Change and Motion: Calculus Made Clear, 2nd Edition

Course No. 177
Professor Michael Starbird, Ph.D.
The University of Texas at Austin
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Course No. 177
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Course Overview

One of the greatest achievements of the human mind is calculus. It justly deserves a place in the pantheon of our accomplishments with Shakespeare's plays, Beethoven's symphonies, and Einstein's theory of relativity.

In fact, most of the differences in the way we experience life now and the way we experienced it at the beginning of the 17th century emerged because of technical advances that rely on calculus. Calculus is a beautiful idea exposing the rational workings of the world; it is part of our intellectual heritage.

The True Genius of Calculus Is Simple

Calculus, separately invented by Newton and Leibniz, is one of the most fruitful strategies for analyzing our world ever devised. Calculus has made it possible to build bridges that span miles of river, travel to the moon, and predict patterns of population change.

Yet for all its computational power, calculus is the exploration of just two ideas—the derivative and the integral—both of which arise from a commonsense analysis of motion. All a 1,300-page calculus textbook holds, Professor Michael Starbird asserts, are those two basic ideas and 1,298 pages of examples, variations, and applications.

Many of us exclude ourselves from the profound insights of calculus because we didn't continue in mathematics. This great achievement remains a closed door. But Professor Starbird can open that door and make calculus accessible to all.

Why You Didn't Get It the First Time

Professor Starbird is committed to correcting the bewildering way that the beauty of calculus was hidden from many of us in school.

He firmly believes that calculus does not require a complicated vocabulary or notation to understand it. Indeed, the purpose of these lectures is to explain clearly the concepts of calculus and to help you see that "calculus is a crowning intellectual achievement of humanity that all intelligent people can appreciate, enjoy, and understand."

He adds: "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.

"In this course, the concepts and insights at the heart of calculus take center stage. The central ideas are absolutely meaningful and understandable to all intelligent people—regardless of the level or age of their previous mathematical experience. Historical events and everyday action form the foundation for this excursion through calculus."

Two Simple Ideas

After the introduction, the course begins with a discussion of a car driving down a road. As Professor Starbird discusses speed and position, the two foundational concepts of calculus arise naturally, and their relationship to each other becomes clear and convincing.

Professor Starbird presents and explores the fundamental ideas, then shows how they can be understood and applied in many settings.

Expanding the Insight

Calculus originated in our desire to understand motion, which is change in position over time. Professor Starbird then explains how calculus has created powerful insight into everything that changes over time. Thus, the fundamental insight of calculus unites the way we see economics, astronomy, population growth, engineering, and even baseball. Calculus is the mathematical structure that lies at the core of a world of seemingly unrelated issues.

As you follow the intellectual development of calculus, your appreciation of its inner workings will deepen, and your skill in seeing how calculus can solve problems will increase. You will examine the relationships between algebra, geometry, trigonometry, and calculus. You will graduate from considering the linear motion of a car on a straight road to motion on a two-dimensional plane or even the motion of a flying object in three-dimensional space.

Designed for Nonmathematicians

Every step is in English rather than "mathese." Formulas are important, certainly, but the course takes the approach that every equation is in fact also a sentence that can be understood, and solved, in English.

This course is crafted to make the key concepts and triumphs of calculus accessible to nonmathematicians. It requires only a basic acquaintance with beginning high-school level algebra and geometry. This series is not designed as a college calculus course; rather, it will help you see calculus around you in the everyday world.

