Lesson 5.6: The Mean Value Theorem and Fundamental Theorem of Calculus Part 2
For questions 1-6, Find the average value of the function over the given interval and all values of x in the
interval for which the function equals its average value.
4( x 2  1)
, [1, 3]
x2
1. f(x) = 4 – x2, [-2, 2]
2. f(x) =
3. f(x) = sinx, [0, π]
4. f(x) = cosx, [0, π/2]
5. f(x) =
9
, [1, 3]
x2
6. f(x) = x2, [1, 4]
x
7. F(x) =  (3t 3  4)dt
1
a. Use Part 2 of the Fundamental Theorem of Calculus to find F’(x)
b. Check the result in part (a) by first integrating and then differentiating.
x
8. F(x) =
 (sin 2t )dt

4
Use Part 2 of the Fundamental Theorem of Calculus to find F’(x)
For questions 9-12, use part 2 of the Fundamental Theorem of Calculus to find the derivatives.
x
d
sin( t 2 ) dt =
9.
dx 1
x
d
10.
e t dt =
dx 0
0
x
11.
d
ln tdt =
dx 1
12.
d
t sec tdt =
dx x
13. The area of an oil spill is increasing at a rate of 25t ft2/s, t seconds after the spill. How much oil spilled
between t = 2 seconds and t = 4 seconds? (Set up the definite integral, then solve)