1. Calculus Warm Up Find derivative 1. ƒ(x) = x3 – (3/2)x2
2. ƒ(x) = 4x3 – 2x2 – 10x

2. Calculus Objective
STW find when a function is increasing and decreasing.

3. Critical Numbers
Find the derivative and set it equal to zero.

4. 1st Derivative Test Take the derivative Find all critical numbers
Make a table
Plug into derivative
Decide if increasing or decreasing

5. INCREASING
If the first derivative is positive (this means the y values of the derivative are positive)

6. Decreasing
If the first derivative is negative (this means the y values of the derivative are negative)

7. ƒ(x) = x3 – (3/2)x2
Critical numbers x = 0, 1

8. ƒ(x) = x3 – (3/2)x2
Critical numbers x = 0, 1

9. ƒ(x) = x3 – (3/2)x2
Critical numbers x = 0, 1

10. Calculus 1. ƒ(x) = x4 – (1/2)x3 2. ƒ(x) = 5x3 –x2 –35
Apply the 1st derivative test
1. ƒ(x) = x4 – (1/2)x3
2. ƒ(x) = 5x3 –x2 –35

11. ƒ(x) = x3 – (3/2)x2
Critical numbers x = 0, 1

12. ƒ(x) =

13. ƒ(x) =

14. ƒ(x) =

15. ƒ(x) =

16. ƒ(x) =

17. Find the second derivative
1. ƒ(x) = 2x4 + 5x2 – 12 2.

18. Soup or No Soup

19. use the first and second derivative to graph a function.
Objective: S.W.B.A.T.
use the first and second derivative to graph a function.

20. First Derivative Test What are critical numbers?
What does the 1st derivative test tell us?
If the derivative of a critical number is 0, what do you know about the graph of the function?
If the derivative of a critical number does not exist, what do you know about the graph of the function?

21. Def: Concavity Informal def – when the graph curves up or down.
Let f be a differentiable on an open interval. We say that the graph of f is concave up if f” is increasing on the interval and concave down if f” is decreasing on the interval.

22. Graphically: Concavity

23. Graphically: Concavity

24. Graphically: Concavity
The graph of f lies above
the tangent lines!

25. Graphically: Concavity
Will hold soup!
The graph of f lies above
the tangent lines!

26. Graphically: Concavity

27. Graphically: Concavity

28. Graphically: Concavity
The graph of f lies below
the tangent lines!

29. Graphically: Concavity
Won’t hold soup!
The graph of f lies below
the tangent lines!

30. Graphically: Concavity
Won’t hold soup!
The graph of f lies below
the tangent lines!

31. Graphically: Concavity
Won’t hold soup!
The graph of f lies below
the tangent lines!

32. Graphically: Concavity
Won’t hold soup!
The graph of f lies below
the tangent lines!

33. Graphically: Concavity
Won’t hold soup!
The graph of f lies below
the tangent lines!

34. Graphically: Concavity
Won’t hold soup!
The graph of f lies below
the tangent lines!

35. Graphically: Concavity
Won’t hold soup!
The graph of f lies below
the tangent lines!

36. Graphically: Concavity
The graph of f lies below
the tangent lines!

37. Test for Concavity
Let f i be a function whose second derivative exists on an open interval I.
1. If f’’(x) > 0, for all x in I, then the graph of f is concave upward.
2. If f’’(x) < 0, for all x in I, then the graph of f is concave downward.

38. Second Derivative Test
Take the 1st derivative
Find the critical Numbers
Take the second Derivative

39. Second Derivative Test
Set the sec. derivative = 0
These points are also critical numbers
Make Chart

40. Second Derivative Test
Pick Test Values
Test the 1st and 2nd derivative
The sign of the 1st derivative tells you Inc/Dec

41. Second Derivative Test
The sign of the 2nd derivative tells you concavity.
Holds Soup +
Doesn’t Hold Soup –

42. Defn: Point of Inflection
Let f be a function whose graph has a tangent line at (c, f(c)). The point (c, f(c)) called a point of inflection if the concavity of f changes from upward to downward or vice-visi.

43. Point of Inflection
If (c, f(c)) is a point of inflection of the graph of of f , then f ’’(c)= f ’’(c) is undefined at x = c.

44. Second Derivative Test
Determine the open intervals on which the graph of f is concave upward and downward.

45. use limits of infinity to find horizontal asymptotes.
Objective: S.W.B.A.T.
use limits of infinity to find horizontal asymptotes.

46. Limit Refresher
Find the limit.

47. Limit Refresher
Find the limit

48. Limit Refresher
Find the limit

49. Defn - Horizontal Asymptote
The line y = L is a horizontal asymptote of the graph of f if

50. Limits at Infinity
If r is a positive rational number and c is any real number, then
If xr is defined when x < 0, then

51. Limits at Infinity
Evaluate:

52. Limits of Infinity
Find the limit.

53. Divide every term by the highest power of x in the denominator.
Limits of Infinity
Start with a function
Divide every term by the highest power of x in the denominator.
Evaluate the limit.

54. Limits of Infinity
Find the following limits.

55. Limits of Infinity
Find each limit.

56. Triggee Limits of Infinity
Find the following limits.

57. Determine the open intervals on which the graph of f is concave upward and downward.

60. P181 #7-21 odd Listen for instructions!
Assignment #??
P181 #7-21 odd Listen for instructions!