Cameos for Calculus

Preface; 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: definitions; 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; References; Index; About the author.