Calculus (Book)

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The textbook Calculus by Tom Apostol introduces calculus.

Calculus
Apostol Calculus V1 Cover.jpg
Information
Author Tom Apostol
Language English
Publisher Wiley
Publication Date 16 January 1991
Pages 666
ISBN-10 0471000051
ISBN-13 978-0471000051

Table of Contents

Chapter/Section # Title Page #
I. INTRODUCTION
Part 1: Historical Introduction
I 1.1 The two basic concepts of calculus 1
I 1.2 Historical background 2
I 1.3 The method of exhaustion for the area of a parabolic segment 3
*I 1.4 Exercises 8
I 1.5 A critical analysis of the Archimedes' method 8
I 1.6 The approach to calculus to be used in this book 10
Part 2: Some Basic Concepts of the Theory of Sets
I 2.1 Introduction to set theory 11
I 2.2 Notations for designating sets 12
I 2.3 Subsets 12
I 2.4 Unions, intersections, complements 13
I 2.5 Exercises 15
Part 3: A set of Axioms for the Real-Number System
I 3.1 Introduction 17
I 3.2 The field axioms 17
*I 3.3 Exercises 19
I 3.4 The order axioms 19
*I 3.5 Exercises 21
I 3.6 Integers and rational numbers 21
I 3.7 Geometric interpretation of real numbers as points on a line 22
I 3.8 Upper bound of a set, maximum element, least upper bound (supremum) 23
I 3.9 The least-Upper-bound axiom (completeness axiom) 25
I 3.10 The Archimedean property of the real-number system 25
I 3.11 Fundamental properties of the supremum and infimum 26
*I 3.12 Exercises 28
*I 3.13 Existence of square roots of nonnegative real numbers 29
*I 3.14 Roots of higher order. Rational powers 30
*I 3.15 Representation of real numbers by decimals 30
Part 4: Mathematical Induction, Summation Notation, and Related Topics
I 4.1 An example of a proof by mathematical induction 32
I 4.2 The principle of mathematical induction 34
*I 4.3 The well-ordering principle 34
I 4.4 Exercises 35
*I 4.5 Proof of the well-ordering principle 37
I 4.6 The summation notation 37
I 4.7 Exercises 39
I 4.8 Absolute values and the triangle inequality 41
I 4.9 Exercises 43
*I 4.10 Miscellaneous exercises involving induction 44
1. THE CONCEPTS OF INTEGRAL CALCULUS
1.1 The basic ideas of Cartesian geometry 48
1.2 Functions. Informal description and examples 50
1.3 Functions. Formal definition as a set of ordered pairs 53
1.4 More examples of real functions 54
1.5 Exercises 56
1.6 The concept of area as a set function 57
1.7 Exercises 60
1.8 Intervals and ordinate sets 60
1.9 Partitions and step functions 61
1.10 Sum and product of step functions 63
1.11 Exercises 63
1.12 The definition of the integral for step functions 64
1.13 Properties of the integral of a step function 66
1.14 Other notations for integrals 69
1.15 Exercises 70
1.16 The integral of more general functions 72
1.17 Upper and lower integrals 74
1.18 The area of an ordinate set expressed as an integral 75
1.19 Informal remarks on the theory and technique of integration 75
1.20 Monotonic and piecewise monotonic functions. Definitions and examples 76
1.21 Integrability of bounded monotonic functions 77
1.22 Calculation of the integral of a bounded monotonic function 79
1.23 Calculation of the integral \(\int_0^b x^p dx\) when \(p\) is a positive integer 79
1.24 The basic properties of the integral 80
1.25 Integration of polynomials 81
1.26 Exercises 83
1.27 Proofs of the basic properties of the integral 84
2. SOME APPLICATIONS OF INTEGRATION
2.1 Introduction 88
2.2 The area of a region between two graphs expressed as an integral 88
2.3 Worked examples 89
2.4 Exercises 94
2.5 The trigonometric functions 94
2.6 Integration formulas for the sine and cosine 94
2.7 A geometric description of the sine and cosine functions 94
