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Calculus: The Two Questions of Calculus


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About this Lesson

  • Type: Video Tutorial
  • Length: 9:29
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 103 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: The Basics (8 lessons, $11.88)
Calculus: An Overview (4 lessons, $2.97)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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The Basics
The Two Questions of Calculus Page [1of 1]
Okay. So, what is calculus all about? Well, basically, calculus just gives us the answers to two basic questions. Okay, so what are the basic questions? Well, I'm going to tell you right now. Now, there's part of the first question, I want to tell you something that actually happened to me yesterday. Trying to get in shape, so I went for a bike ride. Now, let me actually show you this. I went for this long bike ride, oh, it was so long by the way, it nearly killed me. The bike ride, thirty miles long, thirty miles, thirty miles, nearly killed me. And I had my bike; in fact, I started off here. Here's my bike, I had this mountain bike, really pretty by the way, and it actually is purple, in case you're wondering if I'm using the purple pen just to be colorful. But the truth is I am using it to be colorful and my bike is purple. There it is. In fact the mountain bike, really fat tires, you can go anywhere on it. But I was there on the road; okay, I was going down the road. Thirty miles, did the whole trip in an hour and a half. Can you believe that? Well, I hope you can because that's exactly what I did. Impressed? I was.
Now, something interesting happened during this bike ride. Around two-thirds of the way through the bike ride, right around here, there was a road sign, and the road sign read twenty miles an hour. Twenty miles an hour. And it turns out that that road sign was actually located twenty miles from where I started. So, in fact, this distance right here was twenty miles right to there. Did I break the law? Well, what does that mean, "did I break the law?" Well, what I'm asking is at the very instant, at the very moment that I crossed the road sign was I going twenty miles an hour, or was I going above twenty miles an hour, or below? So, I need to find out here, is the rate that I was going, the speed I was going, right at that instant, forget about all the stuff over here. You can go as fast as you want anywhere before or after, but just at that instant the road sign says twenty miles an hour. Now, did I break the law?
Well, that's actually the very first question of calculus, to find out instantaneous velocity, velocity anywhere instant in time. Let me ask you further - enact this for you right now. In fact, I asked the crew -- we don't know each other too well, you and I, but I will tell you a little about the ______. I asked the crew, I wanted to actually animate this because, you know, this is all sort of read, video, you know. So, I wanted to actually animate and actually dramatically show, my goodness, I've ruined the drawing now. You know -- I ruined the sign, by the way. You could go as fast as you want. So much for the first question of calculus. Let's put that right back there. Well, what happened was, I actually wanted to do this for you live, okay, so you could capture something that actually captures me and the bicycle, the ethos of me and my shape and everything. Well, they tried to do this and look what they got. They got this little bear, and they said "well, you know, it's got, you know, brown hair, and so that's the me ethos and the tricycle. My bike is not a tricycle, by the way, in case you're wondering, my bike is a real mountain bike. There it is, but, okay, this is the little thing that they give us.
Okay, but let me ask you seriously, and I want you to watch. And the thing to watch out for here, folks, is watch the velocity of the little bear, Professor Burger, at the very instant that he crosses this thing. Let's try this right now, shall we? This is going to be a live experiment here. So, here I am starting. I'm out of the screen; I kept going, that's how good shape I'm in, right. But that very instant, you see it, there was some speed where I was traveling, at that very second I crossed the twenty mile an hour sign, how fast was I going? That is the very first question of calculus. Find the instantaneous velocity. Okay, now, let's just think about that for one second, and I want actually to sort of wonder, well, how could we even attempt to answer that? Well, I think that you actually have the ability to answer this question, at least to warm up to it, and actually I want to pose it as a warm up question.
So, what do we need to do here? Well, we know distance, we know rate, and we know time. So, what I would like to do is get a relationship between distance, rate and time. So what's the connection? So, here is your actual, this is the first time you actually get to contribute here. We're going to stop everything for a second and now you get to actually come up with a guess. And if you know, great, then just say it. If you don't know make a guess. What I'm looking for, though, is the relationship that links distance, rate and time. Okay. And just click when you're ready.
