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# Honors CalculusCharles R. MacCluer

 TABLE OF CONTENTS:Preface xiAcknowledgments xiiiChapter 1: Functions on Sets1.1 Sets 11.2 Functions 21.3 Cardinality 5Exercises 6Chapter 2: The Real Numbers2.1 The Axioms 122.2 Implications 142.3 Latter-Day Axioms 16Exercises 16Chapter 3: Metric Properties3.1 The Real Line 193.2 Distance 203.3 Topology 213.4 Connectedness 223.5 Compactness 23Exercises 27Chapter 4: Continuity4.1 The Definition 304.2 Consequences 314.3 Combinations of Continuous Functions 334.4 Bisection 364.5 Subspace Topology 37Exercises 38Chapter 5: Limits and Derivatives5.1 Limits 415.2 The Derivative 435.3 Mean Value Theorem 465.4 Derivatives of Inverse Functions 485.5 Derivatives of Trigonometric Functions 50Exercises 53Chapter 6: Applications of the Derivative6.1 Tangents 606.2 Newton's Method 636.3 Linear Approximation and Sensitivity 656.4 Optimization 666.5 Rate of Change 676.6 Related Rates 686.7 Ordinary Differential Equations 696.8 Kepler's Laws 716.9 Universal Gravitation 736.10 Concavity 766.11 Differentials 79Exercises 80Chapter 7: The Riemann Integral7.1 Darboux Sums 897.2 The Fundamental Theorem of Calculus 917.3 Continuous Integrands 927.4 Properties of Integrals 947.5 Variable Limits of Integration 957.6 Integrability 96Exercises 97Chapter 8: Applications of the Integral8.1 Work 1008.2 Area 1028.3 Average Value 1048.4 Volumes 1058.5 Moments 1068.6 Arclength 1098.7 Accumulating Processes 1108.8 Logarithms 1108.9 Methods of Integration 1128.10 Improper Integrals 1138.11 Statistics 1158.12 Quantum Mechanics 1178.13 Numerical Integration 118Exercises 121Chapter 9: Infinite Series9.1 Zeno's Paradoxes 1349.2 Convergence of Sequences 1349.3 Convergence of Series 1369.4 Convergence Tests for Positive Series 1389.5 Convergence Tests for Signed Series 1409.6 Manipulating Series 1429.7 Power Series 1459.8 Convergence Tests for Power Series 1479.9 Manipulation of Power Series 1499.10 Taylor Series 151Exercises 154References 163Index 165Return to Book DescriptionFile created: 4/21/2017 Questions and comments to: webmaster@press.princeton.eduPrinceton University Press