MAT 1234 Calculus I

1. MAT 1234Calculus I Section 3.3 How Derivatives Affect the Shape of a Graph (II) http://myhome.spu.edu/lauw

2. Next • Wednesday Quiz: 3.3,3.4 • Exam II: Next Thursday

3. Preview • We know the critical numbers give the potential local max/min. • How to determine which one is local max/min?

4. Preview • We know the critical numbers give the potential local max/min. • How to determine which one is local max/min? • 30 second summary!

5. Preview • We know the critical numbers give the potential local max/min. • How to determine which one is local max/min? • 30 second summary! • We are going to develop the theory carefully so that it works for all the functions that we are interested in.

6. Preview Part I • Increasing/Decreasing Test • The First Derivative Test Part II • Concavity Test • The Second Derivative Test

7. Definition (a) A function f is called concave upward on an interval I if the graph of f lies above all of its tangents on I. (b) A function f is calledconcave downward on an interval I if the graph of f lies below all of its tangents on I.

8. Concavity f is concave up on I • Potential local min.

9. Concavity f is concave down on I • Potential local max.

10. Concavity Concave down Concave up c f has no local max. or min. f has an inflection point at x=c

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

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

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

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

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

16. Example 3 Find the intervals of concavity and the inflection points

17. Example 3 (a) Find , and the values of such that

18. Example 3 (b) Sketch a diagram of the subintervals formed by the values found in part (a). Make sure you label the subintervals.

19. Example 3 (c) Find the intervals of concavity and inflection point(s). f( )= f has an inflection point at ( , )

20. Expectation • Answer in full sentence • The inflection point should be given by the notation

21. Example 3

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

23. Second Derivative Test Suppose If then f has a local min. at x=c f”(c)>0 f’(c)=0 c

24. Second Derivative Test Suppose If then f has a local max. at x=c f”(c)<0 f’(c)=0 c

25. Example 4 (Example 2 Revisit) Use the second derivative test to find the local max. and local min.

26. Example 4 (Example 2 Revisit) (a) Find the critical numbers of

27. Example 4 (Example 2 Revisit) (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

28. The Second Derivative Test (c) If f’’(c)=0, then no conclusion

29. The Second Derivative Test (c) If f’’(c)=0, then no conclusion

30. The Second Derivative Test (c) If f’’(c)=0, then no conclusion

31. The Second Derivative Test (c) If f’’(c)=0, then no conclusion

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

33. Which Test is Easier? • First Derivative Test • Second Derivative Test

34. Classwork • Do part (a), (d) and (e) only