus
205. Analytical proof of the Fundamental Theorem
206. Areas of plane curves. Rectangular coördinates
207. Area when curve is given in parametric form
208. Areas of plane curves. Polar coördinates
209. Length of a curve
210. Lengths of plane curves. Rectangular coördinates
211. Lengths of plane curves. Polar coördinates
212. Volumes of solids of revolution
213. Areas of surfaces of revolution
214. Miscellaneous applications

CHAPTER XXIX
SUCCESSIVE AND PARTIAL INTEGRATION

215. Successive integration
216. Partial integration
217. Definite double integral. Geometric interpretation
218. Value of a definite double integral over a region
219. Plane area as a definite double integral. Rectangular coördinates
220. Plane area as a definite double integral. Polar coördinates
221. Moment of area
222. Center of area
223. Moment of inertia. Plane areas
224. Polar moment of inertia. Rectangular coördinates
225. Polar moment of inertia. Polar coördinates
226. General method for finding the areas of surfaces
227. Volumes found by triple integration

CHAPTER XXX
ORDINARY DIFFERENTIAL EQUATIONS

228. Differential equations. Order and degree
229. Solutions of differential equations
230. Verifications of solutions
231. Differential equations of the first order and of the first degree
232. Differential equations of the nth order and of the first degree

CHAPTER XXXI
INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

233. Mechanical integration
234. Integral curves
235. The integraph
236. Polar planimeter
237. Area swept over by a line
238. Approximate integration
239. Trapezoidal rule
240. Simpson's rule (parabolic rule)
241. Integrals for reference

INDEX

Notes

This work is in the public domain in the United States because it was published before January 1, 1923. The author died in 1943, so this work is also in the public domain in countries and areas where the copyright term is the author's life plus 60 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works.