1. Graphs and the Derivative Chapter 13

2. Ch. 13 Graphs and the Derivative 13.1 Increasing and Decreasing Functions 13.2 Relative Extrema 13.3 Higher Derivatives, Concavity, and the Second Derivative Test

3. 13.1 Increasing and Decreasing Functions The graph of a typical function may increase on some intervals and decrease on others. The derivative of a function f ’(x) describes whether the graph is increasing (rising from left to right) or decreasing (falling from left to right).  Where f ’(x) > 0, f(x) will increase.  Where f ’(x) < 0, f(x) will decrease.  Where f ’(x) = 0, f(x) is constant.

4. Increasing and Decreasing Functions f(x) f ’(x) < 0, f(x) Decreasing f ’(x) > 0, f(x) Increasing f ’(x) = 0, f(x) is constant

5. Increasing and Decreasing Functions CRITICAL NUMBERS The critical numbers for a function f are those numbers c in the domain of f for which f ’(c) = 0 or f ’(c) does not exist. CRITICAL POINT A critical point is a point whose x -coordinate is the critical number c, and whose y -coordinate is f(c).

6. Increasing and Decreasing Functions critical numbers c c f(c)f(c) f ’(c) = 0

7. Increasing and Decreasing Functions c f(c)f(c) critical points [ c, f ( c )]

8. Increasing and Decreasing Functions Test for increasing or decreasing functions 1. Determine the critical numbers  Calculate the derivative of f(x)  Set f ’(x) equal to 0 2. Locate the critical numbers on a number line to identify “intervals”. 3. Choose a value of x in each interval to determine if f ’(x) > 0 or < 0 in that interval. 4. Use the test above to determine if f is increasing or decreasing in that interval.

9. Increasing and Decreasing Functions Example: f(x) = x 3 + 3 x 2 – 9 x + 4 1. