1. Homework Homework Assignment #24 Read Section 4.4 Page 236, Exercises: 1 – 61 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Example, Page236 Find a point c satisfying the conclusion of the MVT for the given function and the interval. 1. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Example, Page236 Find a point c satisfying the conclusion of the MVT for the given function and the interval. 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Example, Page236 Find a point c satisfying the conclusion of the MVT for the given function and the interval. 9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Example, Page236 13.Determine the intervals on which f (x) is increasing or decreasing, assuming that Figure 12 is the graph of the derivative, f ′(x). The function f (x) is increasing on [0, 2] and [4, 6] since f ′(x) is positive on these intervals and decreasing on [2,4] since f ′(x) is negative on this interval. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Example, Page236 Sketch the graph of the function f (x) whose derivative, f ′(x), has the given description. 17. f ′(x) is negative on (1, 3) and positive everywhere else Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Example, Page236 Use the First Derivative Test to determine whether the function attains a local maximum or minimum at the critical point. 21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point. 25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point. 29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point. 33. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point. 37. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point. 41. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point. 45. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point. 49. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Example, Page236 53. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Example, Page236 57.Sam made two statements that Deborah found dubious. (a) “Although the average velocity for my trip was 70 mph, at no point in time did my speedometer read 70 mph.” (b)“Although a policeman clocked me at 70 mph, my speed- meter never read 65 mph.” In each case, which theorem did Deborah apply to prove Sam’s statement false: the Intermediate Value Theorem or the Mean Value Theorem. Explain. (a)By the MVT, if the average rate of change was 70 mph, at sometime the instantaneous rate of change must also be 70 mph. (b)By the IVT, if the speed varied between 0 and 70 mph, at some point in time it must have been 65 mph. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Example, Page236 61. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

18. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 4: Applications of the Derivative Section 4.4: The Shape of a Graph Jon Rogawski Calculus, ET First Edition

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Concavity is used to describe the manner in which a graph is curving as we proceed from left to right. Observe in Figure 1, that the slope of the segment of the graph is decreasing as we go from left to right on the concave down segments. Similarly, the slope is increasing as we go from left to right on the concave up segments.

20. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 2 offers another way to look at concavity. If the curve is above or up from the tangent line, the curve is concave up. Similarly, if the curve is below or down from the tangent line, the curve is concave down.

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A more formal definition of concavity is given below.

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 3 shows the increase in stock price of two companies over the same time interval. Both companies’ stock currently sells for $75. If the value (stock price) trends continue, which is the better investment?

23. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As noted in Theorem 1, the sign of the second derivative on an interval indicates the concavity of the graph on that interval.

24. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 4, a point of inflection is the point on a curve where the concavity changes from concave up to concave down or concave down to concave up. The second derivative equals zero at a point of inflection. This is formalized in Theorem 2.

25. Example, Page 243 Determine the intervals on which the function is concave up or down and find the points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company