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18.01 Single Variable Calculus, Fall 2007
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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 20
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PROFESSOR: To begin today I want to remind you, I need to write it down on the
board at least twice, of the fundamental theorem of calculus. We called it FTC 1
because it's the first version of the fundamental theorem. We'll be talking about
another version, called the second version, today. And what it says is this: If F' = f,
then the integral from a to b of f (x) dx = F ( b)  F ( a). So that's the fundamental
theorem of calculus. And the way we used it last time was, this was used to evaluate
integrals. Not surprisingly, that's how we used it. But today, I want to reverse that
point of view. We're going to read the equation backwards, and we're going to write
it this way. And we're going to use f to understand capital F. Or in other words, the
derivative. To understand the function.
So that's the reversal of point of view that I'd like to make. And we'll make this point
in various ways. So information about F, about F', gives us information about F. Now,
since there were questions about the mean value theorem, I'm going to illustrate this
first by making a comparison between the fundamental theorem of calculus and the
mean value theorem. So we're going to compare this fundamental theorem of
calculus with what we call the mean value theorem. And in order to do that, I'm
going to introduce a couple of notations. I'll write delta F as F ( b )  F ( a). And
another highly imaginative notation, delta x = b  a. So here's the change in f,
there's the change in x. And then, this fundamental theorem can be written, of
course, right up above there is the formula. And it's the formula for delta F. So this is
what we call the fundamental theorem of calculus.
I'm going to divide by delta x, now. And If I divide by delta x, that's the same thing
as 1 / b  a times the integral from a to b of f ( x) dx. So I've just rewritten the
formula here. And this expression here, on the righthand side, is a fairly important
one. This is the average of f. That's the average value of f. Now, so this is going to
permit me to make the comparison between the mean value theorem, which we
don't have stated yet here. And the fundamental theorem. And I'll do it in the form
of inequalities. So right in the middle here, I'm going to put the fundamental
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 Calculus, Derivative, Mean Value Theorem

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