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24 lectures
 |  Average 31 minutes each
  • 1
    Two Ideas, Vast Implications
    Calculus is a subject of enormous importance and historical impact. It provides a dynamic view of the world and is an invaluable tool for measuring change. Calculus is applicable in many situations, from the trajectory of a baseball to changes in the Dow Jones average or elephant populations. Yet, at its core, calculus is the study of two ideas about motion and change. x
  • 2
    Stop Sign Crime—The First Idea of Calculus—The Derivative
    The example of a car moving down a straight road is a simple and effective way to study motion. An everyday scenario that involves running a stop sign and the use of a camera illustrates the first fundamental idea of calculus: the derivative. x
  • 3
    Another Car, Another Crime—The Second Idea of Calculus—The Integral
    You are kidnapped and driven away in a car. You can't see out the window, but you are able to shoot a videotape of the speedometer. The process by which you can use information about speed to compute the exact location of the car at the end of one hour is the second idea of calculus: the integral. x
  • 4
    The Fundamental Theorem of Calculus
    The moving car scenario illustrates the Fundamental Theorem of Calculus. This states that the derivative and the integral are two sides of the same coin. The insight of calculus, the Fundamental Theorem creates a method for finding a value that would otherwise be hard or impossible to get, even with a computer. x
  • 5
    Visualizing the Derivative—Slopes
    Change is so fundamental to our vision of the world that we view it as the driving force in our understanding of physics, biology, economics—virtually anything. Graphs are a way to visualize the derivative's ability to analyze and quantify change. x
  • 6
    Derivatives the Easy Way—Symbol Pushing
    The derivative lets us understand how a change in one variable affects a dependent quantity. We have studied this relationship with respect to time. But the derivative can be abstracted to many other dependencies, such as that of the area of a circle on the length of its radius, or supply or demand on price. x
  • 7
    Abstracting the Derivative—Circles and Belts
    One of the most useful ways to consider derivatives is to view them algebraically. We can find the derivative of a function expressed algebraically by using a mechanical process, bypassing the infinite process of taking derivatives at each point. x
  • 8
    Circles, Pyramids, Cones, and Spheres
    The description of moving objects is one of the most direct applications of calculus. Analyzing the trajectories and speeds of projectiles has an illustrious history. This includes Galileo's famous experiments in Pisa and Newton's theories that allow us to compute the path and speed of projectiles, from baseballs to planets. x
  • 9
    Archimedes and the Tractrix
    Optimization problems—for example, maximizing the area that can be enclosed by a certain amount of fencing—often bring students to tears. But they illustrate questions of enormous importance in the real world. The strategy for solving these problems involves an intriguing application of derivatives. x
  • 10
    The Integral and the Fundamental Theorem
    Formulas for areas and volumes can be deduced by dividing such objects as cones and spheres into thin pieces. Ancient examples of this method were precursors to the modern idea of the integral. x
  • 11
    Abstracting the Integral—Pyramids and Dams
    Archimedes devised an ingenious method that foreshadowed the idea of the integral in that it involved slicing a sphere into thin sections. Integrals provide effective techniques for computing volumes of solids and areas of surfaces. The image of an onion is useful in investigating how a solid ball can be viewed as layers of surfaces. x
  • 12
    Buffon’s Needle or π from Breadsticks
    The integral involves breaking intervals of change into small pieces and then adding them up. We use Leibniz's notation for the integral because the long S shape reminds us that the definition of the integral involves sums. x
  • 13
    Achilles, Tortoises, Limits, and Continuity
    The integral's strategy of adding up little pieces solves a variety of problems, such as finding the volume of a pyramid or the total pressure on the face of a dam. x
  • 14
    Calculators and Approximations
    The Fundamental Theorem links the integral and the derivative. It shortcuts the integral's infinite process of summing and replaces it by a single subtraction. x
  • 15
    The Best of All Possible Worlds—Optimization
    Calculus is useful in many branches of mathematics. The 18th-century French scientist Georges Louis Leclerc Compte de Buffon used calculus and breadsticks to perform an experiment in probability. His experiment showed how random events can ultimately lead to an exact number. x
  • 16
    Economics and Architecture
    Zeno's Arrow Paradox concerns itself with the fact that an arrow traveling to a target must cover half the total distance, then half the remaining distance, etc. How does it ever get there? The concept of limit solves the problem. x
  • 17
    Galileo, Newton, and Baseball
    The real numbers in toto constitute a smooth, seamless continuum. Viewing the world as continuous in time and space allows us to make mathematical models that are helpful and predictive. x
  • 18
    Getting off the Line—Motion in Space
    Zeno's Arrow Paradox shows us that an infinite addition problem (1/2 + 1/4 + 1/8 + . . .) can result in a single number: 1. Similarly, it is possible to approximate values such as π or the square root of 2 by adding up the first few hundred terms of infinite sum. Calculators use this method when we push the "sin" or square root keys. x
  • 19
    Mountain Slopes and Tangent Planes
    We have seen how to analyze change and dependency according to one varying quantity. But many processes and things in nature vary according to several features. The steepness of a mountain slope is one example. To describe these real-world situations, we must use planes instead of lines to capture the philosophy of the derivative. x
  • 20
    Several Variables—Volumes Galore
    After developing the ideas of calculus for cars moving in a straight line, we have gained enough expertise to apply the same reasoning to anything moving in space—from mosquitoes to planets. x
  • 21
    The Fundamental Theorem Extended
    Calculus plays a central role in describing much of physics. It is integral to the description of planetary motion, mechanics, fluid dynamics, waves, thermodynamics, electricity, optics, and more. It can describe the physics of sound, but can't explain why we enjoy Bach. x
  • 22
    Fields of Arrows—Differential Equations
    Many money matters are prime examples of rates of change. The difference between getting rich and going broke is often determined by our ability to predict future trends. The perspective and methods of calculus are helpful tools in attempts to decide such questions as what production levels of a good will maximize profit. x
  • 23
    Owls, Rats, Waves, and Guitars
    Whether looking at people or pachyderms, the models for predicting future populations all involve the rates of population change. Calculus is well suited to this task. However, the discrete version of the Verhulst Model is an example of chaotic behavior—an application for which calculus may not be appropriate. x
  • 24
    Calculus Everywhere
    There are limits to the realms of applicability of calculus, but it would be difficult to exaggerate its importance and influence in our lives. When considered in all of its aspects, calculus truly has been—and will continue to be—one of the most effective and influential strategies for analyzing our world that has ever been devised. x

