Calculus, Volume I: One-Variable Calculus, with an Introduction to Linear Algebra

CONTENTS


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 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 97
2.7 A geometric description of the sine and cosine functions 102
2.8 Exercises 104
2.9 Polar coordinates 108
2.10 The integral for area in polar coordinates 109
2.11 Exercises 110
2.12 Application of integration to the calculation of volume 111
2.13 Exercises 114
2.14 Application of integration to the concept of work 115
2.15 Exercises 116
2.16 Average value of a function 117
2.17 Exercises 119
2.18 The integral as a function of the upper limit. Indefinite integrals 120
2.19 Exercises 124

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 differentiation to extreme values of functions 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 $b\\eq1$ 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 238
6.11 Exercises 242
6.12 The exponential function 242
6.13 Exponentials expressed as powers of $e$ 244
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 ${\\mathit o}$-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 $+\\infty$ and $-\\infty$. 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

10. SEQUENCES, INFINITE SERIES, IMPROPER INTEGRALS
10.1 Zeno\'s paradox 374
10.2 Sequences 378
10.3 Monotonic sequences of real numbers 381
10.4 Exercises 382
10.5 Infinite series 383
10.6 The linearity property of convergent series 385
10.7 Telescoping series 386
10.8 The geometric series 388
10.9 Exercises 391
*10.10 Exercises on decimal expansions 393
10.11 Tests for convergence 394
10.12 Comparison tests for series of nonnegative terms 394
10.13 The integral test 397
10.14 Exercises 398
10.15 The root test and the ratio test for series of nonnegative terms 399
10.16 Exercises 402
10.17 Alternating series 403
10.18 Conditional and absolute convergence 406
10.19 The convergence tests of Dirichlet and Abel 407
10.20 Exercises 409
*10.21 Rearrangements of series 411
10.22 Miscellaneous review exercises 414
10.23 Improper integrals 416
10.24 Exercises 420

11. SEQUENCES AND SERIES OF FUNCTIONS
11.1 Pointwise convergence of sequences of functions 422
11.2 Uniform convergence of sequences of functions 423
11.3 Uniform convergence and continuity 424
11.4 Uniform convergence and integration 425
11.5 A sufficient condition for uniform convergence 427
11.6 Power series. Circle of convergence 428
11.7 Exercises 430
11.8 Properties of functions represented by real power series 431
11.9 The Taylor\'s series generated by a function 434
11.10 A sufficient condition for convergence of a Taylor\'s series 435
11.11 Power-series expansions for the exponential and trigonometric functions 435
*11.12 Bernstein\'s theorem 437
11.13 Exercises 438
11.14 Power series and differential equations 439
11.15 The binomial series 441
11.16 Exercises 443

12. VECTOR ALGEBRA
12.1 Historical introduction 445
12.2 The vector space of $n$-tuples of real numbers 446
12.3 Geometric interpretation for $n\\leq3$ 448
12.4 Exercises 450
12.5 The dot product 451
12.6 Length or norm of a vector 453
12.7 Orthogonality of vectors 455
12.8 Exercises 456
12.9 Projections. Angle between vectors in $n$-space 457
12.10 The unit coordinate vectors 458
12.11 Exercises 460
12.12 The linear span of a finite set of vectors 462
12.13 Linear independence 463
12.14 Bases 466
12.15 Exercises 467
12.16 The vector space $V_n(C)$ of $n$-tuples of complex numbers 468
12.17 Exercises 470

13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
13.1 Introduction 471
13.2 Lines in $n$-space 472
13.3 Some simple properties of straight lines 473
13.4 Lines and vector-valued functions 474
13.5 Exercises 477
13.6 Planes in Euclidean $n$-space 478
13.7 Planes and vector-valued functions 481
13.8 Exercises 482
13.9 The cross product 483
13.10 The cross product expressed as a determinant 486
13.11 Exercises 487
13.12 The scalar triple product 488
13.13 Cramer\'s rule for solving a system of three linear equations 490
13.14 Exercises 491
13.15 Normal vectors to planes 493
13.16 Linear Cartesian equations for planes 494
13.17 Exercises 496
13.18 The conic sections 497
13.19 Eccentricity of conic sections 500
13.20 Polar equations for conic sections 501
13.21 Exercises 503
13.22 Conic sections symmetric about the origin 504
13.23 Cartesian equations for the conic sections 505
13.24 Exercises 508
13.25 Miscellaneous exercises on conic sections 509

14. CALCULUS OF VECTOR-VALUED FUNCTIONS
14.1 Vector-valued functions of a real variable 512
14.2 Algebraic operations. Components 512
14.3 Limits, derivatives, and integrals 513
14.4 Exercises 516
14.5 Applications to curves. Tangency 517
14.6 Applications to curvilinear motion. Velocity, speed, and acceleration 520
14.7 Exercises 524
14.8 The unit tangent, the principal normal, and the osculating plane of a curve 525
14.9 Exercises 528
14.10 The definition of arc length 529
14.11 Additivity of arc length 532
14.12 The arc-length function 533
14.13 Exercises 535
14.14 Curvature of a curve 536
14.15 Exercises 538
14.16 Velocity and acceleration in polar coordinates 540
14.17 Plane motion with radial acceleration 542
14.18 Cylindrical coordinates 543
14.19 Exercises 543
14.20 Applications to planetary motion 545
14.21 Miscellaneous review exercises 549

15. LINEAR SPACES
15.1 Introduction 551
15.2 The definition of a linear space 551
15.3 Examples of linear spaces 552
15.4 Elementary consequences of the axioms 554
15.5 Exercises 555
15.6 Subspaces of a linear space 556
15.7 Dependent and independent sets in a linear space 557
15.8 Bases and dimension 559
15.9 Exercises 560
15.10 Inner products, Euclidean spaces, norms 561
15.11 Orthogonality in a Euclidean space 564
15.12 Exercises 566
15.13 Construction of orthogonal sets. The Gram-Schmidt process 568
15.14 Orthogonal complements. Projections 572
15.15 Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace 574
15.16 Exercises 576

16. LINEAR TRANSFORMATIONS AND MATRICES
16.1 Linear transformations 578
16.2 Null space and range 579
16.3 Nullity and rank 581
16.4 Exercises 582
16.5 Algebraic operations on linear transformations 583
16.6 Inverses 585
16.7 One-to-one linear transformations 587
16.8 Exercises 589
16.9 Linear transformations with prescribed values 590
16.10 Matrix representations of linear transformations 591
16.11 Construction of a matrix representation in diagonal form 594
16.12 Exercises 596
16.13 Linear spaces of matrices 597
16.14 Isomorphism between linear transformations and matrices 599
16.15 Multiplication of matrices 600
16.16 Exercises 603
16.17 Systems of linear equations 605
16.18 Computation techniques 607
16.19 Inverses of square matrices 611
16.20 Exercises 613
16.21 Miscellaneous exercises on matrices 614

Answers to exercises 617

Index 657