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Calculus: Finding the Average Value of a Function

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  • Type: Video Tutorial
  • Length: 8:19
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 89 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Applications of Integral Calculus (18 lessons, $26.73)

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 http://www.thinkwell.com/student/product/calculus. 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.

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Applications of Integral Calculus
The Average Value of a Function
Finding the Average Value of a Function Page [1 of 2
If you take a class, sometimes you want to figure out what your average is. And so if you have a whole bunch of quizzes, let's say, and they all count the same, how would you compute the average of that? You would add up all those quizzes and you would divide by the total number of quizzes, and that would tell you what your average is. So for example, let's suppose you want to average some numbers. Suppose you want to just average, let's say, 1 and 4. How would you average 1 and 4? What you would do is you would just add those two numbers together and then divide by two, because you have two numbers. And so in this case, you would see or 2.5.
So finding the average of a whole bunch of numbers is not a big deal. In fact, we do it all the time. But what if you want to find the average of infinitely many numbers? For example, suppose I say, let's consider the curve f(x) = x^2, and what I want you to do is I want you to figure out the average of all those numbers as the x ranges from zero to 3 - so average value equals.
Now how in the world would you do that? Well, you might say, what I'll do is I'll just take every point from zero to 3, plug it in here and see what it gives me. Like, if I plug in zero, for example, that's going to give me zero squared which is zero. That's okay. If I plug in 3 into here, I'd see 3^2, which is 9, so there's another point. And I could plug in, like for example, 1 into here, and I'd see 1^2, which is 1. I'd plug in 2 into here and I'd see 2^2, which is 4. So those are fine.
But there are a lot of other points. Like what about a half? Okay, yes, I have to put a half in here and square a half, I see a fourth. Then you also got to put in like - I don't know - . Oh, yes, go in here, so is going to be .
And in fact there are a lot of points I've got to put in. I've got to put in, for example, . I've got to put in there. So ^2 is 2. So I have all these values. In fact, there are so many values, you can't even write them all down. And yet, I want to figure out what the average is among all of these points and get a collection that's even too vast to really understand - certainly too vast to sort of count up and divide by the total number, because the total number's infinite. So how in the world can we possibly resolve this?
Well, again, we go from the discrete just a couple of points to the infinite, tons of points, by figuring out what the analog of sum is. The analog of sum is the integral, where we sum things up. And so in fact, this is a nice application of the integral, trying to find the average value of a function. What value does a function take on, on average, over a particular range.
So let's actually see if we can use calculus and see the idea of an average value of a function on this example. So I want to find the average value of the function x^2 between 0 and 3. So how to proceed? Well, I absolutely have no idea. So that's a weak start.
But let's take a look at a picture then and graph this thing. So I'm going to graph the parabola as best as I can. And this is probably not going to be drawn to scale, I bet. This, let's say, is 1. I can make a 1 anywhere. You know the power I feel knowing that I can just put down anywhere I want and call it want? Not that much power. 1, 2, 3 - so this is not going to be drawn to scale, because see here, 1, this value right here is also 1. So you can see, this is not drawn to scale. The scales will be different here. This would then be 4. And way up here, in life - this is way up here - 9.
Okay, great. Now what I want to do is I want to find the average value of the function. So I want to take all these heights. Here are just three examples of heights: 9, 4 and 1. I want to take all of the heights, and I want to find out what the average height is. Well, let's think about that. What I really want to do is sort of add up all of these things under the curve. So in fact, really what I want to do is find the area under the curve. In fact, if I find the area under the curve, that is tantamount to summing up all these heights. And let me show it to you right now live.
So I want to sum up that length. I want to sum up that length. I want to sum up that length. I want to sum up that length, and this length. Now of course, my magic marker, happily, is pretty thick, but imagine all those points. And I just want to add up all those values. And if I add up all those values, it's just the area under he curve. So that's the sum. But now to find the average, I have to divide by how much I have. Well, how much I have is the length of this interval. And so, in fact, the average value of a function, we now see, between two values should be equal to what?
Well, first to sum up all the values, we just integrate the function from the two points. But then I want to actually now average, so I'd better divide that sum by, in some sense, "how many points I have" - or in this case, the length of the interval - which would just be b minus a. So that's the definition of the average value of the function, which means, what value is take on average, if you would average all these numbers.
So let's work through it. So let's try this example now. If I want to find the average value between zero and 3, what I do is, that average value is equal to 1 divided by this length, which is 3, multiplied by an integral from zero to 3 of just, in this case, this function is the x^2 function - so x^2dx. And what is this integral? Well, that integral is easy to see to be x , which is evaluated from zero to 3. So when I plug in 3, I see 3^3, which is 27, divided by 9, is 3, minus, when I plug in zero, I get zero. So the average value is 3. What does that mean? What does it mean for the average value to be 3?
Well, let's think about it. What it means is that the value that this thing takes on, on average, is 3. So if you look at this picture, for example, it would seem to be right around here - 1,2,3. This is not really drawn to scale. Here's 3. And so what that means is that if you were to take all of this stuff here, that should be able to fit right into here, because it all sort of averages out. So if you now take this stuff put it into here, it should fit in perfectly. Now of course, it doesn't fit in perfectly, but if you cut it up in little pieces, it should fit in, because this is the average value of the function.
So, in fact, I see that from zero to 3, this function, on average - of course, what does that mean? If you average up all these values, it is going to produce the value of 3. So a nice simple application of the integral. It's an application where you can actually now figure out the generalization of figuring out averages of just averages of seven things. You can now use the integral to find the averages of infinitely many things. Neat. I'll see you at the next lecture.

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