# Exercises - Mean Value Theorem for Integrals

1. Determine if the Mean Value Theorem for Integrals applies to the given function over the given interval. If it does, demonstrate how the theorem's hypothesis is met, state clearly the conclusion, and find the average value of the function on the interval. If the theorem does not apply, explain why.

1. $\displaystyle{f(x) = (x+1)^2; \quad [0,2]}$

2. $\displaystyle{f(x) = \sin (3x); \quad [0,\frac{\pi}{9}]}$

3. $\displaystyle{f(x) = x^2; \quad [0,3]}$

4. $\displaystyle{f(x) = \frac{4(x^2+1)}{x^2}; \quad [1,3]}$

5. $\displaystyle{f(x) = \cos x; \quad [0,\pi]}$