2006;
292 pp;
Hardcover

MSC: Primary 26;
Secondary 11; 41

**Print ISBN: 978-0-8218-3750-4
Product Code: ACALC**

List Price: $57.00

AMS Member Price: $45.60

MAA Member Price: $51.30

**Electronic ISBN: 978-1-4704-1113-8
Product Code: ACALC.E**

List Price: $53.00

AMS Member Price: $42.40

MAA Member Price: $47.70

#### Supplemental Materials

# Approximately Calculus

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*Shahriar Shahriari*

2015 Recipient of the MAA University Teaching of Mathematics Award

Is there always a prime number between
\(n\) and \(2n\)? Where, approximately, is the
millionth prime? And just what does calculus have to do with
answering either of these questions? It turns out that calculus has a
lot to do with both questions, as this book can show you.

The theme of the book is approximations. Calculus is a powerful
tool because it allows us to approximate complicated functions with
simpler ones. Indeed, replacing a function locally with a
linear—or higher order—approximation is at the heart of
calculus. The real star of the book, though, is the task of
approximating the number of primes up to a number \(x\). This
leads to the famous Prime Number Theorem—and to the answers to
the two questions about primes.

While emphasizing the role of approximations in calculus, most
major topics are addressed, such as derivatives, integrals, the
Fundamental Theorem of Calculus, sequences, series, and so on.
However, our particular point of view also leads us to many unusual
topics: curvature, Padé approximations, public key
cryptography, and an analysis of the logistic equation, to name a
few.

The reader takes an active role in developing the material by
solving problems. Most topics are broken down into a series of
manageable problems, which guide you to an understanding of the
important ideas. There is also ample exposition to fill in background
material and to get you thinking appropriately about the concepts.

Approximately Calculus is intended for the reader who has
already had an introduction to calculus, but wants to engage the
concepts and ideas at a deeper level. It is suitable as a text for an
honors or alternative second semester calculus course.

#### Readership

Undergraduate students interested in calculus and number theory.

#### Reviews & Endorsements

This fascinating book is a novel approach to undergraduate analysis, which combines most topics in single variable calculus with some elementary number theory. ... It is very well written and fully engages readers in its developments, often beginning with examples and leading them to develop generalizations and, ultimately, theorems and proofs. ... An attractive book, well worth consulting for ideas on presenting topics, or for examples.

-- J.H. Ellison, Choice

The book is very well written and contains many references to articles in journals that are accessible to students ...

-- MAA Reviews

#### Table of Contents

# Table of Contents

## Approximately Calculus

Table of Contents pages: 1 2

- Cover Cover11 free
- Title i2 free
- Copyright iv5 free
- Contents vii8 free
- Preface xiii14 free
- Chapter 1. Patterns and Induction 120 free
- Chapter 2. Divisibility 1736
- Goals 1736
- 2.1. The Division Algorithm and Strange Properties of Positive Integers 1736
- Problems 1938
- 2.2. Writing Mathematics in Paragraphs, Proof by Contradiction, and Irrational Numbers 2241
- Problems 2443
- 2.3. Introducing the planet mod n 2544
- Problems 2645
- 2.4. Additional Problems 2746
- Problems 2746

- Chapter 3. Primes 3352
- Goals 3352
- 3.1. Prime Numbers 3352
- Problems 3453
- 3.2. Formulas for Primes 3857
- Problems 4160
- 3.3. Fermat's Theorem, Pseudo-Primes, and Carmichael Numbers 4261
- Problems 4261
- 3.4. Dynamical Systems and a Proof of Fermat's Theorem 4564
- Problems 5069
- 3.5. Public Key Cryptography 5372
- Problems 5574
- 3.6. Open Conjectures about Primes 5776

- Chapter 4. Derivatives and Approximations of Functions 5978
- Chapter 5. Antiderivatives and Integration 91110
- Goals 91110
- 5.1. Why Can Areas Be Found Using Antiderivatives? A Quick Review of Integration 91110
- Problems 98117
- 5.2. Approximating Integrals: Inscribed and Circumscribed Rectangles 100119
- Problems 103122
- 5.3. Functions Defined by Integrals 105124
- Problems 106125
- 5.4. What if F'(x) = G'(x)? 108127
- Problems 109128

- Chapter 6. Distribution of Primes 111130
- Chapter 7. Log, Exponential, and the Inverse Trigonometric Functions 115134
- Goals 115134
- 7.1. The Natural Log Function and the Distribution of Primes 115134
- Problems 116135
- 7.2. Properties of the Log Function 117136
- Problems 118137
- 7.3. The Exponential Function 120139
- Problems 122141
- 7.4. Inverse Trigonometric Functions 127146
- Problems 128147
- 7.5. Additional Problems 128147
- Problems 128147

- Chapter 8. The Mean Value Theorem and Approximations 131150
- Chapter 9. Linearization Topics 147166
- Chapter 10. Defining Integrals, Areas, and Arclengths 155174
- Chapter 11. Improper Integrals and Techniques of Integration 167186
- Chapter 12. The Prime Number Theorem 175194
- Chapter 13. Local Approximation of Functions and Integral Estimations 183202
- Chapter 14. Sequences and Series Goals 195214
- Goals 195214
- 14.1. Sequences, Convergence, and Mathematical Rigor 195214
- Problems 199218
- 14.2. Series 203222
- Problems 206225
- 14.3. Monotone Bounded Sequences and Limit Properties 208227
- Problems 210229
- 14.4. The nth Term Test and the Comparison Test 213232
- Problems 214233
- 14.5. Euler's Constant and the Alternating Harmonic Series 217236
- Problems 217236
- 14.6. The Integral Test and p-series 218237
- Problems 219238
- 14.7. Additional Problems 221240
- Problems 221240

- Chapter 15. Power Series and Taylor Series 223242
- Goals 223242
- 15.1. Taylor Polynomials and Series 223242
- Problems 225244
- 15.2. Power Series and the Ratio Test 225244
- Problems 226245
- 15.3. Analytic Functions and Convergence of Taylor Series 231250
- Problems 232251
- 15.4. The Interval of Convergence of a Power Series 233252
- Problems 235254
- 15.5. New Power Series from Old 236255
- Problems 238257

- Chapter 16. More On Series 243262
- Chapter 17. Limits of Functions 251270
- Chapter 18. Differential Equations 259278
- Chapter 19. Logical Arguments 271290
- Hints for Selected Problems 277296
- Bibliography 283302
- Index 287306

Table of Contents pages: 1 2