Calculus

Cover......Page 1
Title......Page 2
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
PREFACE......Page 6
CONTENTS......Page 12
PART I. Prologue......Page 15
1 Basic Properties of Numbers......Page 17
2 Numbers of Various Sorts......Page 35
PART II. Foundations......Page 51
3 Functions......Page 53
Appendix. Ordered Pairs......Page 68
4 Graphs......Page 70
Appendix 1. Vectors......Page 89
Appendix 2. The Conic Sections......Page 94
Appendix 3. Polar Coordinates......Page 98
5 Limits......Page 104
6 Continuous Functions......Page 127
7 Three Hard Theorems......Page 134
8 Least Upper Bounds......Page 145
Appendix. Uniform Continuity......Page 156
PART III. Derivatives and Integrals......Page 159
9 Derivatives......Page 161
10 Differentiation......Page 180
11 Significance of the Derivative......Page 199
Appendix. Convexity and Concavity......Page 230
12 Inverse Functions......Page 241
Appendix. Parametric Representation of Curves......Page 255
13 Integrals......Page 264
Appendix. Riemann Sums......Page 293
14 The Fundamental Theorem of Calculus......Page 296
15 The Trigonometric Functions......Page 314
*16 $\\pi$ is Irrational......Page 335
*17 Planetary Motion......Page 341
18 The Logarithm and Exponential Functions......Page 350
19 Integration in Elementary Terms......Page 373
Appendix. The Cosmopolitan Integral......Page 411
PART IV. Infinite Sequences and Infinite Series......Page 417
20 Approximation by Polynomial Functions......Page 419
*21 $e$ is Transcendental......Page 449
22 Infinite Sequences......Page 459
23 Infinite Series......Page 478
24 Uniform Convergence and Power Series......Page 505
25 Complex Numbers......Page 531
26 Complex Functions......Page 546
27 Complex Power Series......Page 560
PART V. Epilogue......Page 583
28 Fields......Page 585
29 Construction of the Real Numbers......Page 592
30 Uniqueness of the Real Numbers......Page 605
Suggested Reading......Page 613
Answers (to selected problems)......Page 623
Glossary of Symbols......Page 669
Index......Page 673