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Your professor

Michael Starbird

About Your Professor

Michael Starbird, Ph.D.
The University of Texas at Austin
Dr. Michael Starbird is Professor of Mathematics and University Distinguished Teaching Professor at The University of Texas at Austin, where he has been teaching since 1974. He received his B.A. from Pomona College in 1970 and his Ph.D. in Mathematics from the University of Wisconsin-Madison in 1974. Professor Starbird's textbook, The Heart of Mathematics: An Invitation to Effective Thinking, coauthored with Edward B. Burger,...
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Reviews

Change and Motion: Calculus Made Clear, 2nd Edition is rated 4.2 out of 5 by 104.
Rated 5 out of 5 by from Great Intro to Calculus I took calculus a long time ago, and wanted to refresh my knowledge. This course was perfect for that. The presentation was clear, and it helped me understand why calculus was so important that it had to be invented twice. If you are looking to improve your calculating skills, this course is not for you, but if you are looking for a clear and concise overview, look no further.
Date published: 2009-08-28
Rated 5 out of 5 by from A Plus I have purchased about 15 Teaching Company courses and have experienced many excellent instructors, but I believe Professor Starbird has to be the most outstanding teacher I have ever had including elementary school through college. I studied Calculus as an engineering student over 50 years ago and although I used it as a tool to solve a few specific problems I never had as clear an understanding as is delivered by Professor Starbird. He presents the subject in extremely clear and simple terms with a very pleasant and unassuming and sometimes amusing manner. It is truly a joy to grasp what to most is such a forbidding and fearful subject.
Date published: 2009-06-12
Rated 4 out of 5 by from My daughter loved it My daughter claims this course augmented her high school math studies and made a significant difference in her grades. She is now at a top-tier university and looks back fondly on this course.
Date published: 2009-06-07
Rated 5 out of 5 by from Calculus in Its Simplest Form If you want to learn calculus, this is the course. If your purpose is to be better in your school calculus, this is the course. All the scary myths of calculus being difficult to understand - eliminated. You will walk away confident about what you learned.
Date published: 2009-05-29
Rated 3 out of 5 by from Good Half a Course The course started out well but by the second half degenerated into discussing applications of calculus without showing the actual mathematics involved. This is doubly frustrating because in the first dozen or so lectures, Professor Starbird does develop the actual calculation of derivatives and integrals. The failure to include the actual mathematics applied to the situations described in the latter lectures seriously undermined the value of the course for me. The professor's lecture presentation is informal but was not the problem in this course. The problem is the lack of substance.
Date published: 2009-05-21
Rated 1 out of 5 by from Someone Has to Say It His baseball analogy was humorous, but playing a stuttering, absent minded professor is not only not entertaining or engaging, it gets in the way. Starbird becomes irksome.
Date published: 2009-05-09
Rated 5 out of 5 by from Lucid & Thoughtful Course Professor Starbird's lectures are extremely intuitive. He explains the core ideas of calculus in a way that makes a lot of sense. Although I had mastered the technical aspects of calculus long before watching this course, I gained a fresh appreciation for its core ideas from Professor Starbird. It would make a helpful supplement to more formal calculus classes taught in high school or college.
Date published: 2009-04-08
Rated 5 out of 5 by from strong base for learning advanced math Buy a text book if you want problem sets. If you want to understand how the many concepts in calculus tie together, buy this. I knew many of the formulas and the mechanics of calculus. However, sometimes I'd be presented a problem to solve (I write software), and I would not even think that I could use calculus as the solution. Computers as by nature deterministic and discrete. Thats how programmers think. Well, now I am Mr. Infinite and continuous. I can design much more interesting software because of this course.
Date published: 2009-03-25
Rated 5 out of 5 by from Best Overview of Calculus I've Seen This is the best overview of Calculus I've encountered. Prof Starbird uses a lot of practical demonstrations and examples to make the concepts of Calculus very clear. The lectures progress from basic ideas through to 3D problems. At times his jokes are a little corny, but that's OK.