Okay. Well, distance equals rate multiplied by time. Did you get that? Was that your guess? If so, terrific. If not, if you made a guess, terrific. The important thing here in life, whether it's calculus or otherwise, is to always take the risk and don't be afraid to make a mistake. Anyway, distance does equal rate times time. And we're after the rate. So if I solve this for rate I would see that rate actually equals distance divided by time. And, you know what? This is, in some sense, the very heart of calculus. This is the heart of calculus. If you understand this basic formula, then you are well on your way to understanding calculus. I know you're saying "No, that can't be, because, gee, that calculus textbook is five inches thick and weighs three tons." Well, the calculus textbook is five inches thick and this really is the heart of calculus. So, great. We're making progress.
Okay, now what's that second question? The first question, finding instantaneous velocity. What's the second question? Well, the second question actually is something completely different, completely different. This is going to be a geometrical question. And the question basically is how do you find the areas of certain things? For example, you all know that if you want to find the area of a rectangle, that's actually not so hard. You just take base and multiply it by height, and, you know there are rectangles in life. As a perfect example, you have bars that you eat to gain energy and stuff. These are great by the way. These are really good ____. And, oh, gees. I've never eaten one of these actually, but it's very chocolaty. Do you actually get energy from eating these things? I think you can get a stomach ache. Well, before I ate it, you see, before I ate it, it was a whole rectangle, now it's a ______. So, you can find the area of those things.
In fact, you can even find the area of slightly more exotic things. For example, you can find the areas of circles. For example, you might have, you know, CD's, right? Do you have CD's? You can wear them as earrings. Here is two CD's, right, and if you wanted to find the area of that, in fact, these are actually a little more interesting than just circles. They're sort of concentric circles, it sort of a region and then there's a little hole inside there. But you could actually find the area of those because you might remember that the area of a circle is Pi R squared. So, those areas, areas of disks, areas of uneaten power bars, those are actually, those are actually pretty easy to find, but why not more exotic areas? Suppose you want to find the area of, let's say something really -- like an amoeba or something. Look at that thing. Now I don't know a formula that gives you the area of that. I don't know a formula for that. Or suppose you want to find the area, a cross section area or an airplane wing. You ever go on an airplane? They're very pretty curves. If you ever take an airplane wing and chop it - well, actually, don't do that because then the plane won't be very good, but if you were to chop it off - because it is very pretty, how do you find the area of exotic regions? Well, that's actually part of the second question.
You can even ask a more general question. What about in three dimensions? Okay, for example, what if you wanted to find the area of something like this. I'll draw you a three dimensional picture. Can you visualize that? That's a three dimensional picture. It's a donut. In fact, I happen to, I happen to have one right here. You see, this is a donut. How would you find, not the area, but how would you find the volume of this? How would you find the volume of a donut? Or how could you find the volume of other funny shaped things? For example, you know, what can you find the volume of? What... okay...thanks, you could find the volume of a football. Okay. How would you find it? There's no formula for that. So, what would you need to find the volume of that? Well, the answer is you need calculus. So, calculus actually will answer the question of how do you find areas of exotic shapes and volumes of exotic looking regions, like uneaten donuts or footballs. So that's the second question of calculus.
Now, remember the first one was finding instantaneous velocity, and the second question is finding areas and volumes of really strange looking things -- completely different questions, completely different questions. Well, now folks I'm going to let you in on what I think is the most interesting secret and the most interesting fact about calculus of all. These two completely different questions turn out to have related answers. Isn't that incredible? Here we have two questions. One question is asking about how fast you're going at one particular instant. And the other question is asking about how do you find areas of really weird things. They're unrelated issues, and yet the answers link them together because, in fact, the answers are almost the same to both these questions. That, I think, is the really cool thing about calculus, the really cool thing.
Okay, well, that's what calculus is all about. We're going to try to answer these two questions and once we answer the two questions, guess what. It's time for the final. Sorry. Okay, anyway what we're going to do next is we're going to take a look at that first question again. We're going to take a look at me on the bicycle and we're going to see if we can make some progress to figuring out how to find instantaneous velocity. Okay. I'll see you in a bit.

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