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18-01F07-L23 - 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 23 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 hold off just a little bit on boiling water. And talk about another application of integrals, and we'll get to the witches' cauldron in the middle. The that I'd like to start with today is average value. This is something that I mentioned a little bit earlier, and there was a misprint on the board, so I want to make sure that we have the definitions straight. And also the reasoning straight. This is one of the most important applications of integrals, one of the most important examples. If you take the average of a bunch of numbers, that looks like this. And we can view this as sampling a function. As we would with Riemann's sum. And what I said last week was that this tends to this expression here, which is called the continuous average. So this guy is the continuous average. Or just the average of f. And I want to explain that, just to make sure that we're all on the same page. In general, if you have a function and you want to interpret the integral, our first interpretation was that it's something like the area under the curve. But average value is another reasonable interpretation. Namely, if you take equally spaced points here, starting with x0, x1, x2, all the way up to xn, which is the left point b, and then we have values y1, which = f ( x1), y2, which = f ( x2), all the way up to yn, which = f ( xn). And again, the spacing here that we're talking about is b - a / n. So remember that spacing, that's going to be the connection that we'll draw. Then the Riemann sum is y1 through yn, the sum of (y1 ... yn) delta x. And that's what tends, as delta x goes to 0, to the integral. . The only change in point of view if I want to write this limiting property, which is right above here, the only change between here and here is that I want to divide by the length of the interval. b - a. So I will divide by b - a here. And divide by b - a over here. And then I'll just check what this thing actually is. Delta x / b - a, what is that factor? Well, if we look over here to what delta x is, if you divide by b - a, it's 1 / n. So the factor delta x / b - a = 1 / n. That's what I put over here, the sum of y1 through yn / n. And as this tends to 0, it's the same as n going to infinity. Those are the same things. The average value
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