# Index:Elements of the Differential and Integral Calculus - Granville - Revised.djvu

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CONTENTS

DIFFERENTIAL CALCULUS

CHAPTER I
COLLECTION OF FORMULAS

SECTION
PAGE

1. Formulas from Algebra, Trigonometry, and Analytic Geometry1
2. Greek alphabet3
3. Rules for signs in the four quadrants3
4. Natural values of the trigonometric functions4
5. Tables of logarithms5

CHAPTER II
VARIABLES AND FUNCTIONS

1. Variables and constants6
2. Interval of a variable6
3. Continuous variation6
4. Functions7
5. Independent and dependent variables7
6. Notation of functions8
7. Values of the independent variable for which a function is defined8

CHAPTER III
THEORY OF LIMITS

1. Limit of a variable11
2. Division by zero excluded12
3. Infinitesimals13
4. The concept of infinity ($\scriptstyle{\infty}$)13
5. Limiting value of a function14
6. Continuous and discontinuous functions14
7. Continuity and discontinuity of functions illustrated by their graphs16
8. Fundamental theorems on limits18
9. Special limiting values20
10. The limit of $\scriptstyle{\frac{\sin x}{x}}$ as $\scriptstyle{x \doteq 0}$21
11. The number $\scriptstyle{e}$22
12. Expressions assuming the form $\scriptstyle{\frac{\infty}{\infty}}$23

CHAPTER IV
DIFFERENTIATION

1. Introduction25
2. Increments25
3. Comparison of increments26
4. Derivative of a function of one variable27
5. Symbols for derivatives28
6. Differentiable functions29
7. General rule for differentiation29
8. Applications of the derivative to Geometry31

CHAPTER V
RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORMS

1. Importance of General Rule34
2. Differentiation of a constant36
3. Differentiation of a variable with respect to itself37
4. Differentiation of a sum37
5. Differentiation of the product of a constant and a function37
6. Differentiation of the product of two functions38
7. Differentiation of the product of any finite number of functions38
8. Differentiation of a function with a constant exponent39
9. Differentiation of a quotient40
10. Differentiation of a function of a function44
11. Differentiation of inverse functions45
12. Differentiation of a logarithm46
13. Differentiation of the simple exponential function48
14. Differentiation of the general exponential function49
15. Logarithmic differentiation50
16. Differentiation of $\scriptstyle{\sin v}$54
17. Differentiation of $\scriptstyle{\cos v}$55
18. Differentiation of $\scriptstyle{\tan v}$56
19. Differentiation of $\scriptstyle{\cot v}$56
20. Differentiation of $\scriptstyle{\sec v}$56
21. Differentiation of $\scriptstyle{\csc v}$57
22. Differentiation of $\scriptstyle{\operatorname{vers} v}$57
23. Differentiation of $\scriptstyle{\operatorname{arc~sin} v}$61
24. Differentiation of $\scriptstyle{\operatorname{arc~cos} v}$62
25. Differentiation of $\scriptstyle{\operatorname{arc~tan} v}$62
26. Differentiation of $\scriptstyle{\operatorname{arc~cot} v}$63
27. Differentiation of $\scriptstyle{\operatorname{arc~sec} v}$63
28. Differentiation of $\scriptstyle{\operatorname{arc~csc} v}$64
29. Differentiation of $\scriptstyle{\operatorname{arc~vers} v}$65
30. Implicit functions69
31. Differentiation of implicit functions69

CHAPTER VI
SIMPLE APPLICATIONS OF THE DERIVATIVE

1. Direction of a curve73
2. Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates76
3. Parametric equations of a curve79
4. Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point83
5. Lengths of polar subtangent and polar subnormal86
6. Solution of equations having multiple roots88
7. Applications of the derivative in mechanics. Velocity90
8. Component velocities91
9. Acceleration92
10. Component accelerations93

CHAPTER VII
SUCCESSIVE DIFFERENTIATION

1. Definition of successive derivatives97
2. Notation97
3. The $\scriptstyle{n\text{th}}$ derivative98
4. Leibnitz's formula for the $\scriptstyle{n\text{th}}$ derivative of a product98
5. Successive differentiation of implicit functions100

