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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Change and Motion: Calculus Made Clear, 2nd Edition is rated 4.1 out of 5 by 107.
Rated 5 out of 5 by from A Survey, Not an Introduction I took this course to reacquaint myself with calculus which I studied many long years ago, but never used in my professional life. Plus this is an effort to work my way through TTC’s mathematics courses. And I have to say that I think Professor Starbird does an outstanding job of presenting his subject. He begins before the beginning, going all the way back to the 5th century BCE and Zeno to introduce some of the ideas of calculus and to help us realize that the “invention” of calculus by Newton and Liebniz was based on thoughts and ideas that had been around and being developed for a long time. The course shines in giving us the background and development of calculus both before and after Newton and Liebniz. But more importantly, it probably would help almost anyone with modest mathematical background understand the basic concepts of calculus. It would not help anyone I think actually solve problems or to be able to even set up problems, perhaps excepting the most basic. And herein lies the one problem I see in this course. Many times the demonstrations of problems involves passing by the fairly simple algebraic manipulation of equations, meaning that anyone who is not able to make the connections from one place to another may become a bit puzzled at best. That is, the course does not put heavy emphasis on actually using math, but often assumes a knowledge that is higher than what I think the target audience is expected to possess. To be fair, the real target audience may well be people like myself with a solid background that has atrophied (and many of the reviews indicate that this may well be the case). Some reviews have not liked Dr. Starbird’s presentation skills. It is true that he often stutters or stammers a bit as he is transitioning from one topic to another or when he is bending down to begin using his computer to set up a demonstration, but I find it much more realistic that reading from a teleprompter and not at all disconcerting. And he does have a quiet, gentle sense of humor. How can one not like his stuffed owl and passel of woodmice as he demonstrates the use of calculus in population change? And my favorite, his commentary and analysis of why calculus is better than sliced bread made me chuckle internally, even if not rolling on the floor laughing. Get this course if you are interested in the whys and wherefores of calculus, but not if your objective is to actually learn the subject. Most highly recommend if you fall into the first category and many who are in the second and wish to prepare for a rigorous course.
Date published: 2020-09-12
Rated 5 out of 5 by from An Interesting Refresher Course I bought this course last week. It has fulfilled ally expectations. I wanted to refresh my memory of Calculus, which I studied at University some 60years ago, and which I had not used for the last 50 years!
Date published: 2020-09-04
Rated 1 out of 5 by from Not a good review I took Calculus over 50 years ago, so I was hoping for a refresher course. This was not it. Lots of repetitive examples barely touching the real math. The content of this entire course should have been covered in 5 lectures, and it wouldn't have been nearly as boring.
Date published: 2020-07-06
Rated 4 out of 5 by from For the Non-mathematician If you are a student and plan to study calculus, this course is not for you. Rather, Professor Bruce Edwards’s—the professor pictured in the catalogue—many courses on calculus and precalculus will brace you for staying up all night and studying for a test and memorizing proofs. (Professor Edwards’s pre-calculus course may have gotten lesser reviews because the subject matter is so difficult to teach and learn.) Professor Starbird offers a calculus survey for people intrigued by the noun, which is used widely as a power synonym for measurement. At the end of Professor Starbird’s 24 lectures, even if you are like many who don’t get algebra or poetry, you will understand the beauty, elegance, and power of calculus. Professor Starbird is a little nuts, with so much activity going on in that extraordinary mind; but he tries harder than anybody, putting his full heart into what he does, and he is one truly fantastic teacher. Even if you don’t want to learn about calculus, you can buy this course to sharpen your teaching skills. He appears to use his children’s or grandchildren’s toys as props. (When trying to make sense of Professor Kung’s course on music and math, differential equations popped up, and though I taught the subject in my youth, I was fortunate to have Professor Starbird’s million-word illustrations by using toy animals.) His teaching ploy of trying to recreate calculus tangibly in the real world is his strength. After blowing up my mind on this subject 37 years ago, it still amazes me how far one can apply calculus. The four stars I award the course factors in some distractions, and the warning that the real calculus course is found under Bruce Edwards. But if you are able to realize the essence of calculus without going crazy, this would be a five-star course for you. You will no longer misuse the term calculus. (Another point of view: on a Miami bus bench where are you can find the most fascinating conversation, in front of the thrift store I had just bought a math textbook, a man told me he had bought Professor Starbird’s course on calculus from TTC and that he really enjoyed it, finally grasping all that that gigantic noun signifies, dissolves, and transmutes.) Calculus is one really cool set of only two tools; and if math is not your thing, and there are many extraordinarily talented people who cannot make heads or tails of mathematics, after this fine course you will understand the power of calculus.
Date published: 2020-06-28
Rated 3 out of 5 by from Confused I viewed the first two lectures and I have to review the second lecture again because i really didn't understand it.
Date published: 2020-05-16
Rated 4 out of 5 by from Great for basic understanding of Calculus I read many of the reviews> I agree w/ almost all of the good ones and bad ones. The overview he presents would be a great help before taking a Cal class. I had to go back over the material numerous times to understand it 100%. His latter lectures were breezed through quickly w/out adequate detail to get to 100%. So I agree he could have developed some of the basics/formulas a little better at times. Overall. It was worth my time and I will now start over to see how well it all fits together.
Date published: 2020-05-16
Rated 5 out of 5 by from The Beauty of Calculus Now Found I wish I had taken this course before any of my calculus education; It would've made life a lot easier. An excellent course!
Date published: 2020-02-11
Rated 2 out of 5 by from I wouldn't advise it I was not too impressed with this. Starbird gets hyper-repetetive on many simple concepts to the point of frustration and then he glosses over more complicated conclusions that he presents, which further obfuscates the watcher. I even counted and average of 5 verbal stutters from him before most ahh, umm, ahh ummm uhhh transitions. But he definitely cherishes his lame jokes, and so he seems to feel, should you. Ugggh, no thanks.
Date published: 2019-11-12
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