2. 4.2 Mean Value Theorem

4. Quick Review

5. What you’ll learn about Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences Essential Questions How can we use the Mean Value Theorem as a tool to connect the average and instantaneous rates of Change?

6. Mean Value Theorem for Derivatives

7. Example Explore the Mean Value Theorem 1. The function f (x) = x 2 is continuous on [0, 2] and differentiable on (0, 2).

8. Example Explore the Mean Value Theorem At some point her instantaneous speed was 79.5 mph. 2.A trucker handed in a ticket at a toll booth showing that in two hours she covered 159 mi on a toll road with a speed limit of 65 mph. The trucker was cited for speeding. Why?

9. Increasing Function, Decreasing Function

10. Corollary: Increasing and Decreasing Functions

11. 3.Where is the function f (x) = 2x 3 – 12x increasing and where is it decreasing? The function is increasing when f ‘ (x) > 0. The function is decreasing when f ‘ (x) < 0. Example Determining Where Graphs Rise or Fall

12. 4.Use analytic methods to find (a) the local extrema, (b) the intervals on which the function is increasing, (c) the intervals on which the function is decreasing. The function is increasing when f ‘ (x) > 0. The function is decreasing when f ‘ (x) < 0. Maximum at x = – 2.Minimum at x = 2. l

13. Corollary: Functions with f’ = 0 are Constant Corollary: Functions with the Same Derivative Differ by a Constant Antiderivative

14. Example Finding Velocity and Position 5.Find the velocity and position functions of a freely falling body for the following set of conditions: The acceleration is 9.8 m/sec 2 and the body falls from rest. Assume that the body is released at time t = 0. Velocity: We know that a(t) = 9.8 and v(0) = 0, so Position: We know that v(t) = 9.8t and s(0) = 0, so

15. Pg. 202, 4.2 #1-37 odd

16. Example Explore the Mean Value Theorem 12. It took 20 sec for the temperature to rise from 0 o F to 212 o F when a thermometer was taken from a freezer and placed in boiling water. Explain why at some moment in that interval the mercury was rising at exactly 10.6 o F/sec.

17. Example Determining Where Graphs Rise or Fall 20. Use analytic methods to find (a) the local extrema, (b) the intervals on which the function is increasing, (c) the intervals on which the function is decreasing.

18. Example Determining Where Graphs Rise or Fall 22. Use analytic methods to find (a) the local extrema, (b) the intervals on which the function is increasing, (c) the intervals on which the function is decreasing.

19. Example Determining Where Graphs Rise or Fall 24. Use analytic methods to find (a) the local extrema, (b) the intervals on which the function is increasing, (c) the intervals on which the function is decreasing.

20. Example Determining Where Graphs Rise or Fall 34. Find all possible functions f with the given derivative.

21. Example Determining Where Graphs Rise or Fall 36. Find the function with the given derivative whose graph passes through the given point P.