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Calculus: Average Rates of Change


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:00
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 119 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

2174 lessons

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Recent Reviews

~ Alexis7

Review for Exam 1

~ Alexis7

Review for Exam 1

The Basics
Average Rates of Change Page [1 of 1]
Let's take a crack at answering that very first question - finding instantaneous velocity. And I want to do that by returning this question of mine. So remember what happened. I was on this mountain bike ride and I went for a 30-mile ride and 20 miles into it, there was this sign that said, "20 miles per hour". And just to show you, the crew wanted to see this again, so here's how it goes. And the question is, how fast was I going at the very instant I crossed that mark? So now watch it. Here I go. I sort of fell of there a little bit, but I got a little bit tired. I have to admit, I'm a little bit old. Anyway, let me remind you that at the time I did that in one-and-a-half hours.
We know time. We know information about distance. We want rate. Well, we saw in the last discussion that rate equals distance over time. Now I'm not going to change that at all, because, folks, the truth is, rate does equal distance over time. But I want to be a little bit more clear on that point. Because actually what rate is is the change in the distance you've traveled divided by the change in time. How far you actually traveled is actually a change in how far you've gone. And how long you've traveled is actually a change in the time. And of course, you knew that already. So you're saying, "Gosh, why is he going on about that?" But I just want to make that point clear that actually, when I say that rate equals distance over time, what I really mean is that rate equals the change in distance divided by the change in time.
Now, we actually have a really neat way of writing that. When we want to talk about change, we use the Greek letter, delta (). Isn't that exciting? So in fact, what I want to think about, and what I'd like you to think about, is actually this fact, that rate equals the change in distance divided by the change in time. It's the exact same thing as this. The delta looks sort of scary, but don't think of it as being scary and don't think of it as like a symbol, because then you might say, "Gee, delta over delta! I can cancel." Well, don't think about that. Think of this as being the change in distance and then divide that by the change in time. That's all we're up against here.
So let's now just throw this one away. That's gone, because we're never going to talk about that again. What we want to do is we want to talk about this one. Well, let's see if we can figure out what the rate was. The rate equals the change in distance. What was the change in distance? Well, I started here and I ended here. That was 30 miles. It was a long bike ride. It really was. So the rate would equal the change in distance, which is 30 miles, divided by the change in time. Well, how long was I traveling? For hours. So I have to divide this by . Now you could write . You could write 1.5. I'm actually going to write it as one big fraction and I guess that fraction would be . So I'm going to divide this whole thing - it's going to look really complicated here - by , and that's the rate.
Now how do you find this? Well, let me just remind you of some of the little stuff from Algebra here. What do you do? If I'm dividing by a fraction, what I was taught was that you invert and multiply. So actually, if you watch this, I'll show you exactly how I think about it. I take this fraction down here and I invert it and I multiply it. So if I actually write that out, what I see is 30 multiplied by - and if I invert that, I see . So I'd see this times . And you notice I can do a little bit of canceling. I can cancel this 3 and make that 1 and make this a 10. And then 10 x 2 = 20. So 20 - what are my units - miles per hour. So 20 miles per hour.
So maybe I didn't break the law. Well, actually, we haven't answered the question yet. Because what we actually found was my rate over this entire journey - this long, long journey. But you see I still don't know how fast I was going at that instant. You see, for example, I could have done the following. I could have started off and started sprinting. I just really wanted to do well. And then I got tired. I'm so old. And then I sort of pedaled along, barely moving, at like 5 miles an hour. But when you average out that 70 miles an hour starting and then the 5 miles an hour way at the end. It turns out it averaged out to 20 miles an hour. So what we actually found was the average velocity or my average rate of change. You see that? By looking at the entire distance and then dividing by the entire time, I found out how fast I was going on average. That doesn't tell me what happens at that particular point. So we're still stuck. However, let's celebrate, we can find average velocities and rates. All you do is take the change in distance and divide it by the change in time.
