Calculus Made Clear - Part I.pdf

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Calculus Made Clear - Part I.pdf




     Change and Motion:

     Calculus Made Clear, 2nd Edition

     Taught by: Professor Michael Starbird, The University of Texas at Austin


     Course Guidebook

     Michael Starbird, Ph.D.

     University Distinguished Teaching Professor of Mathematics, The University of Texas at Austin 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. The professor's mathematical research is in the field 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 significant 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 coauthor 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:

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     Making Light of Weighty Ideas, published by W. W. Norton, 2005. Professor Starbird has produced three previous courses for The Teaching Company, the first 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 find the intrigue and fascination that mathematics can bring.


     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 flow 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 first 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.


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     The Teaching Company


     Table of Contents Change and Motion: Calculus Made Clear 2nd Edition Parti

     Professor Biography Course Scope Lecture One Lecture Two Lecture Three Lecture Four Lecture Five Lecture Six Lecture Seven Lecture Eight Lecture Nine Lecture Ten Lecture Eleven Lecture Twelve Timeline

    Glossary Biographical Notes Bibliography i 1 Two Ideas, Vast Implications 3 Stop Sign Crime?ªThe First Idea of Calculus?ªThe Derivative 8 Another Car, Another Crime?ªThe Second Idea of Calculus?ªThe Integral The Fundamental Theorem of Calculus Visualizing the Derivative?ªSlopes Derivatives the Easy Way?ªSymbol Pushing Abstracting the Derivative?ªCircles and Belts Circles, Pyramids, Cones, and Spheres Archimedes and the Tractrix The Integral and the Fundamental Theorem Abstracting the Integral?ªPyramids and Dams Buffon's Needle or n from Breadsticks 13 18 23 28 33 38 42 46 53 57 62 64 Part II Part II

     Change and Motion: Calculus Made Clear 2nd Edition


     Twenty-five 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 infinite 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 influential ideas in human history. Its impact on our daily lives is incalculable, even with calculus. Economics, population growth, traffic flow, 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 significance of the subject. The concepts and insights at the heart of calculus are absolutely meaningful, understandable, and accessible to all intelligent people?ªregardless 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 way?ªgrowing by accretion of infinitely 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


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     The Teaching Company

     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 traffic flow 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. Our daily experience of life at the beginning of the third millennium contrasts markedly with life in the 17th century. Most of the differences emerged from technical advances that rely on calculus. We live differently now because we can manipulate and control nature better than we could 300 years ago. That practical, predictive understanding of the physical processes of nature is largely enabled by the power and perspective of calculus. Calculus not only provides specific tools that solve practical problems, but it also entails an intellectual perspective on how we analyze the world. Calculus is all around us and

    is a landmark achievement of humans that can be enjoyed and appreciated by all.


     Lecture One Two Ideas, Vast Implications

     Scope: Calculus is the exploration of two ideas that arise from a clear, commonsensical analysis of everyday experience. But explorations of these ideas?ªthe derivative and the integral ?ªhelp us construct the very foundation of what we know of physical reality and how we express that knowledge. 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 aspects of economics, population growth, traffic flow, fluid flow, electricity, baseball, planetary motion, and countless other topics. 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 and describe our world. Calculus has been one of the most effective conceptual tools in human history.



     Calculus is all around us. A. When we're driving down a road and see where we are and how fast we are going.. .that's calculus. B. When we throw a baseball and see where it lands.. .that's calculus. C. When we see the planets and how they orbit around the Sun.. .that's calculus. D. When we lament the decline in the population of the spotted owl.. .that's calculus. E. When we analyze the stock market.. .that's calculus. Calculus is an idea of enormous importance and historical impact. A. Calculus has been extremely effective in allowing people to bend nature to human purpose. B. In the 20th century, calculus has also become an essential tool for understanding social and biological sciences: It occurs every day in the description of economic trends, population growth, and medical treatments. C. The physical world and how it works are described using calculus?ªits terms, its notation, its perspective. D. To understand the history of the last 300 years, we must understand calculus. The technological developments in recent centuries are the story of this time, and many of those developments depend on calculus.

