4.3 Fermat, Tangent Lines, and Extrema

Chapters in this book

1. Frontmatter i
2. Contents ix
3. Preface to the 2021 Edition xiii
4. Preface to the 2007 Paperback Edition xix
5. Preface xxvii
6. 1. Minimums, Maximums, Derivatives, and Computers
7. 1.1 Introduction 1
8. 1.2 When Derivatives Don’t Work 4
9. 1.3 Using Algebra to Find Minimums 5
10. 1.4 A Civil Engineering Problem 9
11. 1.5 The AM-GM Inequality 13
12. 1.6 Derivatives from Physics 20
13. 1.7 Minimizing with a Computer 24
14. 2. The First Extremal Problems
15. 2.1 The Ancient Confusion of Length and Area 37
16. 2.2 Dido’s Problem and the Isoperimetric Quotient 45
17. 2.3 Steiner’s “Solution” to Dido’s Problem 56
18. 2.4 How Steiner Stumbled 59
19. 2.5 A “Hard” Problem with an Easy Solution 62
20. 2.6 Fagnano’s Problem 65
21. 3. Medieval Maximization and Some Modern Twists
22. 3.1 The Regiomontanus Problem 71
23. 3.2 The Saturn Problem 77
24. 3.3 The Envelope-Folding Problem 79
25. 3.4 The Pipe-and-Corner Problem 85
26. 3.5 Regiomontanus Redux 89
27. 3.6 The Muddy Wheel Problem 94
28. 4. The Forgotten War of Descartes and Fermat
29. 4.1 Two Very Different Men 99
30. 4.2 Snell’s Law 101
31. 4.3 Fermat, Tangent Lines, and Extrema 109
32. 4.4 The Birth of the Derivative 114
33. 4.5 Derivatives and Tangents 120
34. 4.6 Snell’s Law and the Principle of Least Time 127
35. 4.7 A Popular Textbook Problem 134
36. 4.8 Snell’s Law and the Rainbow 137
37. 5. Calculus Steps Forward, Center Stage
38. 5.1 The Derivative: Controversy and Triumph 140
39. 5.2 Paintings Again, and Kepler’s Wine Barrel 147
40. 5.3 The Mailable Package Paradox 149
41. 5.4 Projectile Motion in a Gravitational Field 152
42. 5.5 The Perfect Basketball Shot 158
43. 5.6 Halley’s Gunnery Problem 165
44. 5.7 De L’Hospital and His Pulley Problem, and a New Minimum Principle 171
45. 5.8 Derivatives and the Rainbow 179
46. 6. Beyond Calculus
47. 6.1 Galileo’s Problem 200
48. 6.2 The Brachistochrone Problem 210
49. 6.3 Comparing Galileo and Bernoulli 221
50. 6.4 The Euler-Lagrange Equation 231
51. 6.5 The Straight Line and the Brachistochrone 238
52. 6.6 Galileo’s Hanging Chain 240
53. 6.7 The Catenary Again 247
54. 6.8 The Isoperimetric Problem, Solved (at last!) 251
55. 6.9 Minimal Area Surfaces, Plateau’s Problem, and Soap Bubbles 259
56. 6.10 The Human Side of Minimal Area Surfaces 271
57. 7. The Modern Age Begins
58. 7.1 The Fermat/Steiner Problem 279
59. 7.2 Digging the Optimal Trench, Paving the Shortest Mail Route, and Least-Cost Paths through Directed Graphs 286
60. 7.3 The Traveling Salesman Problem 293
61. 7.4 Minimizing with Inequalities (Linear Programming) 295
62. 7.5 Minimizing by Working Backwards (Dynamic Programming) 312
63. Appendix A. The AM-GM Inequality 331
64. Appendix B. The AM-QM Inequality, and Jensen’s Inequality 334
65. Appendix C. “The Sagacity of the Bees” (the preface to Book 5 of Pappus’ Mathematical Collection) 342
66. Appendix D. Every Convex Figure Has a Perimeter Bisector 345
67. Appendix E. The Gravitational Free-Fall Descent Time along a Circle 347
68. Appendix F. The Area Enclosed by a Closed Curve 352
69. Appendix G. Beltrami’s Identity 359
70. Appendix H. The Last Word on the Lost Fisherman Problem 361
71. Appendix I. Solution to the New Challenge Problem 364
72. Acknowledgments 367
73. Index 369