18-01F07-L20

18-01F07-L20 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 20 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: To begin today I want to remind you, I need to write it down on the board at least twice, of the fundamental theorem of calculus. We called it FTC 1 because it's the first version of the fundamental theorem. We'll be talking about another version, called the second version, today. And what it says is this: If F' = f, then the integral from a to b of f (x) dx = F ( b) - F ( a). So that's the fundamental theorem of calculus. And the way we used it last time was, this was used to evaluate integrals. Not surprisingly, that's how we used it. But today, I want to reverse that point of view. We're going to read the equation backwards, and we're going to write it this way. And we're going to use f to understand capital F. Or in other words, the derivative. To understand the function. So that's the reversal of point of view that I'd like to make. And we'll make this point in various ways. So information about F, about F', gives us information about F. Now, since there were questions about the mean value theorem, I'm going to illustrate this first by making a comparison between the fundamental theorem of calculus and the mean value theorem. So we're going to compare this fundamental theorem of calculus with what we call the mean value theorem. And in order to do that, I'm going to introduce a couple of notations. I'll write delta F as F ( b ) - F ( a). And another highly imaginative notation, delta x = b - a. So here's the change in f, there's the change in x. And then, this fundamental theorem can be written, of course, right up above there is the formula. And it's the formula for delta F. So this is what we call the fundamental theorem of calculus. I'm going to divide by delta x, now. And If I divide by delta x, that's the same thing as 1 / b - a times the integral from a to b of f ( x) dx. So I've just rewritten the formula here. And this expression here, on the right-hand side, is a fairly important one. This is the average of f. That's the average value of f. Now, so this is going to permit me to make the comparison between the mean value theorem, which we
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18-01F07-L20 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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