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     E. Why is calculus so effective? Because it resolves some basic issues associated with change and motion. III. Twenty-five hundred years ago, the philosopher Zeno pointed out the paradoxical nature of motion. Zeno's paradoxes confront us with questions about motion. Calculus resolves these ancient conundrums. A. Two of Zeno's paradoxes

    of motion involve an arrow in flight. 1. The first is the arrow paradox: If at every moment, the arrow is at a particular point, then at every moment, it is at rest at a point. 2. The second is the dichotomy paradox: To reach its target, an arrow must first fly halfway, then half the remaining distance, then half the remaining distance, and so on, forever. Because it must move an infinite number of times, it will never reach the target. B. Looking at familiar occurrences afresh provokes insights and questions. Zeno's paradoxes have been extremely fruitful. C. Zeno's paradoxes bring up questions about infinity and instantaneous motion. IV. In this course, we emphasize the ideas of calculus more than the mechanical side. A. But I must add that one of the reasons that calculus has been of such importance for these last 300 years is that it can be used in a mechanical way. It can be used by people who don't understand it. That's part of its power. B. Perhaps we think calculus is hard because the word calculus comes from the Greek word for stones (stones were used for reckoning in ancient times). C. Calculus does have a fearsome reputation for being very hard, and part of the goal of this course is to help you see calculus in a different light. D. In describing his college entrance examinations in his autobiography, Sir Winston Churchill says, "Further dim chambers lighted by sullen, sulphurous fires were reputed to contain a dragon called the 'Differential Calculus.' But this monster was beyond the bounds appointed by the Civil Service Commissioners who regulated this stage of Pilgrim's heavy journey." We will attempt to douse the dragon's fearsome fires. E. Another reason calculus is considered so forbidding is the size of calculus textbooks. To students, a calculus book has 1,200 different pages. But to a professor, it has two ideas and lots of examples, applications, and variations.

     V. Fortunately, the two fundamental ideas of calculus, called the derivative and the integral, come from everyday observations. A. Calculus does not require complicated notation or vocabulary. It can be understood in English. B. We will describe and define simply and understandably those two fundamental ideas in Lectures Two and Three. Both ideas will come about from analyzing a car moving down a straight road and just thinking very clearly about that scenario. C. The viewer is not expected to have any sense whatsoever of the meanings of these ideas now. In fact, I hope these technical terms inspire, if anything, only a foreboding sense of impending terror. That sense will make the discovery that these ideas are commonsensical and even joyful, instead of terrifying, all the sweeter. D. The derivative deals with how fast things are changing (instantaneous change). E. The integral provides a dynamic view of the static world, showing fixed objects growing by accretion (the accumulation of small pieces). F. We can even view apparently static things dynamically. For example, we can view the area

    of a square or the volume of a cube dynamically by thinking of it as growing rather than just being at its final size. G. The derivative and the integral are connected by the Fundamental Theorem of Calculus, which we will discuss in Lecture Four. H. Both of the fundamental ideas of calculus arise from a straightforward discussion of a car driving down a road, but both are applicable in many other settings. VI. The history of calculus spans two and a half millennia. A. Pythagoras invented the Pythagorean Theorem in the 6th century B.C., tV\ and as we know, Zeno posed his paradoxes of motion in the 5 century B.C. B. In the 4th century B.C., Eudoxus developed the Method of Exhaustion, similar to the integral, to study volumes of objects. C. In about 300 B.C., Euclid invented the axiomatic method of geometry. D. In 225 B.C., Archimedes used calculus-like methods to find areas and volumes of geometric objects. E. For many centuries, other mathematicians developed ideas that were important prerequisites for the full development of calculus. 1. In around 1600, Johannes Kepler and Galileo Galilei worked making mathematical formulas that described planetary motion. 2. In 1629, Pierre de Fermat developed methods for finding maxima of values, a precursor to the idea of the derivative.

