18-01F07-L12 - 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 12 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: In the twelfth lecture, we're going to talk about maxima and minima. Let's finish up what we did last time. We really only just started with maxima and minima. And then we're going to talk about related rates. So, right now I want to give you some examples of max-min problems. And we're going to start with a fairly basic one. So what's the thing about max-min problems? The main thing is that we're asking you to do a little bit more of the interpretation of word problems. So many of the problems are expressed in terms of words. And so, in this case, we have a wire which is length 1. Cut into two pieces. And then each piece encloses a square. Sorry, encloses a square. And the problem - so this is the setup. And the problem is to find the largest area enclosed. So here's the problem. Now, in all of these cases, in all these cases, there's a bunch of words. And your job is typically to draw a diagram. So the first thing you want to do is to draw a diagram. In this case, it can be fairly schematic. Here's your unit length. And when you draw the diagram, you're going to have to pick variables. So those are really the two main tasks. To set up the problem. So you're drawing a diagram. This is like word problems of old, in grade school through high school. Draw a diagram and name the variables. So we'll be doing a lot of that today. So here's my unit length. And I'm going to choose the variable x to be the length of one of the pieces of wire. And that makes the other piece 1 - x. And that's pretty much the whole diagram, except that there's something that we did with the wire after we cut it in half. Namely, we built two little boxes out of it. Like this, these are our squares. And their side lengths are x / 4 and (1 - x) / 4. So, so far, so good. And now we have to think, well, we want to find the largest area. So I need a formula for area in terms of variables that I've described. And so that's the last thing. I'll give the letter a as the label for the area. And then the area is just the square of x/4 + the square of 1 - x. Whoops, that strange 2 got in here. Over 4. So far, so good. Now, The instinct that you'll have, and I'm going to yield to that instinct, is we should charge ahead and just differentiate. Alright? That's alright.
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