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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 9
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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