Math 180

1. Math 180 4.3 – Monotonic Functions and the First Derivative Test

2. Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.)

3. Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.)

4. Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.) increasing

5. Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.) increasing

6. Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.) increasing decreasing

7. Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.) increasing decreasing

8. If is increasing or decreasing on an interval, it is said to be ___________on that interval.

9. If is increasing or decreasing on an interval, it is said to be ___________on that interval. monotonic

10. Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When is undefined

11. Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When is undefined In other words, where might change signs?

12. Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When is undefined In other words, where might change signs?

13. Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When is undefined In other words, where might change signs?

14. 1. When

15. 1. When

16. 1. When

17. 1. When

18. 1. When

19. 2. When is undefined

20. Ex 1. Identify the intervals of increase and decrease for

21. Ex 1. Identify the intervals of increase and decrease for

22. Critical points (where or undefined) give us a list of -values where might have a local max or local min. The First Derivative Test for Local Extrema tells us that if the sign of changes across a critical point, then we’ve got a local max or a local min.

23. Critical points (where or undefined) give us a list of -values where might have a local max or local min. The First Derivative Test for Local Extrema tells us that if the sign of changes across a critical point, then we’ve got a local max or a local min.

24. First Derivative Test for Local Extrema Let be a critical point. Local max at

25. First Derivative Test for Local Extrema Let be a critical point. Local max at Local min at

26. First Derivative Test for Local Extrema Let be a critical point. Not a local extremum

27. First Derivative Test for Local Extrema Let be a critical point. Not a local extremum Not a local extremum

28. Ex 2. Find all critical points for . Then find all local maxima and minima.

29. Ex 2. Find all critical points for . Then find all local maxima and minima.

30. Ex 3. Find the critical points of . Identify the intervals on which is increasing and decreasing. Find the function’s local and absolute extreme values.

31. Ex 3. Find the critical points of . Identify the intervals on which is increasing and decreasing. Find the function’s local and absolute extreme values.