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     In the 1630s, Bonaventura Cavalieri developed the "Method of Indivisibles," and later, Rene Descartes established the Cartesian Coordinate System, a connection between algebra and geometry. F. The two mathematicians whose names are associated with the invention or discovery of calculus are Isaac Newton and Gottfried Wilhelm von Leibniz. They independently developed calculus in the 1660s and 1670s. G. From the time of the invention of calculus, other people contributed variations on the idea and developed applications of calculus in many areas of life. 1. Johann and Jakob Bernoulli were two of eight Bernoullis who were involved in developing calculus. 2. Leonhard Euler developed extensions of calculus, especially infinite series. 3. Joseph Louis Lagrange worked on calculus variations, and Pierre Simon de Laplace worked on partial differential equations and applied calculus to probability theory. 4. Jean Baptiste Joseph Fourier invented ways to approximate certain kinds of dependencies, and Augustin Louis Cauchy developed ideas about infinite series and tried to formalize the idea of limit. 5. In the 1800s, Georg Friedrich Bernhard Riemann developed the modern definition of the integral, one of the two ideas of calculus. 6. In the middle of the 1850s?ªabout 185 years after Newton and Leibniz invented calculus?ªKarl Weierstrass formulated the rigorous definition of limit that we know today. VII. Here is an overview of the lectures. A. In Lectures Two, Three, and Four, we will introduce the basic ideas of calculus in the context of a moving car and discuss the connection

    between those ideas. B. Then, we have a series of lectures describing the meaning of the derivative graphically, algebraically, and in many applications. C. Following that, we have a similar series of lectures showing the integral from graphical, algebraic, and application points of view. D. The last half of the course demonstrates the richness of these two ideas by showing examples of their extensions, variations, and applications. E. The purpose of these lectures is to explain clearly the concepts of calculus and to convince the viewer that calculus can be understood from simple scenarios. 1. Calculus is so effective because it deals with change and motion and allows us to view our world as dynamic rather than static.


     Calculus provides a tool for measuring change, whether it is change in position, change in temperature, change in demand, or change in population. F. Calculus is intrinsically intriguing and beautiful, as well as important. G. Calculus is a crowning intellectual achievement of humanity that all intelligent people can appreciate, understand, and enjoy. Readings: Cajori, Florian. "History of Zeno's Arguments on Motion," The American Mathematical Monthly, Vol. 22, Nos. 1-9 (1915). Churchill, Winston Spencer. My Early Life: A Roving Commission. Any standard calculus textbook. Questions to Consider: 1. Find things or ideas in your world that you usually view as complete and fixed and think about them dynamically. That is, view their current state as the result of a growing or changing process. 2. Explore the idea of function by describing dependent relationships between varying quantities that you see in everyday life. For example, in what way is the amount of money in a savings account dependent on interest rate and time? Or, for a less quantitative example, how is happiness a function of intellectual stimulation, exercise, rest, and other variables? 3. Think of a scenario from your daily life that interests you. Keep it in mind as you progress through the lectures?ªcan calculus be applied to understand and analyze it?


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     Lecture Two Stop Sign Crime?ª The First Idea of Calculus?ªThe Derivative

     Scope: Change is a fundamental feature of our world: temperature, pressure, the stock market, the population?ªall change. But the most basic example of change is motion?ªa change in position with respect to time. We will start with a simple example of motion as our vehicle for developing an effective way to analyze change. Specifically, suppose we run a stop sign, but in preparation for potential citations, we have a camera take a picture of our car neatly lined up with the

    stop sign at the exact instant that we were there. We show this photograph to the officer and ask to have the ticket dismissed, presenting the photograph as evidence. The officer responds by analyzing our motion in a persuasive way that illustrates the first of the two fundamental ideas of calculus?ªthe derivative. We get the ticket but can take some solace in resolving one of Zeno's paradoxes.

