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[music].
A classic problem about the mean theorem

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problem that's often asked in Calculus
courses, well, that classic problem goes

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like this.
Given a function f and points a and b, I

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want to find some other point c, so that f
prime of c is equal to f of b minus f of a

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all over b minus a.
The mean value theorem promises us that as

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long as f is a nice enough function, nice
enough meaning it's differentiable on the

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open interval a b, continuous on the
closed interval a b.

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That as long as f is a nice enough
function, then that's possible.

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I can find that point c, so that the
derivative of f at the point c is equal to

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f of b minus f of a over b minus a.
Let, let's do an example where I try to

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find that point c.
So let's do an example.

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I'll have f of x, just making up some
example, that'll be x cubed and it will be

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on the interval where a equal 1 and b
equals 4.

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So really, what I'm trying to do is find
some point c so that the derivative of the

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function at the point c is f of b minus f
of a over b minus a.

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Now, in this case, f of b is 4 cubed,
that's 64.

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F of a is 1 cubed, that's 1.
And b minus a, that's 4 minus 1.

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This is 63 over 3.
That's 21.

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So I'm looking for a point c, where the
derivative is equal to 21.

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Now, can I do that?
Yeah.

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I just gotta differentiate this function.
So here we go.

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I'll differentiate this function, and I
get that the derivative of x cube is 3 x

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squared, and I'm looking for which value
of x is that equal to 21?

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We'll divide both sides by 3.
I'm looking for a value of x, where x

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squared is 7.
And so divide both sides by 3.

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And my x should be between a and b, right?
So in that case, x is the positive square

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root of 7.
And what I've really accomplished here, if

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I make a little graph, here's the x y
plane, and I'll draw kind of a made up

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graph of the cubed function.
All right, here's the point 1, say.

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Here's the point 4.
Here is 1,1.

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Here's 4,64.
Right?

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The slope of the line here, all right, is
21.

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All right, the slope here is 21.
And what I've really managed to do here is

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find that at the square root of 7, the
slope of the tangent line is exactly the

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same as the slope of the line through 1,
1, and 4, 4 cubed.

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It's super important to emphasize that
although these kinds of problems are fun,

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and the mean value theorem guarantees that
for nice enough functions we can find such

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a c, yeah, these are fun problems.
The mean value theorem guarantees that

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it's possible.
But that's not the point of the mean value

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theorem.
The mean value theorem is more than just a

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computational result.
It's really an existence theorem.

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It's important to emphasize the power of
these existence results.

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Something like the mean value theorem
tells you that something exists without

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you having to actually go out into the
world and find that thing.

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Right?
And having you do a ton of problems where

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you find the point c really would
undermine that important lesson, right?

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The power of the mean value theorem lies
not in the fact that you can actually go

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out and compute the value c, right?
The power lies in the fact, the mean value

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theorem tells you that it's possible, that
you know there's a value of c out there,

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without you having actually go and find
it.

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And because of that, it opens up the door
to all these interesting conceptual

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applications where we're relating, say,
positive derivative on an interval to the

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fact the function's values are increasing.
And that's really the power of the mean

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value theorem.
It's a conceptual result that helps you

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relate Information about the derivative to
information about the original function.

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.