Honors Calculus / Edition 1

by Charles R. MacCluer

Overview

This is the first modern calculus book to be organized axiomatically and to survey the subject's applicability to science and engineering. A challenging exposition of calculus in the European style, it is an excellent text for a first-year university honors course or for a third-year analysis course. The calculus is built carefully from the axioms with all the standard results deduced from these axioms. The concise construction, by design, provides maximal flexibility for the instructor and allows the student to see the overall flow of the development. At the same time, the book reveals the origins of the calculus in celestial mechanics and number theory.


The book introduces many topics often left to the appendixes in standard calculus textbooks and develops their connections with physics, engineering, and statistics. The author uses applications of derivatives and integrals to show how calculus is applied in these disciplines. Solutions to all exercises (even those involving proofs) are available to instructors upon request, making this book unique among texts in the field.

• Focuses on single variable calculus
• Provides a balance of precision and intuition
• Offers both routine and demanding exercises

Product Details

ISBN-13:	9780691125336
Publisher:	Princeton University Press
Publication date:	03/26/2006
Edition description:	New Edition
Pages:	200
Product dimensions:	7.00(w) x 10.00(h) x (d)

About the Author

Charles R. MacCluer is Professor of Mathematics and director of the industrial mathematics program at Michigan State University. His first interest was algebraic number theory but later turned to the more practical disciplines of control theory, signal processing, building science, and industrial problems. He is the author of Industrial Mathematics, Boundary Value Problems and Fourier Expansions, and Calculus of Variations.

Table of Contents

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

Editorial Reviews

"MacCluer's book . . . is calculus 'done right.' And in a mere 162 pages and 1.1 lb! . . . The approach is axiomatic, but the writing style is breezy and inviting. The student will see right away that this is not simply a rehash of material seen in high school—there are some genuinely new ideas here. These include topology, compactness, quantum mechanics, differential equations, and a host of other fascinating topics. . . . This could be an exciting book from which to teach. This is a book that will allow the instructor to build a fascinating course in a variety of different ways. Teaching from this book should be a joy for all."—-Steven G. Krantz, UMAP Journal

From the Publisher

MacCluer's book . . . is calculus 'done right.' And in a mere 162 pages and 1.1 lb! . . . The approach is axiomatic, but the writing style is breezy and inviting. The student will see right away that this is not simply a rehash of material seen in high school—there are some genuinely new ideas here. These include topology, compactness, quantum mechanics, differential equations, and a host of other fascinating topics. . . . This could be an exciting book from which to teach. This is a book that will allow the instructor to build a fascinating course in a variety of different ways. Teaching from this book should be a joy for all.

UMAP Journal - Steven G. Krantz

MacCluer's book . . . is calculus 'done right.' And in a mere 162 pages and 1.1 lb! . . . The approach is axiomatic, but the writing style is breezy and inviting. The student will see right away that this is not simply a rehash of material seen in high school—there are some genuinely new ideas here. These include topology, compactness, quantum mechanics, differential equations, and a host of other fascinating topics. . . . This could be an exciting book from which to teach. This is a book that will allow the instructor to build a fascinating course in a variety of different ways. Teaching from this book should be a joy for all.
— Steven G. Krantz

UMAP Journal