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 98.
Rated 5 out of 5 by from Conceptual Overview of Calculus Professor Starbird has a passion for math and his aim is to provide an overview of calculus concepts to non-mathematicians. The first half of the course provides an overview of the two fundamental concepts of calculus, i.e., the derivative and the integral, and how they are related via the Fundamental Theorem of Calculus. The second half of the course shows some of the applications of these concepts, e.g., optimization, economics, calculating volumes, etc. In my experience, if you’ve taken a calculus course, then this course will add some conceptual understanding to the formula-based instruction you likely received. However, if you haven’t taken a calculus course, you will gain a conceptual understanding that is foundational to this subject, but you will not be able to apply these methods without further study. If you desire to learn the mechanics of calculating a derivative or integral, definitely check out Professor Bruce Edwards' course, "Understanding Calculus." I do think these courses complement each other very nicely.
Date published: 2017-03-31
Rated 5 out of 5 by from Clear explanation of topic My college calculus course was a disappointment. When I had a question I was shut down with "it's intuitively obvious!" Dr. Starbird is a wonderful teacher with a firm grasp of the subject matter and a clear delivery. His kindness and concern for the student's struggles is refreshing.
Date published: 2017-02-11
Rated 4 out of 5 by from Excellent Overview I earned a Masters degree in math many years ago. So my motivation for purchasing this course was to refresh my memory about a subject (calculus) that I was already familiar with. The videos not only refreshed my memory on the basic concepts, but also provided many new insights. I especially liked the historical references in the course.
Date published: 2017-01-06
Rated 5 out of 5 by from Finally! Calculus made understandable! Years and years of math course and I learned nothing. Professor Starbird FINALLY made calculus intelligible. Thank you! And as a bonus, this got me excited about looking at other Great Courses. I've been hooked ever since.
Date published: 2016-10-10
Rated 5 out of 5 by from A Great Course The lecture series definitely clarify Calculus and its applications.
Date published: 2016-10-08
Rated 5 out of 5 by from A Great Introduction to the Concepts of Calculus Why would a retiree like me, who's never used calculus in his life (but studied a bit of it in graduate business school), take a course like this? Well, I had two reasons: (1) I had previously taken Professor Starbird's Probability and Statistics courses and had enjoyed his clear teaching style and wonderful examples in those courses very much, and (2) my interest in science has shown me that many scientists and philosophers have found it remarkable that the natural world is so amenable to the tools of calculus. So, with virtually no background in the subject, I trusted that the professor could give me a clear understanding of the concepts of calculus, and he delivered. It was a very stimulating course, and you don't have to be a mathematician or engineer (I'm not) to get a lot of benefit from this course, but simply must possess an inquiring mind. Among the highlights of the course, at least for me, were: (1) the clear explanations of the derivative and the integral and concrete examples supporting both "sides" of calculus, (2) Archimedes ingeniously using a kind of pre-calculus to calculate the volume of a sphere, (3) optimizing in economics by finding the derivative where it's zero to locate maxima and minima for profits, costs, etc., and (4) the marvelous calculus that outfielders intuitively perform when they judge the velocity, height, and arc of a fly ball and run right to the spot where the ball comes down. Professor Starbird's verbal comments are also very insightful. He says the integral represents "the accumulation of change to a net effect" (Lecture 21), and in his final lecture he says: "that the story of calculus is the story of the intellectual conquest of infinity." I liked that.
Date published: 2016-09-01
Rated 5 out of 5 by from Surprisingly engaging I thought sure this was going to be difficult to understand, but it was wonderful. I loved the instructor's wry humor and he does every thing he can to make this topic interesting. I've not completed the course yet, but I look forward to doing so.
Date published: 2016-08-03
Rated 5 out of 5 by from Calculus I have a BS in Math and never heard calculus explained so well. The professor makes calculus not only understandable, but shows how it effects our everyday lives. We may not use calculus to solve problems every day, but we use the tools and ideas of calculus in making many decisions every day, Anyone, whether student or curious individual, can benefit greatly from this course.
Date published: 2016-08-01
Rated 5 out of 5 by from Math Is Fun I've loved math all my 74 years. Starbird presents math in a way that is understandable, and fun, to persons of all levels of math knowledge and intuition. The owls eating the wood rats is hilarious. ... Thanks Michael, Al
Date published: 2016-06-09
Rated 4 out of 5 by from "Great Courses" is a "great find" for me! This course is an excellent value. The instructor is engaging and presents material in a very simple and straight forward way. My only complaint is that I could not download the course using my windows 10 system. Streaming is working fine but I would like to be able to use the course when I am in an area where streaming is not available.
Date published: 2016-02-01
Rated 4 out of 5 by from Calculus Made Clear, 2nd Edition I purchased this for my new bride. We are both abt 70, and she is a retired high school mathematics teacher. She is once again tutoring students (loves to do that!) and wanted a refresher on Calc. She is thrilled by the explainations offered by the Prof. I am a retired engineering manager, and during our courtship would exchange off-color text messages written in code, based on Calc. When a good thing in her life was getting even better, she would relay that as the 2nd derivative of ABCD was positive. How nerdy is that?!! Her children and students could not understand them when she left her phone on a table; thus a big success for us. Now, after getting halfway through this course she is speaking to me in calc codes. Doesn't get much funnier than that; or more weird. We are expecting to hear or see that soon on The Big Bang. Your course deserves 6 stars out of 5.