     C. Soon thereafter, he is pulled over by Officers Newton and Leibniz. (The corny names will make memorable how the roles in this drama relate to Zeno's paradox and the invention of calculus.) D. The driver, Zeno, protests by showing a still picture of his car exactly at the stop sign at the exact moment, 1 minute after the hour, when he is supposed to have been running the stop sign. On this street, time is measured by stating the minutes only. E. Zeno claims that there could be no violation because the car was in one place at that moment. III. Officers Newton and Leibniz produce additional evidence. A. The officers produce a still photo of the car at 2 minutes after the hour clearly showing the car 1 mile beyond the stop sign. B. Zeno argues, "So what? At 1 minute, I was stopped at the stop sign." C. Newton: "But you must admit your average velocity between 1 and 2 minutes was 1 mile per minute." 1. To compute the average velocity during any interval of time, you need to know the position of the car at the beginning, the position at the end, and the amount of time that passed. 2. The average velocity is change in position divided by change in time, that is, how far you went divided by how long it took. 3. The average velocity does not rely on what happened between those two moments?ªjust on where the car is at the beginning and at the end. D. Again, Zeno says, "So what?" E. Officers Newton and Leibniz produce an infinite amount of additional evidence, all incriminating. They note that Zeno was at the 1.1 mile marker at 1.1 minute, at the 1.01 mile marker at 1.01 minute, and so on, all of it proving that Zeno's velocity was 1 mile per minute. IV. The idea of instantaneous velocity is the result of an infinite amount of data. A. All the evidence is that, on every even incredibly tiny interval of time, the average velocity was 1 mile per minute. B. The cumulative effect of all the evidence?ªan infinite number of intervals?ªleads to the idea of instantaneous velocity. V. Knowing the position of a car at every moment allows us to compute the velocity at every moment. This can be illustrated with a car whose velocity is increasing. A. Let's now consider the example where at every time, measured in minutes and denoted by the letter t, we are at mileage marker t2 miles. For example, at time 1, the car is at position 1, but at time 2, it is at 0 0 position 4 (2 ), and at time 0.5, it is at position 0.25 (0.5 ).


     I. Calculus has two fundamental ideas (called the derivative and the integral)?ªone centered on a method for analyzing change; the other,

    on a method of combining pieces to get the whole. A. Both of the fundamental concepts of calculus arise from analyzing simple situations, such as a car moving down a straight road. B. This lecture presents an everyday scenario that leads to one of the two ideas of calculus?ªthe derivative.

     II. The following stop-sign scenario is a modern-day enactment of one of Zeno's paradoxes of motion. A. Let us suppose we have a car driving on a road, and there is a mileage marker at every point along the road. Such a simple scenario can be represented in a graph. 1. The horizontal axis is the time axis. 2. The vertical axis tells us the position of the car at each moment of time. 3. For the sake of arithmetic simplicity, we will talk about measuring the velocity (speed) of the car in miles per minute. Therefore, the vertical axis of our graph is in miles and the horizontal axis is in minutes. B. Suppose Zeno is driving this car, and he goes through a stop sign without slowing down.


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     B. If we know where the car is at every time during an hour, we can tell how fast it was going at any selected moment by doing the infinite process of finding instantaneous velocity. C. Let's apply that infinite process to this moving car at several different times t. 1. First, if time / = 2 minutes, the position of the car is 22 = 4. Then, if time t=l minute, the position of the car is I 2 = 1. Subtracting 1 from 4 and dividing by 1 minute, we have 3 miles per minute. In other words, by looking at where the car was 1 minute after the 1minute mark, we find that the average velocity was 3 miles per minute. 2. However, when we look at shorter intervals of time, we get a different story, as shown in the chart below. We will find the positions of the car at various nearby times, such as 1.1, 1.01, 1.001, and 0.99 minutes, and compute our average velocity between time 1 and those times.

     Position :p(t) -f Initial Time Final Time Average Velocity (mi/min) 2.4 0.7 1.7 1.5 0.7 0.8 1.41 0.7 0.71 1.401 0.701 0.7 1.399 0.699 0.7 Instantaneous velocity at t = 0.7 is 1.4 mi/min

     G. If welook at the same question for other times, such as 1.4, 2, or 3 minutes, we have similar results for the instantaneous velocity. H. We have been computing instantaneous velocities at various times for a car that is moving in such a manner that at every time t minutes, the car is at mileage marker f miles. If we look at a chart of all our examples of instantaneous velocity, we see a pattern that indicates that at every time t, the velocity is 2t miles per minute.

     Position :p(t) ?ª f Time (min) 0.7 1 1.4 2 3

     Position :p(t) = ? Initial Time Final Time Average Velocity (mi/min)

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