1. Sec 2 Cont: Mean Value Theorem (MVT)

2. Review: Extreme Value Theorem
Critical Numbers are any ________ or ________.
Critical Numbers can be found by using the equation ________.
The EVT finds ___________ maximums and minimums.
To find any extreme using the EVT, you must test the ________ ________ , where the function is ___________ and the _____ ________.

3. Example 1: Extreme Value Theorem
The volume of a 3-dimensional figure is given by the function y = x³ - 9x + 1 where the length of each side (x) is between .2 and 3. What length will give the greatest volume for the figure?

4. Review: Rolle’s Theorem
For Rolle’s Theorem to work the function must be ____________ and ____________.
If f(a) = f(b), the there must be a ________ _________ on the interval (a, b).
With Rolle’s Theorem, the _________ are not tested because the function is differentiable on an _________ interval.

5. Ex 2: Rolle’s Theorem Let f(x) = x² - 5x + 4 on the interval [1, 4].
A. Can Rolle’s Theorem be applied?
B. If so, find the value of c at which there is a critical
point.

6. Mean Value Theorem (MVT)
If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number, c, in (a, b) such that
The MVT says that the slope of a tangent line on a curve is equal to the slope of the secant line on the same curve at a particular point. 

7. Ex 3: Slope of the Tangent Line
What value of c in the open interval (0, 4) satisfies the MVT for ?

8. Ex 4: MVT
Given , find all values of c in the open interval (1,4) such that

9. Ex 2: Finding an Instantaneous Rate of Change
Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first car, its speed is clocked at 55 mph. Four minutes later, the truck passes the 2nd patrol car at 50 mph. Prove that the truck must have exceeded the speed limit (55 mph) at some time during the 4 minutes.

10. HOMEWORK
Pg 172 # , 31 – 38 odds,