**AMS/MAA Textbooks**

Volume: 64;
2020;
571 pp;
Softcover

MSC: Primary 26; 00;

**Print ISBN: 978-1-4704-5588-0
Product Code: TEXT/64**

List Price: $99.00

AMS Member Price: $74.25

MAA Member Price: $74.25

**Electronic ISBN: 978-1-4704-6305-2
Product Code: TEXT/64.E**

List Price: $99.00

AMS Member Price: $74.25

MAA Member Price: $74.25

#### You may also like

#### Supplemental Materials

# Calculus From Approximation to Theory

Share this page
*Dan Sloughter*

MAA Press: An Imprint of the American Mathematical Society

Calculus from Approximation to Theory takes a fresh and
innovative look at the teaching and learning of calculus. One way to
describe calculus might be to say it is a suite of techniques that
approximate curved things by flat things and through a limiting
process applied to those approximations arrive at an exact answer.
Standard approaches to calculus focus on that limiting process as the
heart of the matter. This text places its emphasis on the
approximating processes and thus illuminates the motivating ideas and
makes clearer the scientific usefulness, indeed centrality, of the
subject while paying careful attention to the theoretical foundations.
Limits are defined in terms of sequences, the derivative is defined
from the best affine approximation, and greater attention than usual
is paid to numerical techniques and the order of an approximation.
Access to modern computational tools is presumed throughout and the
use of these tools is woven seamlessly into the exposition and
problems. All of the central topics of a yearlong calculus course are
covered, with the addition of treatment of difference equations, a
chapter on the complex plane as the arena for motion in two
dimensions, and a much more thorough and modern treatment of
differential equations than is standard.

Dan Sloughter is Emeritus Professor of Mathematics at Furman
University with interests in probability, statistics, and the
philosophy of mathematics and statistics. He has been involved in
efforts to reform calculus instruction for decades and has published
widely on that topic. This book, one of the results of that work, is
very well suited for a yearlong introduction to calculus that focuses
on ideas over techniques.

#### Readership

Undergraduate students interested in calculus.

#### Table of Contents

# Table of Contents

## Calculus From Approximation to Theory

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents v7
- Preface ix11
- Chapter 1. Sequences and Limits 113
- Chapter 2. Functions and Their Properties 5365
- Chapter 3. Derivatives and Best Affine Approximations 111123
- 3.1. Best affine approximations 111123
- Problems 3.1 119131
- 3.2. Derivatives and rates of change 120132
- Problems 3.2 127139
- 3.3. Differentiation of rational functions 130142
- Problems 3.3 136148
- 3.4. Differentiation of compositions 137149
- Problems 3.4 144156
- 3.5. Differentiation of trigonometric functions 146158
- Problems 3.5 152164
- 3.6. Newton’s method 154166
- Problems 3.6 159171
- 3.7. The Mean Value Theorem 160172
- Problems 3.7 169181
- 3.8. Finding maximum and minimum values 171183
- Problems 3.8 183195
- 3.9. The geometry of graphs 185197
- Problems 3.9 192204

- Chapter 4. Integrals 195207
- 4.1. The definite integral 195207
- Problems 4.1 215227
- 4.2. Numerical approximations 217229
- Problems 4.2 229241
- 4.3. The Fundamental Theorem of Calculus 231243
- Problems 4.3 239251
- 4.4. Using the Fundamental Theorem 241253
- Problems 4.4 249261
- 4.5. More techniques of integration 251263
- Problems 4.5 256268
- 4.6. Improper integrals 258270
- Problems 4.6 269281
- 4.7. More on area 272284
- Problems 4.7 277289
- 4.8. Distance, position, and length 278290
- Problems 4.8 286298

- Chapter 5. Taylor Polynomials and Series 289301
- 5.1. Polynomial approximations 289301
- Problems 5.1 299311
- 5.2. Taylor’s theorem 300312
- Problems 5.2 306318
- 5.3. Infinite series revisited 307319
- Problems 5.3 315327
- 5.4. The comparison test 316328
- Problems 5.4 321333
- 5.5. The ratio test 322334
- Problems 5.5 325337
- 5.6. Absolute convergence 326338
- Problems 5.6 330342
- 5.7. Power series 332344
- Problems 5.7 340352
- 5.8. Taylor series 342354
- Problems 5.8 348360
- 5.9. Some limit calculations 349361
- Problems 5.9 355367

- Chapter 6. More Transcendental Functions 357369
- 6.1. The exponential function 357369
- Problems 6.1 365377
- 6.2. The natural logarithm function 367379
- Problems 6.2 373385
- 6.3. Models of growth and decay 375387
- Problems 6.3 383395
- 6.4. Integration of rational functions 385397
- Problems 6.4 390402
- 6.5. Inverse trigonometric functions 391403
- Problems 6.5 399411
- 6.6. Trigonometric substitutions 400412
- Problems 6.6 408420
- 6.7. Hyperbolic functions 409421
- Problems 6.7 417429

- Chapter 7. The Complex Plane 419431
- Chapter 8. Differential Equations 463475
- 8.1. Numerical solutions 463475
- Problems 8.1 471483
- 8.2. Separation of variables 474486
- Problems 8.2 480492
- 8.3. First-order linear equations 482494
- Problems 8.3 486498
- 8.4. Second-order linear equations 488500
- Problems 8.4 494506
- 8.5. Pendulums and mass-spring systems 495507
- Problems 8.5 504516
- 8.6. Phase planes 505517
- Problems 8.6 515527
- 8.7. Power series solutions 519531
- Problems 8.7 527539

- Appendix A. Answers to Selected Problems 529541
- Index 567579
- Back Cover Back Cover1584