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18-01F07-L19 - 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 19 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: Today we're going to continue with integration. And we get to do the probably the most important thing of this entire course. Which is appropriately named. It's called the fundamental theorem of calculus. And we'll be abbreviating it FTC and occasionally I'll put in a 1 here, because there will be two versions of it. But this is the one that you'll be using the most in this class. The fundamental theorem of calculus says the following. It says that if F' = f, so F' ( x) = little f ( x), there's a capital F and a little f, then the integral from a to b of f ( x) = F ( b) - F (a). That's it. That's the whole theorem. And you may recognize it. Before, we had the notation that f was the antiderivative, that is, capital F was the integral of f(x). We wrote it this way. This is this indefinite integral. And now we're putting in definite values. And we have a connection between the two uses of the integral sign. But with the definite values, we get real numbers out instead of a function. Or a function up to a constant. So this is it. This is the formula. And it's usually also written with another notation. So I want to introduce that notation to you as well. So there's a new notation here. Which you'll find very convenient. Because we don't always have to give a letter f to the functions involved. So it's an abbreviation. For right now there'll be a lot of f's, but anyway. So here's the abbreviation. Whenever I have a difference between a function at two values, I also can write this as F ( x) with an a down here and a b up there. So that's the notation that we use. And you can also, for emphasis, and this sometimes turns out to be important, when there's more than one variable floating around in the problem. To specify that the variable is x. So this is the same thing as x = a. And x = b. It indicates where you want to plug in, what you want to plug in. And now you take the top value minus the bottom value. So F ( b) - F(a). So this is just a notation, and in that notation, of course, the theorem can be written with this set of symbols here. Equally well.
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