2.8 Exercises 94
2.9 Polar coordinates 94
2.10 The integral for area in polar coordinates 94
2.11 Exercises 94
2.12 Application of integration to the calculation of volume 94
2.13 Exercises 94
2.14 Application of integration to the calculation of work 94
2.15 Exercises 94
2.16 Average value of a function 94
2.17 Exercises 94
2.18 The integral as a function of the upper limit. Indefinite integrals 94
2.19 Exercises 94
3. CONTINUOUS FUNCTIONS
3.1 Informal description of continuity 126
3.2 The definition of the limit of a function 127
3.3 The definition of continuity of a function 130
3.4 The basic limit theorems. More examples of continuous functions 131
3.5 Proofs of the basic limit theorems 135
3.6 Exercises 138
3.7 Composite functions and continuity 140
3.8 Exercises 142
3.9 Bolzano's theorem for continuous functions 142
3.10 The intermediate-value theorem for continuous functions 144
3.11 Exercises 145
3.12 The process of inversion 146
3.13 Properties of functions preserved by inversion 147
3.14 Inverses of piecewise monotonic functions 148
3.15 Exercises 149
3.16 The extreme-value theorem for continuous functions 150
3.17 The small-span theorem for continuous functions (uniform continuity) 152
3.18 The integrability theorem for continuous functions 152
3.19 Mean-value theorems for integrals of continuous functions 154
3.20 Exercises 155
4. DIFFERENTIAL CALCULUS
4.1 Historical introduction 156
4.2 A problem involving velocity 157
4.3 The derivative of a function 159
4.4 Examples of derivatives 161
4.5 The algebra of derivatives 164
4.6 Exercises 167
4.7 Geometric interpretation of the derivative as a slope 169
4.8 Other notations for derivatives 171
4.9 Exercises 173
4.10 The chain rule for differentiating composite functions 174
4.11 Applications of the chain rule. Related rates and implicit differentiation 176
4.12 Exercises 179
4.13 Applications of the differentiation to extreme values of cuntions 181
4.14 The mean-value theorem for derivatives 183
4.15 Exercises 186
4.16 Applications of the mean-value theorem to geometric properties of functions 187
4.17 Second-derivative test for extrema 188
4.18 Curve sketching 189
4.19 Exercises 191
4.20 Worked examples of extremum problems 191
4.21 Exercises 194
4.22 Partial derivatives 196
4.23 Exercises 201
5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
5.1 The derivative of an indefinite integral. The first fundamental theorem of calculus 202
5.2 The zero-derivative theorem 204
5.3 Primitive functions and the second fundamental theorem of calculus 205
5.4 Properties of a function deduced from properties of its derivative 207
5.5 Exercises 208
5.6 The Leibniz notation for primitives 210
5.7 Integration by substitution 212
5.8 Exercises 216
5.9 Integration by parts 217
5.10 Exercises 220
5.11 Miscellaneous review exercises 222
6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS
6.1 Introduction 226
6.2 Motivation for the definition of the natural logarithm as an integral 227
6.3 The definition of the logarithm. Basic properties 229
6.4 The graph of the natural logarithm 230
6.5 Consequences of the functional equation \(L(ab) = L(a) + L(b)\) 230
6.6 Logarithms referred to any positive base \(b \ne 1\) 232
6.7 Differentiation and integration formulas involving logarithms 233
6.8 Logarithmic differentiation 235
6.9 Exercises 236
6.10 Polynomial approximations to the logarithm 236
6.11 Exercises 242
6.12 The exponential function 242
6.13 Exponentials expressed as powers of e 242
6.14 The definition of \(e^x\) for arbitrary real x 244
6.15 The definition of \(a^x\) for \(a > 0\) and x real 245
