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Cameos for Calculus: Visualization in the First-Year Course

Roger B. Nelsen
MAA Press
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MAA Press Classroom Resource Materials
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mehdi Hassani
, on

Visualizing mathematical ideas usually reduces the complexity of topics and therefore have educational value. This plays essential rule in courses such as calculus, which are both fundamental and scheduled for first-year students. Sometimes, a suitable graph or graphic contains the whole idea of a proof. These are known as “proofs without words.”

After his several books gathering Proofs without Words, the author has now written a book on calculus, combining similar ideas with some exact arguments. The book under review is an interesting and pretty collection of proofs of material from the first-year course, all based on visualizing ideas. They include several theorems about limits and differentiation (Part I), integration (Part II), infinite series (Part III), and several additional topics (Part IV). The final part of the book is an appendix (Part V) covering some pre-calculus topics. Finally, the author gives solutions to the exercises, which are at the end of many of the chapters.

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 is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.

Part I: Limits and Differentiation
1. The limit of (sin t)/t
2. Approximating π with the limit of (sin t)/t
3. Visualizing the derivative
4. The product rule
5. The quotient rule
6. The chain rule
7. The derivative of the sine
8. The derivative of the arctangent
9. The derivative of the arcsine
10. Means and the mean value theorem
11. Tangent line inequalities
12. A geometric illustration of the limit for e
13. Which is larger, eπ or πe? ab or ba?
14. Derivatives of area and volume
15. Means and optimization
Part II: Integration
16. Combinatorial identities for Riemann sums
17. Summation by parts
18. Integration by parts
19. The world’s sneakiest substitution
20. Symmetry and integration
21. Napier’s inequality and the limit for e
22. The nth root of n? and another limit for e
23. Does shell volume equal disk volume?
24. Solids of revolution and the Cauchy-Schwarz inequality
25. The midpoint rule is better than the trapezoidal rule
26. Can the midpoint rule be improved?
27. Why is Simpson’s rule exact for cubics?
28. Approximating π with integration
29. The Hermite-Hadamard inequality
30. Polar area and Cartesian area
31. Polar area as a source of antiderivatives
32. The prismoidal formula
Part III Infinite Series
33. The geometry of geometric series
34. Geometric differentiation of geometric series
35. Illustrating a telescoping series
36. Illustrating applications of the monotone sequence theorem
37. The harmonic series and the Euler-Mascheroni constant
38. The alternating harmonic series
39. The alternating series test
40. Approximating π with Maclaurin series
Part IV Additional Topics
41. The hyperbolic functions I: Definintions
42. The hyperbolic functions II: Are they circular?
43. The conic sections
44. The conic sections revisited
45. The AM-GM inequality for π positive numbers
Part V: Appendix: Some Precalculus Topics
46. Are all parabolas similar?
47. Basic trigonometric identities
48. The addition formulas for the sine and cosine
49. The double angle formulas
50. Completing the square
Solutions to the Exercises
About the Author