The Mean Value Theorem and Its Consequences
Read this section to learn about the Mean Value Theorem and its consequences. Work through practice problems 1-3.
Some Consequences of the Mean Value Theorem
If the Mean Value Theorem was just an isolated result about the existence of a particular point , it would not be very important or useful. However, the Mean Value Theorem is the basis of several results about the behavior of functions over entire intervals, and it is these consequences which give it an important place in calculus for both theoretical and applied uses.
The next two corollaries are just the first of many results which follow from the Mean Value Theorem.
We already know, from the Main Differentiation Theorem, that the derivative of a constant function. is always , but can a nonconstant function have a derivative which is always ? The first corollary says no.
Proof: Assume for all in an interval , and pick any two points and in the interval. Then, by the Mean Value Theorem, there is a number
between and so that . By our assumption, for all in
so we know that and we can conclude that and .
But and were any two points in , so the value of is the same for any two values of in , and is a constant function on the interval .
We already know that if two functions are parallel (differ by a constant), then their derivatives are equal, but can two nonparallel functions have the same derivative? The second corollary says no.
Corollary 2: If for all in an interval , then , a constant, for all in , so the graphs of and are "parallel" on the interval .
Proof: This corollary involves two functions instead of just one, but we can imitate the proof of the Mean Value Theorem and introduce a new function . The function
is differentiable, and for all in , so, by Corollary is a constant function and for all
in the interval. Then .
We will use Corollary 2 hundreds of times in Chapters 4 and 5 when we work with "integrals". Typically you will be given the derivative of a function, , and asked to find all functions which have that derivative. Corollary 2 tells
us that if we can find one function which has the derivative we want, then the only other functions which have the same derivative are : once you find one function with the right derivative, you
have essentially found all of them.
Example 2: (a) Find all functions whose derivatives equal .
(b) Find a function with and .
Solution: (a) We can recognize that if then so one function with the derivative we want is . Corollary 2 guarantees that every function whose
derivative is has the form . The only functions with derivative have the form .
(b) Since , we know that must have the form , but this is a whole "family" of functions (Fig. 8), and we want to find one member of the family . We know that so we want to find the member of the family
which goes through the point . All we need to do is replace the with 5 and the with 3 in the formula , and then solve for the value of so . The function we want is .
Fig. 8
Practice 3: Restate Corollary 2 as a statement about the positions and velocities of two cars.