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

1. Introduction103
2. Increasing and decreasing functions106
3. Tests for determining when a function is increasing and when decreasing108
4. Maximum and minimum values of a function109
5. First method for examining a function for maximum and minimum values111
6. Second method for examining a function for maximum and minimum values112
7. Definition of points of inflection and rule for finding points of inflection125
8. Curve tracing128

CHAPTER IX
DIFFERENTIALS

1. Introduction131
2. Definitions131
3. Infinitesimals132
4. Derivative of the arc in rectangular coördinates134
5. Derivative of the arc in polar coördinates135
6. Formulas for finding the differentials of functions137
7. Successive differentials139

CHAPTER X
RATES

1. The derivative considered as the ratio of two rates141

CHAPTER XI
CHANGE OF VARIABLE

1. Interchange of dependent and independent variables148
2. Change of the dependent variable149
3. Change of the independent variable150
4. Simultaneous change of both independent and dependent variables152

CHAPTER XII

1. Curvature155
2. Curvature of a circle155
3. Curvature at a point156
4. Formulas for curvature159
6. Circle of curvature161

CHAPTER XIII
THEOREM OF MEAN VALUE. INDETERMINATE FORMS

1. Rolle's Theorem164
2. The Theorem of Mean Value165
3. The Extended Theorem of Mean Value166
4. Maxima and minima treated analytically167
5. Indeterminate forms170
6. Evaluation of a function taking on an indeterminate form170
7. Evaluation of the indeterminate form $\scriptstyle{\frac{0}{0}}$171
8. Evaluation of the indeterminate form $\scriptstyle{\frac{\infty}{\infty}}$174
9. Evaluation of the indeterminate form $\scriptstyle{0\cdot\infty}$174
10. Evaluation of the indeterminate form $\scriptstyle{\infty-\infty}$175
11. Evaluation of the indeterminate forms $\scriptstyle{0^0,~1^\infty,~\infty^0}$176

CHAPTER XIV
CIRCLE OF CURVATURE. CENTER OF CURVATURE

1. Circle of curvature. Center of curvature178
2. Second method for finding center of curvature180
3. Center of curvature the limiting position of the intersection of normals at neighboring points181
4. Evolutes182
5. Properties of the evolute186
6. Involutes and their mechanical construction187

CHAPTER XV
PARTIAL DIFFERENTIATION

1. Continuous functions of two or more independent variables190
2. Partial derivatives191
3. Partial derivatives interpreted geometrically192
4. Total derivatives194
5. Total differentials197
6. Differentiation of implicit functions198
7. Successive partial derivatives202
8. Order of differentiation immaterial203

CHAPTER XVI
ENVELOPES

1. Family of curves. Variable parameter205
2. Envelope of a family of curves depending on one parameter205
3. The evolute of a given curve considered as the envelope of its normals208
4. Two parameters connected by one equation of condition209

CHAPTER XVII
SERIES

1. Introduction212
2. Infinite series213
3. Existence of a limit215
4. Fundamental test for convergence216
5. Comparison test for convergence217
6. Cauchy's ratio test for convergence218
7. Alternating series220
8. Absolute convergence220
9. Power series223

CHAPTER XVIII
EXPANSION OF FUNCTIONS

1. Introduction227
2. Taylor's Theorem and Taylor's Series228
3. Maclaurin's Theorem and Maclaurin's Series230
4. Computation by series234
5. Approximate formulas derived from series. Interpolation237
6. Taylor's Theorem for functions of two or more variables240
7. Maxima and minima of functions of two independent variables243

CHAPTER XIX
ASYMPTOTES. SINGULAR POINTS

1. Rectilinear asymptotes249
2. Asymptotes found by method of limiting intercepts249
3. Method of determining asymptotes to algebraic curves250
4. Asymptotes in polar coördinates254
5. Singular points255
6. Determination of the tangent to an algebraic curve at a given point by inspection255
7. Nodes258
8. Cusps259
9. Conjugate or isolated points260
10. Transcendental singularities260

CHAPTER XX
APPLICATIONS TO GEOMETRY OF SPACE

1. Tangent line and normal plane to a skew curve whose equations are given in parametric form262
2. Tangent plane to a surface264
3. Normal line to a surface266
4. Another form of the equations of the tangent line to a skew curve268
5. Another form of the equation of the normal plane to a skew curve269

