# Calculus/Some Important Theorems/Solutions

# Rolle's Thoerem[edit]

- Factor the expression to obtain . and are our two endpoints. We know that and are the same, thus that satisfies the first part of Rolle's theorem ().

2: Now by Rolle's Theorem, we know that somewhere between these points, the slope will be zero. Where? Easy: Take the derivative.

Thus, at , we have a spot with a slope of zero. We know that (or 1.5) is between 0 and 3. Thus, Rolle's Theorem is true for this (as it is for all cases).

# Mean Value Theorem[edit]

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

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.