6.16 Differentiation and integration formulas involving exponentials 245
6.17 Exercises 248
6.18 The hyperbolic functions 251
6.19 Exercises 251
6.20 Derivatives of inverse functions 252
6.21 Inverses of the trigonometric functions 253
6.22 Exercises 256
6.23 Integration by partial fractions 258
6.24 Integrals which can be transformed into integrals of rational functions 264
6.25 Exercises 267
6.26 Miscellaneous review exercises 268
7. POLYNOMIAL APPROXIMATIONS TO FUNCTIONS
7.1 Introduction 272
7.2 The Taylor polynomials generated by a function 273
7.3 Calculus of Taylor polynomials 275
7.4 Exercises 278
7.5 Taylor's formula with remainder 278
7.6 Estimates for the error in Taylor's formula 280
7.7 Other forms of the remainder in Taylor's formula 283
7.8 Exercises 284
7.9 Further remarks on the error in Taylor's formula. The o-notation 286
7.10 Applications to indeterminate forms 289
7.11 Exercises 290
7.12 L'Hopital's rule for the indeterminate form 0/0 292
7.13 Exercises 295
7.14 The symbols \(+\inf\) and \(-\inf\). Extension of L'Hopital's rule 296
7.15 Infinite limits 298
7.16 The behavior of log\(x\) and \(e^x\) for large \(x\) 300
7.17 Exercises 303
8. INTRODUCTION TO DIFFERENTIAL EQUATIONS
8.1 Introduction 305
8.2 Terminology and notation 306
8.3 A first-order differential equation for the exponential function 307
8.4 First-order linear differential equations 308
8.5 Exercises 311
8.6 Some physical problems leading to first-order linear differential equations 313
8.7 Exercises 319
8.8 Linear equations of second order with constant coefficients 322
8.9 Existence of solutions of the equation \(y^{} + by = 0\) 323
8.10 Reduction of the general equation to the special case \(y^{} + by = 0\) 324
8.11 Uniqueness theorem for the equation \(y^{} + by = 0\) 324
8.12 Complete solution of the equation \(y^{} + by = 0\) 326
8.13 Complete solution of the equation \(y^{} + ay^' + by = 0\) 326
8.14 Exercises 328
8.15 Nonhomogeneous linear equations of second order with constant coefficients 329
8.16 Special methods for determining a particular solution of the nonhomogeneous equation \(y^{} + ay^' + by = R\) 332
8.17 Exercises 333
8.18 Examples of physical problems leading to linear second-order equations with constant coefficients 334
8.19 Exercises 339
8.20 Remarks concerning nonlinear differential equations 339
8.21 Integral curves and direction fields 341
8.22 Exercises 344
8.23 First-order separable equations 345
8.24 Exercises 347
8.25 Homogeneous first-order equations 347
8.26 Exercises 350
8.27 Some geometrical and physical problems leading to first-order equations 351
8.28 Miscellaneous review exercises 355
9. COMPLEX NUMBERS
9.1 Historical introduction 358
9.2 Definitions and field properties 358
9.3 The complex numbers as an extension of the real numbers 360
9.4 The imaginary unit \(i\) 361
9.5 Geometric interpretation. Modulus and argument 362
9.6 Exercises 365
9.7 Complex exponentials 366
9.8 Complex-valued functions 368
9.9 Examples of differentiation and integration formulas 369
9.10 Exercises 371
PART IV: MISCELLANEOUS
Chapter 13: Functions
1 Definition of a function 313
2 Polynomial functions 318
3 Graphs of functions 330
4 Exponential function 333
5 Logarithms 338
Chapter 14: Mappings
1 Definition 345
2 Formalism of mappings 351
3 Permutations 359
Chapter 15: Complex Numbers
1 The complex plane 375
2 Polar form 380
Chapter 16: Induction and Summations
1 Induction 383
2 Summations 388
3 Geometric series 396
Chapter 17: Determinants
1 Matrices 401
2 Determinants of order 2 406
3 Properties of 2 x 2 determinants 409
4 Determinants of order 3 414
5 Properties of 3 x 3 determinants 418
6 Cramer's Rule 424
Index 429