# Calculus/Some Important Theorems/Solutions

# Rolle's Thoerem[edit]

1. Show that Rolle's Theorem holds true between the x-intercepts of the function .

# Mean Value Theorem[edit]

2. Show that , where is the function that was defined in the proof of Cauchy's Mean Value Theorem.

3. Show that the Mean Value Theorem follows from Cauchy's Mean Value Theorem.

4. Find the that satisfies the Mean Value Theorem for the function with endpoints and .

1: Using the expression from the mean value theorem

insert values. Our chosen interval is . So, we have

2: By the Mean Value Theorem, we know that somewhere in the interval exists a point that has the same slope as that point. Thus, let us take the derivative to find this point .

Now, we know that the slope of the point is 4. So, the derivative at this point is 4. Thus, . So

5. Find the point that satisifies the mean value theorem on the function and the interval .

1: We start with the expression:

so,

(Remember, sin(π) and sin(0) are both 0.)

2: Now that we have the slope of the line, we must find the point x = c that has the same slope. We must now get the derivative!

The cosine function is 0 at (where is an integer). Remember, we are bound by the interval , so is the point that satisfies the Mean Value Theorem.