Honors Calculus Book

Preface xi Acknowledgments xiii

Chapter 1: Functions on Sets
1.1 Sets 1
1.2 Functions 2
1.3 Cardinality 5
Exercises 6

Chapter 2: The Real Numbers
2.1 The Axioms 12
2.2 Implications 14
2.3 Latter-Day Axioms 16
Exercises 16

Chapter 3: Metric Properties
3.1 The Real Line 19
3.2 Distance 20
3.3 Topology 21
3.4 Connectedness 22
3.5 Compactness 23
Exercises 27

Chapter 4: Continuity
4.1 The Definition 30
4.2 Consequences 31
4.3 Combinations of Continuous Functions 33
4.4 Bisection 36
4.5 Subspace Topology 37
Exercises 38

Chapter 5: Limits and Derivatives
5.1 Limits 41
5.2 The Derivative 43
5.3 Mean Value Theorem 46
5.4 Derivatives of Inverse Functions 48
5.5 Derivatives of Trigonometric Functions 50
Exercises 53

Chapter 6: Applications of the Derivative
6.1 Tangents 60
6.2 Newton's Method 63
6.3 Linear Approximation and Sensitivity 65
6.4 Optimization 66
6.5 Rate of Change 67
6.6 Related Rates 68
6.7 Ordinary Differential Equations 69
6.8 Kepler's Laws 71
6.9 Universal Gravitation 73
6.10 Concavity 76
6.11 Differentials 79
Exercises 80

Chapter 7: The Riemann Integral
7.1 Darboux Sums 89
7.2 The Fundamental Theorem of Calculus 91
7.3 Continuous Integrands 92
7.4 Properties of Integrals 94
7.5 Variable Limits of Integration 95
7.6 Integrability 96
Exercises 97

Chapter 8: Applications of the Integral
8.1 Work 100
8.2 Area 102
8.3 Average Value 104
8.4 Volumes 105
8.5 Moments 106
8.6 Arclength 109
8.7 Accumulating Processes 110
8.8 Logarithms 110
8.9 Methods of Integration 112
8.10 Improper Integrals 113
8.11 Statistics 115
8.12 Quantum Mechanics 117
8.13 Numerical Integration 118
Exercises 121

Chapter 9: Infinite Series
9.1 Zeno's Paradoxes 134
9.2 Convergence of Sequences 134
9.3 Convergence of Series 136
9.4 Convergence Tests for Positive Series 138
9.5 Convergence Tests for Signed Series 140
9.6 Manipulating Series 142
9.7 Power Series 145
9.8 Convergence Tests for Power Series 147
9.9 Manipulation of Power Series 149
9.10 Taylor Series 151
Exercises 154

References 163
Index 165