Date published: 2009-03-14
Rated 5 out of 5 by from Calculus Made Clear Dr. Michael Starbird is the most exemplary teacher I have ever seen in my life. I can tell he loves his subject. He knows his subject inside and out. I've never been within ten feet of calculus when I took higher math in High School. I struggled in Algebra I. Dr. Starbird is so good, that he was making Calculus clear to me! He makes an overwise terrifing subject for so many of the rest of us, nothing but interesting, exciting and maybe I'd be willing to jump into higher math once and for all again with a new attitude seeing how things have progressed in the teaching of the subject. His trick is explaining everything in minute detail. He also explains the subject that used to be so far out of reach more accessible and familiar with our everyday lives by showing its application. He also seems to be reviewing some Algebra and Trigonometry in order to bring those people up to speed to 'slip it to you,' the Calculus. It's like a plot to convert the masses. It's a rush. He does a good job of it.
Date published: 2009-02-26
Rated 5 out of 5 by from Great Conceptual Overview If you are looking for a regular college calculus class, this isn't it. If you want to know why you're doing what you're doing in your college class, this will be enormously helpful. Calculus Made Clear is a class much more about why and how calculus works, and less about how to do it, though you will work through some examples. If you're interested in understanding the concepts and enjoy knowing about some of the interesting math history that goes along with it, you'll think this class is great. Going back to college after many years out of school, Calculus Made Clear has kept me interested and helped me learn because understanding concepts helps me remember the mechanics. Michael Starbird is perhaps not the most dynamic teacher, but he keeps your interest with lots of examples and props and models. He does repeat some concepts, but for me, it was useful repetition. I also found his style somewhat unique--all of the concepts are taught conceptually, algebraically and graphically, which helped to crystallize the concepts.
Date published: 2009-02-25
Rated 5 out of 5 by from I took two calculus courses in college 50 years ago. Enjoyed them greatly. I enjoy solving puzzles and that's what those college courses provided - the rules and methods for using derivatives and integrals. I had no knowledge of the concepts behind the rules and methods. This course provided what I should have learned 50 years ago. Amazingly, I found my old "Analytic Geometry and Calculus" text book. I started at page 1 and now I'm up to page 176. I love it now more than ever because now I understand what I'm doing.
Date published: 2009-02-23
Rated 4 out of 5 by from Calculus Concepts First My first exposure to calculus was six eight-o'clock classes a week at UC Berkeley in 1947, and I progressed from there. Understanding the concepts took much longer than learning the methods; expertise came before understanding, if you will, and, therefore, the understanding just wasn't good enough until later. Starbird's course provides great preparation for the formal study of calculus. And I was vastly entertained by it, though sometimes thinking faster than his presentation. I understand that calculus is now taught in high school, I suppose superseding the play with trigonometric functions that used to be so important for working out new integration formulas. (And which I was bad at; couldn't make a mental picture of what had become abstractions without physical meaning.) I think that the present educational problem is how to work a course such as Starbird's into the teaching sequence for the student before teaching the methods of calculus. For a student who expects to use calculus as an engineer or scientist, and therefore probably has the aptitude, I think that the duration could be halved.
Date published: 2009-02-21
Rated 5 out of 5 by from Made the concepts of Calculus clear to me I took two semesters of Calculus in college but never really understood the principles like now after this course! The presentation is good and this would be a great course for a high school senior who is talking Calculus in high school or going to take the class in College. Though that the class could have had a few more example problems.
Date published: 2009-02-03
Rated 3 out of 5 by from Great on ideas of the Calculus. Short on "doing" I was a bit disappointed with this course. I expected concrete examples with problem sets and exercises. Instead what I got was an explanation of great ideas of the Calculus. This is good but if you wish to actually LEARN calculus then this course will not meet your requirement. However, if you wish to know the general conceptual underpinnings of calculus then the course is great.
