[music]. A classic problem about the mean theorem
problem that's often asked in Calculus courses, well, that classic problem goes
like this. Given a function f and points a and b, I
want to find some other point c, so that f prime of c is equal to f of b minus f of a
all over b minus a. The mean value theorem promises us that as
long as f is a nice enough function, nice enough meaning it's differentiable on the
open interval a b, continuous on the closed interval a b.
That as long as f is a nice enough function, then that's possible.
I can find that point c, so that the derivative of f at the point c is equal to
f of b minus f of a over b minus a. Let, let's do an example where I try to
find that point c. So let's do an example.
I'll have f of x, just making up some example, that'll be x cubed and it will be
on the interval where a equal 1 and b equals 4.
So really, what I'm trying to do is find some point c so that the derivative of the
function at the point c is f of b minus f of a over b minus a.
Now, in this case, f of b is 4 cubed, that's 64.
F of a is 1 cubed, that's 1. And b minus a, that's 4 minus 1.
This is 63 over 3. That's 21.
So I'm looking for a point c, where the derivative is equal to 21.
Now, can I do that? Yeah.
I just gotta differentiate this function. So here we go.
I'll differentiate this function, and I get that the derivative of x cube is 3 x
squared, and I'm looking for which value of x is that equal to 21?
We'll divide both sides by 3. I'm looking for a value of x, where x
squared is 7. And so divide both sides by 3.
And my x should be between a and b, right? So in that case, x is the positive square
root of 7. And what I've really accomplished here, if
I make a little graph, here's the x y plane, and I'll draw kind of a made up
graph of the cubed function. All right, here's the point 1, say.
Here's the point 4. Here is 1,1.
Here's 4,64. Right?
The slope of the line here, all right, is 21.
All right, the slope here is 21. And what I've really managed to do here is
find that at the square root of 7, the slope of the tangent line is exactly the
same as the slope of the line through 1, 1, and 4, 4 cubed.
It's super important to emphasize that although these kinds of problems are fun,
and the mean value theorem guarantees that for nice enough functions we can find such
a c, yeah, these are fun problems. The mean value theorem guarantees that
it's possible. But that's not the point of the mean value
theorem. The mean value theorem is more than just a
computational result. It's really an existence theorem.
It's important to emphasize the power of these existence results.
Something like the mean value theorem tells you that something exists without
you having to actually go out into the world and find that thing.
Right? And having you do a ton of problems where
you find the point c really would undermine that important lesson, right?
The power of the mean value theorem lies not in the fact that you can actually go
out and compute the value c, right? The power lies in the fact, the mean value
theorem tells you that it's possible, that you know there's a value of c out there,
without you having actually go and find it.
And because of that, it opens up the door to all these interesting conceptual
applications where we're relating, say, positive derivative on an interval to the
fact the function's values are increasing. And that's really the power of the mean
value theorem. It's a conceptual result that helps you
relate Information about the derivative to information about the original function.
.