1. Warm Up – NO CALCULATOR Let f(x) = x2 – 2x.
Determine the average rate of change of f(x) over the interval [-1, 4].
Determine the value of
(Check your answer using
your calculator)

2. Mean Value Theorem for Integrals Average Value 2nd Fundamental Theorem of Calculus

3. Mean Value Theorem for Integrals If f is continuous on [a,b] then there is a certain point (c, f(c)) between a and b so if you draw a rectangle whose length is the interval [a,b] and whose height is f(c), the area of the rectangle will be exactly the area beneath the function on [a,b]. a b c

4. In other words…
*If f is continuous on [a,b], then there exists a number c in the open interval (a,b) such that .
Area under the curve from a to b
Area of the rectangle formed

5. Example 1:
Find the value of f(c) guaranteed by MVT for integration for the function
f(x) = x3 – 4x2 + 3x + 4 on [1,4]
Explain the relationship of this value to the graph of f(x)?

6. Example 2
Find the value of f(c) guaranteed by MVT for integrals on the interval [1,9] for

7. Solving for f(c) gives the formula for average value.
The f(c) value you found in both examples is called the average value of f.
Solving for f(c) gives the formula for average value.

8. Example 3:
Find the average value of f(x) = 3x2 – 2x on the interval [1,4] and all values of x in the interval for which the function equals its average value.

9. Taking the derivative of a definite integral whose lower bound is a number and whose upper bound contains a variable.

10. The long way…

11. The 2nd Fundamental Theorem of Calculus:
If f(x) is continuous and differentiable,

12. Here’s what you REALLY do…

13. Your turn…

14. If

15. f
Let f be defined on the closed interval [-5,5]. The graph of f consisting of two line segments and two semicircles, is shown above.

16. Let g be the function given by
Find g(2)
Find g’(2)
Find g”(2)

17. On what intervals, if any, is g increasing?
g(x)= f
On what intervals, if any, is g increasing?
Find the x-coordinate of each point of inflection of the graph g on the open interval (-5,5). Justify your answer.

18. Find the average rate of change of g on the interval [-5,5].
g(x)= f
Find the average rate of change of g on the interval [-5,5].