Date published: 2009-01-21
Rated 4 out of 5 by from Understanding Calculus Conceptually Following his 'Joy of Thinking' contribution (with Ed Burger) I wasn't thrilled at the idea of another Starbird course because I find his unprepared style of verbal presentation quite annoying. As I've said in my review of that previous course he is no doubt great in the flesh but he needs to rehearse more so that his style of explanation is less faltering and repetitious. Having given him the thumbs down for the other course I am relieved to say I found this one much better. Like DNAunion I would have liked a course that provided me with even more practical assistance but, like AZMorg I feel this conceptual overview sets me up well for a more detailed examination which hopefully TTC will address (as it has with its other basic courses). Starbird's delivery can still be annoying (his presentation with the toy car props at the beginning is painful) but as he moves more and more deeply into aspects of the subject he resorts to more online graphics and these graphics greatly assist his delivery.
Date published: 2009-01-06
Rated 5 out of 5 by from Simply - Effective This review consists almost solely of an example. I took calculus in college, but that was back in the early 60s. I wanted to get back into math, so I got this course, and went through it several times before I was sure I "got it". It's not that the material is totally mind-boggling. It's just that a lot of it is non-intuitive and requires some repitition and visualization before it sinks in (to me, at least). Anyway, some time later I got Prof. Starbird's Statistics course. In one lecture, he offered math-added information to a topic that used calculus. Now, Prof. Starbird didn't spend much time at all on that panel of math, but I looked at it and thought, "You know, I should be able to understand that." So, I backed up the DVD, listened again to the professor's brief explanation and froze the screen on that panel. It took me a few minutes, but I was able to completely understand every single equation and derivation in that panel, and was thus able to get a much better understanding of that topic. Without this calculus course, that panel might as well have been in Farsi. If what you're looking for is a conceptual view (not overview) of calculus - looking down from the top - this is it. In addition to some "how", there's an awful lot of "who", "when" and, most importantly "why". If you are interested in problem solving, or if you're planning to take a formal calculus course, take this course first. It'll make that effort a much easier step.
Date published: 2009-01-02
Rated 3 out of 5 by from I don't believe this course prepares a person to actually do calculus. I would prefer a course that gives more calculus and examples of calculus problems as opposed to overly long, basically mathless explanations of ideas.
Date published: 2008-12-27
Rated 5 out of 5 by from Best calculus course I have ever seen! I am a math teacher.
Date published: 2008-10-17
Rated 5 out of 5 by from Prof. Starbird has much enthusiasm for the subject. Visual aids are excellent. I think there is math background needed for the course- and a little understated in course summary.
Date published: 2008-10-17
Rated 5 out of 5 by from As a Math teacher, I appreciated how clean the course was designed and how concepts were explained. I will use many ideas in my won courses.
Date published: 2008-10-17
Rated 5 out of 5 by from If I had taken this course prior to starting on my engineering degrees it would have made a world of difference.
Date published: 2008-10-17
Rated 5 out of 5 by from Consider offering a problems notebook with this course. Please. Transcripts are useful. Professors are superb.
Date published: 2008-10-17
Rated 5 out of 5 by from These lectures need more graphics. Just looking at the prof is not enough more pix and more video segments to illustrate
Date published: 2008-10-17
Rated 4 out of 5 by from I have an engineering degree from a very competitive school. Where was Prof Starbird when I needed him as a freshman?
Date published: 2008-10-17
Rated 5 out of 5 by from After years of seeking the true use and understanding of calculus and taking courses multiple times-this series provided the insight with clear explanations and examples. Well done!
Date published: 2008-10-17
Rated 5 out of 5 by from Why wasn't calculus taught like this when I was in college? Pfrofessor Starbird makes it clear and fun.
Date published: 2008-10-17
Rated 5 out of 5 by from Would like to have had a few more questions on problems to work through but Dr Starbird is a wonderful teacher - would have liked to encounter more like him in my undergrad/grad classes.
Date published: 2008-10-17
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