When Least Is Best

by Nahin, Paul J.

• ISBN13: 9780691070780
• ISBN10: 0691070784
• Format: Hardcover
• Copyright: 2003-11-24
• Publisher: Princeton Univ Pr

Summary

What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area? By combining the mathematical history of extrema with contemporary examples, Paul J. Nahin answers these intriguing questions and more in this engaging and witty volume. He shows how life often works at the extremes--with values becoming as small (or as large) as possible--and how mathematicians over the centuries have struggled to calculate these problems of minima and maxima. From medieval writings to the development of modern calculus to the current field of optimization, Nahin tells the story of Dido's problem, Fermat and Descartes, Torricelli, Bishop Berkeley, Goldschmidt, and more. Along the way, he explores how to build the shortest bridge possible between two towns, how to shop for garbage bags, how to vary speed during a race, and how to make the perfect basketball shot. Written in a conversational tone and requiring only an early undergraduate level of mathematical knowledge,When Least Is Bestis full of fascinating examples and ready-to-try-at-home experiments. This is the first book on optimization written for a wide audience, and math enthusiasts of all backgrounds will delight in its lively topics.

Author Biography

Paul J. Nahin is Professor of Electrical Engineering at the University of New Hampshire.

Table of Contents

Preface	xiii
Minimums, Maximums, Derivatives, and Computers	1	(36)
Introduction	1	(3)
When Derivatives Don't Work	4	(1)
Using Algebra to Find Minimums	5	(4)
A Civil Engineering Problem	9	(4)
The AM-GM Inequality	13	(7)
Derivatives from Physics	20	(4)
Minimizing with a Computer	24	(13)
The First External Problems	37	(34)
The Ancient Confusion of Length and Area	37	(8)
Dido's Problem and the Isoperimetric Quotient	45	(11)
Steiner's ``Solution'' to Dido's Problem	56	(3)
How Steiner Stumbled	59	(3)
A ``Hard'' Problem with an Easy Solution	62	(3)
Fagnano's Problem	65	(6)
Medieval Maximization and Some Modern Twists	71	(28)
The Regiomontanus Problem	71	(6)
The Saturn Problem	77	(2)
The Envelope-Folding Problem	79	(6)
The Pipe-and-Corner Problem	85	(4)
Regiomontanus Redux	89	(5)
The Muddy Wheel Problem	94	(5)
The Forgotten War of Descartes and Fermat	99	(41)
Two Very Different Men	99	(2)
Snell's Law	101	(8)
Fermat, Tangent Lines, and Extrema	109	(5)
The Birth of the Derivative	114	(6)
Derivatives and Tangents	120	(7)
Snell's Law and the Principle of Least Time	127	(7)
A Popular Textbook Problem	134	(3)
Snell's Law and the Rainbow	137	(3)
Calculus Steps Forward, Center Stage	140	(60)
The Derivative: Controversy and Triumph	140	(7)
Paintings Again, and Kepler's Wine Barrel	147	(2)
The Mailable Package Paradox	149	(3)
Projectile Motion in a Gravitational Field	152	(6)
The Perfect Basketball Shot	158	(7)
Halley's Gunnery Problem	165	(6)
De L'Hospital and His Pulley Problem, and a New Minimum Principle	171	(8)
Derivatives and the Rainbow	179	(21)
Beyond Calculus	200	(79)
Galileo's Problem	200	(10)
The Brachistochrone Problem	210	(11)
Comparing Galileo and Bernoulli	221	(10)
The Euler-Lagrange Equation	231	(7)
The Straight Line and the Brachistochrone	238	(2)
Galileo's Hanging Chain	240	(7)
The Catenary Again	247	(4)
The Isoperimetric Problem, Solved (at last!)	251	(8)
Minimal Area Surfaces, Plateau's Problem, and Soap Bubbles	259	(12)
The Human Side of Minimal Area Surfaces	271	(8)
The Modern Age Begins	279	(52)
The Fermat/Steiner Problem	279	(7)
Digging the Optimal Trench, Paving the Shortest Mail Route, and Least-Cost Paths through Directed Graphs	286	(7)
The Traveling Salesman Problem	293	(2)
Minimizing with Inequalities (Linear Programming)	295	(17)
Minimizing by Working Backwards (Dynamic Programming)	312	(19)
Appendix A. The AM-GM Inequality	331	(3)
Appendix B. The AM-QM Inequality, and Jensen's Inequality	334	(8)
Appendix C. ``The Sagacity of the Bees''	342	(3)
Appendix D. Every Convex Figure Has a Perimeter Bisector	345	(2)
Appendix E. The Gravitational Free-Fall Descent Time along a Circle	347	(5)
Appendix F. The Area Enclosed by a Closed Curve	352	(7)
Appendix G. Beltrami's Identity	359	(2)
Appendix H. The Last Word on the Lost Fisherman Problem	361	(4)
Acknowledgments	365	(2)
Index	367