CHAPTER XXI
CURVES FOR REFERENCE

INTEGRAL CALCULUS

CHAPTER XXII
INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS

1. Integration279
2. Constant of integration. Indefinite integral281
3. Rules for integrating standard elementary forms282
4. Trigonometric differentials298
5. Integration of expressions containing $\scriptstyle{\sqrt{a^2-x^2}}$ or $\scriptstyle{\sqrt{x^2\pm a^2}}$ by a trigonometric substitution304

CHAPTER XXIII
CONSTANT OF INTEGRATION

1. Determination of the constant of integration by means of initial conditions307
2. Geometrical signification of the constant of integration307
3. Physical signification of the constant of integration309

CHAPTER XXIV
THE DEFINITE INTEGRAL

1. Differential of an area314
2. The definite integral314
3. Calculation of a definite integral316
4. Calculation of areas318
5. Geometrical representation of an integral319
6. Mean value of $\scriptstyle{\phi(x)}$320
7. Interchange of limits320
8. Decomposition of the interval321
9. The definite integral a function of its limits321
10. Infinite limits321
11. When $\scriptstyle{y=\phi(x)}$ is discontinuous322

CHAPTER XXV
INTEGRATION OF RATIONAL FRACTIONS

1. Introduction325
2. Case I325
3. Case II327
4. Case III329
5. Case IV331

CHAPTER XXVI
INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

1. Introduction335
2. Differentials containing fractional powers of $\scriptstyle{x}$ only335
3. Differentials containing fractional powers of $\scriptstyle{a+bx}$ only336
4. Change in limits corresponding to change in variable336
5. Differentials containing no radical except $\scriptstyle{\sqrt{a+bx+x^2}}$338
6. Differentials containing no radical except $\scriptstyle{\sqrt{a+bx-x^2}}$338
7. Binomial differentials340
8. Conditions of integrability of binomial differentials341
9. Transformation of trigonometric differentials343
10. Miscellaneous substitutions345

CHAPTER XXVII
INTEGRATION BY PARTS. REDUCTION FORMULAS

1. Formula for integration by parts347
2. Reduction formulas for binomial differentials350
3. Reduction formulas for trigonometric differentials356
4. To find $\scriptstyle{\int e^{ax}\sin{nx}dx}$ and $\scriptstyle{\int e^{ax}\cos{nx}dx}$359

CHAPTER XXVIII
INTEGRATION A PROCESS OF SUMMATION

1. Introduction361
2. The fundamental theorem of Integral Calculus361
3. Analytical proof of the Fundamental Theorem364
4. Areas of plane curves. Rectangular coördinates365
5. Area when curve is given in parametric form368
6. Areas of plane curves. Polar coördinates370
7. Length of a curve372
8. Lengths of plane curves. Rectangular coördinates373
9. Lengths of plane curves. Polar coördinates375
10. Volumes of solids of revolution377
11. Areas of surfaces of revolution381
12. Miscellaneous applications385

CHAPTER XXIX
SUCCESSIVE AND PARTIAL INTEGRATION

1. Successive integration393
2. Partial integration395
3. Definite double integral. Geometric interpretation396
4. Value of a definite double integral over a region400
5. Plane area as a definite double integral. Rectangular coördinates402
6. Plane area as a definite double integral. Polar coördinates406
7. Moment of area408
8. Center of area408
9. Moment of inertia. Plane areas410
10. Polar moment of inertia. Rectangular coördinates410
11. Polar moment of inertia. Polar coördinates411
12. General method for finding the areas of surfaces413
13. Volumes found by triple integration417

CHAPTER XXX
ORDINARY DIFFERENTIAL EQUATIONS

1. Differential equations. Order and degree421
2. Solutions of differential equations422
3. Verifications of solutions423
4. Differential equations of the first order and of the first degree424
5. Differential equations of the $\scriptstyle{n\text{th}}$ order and of the first degree432

CHAPTER XXXI
INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

1. Mechanical integration443
2. Integral curves443
3. The integraph445
4. Polar planimeter446
5. Area swept over by a line446
6. Approximate integration448
7. Trapezoidal rule448
8. Simpson's rule (parabolic rule)449
9. Integrals for reference451