And in fact right now, let me just stop for a second and I want to have you just try, on your own, to try one of these problems right now. So try it and then just, when you're done, click and we'll move on.
Well, I hope you had fun with that. The problem of course, as I mentioned, was that finding these kind of rates, those are average rates. It doesn't tell me what's going on at this point. What we want to find is we want to find out the instantaneous rate of change - the rate of change at that one instant - not over the whole trip. At that instant, was I breaking the law? So how would I do that?
Let's try to use the formula once again. We try to use the formula now for the instantaneous rate, just at that point. Let's try it. Well, rate equals - now remember what the formula is. The formula, I remind you, is that rate equals the change in distance divided by the change in time. Now at this instant in time - at that very instant in time, at that very second across the sign - what was my change in distance, at that very instant? Zero! Because it's that moment I cross that sign. There's no change in distance. So this would be a zero on top, and divide through by the change in time.
Well, what was the change in time at an instant? Well, there is no change in time in an instant. It's zero. There's no time change. So I'd divide by zero. Zero over zero - whoops! This is a problem. This is a big problem. Now I want to come back to all this, but I want to, first of all, just show you why zero over zero can hurt. Okay?
So here we go. Zero over zero, let me tell you the sad story about zero over zero. In fact, one of the people in the staff thought it would be cute to actually have sort of an edible zero over zero, so they took me two donuts from the last little thing here. And you see zero over zero. But this is bad. This is bad, and I don't want you ever to think about zero over zero as being something edible and tasty. I'm not even going to bite into this, folks, because you don't want to bite into zero over zero, and I'm going to show you why. This is why you don't want to bite into zero over zero. What does it equal?
On the one hand, someone might come along and say, "Hey, whenever you have zero divided by anything, that always equals zero." So maybe zero over zero equals zero. But now, on the other hand, someone might come along and remember that if you take a number and divide by zero, it's undefined. So someone might say, "Wait a minute, anything divided by zero is undefined." And someone else, some other person, might say, "Wait a minute, I remember if we have anything over itself, you can cancel and you're just left with 1. So that would seem to indicate that zero equals 1 equals undefined. Well, no. This is a big problem, big problem, big problem. You cannot divide by zero over zero. Zero over zero is not a number, so this is wrong and this is wrong. It's just not a number.
Okay, so now if you think back to how we got to this, how did we get to zero over zero by the way? You might have forgotten, but I'm going to remind you because I happen to remember. We got to zero over zero because we were trying to find the instantaneous rate of change right at this point. And what we saw was there was zero change in distance, zero change in time, and so we got something that's garbage. So you can see that there's a little bit more to this than just rate equals change in distance over change in time, because if we just try to plug in, what do we see? Well, we get sort of garbage, so we can't just plug in.
Let me ask you something else. And let me tell you right now what it is we're going to do. What we're going to do is, instead of trying to plug in the zero over zero, what we're going to do is we're going to inch up to it. We're just going to inch up to it really, really slowly and we're going to see if we can see what the answer is without actually plugging in the zero over zero.
So on the one hand, you might say, "Well, gee, there's more to calculus," which is good, because remember half of the course is just to answer that first question. If I answered it right for you right now, you'd have been done and we'd be done and we'd be finished. So no, no, no. So there's a little bit more to it than just plugging in. But we have to celebrate what we can do. We can't answer this question yet, finding instantaneous rate of change. But let's just take sort of solace in the fact that we can find average velocities and average rates. So we can do something and we need to celebrate that, because that's going to be important for figuring out how to do the slightly more interesting problem, which still remains unanswered. So what we're going to do next is we're going to actually start to build up some mathematical ideas that are going to allow us to come to grips with this notion of plugging in the zero over zero. Without doing that, instead we're going to actually inch up to the zero over zero. So that's going to be what we do next. But first, why don't you try some of these things on your own and see how you fare? Okay, see you soon.

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