Change and Motion: Calculus Made Clear, 2nd Edition Michael Starbird, Ph.D. PUBLISHED BY: THE GREAT COURSES Corporate Headquarters 4840 Westfi elds Boulevard, Suite 500 Chantilly, Virginia 20151-2299 Phone: 1-800-832-2412 Fax: 703-378-3819 www.thegreatcourses.com Copyright © The Teaching Company, 2006 Printed in the United States of America This book is in copyright. All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of The Teaching Company. Michael Starbird, Ph.D. University Distinguished Teaching Professor of Mathematics The University of Texas at Austin P rofessor Michael Starbird is Professor of Mathematics and a University Distinguished Teaching Professor at The University of Texas at Austin. He received his B.A. degree from Pomona College in 1970 and his Ph.D. in mathematics from the University of Wisconsin, Madison, in 1974. That same year, he joined the faculty of the Department of Mathematics of The University of Texas at Austin, where he has stayed, except for leaves as a Visiting Member of the Institute for Advanced Study in Princeton, New Jersey; a Visiting Associate Professor at the University of California, San Diego; and a member of the technical staff at the Jet Propulsion Laboratory in Pasadena, California. Professor Starbird served as Associate Dean in the College of Natural Sciences at The University of Texas at Austin from 1989 to 1997. He is a member of the Academy of Distinguished Teachers at UT. He has won many teaching awards, including the Mathematical Association of America’s Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics, which is awarded to three professors annually from among the 27,000 members of the MAA; a Minnie Stevens Piper Professorship, which is awarded each year to 10 professors from any subject at any college or university in the state of Texas; the inaugural award of the Dad’s Association Centennial Teaching Fellowship; the Excellence Award from the Eyes of Texas, twice; the President’s Associates Teaching Excellence Award; the Jean Holloway Award for Teaching Excellence, which is the oldest teaching award at UT and is presented to one professor each year; the Chad Oliver Plan II Teaching Award, which is student-selected and awarded each year to one professor in the Plan II liberal arts honors program; and the Friar Society Centennial Teaching Fellowship, which is awarded to one professor at UT annually and includes the largest monetary teaching prize given at UT. Also, in 1989, Professor Starbird was the Recreational Sports Super Racquets Champion. iii The professor’s mathematical research is in the (cid:191) eld of topology. He recently served as a member-at-large of the Council of the American Mathematical Society and on the national education committees of both the American Mathematical Society and the Mathematical Association of America. Professor Starbird is interested in bringing authentic understanding of signi(cid:191) cant ideas in mathematics to people who are not necessarily mathematically oriented. He has developed and taught an acclaimed class that presents higher-level mathematics to liberal arts students. He wrote, with co-author Edward B. Burger, The Heart of Mathematics: An invitation to effective thinking, which won a 2001 Robert W. Hamilton Book Award. Professors Burger and Starbird have also written a book that brings intriguing mathematical ideas to the public, entitled Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, published by W.W. Norton, 2005. Professor Starbird has produced three previous courses for The Teaching Company, the (cid:191) rst edition of Change and Motion: Calculus Made Clear; Meaning from Data: Statistics Made Clear; and with collaborator Edward Burger, The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas. Professor Starbird loves to see real people (cid:191) nd the intrigue and fascination that mathematics can bring. (cid:374) ii Table of Contents INTRODUCTION Professor Biography ............................................................................i Acknowledgments .............................................................................vi Course Scope .....................................................................................1 LECTURE GUIDES LECTURE 1 Two Ideas, Vast Implications ..............................................................4 LECTURE 2 Stop Sign Crime—The First Idea of Calculus—The Derivative ........12 LECTURE 3 Another Car, Another Crime— The Second Idea of Calculus—The Integral.....................................18 LECTURE 4 The Fundamental Theorem of Calculus ...........................................23 LECTURE 5 Visualizing the Derivative—Slopes ...................................................28 LECTURE 6 Derivatives the Easy Way—Symbol Pushing ...................................34 LECTURE 7 Abstracting the Derivative—Circles and Belts ..................................39 LECTURE 8 Circles, Pyramids, Cones, and Spheres ...........................................44 LECTURE 9 Archimedes and the Tractrix .............................................................48 iii Table of Contents LECTURE 10 The Integral and the Fundamental Theorem ....................................52 LECTURE 11 Abstracting the Integral—Pyramids and Dams.................................58 LECTURE 12 Buffon’s Needle or (cid:652) from Breadsticks .............................................62 LECTURE 13 Achilles, Tortoises, Limits, and Continuity ........................................67 LECTURE 14 Calculators and Approximations .......................................................72 LECTURE 15 The Best of All Possible Worlds—Optimization ................................76 LECTURE 16 Economics and Architecture .............................................................80 LECTURE 17 Galileo, Newton, and Baseball .........................................................85 LECTURE 18 Getting off the Line—Motion in Space ..............................................89 LECTURE 19 Mountain Slopes and Tangent Planes ..............................................92 LECTURE 20 Several Variables—Volumes Galore ................................................96 LECTURE 21 The Fundamental Theorem Extended ............................................100 LECTURE 22 Fields of Arrows—Differential Equations ........................................105 iv Table of Contents LECTURE 23 Owls, Rats, Waves, and Guitars .....................................................111 LECTURE 24 Calculus Everywhere ......................................................................115 SUPPLEMENTAL MATERIAL Timeline ..........................................................................................119 Glossary .........................................................................................122 Biographical Notes .........................................................................127 Bibliography ....................................................................................133 v Acknowledgments I want to thank Alex Pekker for his excellent help with every aspect of this second edition of the calculus course. Alex collaborated with me substantially on the design of the whole course, on the examples and (cid:192) ow of the individual lectures, on the design of the graphics, and on the written materials. Thanks also to Professor Katherine Socha for her work on the (cid:191) rst edition of this course and for her help during the post-production process of the second edition. Thanks to Alisha Reay, Pam Greer, Lucinda Robb, Noreen Nelson, and others from The Teaching Company not only for their excellent professional work during the production of this series of lectures but also for creating a supportive and enjoyable atmosphere in which to work. Thanks to my wife, Roberta Starbird, for her design and construction of several of the props. Finally, thanks to Roberta and my children, Talley and Bryn, for their special encouragement. (cid:374) vi Change and Motion: Calculus Made Clear, 2nd Edition Scope: T wenty-(cid:191) ve hundred years ago, the Greek philosopher Zeno watched an arrow speeding toward its target and framed one of the most productive paradoxes in the history of human thought. He posed the paradox of motion: Namely, at every moment, the arrow is in only one place, yet it moves. This paradox evokes questions about the in(cid:191) nite divisibility of position and time. Two millennia later, Zeno’s paradox was resolved with the invention of calculus, one of the triumphs of the human intellect. Calculus has been one of the most in(cid:192) uential ideas in human history. Its impact on our daily lives is incalculable, even with calculus. Economics, population growth, traf(cid:191) c (cid:192) ow, money matters, electricity, baseball, cosmology, and many other topics are modeled and explained using the ideas and the language of calculus. Calculus is also a fascinating intellectual adventure that allows us to see our world differently. The deep concepts of calculus can be understood without the technical background traditionally required in calculus courses. Indeed, frequently, the technicalities in calculus courses completely submerge the striking, salient insights that compose the true signi(cid:191) cance of the subject. The concepts and insights at the heart of calculus are absolutely meaningful, understandable, and accessible to all intelligent people—regardless of the level or age of their previous mathematical experience. Calculus is the exploration of two ideas, both of which arise from a clear, commonsensical analysis of our everyday experience of motion: the derivative and the integral. After an introduction, the course begins with a discussion of a car driving down a road. As we discuss velocity and position, these two foundational concepts of calculus arise naturally, and their relationship to each other becomes clear and convincing. Calculus directly describes and deals with motion. But the ideas developed there also present us with a dynamic view of the world based on a clear analysis of change. That perspective lets us view even such static objects as circles in a dynamic 1 way—growing by accretion of in(cid:191) nitely thin layers. The pervasive nature of change makes calculus extremely widely applicable. The course proceeds by exploring the rich variations and applications of the two fundamental ideas of calculus. After the introduction in the setting of motion, we proceed to develop the concepts of calculus from several points of view. We see the ideas geometrically and graphically. We interpret calculus ideas in terms of familiar formulas for areas and volumes. We see how the ideas developed in the simple setting of a car moving in a straight line can be extended to apply to motion in space. Among the many variations of the concepts of calculus, we see how calculus describes the contours of mountains and other three-dimensional objects. Finally, we explore the use of calculus in describing the physical, biological, and even architectural worlds. One of the bases for the power of calculus lies in the fact that many questions in many subjects are equivalent when viewed at the appropriate level of abstraction. That is, the mathematical structures that one creates to study and model motion are identical, mathematically, to the structures that model phenomena from biology to economics, from traf(cid:191) c (cid:192) ow to cosmology. By looking at the mathematics itself, we strip away the extraneous features of the questions and focus on the underlying relationships and structures that govern the behavior of the system in question. Calculus is the mathematical structure that lies at the core of a world of seemingly unrelated issues. It is in the language of calculus that scientists describe what we know of physical reality and how we express that knowledge. The language of calculus contains its share of mathematical symbols and terminology. However, we will see that every calculus idea and symbol can be understood in English, not requiring “mathese.” We will not eschew formulas altogether, but we will make clear that every equation is an English sentence that has a meaning in English, and we will deal with that meaning in English. Indeed, one of the principal goals of this series of lectures is to have viewers understand the concepts of calculus as meaningful ideas, not as the manipulation of meaningless symbols. e p o c S 2