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There are several important theorems that help to describe derivatives in calculus. A lot of the time, looking a the curve of a function that is described will help us literally see what is going on. A picture is worth a thousand words, right? We'll talk about Rolle's Theorem and the Mean Value Theorem.
Rolle's Theorem says:
Let f be a function that
Then there is some c in the open interval (a, b) with f ' (c) = 0.
Sometimes the third condition is stated as f (a) = f (b) = 0, but for the proof, it doesn't matter.
In pictures, we're saying suppose f is a nice smooth function with the same starting and ending height:
If f increases or decreases from its starting height, it needs to turn around and come back in order to end at the same height it started at:
Since f is a nice smooth differentiable function, its derivative at that turn-around point must be 0:
If f doesn't go up or down from its starting point, then f is constant:
In this case, f ' (c) is 0 for every value of c in the interval (a, b).
Rolle's Theorem is reminiscent of the Intermediate Value Theorem. Rolle's Theorem says if f satisfies some assumptions (more mathematically known as hypotheses) then f ' will be zero at some point in (a, b). We could have a constant function, in which case f ' will be 0 infinitely many times:
We could have a function that turns around once:
Or we could have a function that turns around many times:
Rolle's Theorem doesn't tell us where or how many times f ' will be zero; it tells us f ' must be zero at least once if the hypotheses are all satisfied.
Suppose f is not continuous on [a, b]. Then there doesn't need to be any c in (a, b) with f ' (c) = 0. Here's an example:
This function is not continuous. At the point of discontinuity, f ' doesn't exist. At all other points in the interval, f ' is positive:
There is no point c in (a, b) where f ' (c) = 0.
We found earlier that the derivative of the absolute value function doesn't exist at 0. When x is negative the slope of the absolute value function is -1; when x is positive the slope of the absolute value function is 1:
There is no value of c anywhere, in any interval (a, b), with f ' (c) = 0. The derivative of the absolute value function isn't 0 anywhere.
If a function fails any of the hypotheses, we aren't allowed to use Rolle's Theorem.
After arriving at our Aunt Petunia's house for Thansgiving, we figured out that our average speed for the trip was 60 mph. Before diving into the sweet potato casserole, though, we asked ourselves a very important question. Does this mean that there must have been a point in the trip when we were driving exactly 60 mph?
Think of this way. Our speed increases continuously; we can't just jump from 30 to 40 mph without hitting every speed in between. This leaves us with a few options:
Maybe we drove 60 mph for the whole trip? In this case, there definitely was a time we were driving exactly 60 mph.
Another option is that we were driving less than 60 mph during the trip. We couldn't have been doing this for the whole trip, though, since our average speed was 60 mph. This means that at some point we would have had to speed up and go more than 60 mph for everything to balance out. When we were accelerating, we must have hit 60 mph at some point.
The last option is that we were driving more than 60 mph during the trip. Like before, we couldn't have been doing this for the whole trip, otherwise our average speed would have been more than 60 mph. In order for everything to balance out, we would have needed to slow down to a speed less than 60 mph, and pass 60 mph along the way.
The scenario we just described is an intuitive explanation of the Mean Value Theorem.
The Mean Value Theorem is a glorified version of Rolle's Theorem.
More formally, The Mean Value Theorem states:
If f is continuous on [a, b], and f is differentiable on (a, b), then there is some c in (a, b) with
is the slope of the secant line between the points (a, f(a)) and (b, f(b)), or the "average speed" of the function between these points if we want to think of it that way.
As with the Intermediate Value Theorem and Rolle's Theorem, the conclusion of the Mean Value Theorem leaves a lot of things out. It doesn't tell us the value of c where f ' (c) equals the slope of the secant line. It doesn't tell us how many such values of c there are. All the Mean Value Theorem does is guarantee that if all the hypotheses are met, at least one such c exists.
As with all the other theorems, if the necessary hypotheses aren't met we aren't allowed to use the Mean Value Theorem.