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The How and Why of One Variable Calculus

Amol Sasane
John Wiley
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BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

This is not a typical calculus book. The title is a bit of a tip-off, although the emphasis should probably be more on the “why” than the “how”. The author says that his book is intended as a textbook for honors calculus and that high school mathematics provides an adequate background. He notes that the book might also be used as supplementary reading for a regular methods-based calculus course or as a text for a transition-to-analysis class.

The book begins with an extended treatment of the real numbers: field axioms, order axioms, the least upper bound property and a brief look at Dedekind cuts. Following this are chapters on sequences, continuity, differentiation, integration and series. All the classic theorems of basic calculus are carefully stated and proved. The usual methods and techniques of calculus are not absent, just downplayed.

The pace is very deliberate. The author lays the groundwork for theorems carefully with examples and exercises. Proofs are clear and full of detail. The author tells students that they need not fully understand the theorems on first reading. He tells them that they should review the examples and exercises that follow, try to understand why the theorems are important, and then go back and study the proofs.

The author includes several topics that don’t often appear in calculus books. An extended treatment of the real numbers goes well beyond the usual quick overview. Sequences appear early. The chapter on continuity discusses how continuous functions preserve convergent sequences and connectedness and why uniform convergence and uniform continuity are important. One consequence of this is that it takes about 125 pages to get to the derivative. The concluding chapter on series has material that is more sophisticated than the more basic treatment that once appeared late in first year calculus.

Both examples and exercises have a broad range of difficulty and sophistication. Included, for instance, is a painstakingly complete proof that constant functions are continuous, but there is also an extended treatment of the Cantor set and proofs that the Cantor set has zero “length” and that the indicator function of the Cantor set is Riemann integrable.

The exercises are plentiful, well-selected and well-constructed. Detailed solutions to all exercises are provided; they fill the last 150 pages of the book. The author admonishes students to avoid peeking at solutions until they have made serious efforts at solving them. Whether this is idealistic or naïve, it means that instructors may need to provide separate exercises of their own.

A lot of lovely mathematics appears here, presented in a way that many of us would have loved as students. So it puzzles me that a short introductory section called “Why study calculus?” mostly cites four applications that range from somewhat interesting to …well… dull. A student capable of using this book should be inspired by the wonderful mathematics. Perhaps the author could provide a few dazzling applications too.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Preface ix

Introduction xi

Preliminary notation xv

1 The real numbers 1

1.1 Intuitive picture of R as points on the number line 2

1.2 The field axioms 6

1.3 Order axioms 8

1.4 The Least Upper Bound Property of R 9

1.5 Rational powers of real numbers 20

1.6 Intervals 21

1.7 Absolute value | · | and distance in R 23

1.8 (∗) Remark on the construction of R 26

1.9 Functions 28

1.10 (∗) Cardinality 40

Notes 43

2 Sequences 44

2.1 Limit of a convergent sequence 46

2.2 Bounded and monotone sequences 54

2.3 Algebra of limits 59

2.4 Sandwich theorem 64

2.5 Subsequences 68

2.6 Cauchy sequences and completeness of R 74

2.7 (∗) Pointwise versus uniform convergence 78

Notes 85

3 Continuity 86

3.1 Definition of continuity 86

3.2 Continuous functions preserve convergence 91

3.3 Intermediate Value Theorem 99

3.4 Extreme Value Theorem 106

3.5 Uniform convergence and continuity 111

3.6 Uniform continuity 111

3.7 Limits 115

Notes 124

4 Differentiation 125

4.1 Differentiable Inverse Theorem 136

4.2 The Chain Rule 140

4.3 Higher order derivatives and derivatives at boundary points 144

4.4 Equations of tangent and normal lines to a curve 148

4.5 Local minimisers and derivatives 157

4.6 Mean Value, Rolle’s, Cauchy’s Theorem 159

4.7 Taylor’s Formula 167

4.8 Convexity 172

4.9 00 form of l’Hôpital’s Rule 180

Notes 182

5 Integration 183

5.1 Towards a definition of the integral 183

5.2 Properties of the Riemann integral 198

5.3 Fundamental Theorem of Calculus 210

5.4 Riemann sums 226

5.5 Improper integrals 232

5.6 Elementary transcendental functions 245

5.7 Applications of Riemann Integration 278

Notes 296

6 Series 297

6.1 Series 297

6.2 Absolute convergence 305

6.3 Power series 320

Appendix 335

Notes 337

Solutions 338

Solutions to the exercises from Chapter 1 338

Solutions to the exercises from Chapter 2 353

Solutions to the exercises from Chapter 3 369

Solutions to the exercises from Chapter 4 388

Solutions to the exercises from Chapter 5 422

Solutions to the exercises from Chapter 6 475

Bibliography 493

Index 495