18-01F07-L21 - 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 21 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: One correction from last time. Sorry to say, I forgot a very important factor when I was telling you what an average value is. If you don't put in that factor, it's only half off on the exam problem that will be given on this. So I would have gotten half off for missing out on this factor, too. So remember you have to divide by n here, certainly when you're integrating over to n, the Riemann sum is the numerator here. And if I divide by n on that side, I've got to divide by n on the other side. This was meant to illustrate this idea that we're dividing by the total here. And we are going to be talking about average value in more detail. Not today, though. So this has to do with average value. And we'll discuss it in considerable detail in a couple of days, I guess. Now, today I want to continue. I didn't have time to finish my discussion of the Fundamental Theorem of Calculus 2. And anyway it's very important to write it down on the board twice, because you want to see it at least twice. And many more times as well. So let's just remind you, the second version of the Fundamental Theorem of Calculus says the following. It says that the derivative of an integral gives you the function back again. So here's the theorem. And the way I'd like to use it today, I started this discussion last time. But we didn't get into it. And this is something that's on your problem set along with several other examples. Is that we can use this to solve differential equations. And in particular, for example, we can solve the equation y' = 1 / x with this formula. Namely, using an integral. L ( x ) is the integral from 1 to x of dt / t. The function f ( t ) is just 1 / t. Now, that formula can be taken to be the starting place for the derivation of all the properties of the logarithm function. So what we're going to do right now is we're going to take this to be the definition of the logarithm. And if we do that, then I claim that we can read off the properties of the logarithm just about as easily as we could before. And so I'll illustrate that now. And there are a few other examples of this where somewhat more unfamiliar functions come up. This one is one that in
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