Cauchy's Mean Value Theorem

Introduciton

Cauchy's Mean Value Theorem is an important part in proving l'Hospital's Rule and as such, it is important to have a basic understanding of the Theorem. Proving Cauchy's Mean Value Theorem is very similar to proving the Mean Value Theorem and we will address this later in the activity. Before that, we begin by introducing the theorem.

Cauchy's Mean Value Theorem

Suppose that the functions and are continuous on and differentiable on , and for all in . Then there is a number in such that

Now, before we prove the theorem, let us look at an example to build some intuition.

Example 1

Let and and consider the interval . First, we see that since and are both continuous for all , they are continuous on the interval . Second, their derivatives and are also continuous for all , and are therefore continuous on the interval . Finally, we see that for all , and thus clearly on . Thus, there exists a number in the interval such that

Replacing the values into the functions and simplifying, we have

So, we have

Since we cannot solve this for directly, we will try to estimate the value using the graphic below. Use the slider to move the point closer to the point of intersection and then use the input box above to get a closer approximation.

Question 1

Estimate the value of that satisfies the conclusion of Cauchy's Mean Value Thorem. If you can only find a possible range of values, give a range of possible values for .

Question 2

Use Newton's Method to find an approximation for the value of accurate to eight decimal places. How many more steps will it require for you to find an approximation accurate to 12 decimal places?

Proof

Suppose that and are functions that are continuous on , differentiable on , and for all on . First, we note that the denominator on the right-hand side is not zero. If then and by Rolle's Theorem there exists a number in the interval such that . This contradicts the hypothesis that for all in . Thus, . Next, we define a new function

This new function is continuous on , differentiable on . Additionally, this new function satisfies the condition that since,

and

Therefore, by Rolle's Theorem there exists a number in such that

Reorganizing terms gives us

This completes the proof.

Question 3

Find the value of so that the function

satisfies the condition .

Question 4

Use this new function to prove Cauchy's Mean Value Theorem.