Honors Calculus

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