1. Derivatives and Graphing

2. Increasing and Decreasing
Where a function is increasing...

3. Increasing and Decreasing
…tangent lines have positive slope.

4. Increasing and Decreasing
So the first derivative is positive.

5. Increasing and Decreasing
Where the function is decreasing...

6. Increasing and Decreasing
…tangent lines have negative slope.

7. Increasing and Decreasing
So the first derivative is negative.

8. Increasing and Decreasing
Where a function changes from increasing to decreasing…

9. Increasing and Decreasing
…or from decreasing to increasing…

10. Increasing and Decreasing
A function may have local maximum or minimum values.

11. Increasing and Decreasing
If the first derivative exists at these points, it is zero.

12. Increasing and Decreasing
Local max
Local min
Local extrema may also exist where the derivative is undefined.

13. Identify the open intervals on which the function is increasing or decreasing.

14. Identify the open intervals on which the function is increasing or decreasing.

15. Find the critical numbers, find the open intervals on which the function is increasing or decreasing and locate all relative extrema

16. Find the critical numbers, find the open intervals on which the function is increasing or decreasing and locate all relative extrema

17. Concavity inflection point inflection point concave down concave down
concave up

18. Concavity Where the curve is concave down...
...slopes of tangent lines are decreasing.

19. Concavity Since the first derivative is decreasing...
concave down
…the second derivative is negative.

20. Concavity Where the curve is concave up...
concave down
concave up
...slopes of tangent lines are increasing.

21. Concavity Since the first derivative is increasing...
concave down
concave up
…the second derivative is positive.

22. Concavity
concave down
At the inflection points, the second derivative (if it exists) is 0.
concave down
concave up

23. Concavity
concave down
concave down
concave up

24. Summary

25. Summary

26. Identify the open intervals on which the function is concave upward and the intervals where the function is concave down.

27. Identify the open intervals on which the function is concave upward and the intervals where the function is concave down.

28. Find all relative extrema.

29. Find all relative extrema.