Cameos for Calculus

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To the benefit of both teachers and students, Cameos for Calculus presents fifty short visual and geometric enhancements to the first-year calculus course. These range from mainstream topics, such as the derivative and combinatorial formulas used to compute Riemann sums, to equally accessible material less often encountered in the course, such as the Cauchy-Schwarz inequality and the Euler-Mascheroni constant. The fifty cameos are grouped into five sections: Limits and Differentiation; Integration; Infinite Series; Additional Topics; and an appendix on precalculus topics. Many of these also include exercises with solutions to enhance learning. By gathering various cameos, including many adapted from journal articles and some created for this book, Nelsen presents a collection accessible to teachers of calculus, both for use in the classroom and as supplementary explorations for students.

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Preface; Part I. Limits and Differentiation: 1. The limit of (sin t)/t; 2. Approximating Ï 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, eÏ or Ï e? 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 Ï 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 Ï with Maclaurin series; Part IV. Additional Topics: 41. The hyperbolic functions I: definitions; 42. The hyperbolic functions II: are they circular?; 43. The conic sections; 44. The conic sections revisited; 45. The AM-GM inequality for Ï 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; References; Index; About the author.

Roger B. Nelsen received his BA in Mathematics from DePauw University, Indiana, in 1964, and his PhD in Mathematics from Duke University, North Carolina, in 1969. Nelsen was elected to Phi Beta Kappa and Sigma Xi, and taught mathematics and statistics at Lewis and Clark College, Oregon, for forty years before his retirement in 2009. His previous books include Proofs Without Words (1993), An Introduction to Copulas (1999, 2nd edition 2006), Proofs Without Words II, (2000), Math Made Visual (with Claudi Alsina, 2006), When Less Is More (with Claudi Alsina, 2009), Charming Proofs (with Claudi Alsina, 2010), The Calculus Collection (with Caren Diefenderfer, 2010), Icons of Mathematics (with Claudi Alsina, 2011), College Calculus (with Michael Boardman, 2015) and A Mathematical Space Odyssey (with Claudi Alsina, 2015).

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