18-01F07-L11 - 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 11 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: OK, we're ready to start the eleventh lecture. We're still in the middle of sketching. And, indeed, one of the reasons why we did not talk about hyperbolic functions is this that we're running just a little bit behind. And we'll catch up a tiny bit today. And I hope all the way on Tuesday of next week. So let me pick up where we left off, with sketching. So this is a continuation. I want to give you one more example of how to sketch things. And then we'll go through it systematically. So the second example that we did as one example last time, is this. The function is x + 1 / x + 2. And I'm going to save you the time right now. This is very typical of me, especially if you're in a hurry on an exam, I'll just tell you what the derivative is. So in this case, it's 1 / (x + 2)^2. Now, the reason why I'm bringing this example up, even though it'll turn out to be a relatively simple one to sketch, is that it's easy to fall into a black hole with this problem. So let me just show you. This is not equal to 0. It's never equal to 0. So that means there are no critical points. At this point, students, many students who have been trained like monkeys to do exactly what they've been told, suddenly freeze and give up. Because there's nothing to do. So this is the one thing that I have to train out of you. You can't just give up at this point. So what would you suggest? Can anybody get us out of this jam? Yeah. STUDENT: [INAUDIBLE] PROFESSOR: Right. So the suggestion was to find the x values where f (x) is undefined. In fact, so now that's a fairly sophisticated way of putting the point that I want to make, which is that what we want to do is go back to our precalculus skills. And just plot points. So instead, you go back to precalculus and you just plot some points. It's a perfectly reasonable thing. Now, it turns out that the most important point to plot is the one that's not there. Namely, the value of x = - 2. Which is just what was suggested. Namely, we plot the points where the function is not defined. So how do we do that? Well, you have to think about it for a second and I'll
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This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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18-01F07-L11 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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