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

. Then there is a number

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

. 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

. 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

. 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

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.

Cauchy's Mean Value Theorem

Introduciton