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18-01F07-L09

# 18-01F07-L09 - 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 Transcript – Lecture 9 The following content is provided under a Creative Commons license. Your support will help MIT OpenCorseWare 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. PROF. JERISON: We're starting a new unit today. And, so this is Unit 2, and it's called Applications of Differentiation. OK. So, the first application, and we're going to do two today, is what are known as linear approximations. Whoops, that should have two p's in it. Approximations. So, that can be summarized with one formula, but it's going to take us at least half an hour to explain how this formula is used. So here's the formula. It's f(x) is approximately equal to f(x0) f'(x)( x - x0). Right? So this is the main formula. For right now. Put it in a box. And let me just describe what it means, first. And then I'll describe what it means again, and several other times. So, first of all, what it means is that if you have a curve, which is y = f(x), it's approximately the same as its tangent line. So this other side is the equation of the tangent line. So let's give an example. I'm going to take the function f(x), which is ln x, and then its derivative is 1 / x. And, so let's take the base point x0 = 1. That's pretty much the only place where we know the logarithm for sure. And so, what we plug in here now, are the values. So f (1) is the ln of 0. Or, sorry, the ln of 1, which is 0. And f'(1), well, that's 1/1, which is 1. So now we have an approximation formula which, if I copy down what's right up here, it's going to be ln x is approximately, so f(0) is 0, right? 1 (x - 1). So I plugged in here, for x0, three places. I evaluated the coefficients and this is the dependent variable. So, all told, if you like, what I have here is that the logarithm of x is approximately x - 1. And let me draw a picture of this. So here's the graph of ln x. And then, I'll draw in the tangent line at the place that we're considering, which is x = 1. So here's the tangent line. And I've separated a little bit, but really I probably should have drawn it a little closer there, to show you the whole point is that these two are nearby. But they're not nearby everywhere. So this is the line y = x - 1. Right, that's the tangent line. They're nearby only when x is near 1. So say in this little realm here. So when x is approximately 1, this is true. Once you get a little farther away, this straight line, this straight green line will separate from the graph. But near this place they're close
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