18-01F07-L16

# 18-01F07-L16 - 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 16 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: And this last little bit is something which is not yet on the Web. But, anyway, when I was walking out of the room last time, I noticed that I'd written down the wrong formula for c1 - c1. There's a misprint, there's a minus sign that's wrong. I claimed last time that c1 - c2 was + 1/2. But, actually, it's - 1/2. If you go through the calculation that we did with the antiderivative of sine x cosine x, we get these two possible answers. And if they're to be equal, then if we just subtract them we get c1 - c2 + 1/2 = 0. So c1 - c2 = 1/2. So, those are all of the correction. Again, everything here will be on the Web. But just wanted to make it all clear to you. So here we are. This is our last day of the second unit, Applications of Differentiation. And I have one of the most fun topics to introduce to you. Which is differential equations. Now, we have a whole course on differential equations, which is called 18.03. And so we're only going to do just a little bit. But I'm going to teach you one technique. Which fits in precisely with what we've been doing already. Which is differentials. The first and simplest kind of differential equation dy/dx = some function, f (x). Now, that's a perfectly good differential equation. And we already discussed last time that the solution; that is, the function y, is going to be the antiderivative, or the integral, of x. Now, for the purposes of today, we're going to consider this problem to be solved. That is, you can always do this. You can always take antiderivatives. And for our purposes now, that is for now, we only have one technique to find antiderivatives. And that's called substitution. It has a very small variant, which we called advanced guessing. And that works just as well. And that's basically all that you'll ever need to do. As a practical matter, these are the ones you'll face for now. Ones that you can actually see what the answer is, or you'll have to make a substitution. Now, the first tricky example, or the first maybe interesting example of a differential equation, which I'll call Example 2, is
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18-01F07-L16 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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