1. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 5 illustrates the relationship between the concavity of the graph of y = cos x and the graph of its second derivative.

2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The table on the next slide shows how to check if the points where f ″ = 0 are points of inflection.

3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The information from the table above would normally be displayed in a sign chart, which is constructed as follows:

4. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Just as a critical point occurs where f ′ does not exist, so a point of inflection may exist at a point where f ″ does not exist, as illustrated in Figure 7.

5. 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

6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company There is a distinct relationship between the graphs of the parent function, its first derivative, and its second derivative. This relationship may be seen in Figure 8.

7. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 9, the sign of the second derivative at a critical point tells us whether the critical point is a local minimum or a local maximum.

8. Example, Page 243 20. Through her website, Leticia has been selling bean bag chairs with monthly sales as recorded below. In a report to her investors, she states, “Sales reached a point of inflection when I started using pay-per-click advertising.” In which month did that occur? Explain. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Month12345678 Sales220303538446090

9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Our observation in Figure 9 is formalized in Theorem 3. Theorem 3 gives us a second means of justifying that a minimum or maximum occurs at a critical point.

10. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 10 illustrates using the Second Derivative Test to find a local maximum.

11. Example, Page 243 Find the critical points of f (x) and use the Second Derivative Test to determine whether each corresponds to a local minimum or maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Example, Page 243 Find the critical points of f (x) and use the Second Derivative Test to determine whether each corresponds to a local minimum or maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Use the second derivative test to find the local extrema of f (x) = x 5 – 5x 4.

14. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The signs of the first and second derivative provide us specific information about the behavior of the parent function that is summarized in the following table:

15. Example, Page 243 52. Water is pumped into a sphere at a constant rate. Let h (t) be the water level at time t. Sketch the graph of h (t). Where does the point of inflection occur? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Example, Page 243 Sketch the graph of a function satisfying the given condition. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Homework Homework Assignment #25 Read Section 4.5 Page 243, Exercises: 1 – 57 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company