Why is that? Well, this part right over here gives you the area, and then you divide the area by the base, and you get the mean height. Or another way you could say it is, if you were to take the height right over here, multiply it times the base, you get a rectangle that has the exact same area as the area under the curve. Well, this is useful, because this is exactly what we got as the derivative of f prime of x. So there must exist a c such that f of c is equal to this stuff, or we could say that the limit, and let me rewrite all of this down a new color. So there exists a c in the interval x to x plus delta x, where f prime of x, which we know is equal to this, we can now say is now equal to the limit as delta x approaches 0. And instead of writing this, we know that there's some c that's equal to all of this business, of f of c. Now we're in the home stretch.