MAT 1221 Survey of Calculus

1. MAT 1221Survey of Calculus Section 3.3 Concavity and the Second Derivative Test http://myhome.spu.edu/lauw

2. Expectations • Check your algebra. • Check your calculator works • Formally answer the question with the expected information

3. 1 Minute… • You can learn all the important concepts in 1 minute.

4. 1 Minute… • Critical numbers – give the potential local max/mins

5. 1 Minute… • Critical numbers – give the potential local max/mins • If the graph is“concave down”at a critical number, it has a local max

6. 1 Minute… • Critical numbers – give the potential local max/mins • If the graph is“concave up”at a critical number, it has a local min

7. 1 Minute… • You can learn all the important concepts in 1 minute. • We are going to develop the theory carefully so that it works for all the functions that we are interested in. • There are a few definitions…

8. New - 2014 • I will try to streamline this a bit and see if it can go better than covering all the details.

9. Preview • Define • Second Derivatives • Concavities • Find the intervals of concave up and concave down • The Second Derivative Test

10. Second Derivative

11. Second Derivative

12. Higher Derivatives Given a function which is a function.

13. Higher Derivatives Given a function

14. Concave Up (a) A function is called concave upward on an interval if the graph of lies above all of its tangents on . (b) A function is calledconcave downward on an interval if the graph of lies below all of its tangents on .

15. Concavity is concave up on • Potential local min.

16. Concavity is concave down on • Potential local max.

17. Concavity y Concave down Concave up x c has no local max. or min. has an inflection point at

18. Definition • An inflection point is a point where the concavity changes (from up to down or from down to up)

19. Concavity Test (a) If on an interval , then is concave upward on . (b) If on an interval , then f is concave downward on .

20. Concavity Test (a) If on an interval , then is concave upward on . (b) If on an interval , then f is concave downward on . Why? (Hint: )

21. Why? implies is increasing. i.e. the slope of tangent lines is increasing.

22. Why? implies is decreasing. i.e. the slope of tangent lines is decreasing.

23. Example 1 Find the intervals of concavity and the inflection points

24. Example 1 1. Find , and the values of such that

25. Example 1 2. Sketch a diagram of the subintervals formed by the values found in step 1. Make sure you label the subintervals.

26. Example 1 3. Find the intervals of concavity and inflection point.

27. Example 1

28. The Second Derivative Test Suppose is continuous near . (a) If and , then has a local minimum at c. (b) If and , then f has a local maximum at . (c) If , then no conclusion (use 1st derivative test)

29. Second Derivative Test Suppose If then has a local min at y x c

30. Second Derivative Test Suppose If then has a local max at y x c

31. The Second Derivative Test (c) If , then no conclusion

32. The Second Derivative Test If , then no conclusion

33. The Second Derivative Test If , then no conclusion

34. The Second Derivative Test If , then no conclusion

35. The Second Derivative Test Suppose is continuous near . (a) If and , then has a local minimum at c. (b) If and , then f has a local maximum at . (c) If , then no conclusion (use 1st derivative test)

36. Example 2 Use the second derivative test to find the local max. and local min.

37. Example 2 (a) Find the critical numbers of

38. Example 2 (b) Use the Second Derivative Test to find the local max/min of • The local max. value of f is • The local min. value of f is

39. Review • Example 1 & 2 illustrate two different but related problems. • 1. Find the intervals of concavity and inflection points. • 2. Find the local max. /min. values

40. Expectations • Follow the steps to solve the two problems