**Classroom Resource Materials**

Volume: 49;
2015;
169 pp;
Hardcover

**Print ISBN: 978-0-88385-788-5
Product Code: CLRM/49**

List Price: $50.00

AMS Member Price: $37.50

MAA Member Price: $37.50

**Electronic ISBN: 978-1-61444-120-5
Product Code: CLRM/49.E**

List Price: $50.00

AMS Member Price: $37.50

MAA Member Price: $37.50

# Cameos for Calculus: Visualization in the First-Year Course

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*Roger B. Nelsen*

MAA Press: An Imprint of the American Mathematical Society

A thespian or cinematographer might define a
cameo as a brief appearance of a known figure, while a gemologist or
lapidary might define it as a precious or semiprecious stone. This
book presents fifty short enhancements or supplements (the cameos) for
the first-year calculus course in which a geometric figure briefly
appears. Some of the cameos illustrate mainstream topics such as the
derivative, combinatorial formulas used to compute Riemann sums, or
the geometry behind many geometric series. Other cameos present topics
accessible to students at the calculus level but not usually
encountered in the course, such as the Cauchy-Schwarz inequality, the
arithmetic mean-geometric mean inequality, and the Euler-Mascheroni
constant.

There are fifty cameos in the book, grouped into five
sections: Part I. Limits and Differentiation, Part II. Integration, Part
III. Infinite Series, Part IV. Additional Topics, and Part V. Appendix:
Some Precalculus Topics. Many of the cameos include exercises, so
Solutions to all the Exercises follows Part V. The book concludes with
references and an index. Many of the cameos are adapted from articles
published in journals of the MAA, such as The American Mathematical
Monthly, Mathematics Magazine, and The College Mathematics
Journal. Some come from other mathematical journals, and some were
created for this book. By gathering the cameos into a book the author hopes
that they will be more accessible to teachers of calculus, both for
use in the classroom and as supplementary explorations for
students.

#### Reviews & Endorsements

Visualizing mathematical ideas usually reduces the complexity of topics and therefore has educational value. This plays an essential role in courses such as calculus, which are both fundamental and scheduled for first-year students. … The book under review is an interesting and pretty collection of proofs of material from the first-year course, all based on visualizing ideas. … This is not a standard textbook, but it is a very useful complement for both students and instructors in a first-year calculus course.

-- Mehdi Hassani, MAA Reviews

… This unique, student-friendly text should be required reading for anyone enrolling in a first-year calculus course, especially for those who are math challenged. …

-- D. J. Gougeon, CHOICE Connect

… Even the most experienced calculus instructor will likely find something new and useful in this slim volume.

-- CMS Notices

# Table of Contents

## Cameos for Calculus: Visualization in the First-Year Course

- Cover cov11
- Half title i2
- Copyright ii3
- Title iii4
- Series iv5
- Dedication vii8
- Preface ix10
- Contents xi12
- Part I Limits and Differentiation 116
- 1 The limit of (sin t)/t 318
- 2 Approximating π with the limit of (sint)/t 520
- 3 Visualizing the derivative 722
- 4 The product rule 924
- 5 The quotient rule 1126
- 6 The chain rule 1328
- 7 The derivative of the sine 1530
- 8 The derivative of the arctangent 1732
- 9 The derivative of the arcsine 1934
- 10 Means and the mean value theorem 2136
- 11 Tangent line inequalities 2338
- 12 A geometric illustration of the limit for e 2742
- 13 Which is larger, eπ or πe? ab or ba? 2944
- 14 Derivatives of area and volume 3146
- 15 Means and optimization 3348

- Part II Integration 3954
- 16 Combinatorial identities for Riemann sums 4156
- 17 Summation by parts 4762
- 18 Integration by parts 5166
- 19 The world's sneakiest substitution 5368
- 20 Symmetry and integration 5772
- 21 Napier's inequality and the limit for e 6176
- 22 The nth root of n! and another limit for e 6580
- 23 Does shell volume equal disk volume? 6782
- 24 Solids of revolution and the Cauchy-Schwarz inequality 7186
- 25 The midpoint rule is better than the trapezoidal rule 7590
- 26 Can the midpoint rule be improved? 7792
- 27 Why is Simpson's rule exact for cubics? 7994
- 28 Approximating π with integration 8196
- 29 The Hermite-Hadamard inequality 8398
- 30 Polar area and Cartesian area 87102
- 31 Polar area as a source of antiderivatives 89104
- 32 The prismoidal formula 91106

- Part III Infinite Series 93108
- 33 The geometry of geometric series 95110
- 34 Geometric differentiation of geometric series 99114
- 35 Illustrating a telescoping series 101116
- 36 Illustrating applications of the monotone sequence theorem 103118
- 37 The harmonic series and the Euler-Mascheroni constant 107122
- 38 The alternating harmonic series 111126
- 39 The alternating series test 113128
- 40 Approximating π with Maclaurin series 115130

- Part IV Additional Topics 119134
- Part V Appendix: Some Precalculus Topics 139154
- Solutions to the Exercises 151166
- References 163178
- Index 167182
- About the Author 171186

- Cover cov1187
- Half title i188
- Copyright ii189
- Title iii190
- Series iv191
- Dedication vii194
- Preface ix196
- Contents xi198
- Part I Limits and Differentiation 1202
- 1 The limit of (sin t)/t 3204
- 2 Approximating π with the limit of (sint)/t 5206
- 3 Visualizing the derivative 7208
- 4 The product rule 9210
- 5 The quotient rule 11212
- 6 The chain rule 13214
- 7 The derivative of the sine 15216
- 8 The derivative of the arctangent 17218
- 9 The derivative of the arcsine 19220
- 10 Means and the mean value theorem 21222
- 11 Tangent line inequalities 23224
- 12 A geometric illustration of the limit for e 27228
- 13 Which is larger, eπ or πe? ab or ba? 29230
- 14 Derivatives of area and volume 31232
- 15 Means and optimization 33234

- Part II Integration 39240
- 16 Combinatorial identities for Riemann sums 41242
- 17 Summation by parts 47248
- 18 Integration by parts 51252
- 19 The world's sneakiest substitution 53254
- 20 Symmetry and integration 57258
- 21 Napier's inequality and the limit for e 61262
- 22 The nth root of n! and another limit for e 65266
- 23 Does shell volume equal disk volume? 67268
- 24 Solids of revolution and the Cauchy-Schwarz inequality 71272
- 25 The midpoint rule is better than the trapezoidal rule 75276
- 26 Can the midpoint rule be improved? 77278
- 27 Why is Simpson's rule exact for cubics? 79280
- 28 Approximating π with integration 81282
- 29 The Hermite-Hadamard inequality 83284
- 30 Polar area and Cartesian area 87288
- 31 Polar area as a source of antiderivatives 89290
- 32 The prismoidal formula 91292

- Part III Infinite Series 93294
- 33 The geometry of geometric series 95296
- 34 Geometric differentiation of geometric series 99300
- 35 Illustrating a telescoping series 101302
- 36 Illustrating applications of the monotone sequence theorem 103304
- 37 The harmonic series and the Euler-Mascheroni constant 107308
- 38 The alternating harmonic series 111312
- 39 The alternating series test 113314
- 40 Approximating π with Maclaurin series 115316

- Part IV Additional Topics 119320
- Part V Appendix: Some Precalculus Topics 139340
- Solutions to the Exercises 151352
- References 163364
- Index 167368
- About the Author 171372
- Back cover 172373