1. Review Derivatives When you see the words… This is what you know…  f has a local (relative) minimum at x = a  f(a) is less than or equal to every other y-value in some interval containing x = a 1

2. Review Derivatives When you see the words… This is what you know…  The difference quotient for f at x = a .. 2

3. Review Derivatives When you see the words… This is what you know…  Definition of derivative f’(x) = .. 3

4. Review Derivatives When you see the words… This is what you know…  If f is differentiable at x = a  Then f is continuous at x = a 4

5. Review Derivatives When you see the words… This is what you know…  Rolle’s Theorem  If f is continuous on [a,b],  differentiable on (a,b)  and f(a) = f(b),  then for some c on (a,b),  f’( c) = 0 5

6. Review Derivatives When you see the words… This is what you know…  Mean Value Theorem  If f is continuous on [a,b],  differentiable on (a,b)  then for some c on (a,b) 6

7. Review Derivatives When you see the words… This is what you know…  Mean Value Theorem (as rates of change)  If f is continuous on [a,b],  differentiable on (a,b)  then for some point in the interval  the instantaneous rate of change = average rate of change 7

8. Review Derivatives When you see the words… This is what you know…  If f is differentiable at x = a, then …… 8

9. Review Derivatives When you see the words… This is what you know… ..  k * f’ 9

10. Review Derivatives When you see the words… This is what you know… ..  f’ g’ 10

11. Review Derivatives When you see the words… This is what you know… ..  f * g’ + g * f’ 11

12. Review Derivatives When you see the words… This is what you know… .. 12

13. Review Derivatives When you see the words… This is what you know… .. 13

14. Review Derivatives When you see the words… This is what you know… .. 14

15. Review Derivatives When you see the words… This is what you know… .. 15

16. Review Derivatives When you see the words… This is what you know… .. 16

17. Review Derivatives When you see the words… This is what you know… .. 17

18. Review Derivatives When you see the words… This is what you know… .. 18

19. Review Derivatives When you see the words… This is what you know… .. 19

20. Review Derivatives When you see the words… This is what you know…  Average rate of change of f on [a,b] .. 20

21. Review Derivatives When you see the words… This is what you know…  f’(a) = 0  The graph of f has a horizontal tangent at x=a 21

22. Review Derivatives When you see the words… This is what you know…  Instantaneous rate of change of f at x = a  f’(a) 22

23. Review Derivatives When you see the words… This is what you know…  Critical points of f  At endpoints of the domain  Where f’(x) does not exist  Where f’(x) = 0 23

24. Review Derivatives When you see the words… This is what you know…  f’(x) > 0 for a < x < b  f is increasing on the interval a < x < b 24

25. Review Derivatives When you see the words… This is what you know…  f’(x) < 0 for a < x < b  f is decreasing on the interval a < x < b 25

26. Review Derivatives When you see the words… This is what you know…  f” (x) > 0 for a < x < b  The graph of f is concave upward on the interval a < x < b 26

27. Review Derivatives When you see the words… This is what you know…  f” (x) < 0 for a < x < b  The graph of f is concave downward on the interval a < x < b 27

28. Review Derivatives When you see the words… This is what you know…  f’ (x) is increasing for a < x < b  The graph of f is concave upward on the interval a < x < b 28

29. Review Derivatives When you see the words… This is what you know…  f’ (x) is decreasing for a < x < b  The graph of f is concave downward on the interval a < x < b 29

30. Review Derivatives When you see the words… This is what you know…  The second derivative test f’(a) = 0 and f”(a) < 0  f has a maximum at x = a 30

31. Review Derivatives When you see the words… This is what you know…  The second derivative test f’(a) = 0 and f”(a) > 0  f has a minimum at x = a 31

32. Review Derivatives When you see the words… This is what you know…  The graph of f changes concavity  The graph of f has a point of inflection 32

33. Review Derivatives When you see the words… This is what you know…  At x = a, f is continuous and changes from increasing to decreasing  f has a maximum at x = a 33

34. Review Derivatives When you see the words… This is what you know…  At x = a, f is continuous and changes from decreasing to increasing  f has a minimum at x = a 34

35. Review Derivatives When you see the words… This is what you know…  Velocity, speed, and acceleration  Velocity:  Speed: | v(t) |  Accleration: 35

36. Review Derivatives When you see the words… This is what you know…  Increasing speed  Velocity and acceleration have the same sign 36

37. Review Derivatives When you see the words… This is what you know…  Decreasing speed  Velocity and acceleration have the opposite signs 37

38. Review Derivatives When you see the words… This is what you know…  The normal line to a curve at x = a is  Perpendicular to the line tangent to the curve at x = a 38

39. Review Derivatives When you see the words… This is what you know…  An object in motion along a line reverses direction when…  The sign of the object’s velocity changes 39

40. Review Derivatives When you see the words… This is what you know…  An object is at rest when…  v(t) = 0 40

41. Review Derivatives When you see the words… This is what you know…  f has a global (absolute) maximum at x = a  f(a) is greater than or equal to every other y-value of f 41

42. Review Derivatives When you see the words… This is what you know…  f has a global (absolute) minimum at x = a  f(a) is less than or equal to every other y-value of f 42

43. Review Derivatives When you see the words… This is what you know…  f has a local (relative) maximum at x = a  f(a) is greater than or equal to every other y-value in some interval containing x = a 43

44. Review Derivatives When you see the words… This is what you know… ..  Cos x 44

45. Review Derivatives When you see the words… This is what you know… ..  -sin x 45