The Mean Value Theorem
Here’s a nice technical result we may have call for from time to time: a higher-dimensional version of the differential mean value theorem. Remember that this says that if we’ve got a function continuous on the closed interval and differentiable on its interior, there is some point in the middle where the derivative of the function is the same as the average — the mean — rate of change of the function over the interval. In more than one dimension we’re going to modify this a bit to make it clearer what it means.
First of all, instead of talking about the closed interval , we’re going to use the closed straight line segment. That is, the collection of all the points between and in a straight line, and including the endpoints. We first look at the total displacement from one point to the other. Then we start at and move some portion of this displacement towards . That is, the closed line segment consists of all points of the form for in the closed interval . Setting gives us the point , and gives us the point . Similarly, the open line segment consists of all points of the form for in the open interval .
Next, we have to be clear about the average rate of change. As we move from to , the value of the function changes by . It takes a displacement of to get there, so on average the rate of change is
Finally, we don’t just have a single value for the instantaneous rate of change, we have a differential . But we can use it to find directional derivatives. Specifically, we’ll consider the derivative of in the direction pointing from to . We’ll pick out this direction with the unit vector we get by normalizing the displacement
So the mean value theorem will tell us that if is differentiable in some open region that contains the whole closed line segment . Then there is some point in the open line segment so that the average rate of change of from to is equal to the directional derivative of at in the direction pointing from to :
or, more simply
We’ll get at this by changing to a function of one variable so we can bring the one-dimensional version to bear. To that end, we define for in the closed interval . Then , and we can also show that is differentiable everywhere inside the interval. Indeed, we can evaluate the difference quotient
Taking the limit as approaches , we find
which exists since is differentiable.
So our old differential mean-value theorem tells us that there is some so that
where is a point in the open line segment .