Date published: 2016-01-31
Rated 5 out of 5 by from Very enjoyable and entertaining too. Prof. Starbird was entertaining as he presented the lectures via streaming download; the disks and other material arrived shortly after I completed the course online. He was correct, in that, previous high school and college calculus courses only taught the mechanics of using derivatives and the integral functions, and I never had any problems doing so, but I had a serious concern that I may have missed something along the way in my career that could help me to understand why I did these thing. Prof. Starbird filled in some blanks that I had.
Date published: 2016-01-28
Rated 4 out of 5 by from A GOOD PLACE TO CATCH UP I think this is a very good course. I think it will present all the required information to prepare the student for the required expectations. However I think I should have asked for some local advise from someone who knows my background in math. It seems like I should have started with a course a little more primary. This course seems like a rather mid subject course. I intend to continue with this course and then explore what other courses are available in Calculus. This is a subject that I have wanted ever since I wasn't able to acquire it in 1955 at U of I. Clifford (Gene) Dawdy
Date published: 2016-01-22
Rated 4 out of 5 by from My review of "Change and Motion: Calculus Made Clear" by Michael P. Starbird: Professor Starbird does a good job of connecting calculus to real-world applications. He is very knowledgeable and knows how the subject is used in areas such as physics, economics, business, biology, and so on. This makes the course meaningful and enjoyable.
Date published: 2015-05-23
Rated 5 out of 5 by from Great Course, Well Presented As one of those people who faked my way through calculus by learning enough of the mechanics to get good grades, I really appreciated the way this course was presented. The focus on understanding the underlying concepts in the math did slow the pace a bit, but just enough to give you a chance to internalize the ideas. This course could work equally well as a review, or as an intro. I think this series of lectures is also a great way to gain an understanding of calculus even if you might need to brush up on your algebra, geometry, and trigonometry.
Date published: 2015-05-20
Rated 5 out of 5 by from Change and Motion: Calculus Made Clear 1st edition I purchased the course several years ago and--without having previously studied calculus--watched it two or three times without getting much out of it. According to Starbird, the presenter, it was a "conceptual approach", which is exactly what thought I wanted. I am a mathematical layman and not interested in the practical applications of calculus, but I merely wanted to know more about the subject in general. The video didn't help much. Then I took two courses in calculus. The courses emphasized techniques and algorithms, and skimmed over the concepts. A few years later when I returned to the Starbird videos, having had that good dose of calc vocabulary and technique, the whole thing made sense. I highly recommend the video series as long as the viewer does some pencil and paper calculus as well.
Date published: 2015-03-29
Rated 4 out of 5 by from Nice review of my calculus from many years ago in college.
Date published: 2015-03-16
Rated 5 out of 5 by from Great teacher I took this course because I hated calculus in college. I really needed to see if I could finally understand it and be able to utilize it. I am only part way through the course but I understand everything so far. That is a credit to our very qualified instructor. His lectures are very well presented in a way that is interesting, relevant and easy to understand.
Date published: 2015-03-14
Rated 5 out of 5 by from Change and Motion: Calculus Made Clear, 2nd Editio Wish it was this understandable when I took it in College. Professor does a great job making it easy to understand and retain....
Date published: 2015-02-26
Rated 5 out of 5 by from Finally, a course that makes sense of Calculus! I've been trying to make sense of calculus text books, and have watched other calculus courses. But I still didn't understand why you'd want to take a limit, what the derivative was, or how integral calculus differed. This course is just what I needed! This course provides an overview and higher level explanations that helped me understand why calculus is used, what kinds of problems it solves, exactly why taking the limit is important, and why you'd use integral calculus. You won't be able to solve calculus problems after this course, but you will understand the high level view. Now, I can turn back to the textbooks and make sense of those problems and how to work them. I wish I had taken this course before trying to make sense of working problems, but I'm glad I did get a chance to watch all of these lectures. This is a great course for people who need the conceptual understanding first, and then can later get down to the nitty-gritty of problem-solving. I have the other calculus course on problem solving through great courses, so now I'm going back to that one as I'm confident I'll be able to make sense of it now. I found this course enjoyable as well as informative. Hard to believe I used to hate math so much. My love of science has brought on the desire to learn math now in my 50's, whereas when I was younger I really detested it. Math is fascinating and I'm grateful for these courses!
Date published: 2014-11-30
Rated 5 out of 5 by from What I did not understand before! Even though I was a Physics major in college and took Calculus then, I did not fully understand the concept. I probably did not take the time to fully try to understand - just get the problems done, because that is what is on the exams. And I am sure that professors did not really want to be teaching us, so did not do a good job of ensuring I got the full understanding. But now Professor Starbird really enjoys teaching and wants to get the real meaning of Calculus across in his lectures, and now I at 68 years old really want to take the time to understand calculus and do not have to worry about exams to take. It is the best of both worlds, excellent teaching on the Great Courses part and true desire to learn on my part! Excellent Course!
