Calculus: A Rigorous First Course

by Daniel J. Velleman

Overview

Designed for undergraduate mathematics majors, this rigorous and rewarding treatment covers the usual topics of first-year calculus: limits, derivatives, integrals, and infinite series. Author Daniel J. Velleman focuses on calculus as a tool for problem solving rather than the subject's theoretical foundations. Stressing a fundamental understanding of the concepts of calculus instead of memorized procedures, this volume teaches problem solving by reasoning, not just calculation. The goal of the text is an understanding of calculus that is deep enough to allow the student to not only find answers to problems, but also achieve certainty of the answers' correctness.
No background in calculus is necessary. Prerequisites include proficiency in basic algebra and trigonometry, and a concise review of both areas provides sufficient background. Extensive problem material appears throughout the text and includes selected answers. Complete solutions are available to instructors.

Product Details

ISBN-13:	9780486809366
Publisher:	Dover Publications
Publication date:	01/18/2017
Series:	Aurora: Dover Modern Math Originals
Pages:	736
Product dimensions:	6.10(w) x 9.20(h) x 1.50(d)

About the Author

Daniel J. Velleman is Professor of Mathematics at Amherst College. His other books include How to Prove It: A Structured Approach.

Table of Contents

Preface ix

1 Preliminaries 1

1.1 Numbers and Sets 1

1.2 Graphs in the Plane 11

1.3 Functions 18

1.4 Combining Functions 28

2 Limits 41

2.1 What is Calculus About? 41

2.2 What Does "Limit" Mean? 51

2.3 Limits by the Definition 66

2.4 Limit Theorems 72

2.5 Variations on Limits 84

2.6 Limits of Compositions 97

2.7 Continuity 104

2.8 Sequences and the Nested Interval Theorem 119

2.9 Monotone Sequences and the Completeness of the Real Numbers 131

3 Derivatives 140

3.1 Rates of Change and Slopes 140

3.2 Derivatives 150

3.3 Derivative Rules 160

3.4 The Chain Rule 173

3.5 Implicit Differentiation 184

4 Applications of Differentiation 193

4.1 Related Rates 193

4.2 The Mean Value Theorem 206

4.3 Increasing and Decreasing Functions 214

4.4 Concavity 225

4.5 Sophisticated Graphing 235

4.6 Optimization Problems 245

4.7 Maxima and Minima on Finite Closed Intervals 254

4.8 L'Hôpital's Rule 261

4.9 Antiderivatives 273

5 Integrals 283

5.1 Summations 283

5.2 Accumulation and Area 290

5.3 Definite Integrals 301

5.4 The Fundamental Theorems of Calculus 310

5.5 Integration by Substitution 321

5.6 Proofs of Theorems 330

6 Applications of Integration 339

6.1 Area Between Curves 339

6.2 Volume by Disks, Washers, and Slices 348

6.3 Volume by Cylindrical Shells 358

6.4 Work 365

6.5 Center of Mass 369

7 Inverse Functions, the Natural Logarithm, and the Exponential Function 376

7.1 Inverse Functions 376

7.2 Calculus with Inverse Functions 385

7.3 The Natural Logarithm 391

7.4 The Exponential Function 400

7.5 The Inverse Trigonometric Functions 412

7.6 L'Hôpital's Rule Again 425

8 Techniques of Integration 433

8.1 Partial Fractions 433

8.2 Integration by Parts 442

8.3 Trigonometric Integrals 450

8.4 Substitution with Inverse Functions 461

8.5 Trigonometric Substitutions 467

8.6 Numerical Integration 476

8.7 Improper Integrals 487

9 Parametric Equations and Polar Coordinates 500

9.1 Parametric Equations 500

9.2 Are Length 512

9.3 Surface Area 519

9.4 Polar Coordinates 527

9.5 Areas in Polar Coordinates 538

10 Infinite Series and Power Series 543

10.1 Infinite Series 543

10.2 Convergence Tests 553

10.3 The Comparison and Limit Comparison Tests 567

10.4 The Ratio and Root Tests 576

10.5 Absolute Convergence and the Alternating Series Test 583

10.6 Power Series 594

10.7 Calculus with Power Series 604

10.8 Taylor Series 616

10.9 The Binomial Series 630

10.10 Taylor Polynomials and the Taylor Remainder 639

10.11 Proof of Theorem 10.7.1 649

Appendix: Answers to Odd-Numbered Exercises 662

Index 704