Date published: 2014-11-24
Rated 5 out of 5 by from Excellent Memory Refresher I took Calculus in high school back in 1983 and after college did not use it on a regular basis. This course is exactly at the right level for me. The topic is familiar and the course is able to refresh my memory on a topic I was once good at. The professor is interesting to listen to and the examples all make sense. I appreciate that the course helps one really understand the topic and covers both theory and mechanics. The course has awakened my love of mathematics and I hope to carry on to more advanced math topics after I finish this course. Thumbs up!
Date published: 2014-11-13
Rated 5 out of 5 by from Insight and interesting applications of Calculus This course delivers exactly what it promises. It will give you an understanding of differential and integral calculus. If you would like to know what calculus is good for, learn a little of its historical development and see some applications you will enjoy Dr. Starbird's course. While it will help you get the big picture while taking a traditional calculus course it is not a tutorial or help with solving problems. This would be particularly helpful for people who aren't that interested in math but want to know about calculus to better understand physics or microeconomics. It would be a great companion for the new electronics course, the probability or statistics courses or any of the physics lectures. Although I have used calculus for years, my understanding of the Fundamental theorem was enhanced by his approach to explaining it. Lecture 8 on circles and geometric solids was very informative. I have studied Buffon's Needle twice from two professors and two universities and while I could do the calculations and get an 'A' I never really understood how it works. Dr. Starbird made this perfectly clear. The big picture approach gave me an intuitive understanding of the concept to complement the theory. The last few lectures go into some very advanced territory for a first class. Entire classes are taught in multivariable caculus, differential equations and Fourier series so enjoy the high level view of these topics but don't expect to master them in one half hour lecture. I enjoy Dr. Starbird's laid back style and his sense of humor very much. I quickly grow tired of the meaningless hand gestures and fake enthusiasm of some over-trained speakers. It's refreshing to learn from someone who acts like a real person while sharing his knowledge and passion for math.
Date published: 2014-07-27
Rated 5 out of 5 by from Best presentation of Calculus I've seen I've studied Calculus with the Open University in England, as well as going through a couple of DIY books. This course not only taught me, better, what I already new, it taught me why it all works from the ground up. I found the Professor to be extremely watchable, clear, and concise and his presentation to be excellent. I would highly recommend this course to anyone who needs or wishes to learn Calculus and its applications.
Date published: 2014-06-02
Rated 1 out of 5 by from May I please have my money back . . . Poor sense of humor, terrible style of story telling, not engaging. Full of mumbo jumbo, boring quotes. I Know Professor Starbird is doing his best, however, unknowing adding complexity and un-intentionally making the subject most boring - so painful to sit thru that it hurts. I was so excited with the descriptions " Calculus Made Clear " oh dear what a let down. I regret - do not recommend.
Date published: 2014-04-21
Rated 5 out of 5 by from Excellent introduction to calculus Job done! As the title promises, the professor explains it all, clearly, succinctly, and in an engaging, amusing manner. A solid course, easy to recommend, particularly for those with no prior knowledge of the world of calculus. Strongly recommended.
Date published: 2014-02-20
Rated 5 out of 5 by from Very useful intro I've gone through this DVD at least four times. Each time I catch another tidbit of info that helps tie this all together. The professor is low key and easy to follow. There is a lot of material and it takes me several viewings before I can fully grasp the concepts. But the concepts are there, just presented quickly. This is a very good DVD that will give me the foundation to move on to more advanced calculus courses. Oh, I'm 67.
Date published: 2014-02-09
Rated 5 out of 5 by from A well done refresher While not minimizing the difficulty of the material Prof. Starbird's course is an enjoyable presentation of a difficult subject. He mixes the nuts and bolts of calculus with interesting real world examples. Perhaps someone who has never taken calculus would struggle to completely understand the material but this IS calculus after all so no apologies for it being difficult even as a refresher some 40 years later for me. Good job. Will definitely look into other courses by this professor.
Date published: 2014-01-30
Rated 5 out of 5 by from Change and Motion : Calculus made clear Apparently here in NC the public high schools have changed to longer classes and are in the habit of totally dropping key subjects such as Math and English for whole semesters. It beats me as to how this can be a better system but, anyway, the reason I bought this course was to teach our grandson in the evenings in those "down" semesters. We typically went through 2 lectures each evening (one evening per week) and I have to say this course is excellent. It logically explains the origin of the Calculus and the practical applications. Professor Starbird is the sort of professor I wish I'd had to teach me calculus long ago. I know that our grandson is now well versed in this subject whether or not the NC system ever gets around to teaching it to him.
Date published: 2014-01-30
Rated 3 out of 5 by from Completed and glad Am 40 years from my last calculus course and pushed to finish this one. Have completed several of his other courses but at times lost his thread in this one. The basic concepts and importance do however come through in the end.
Date published: 2013-04-28
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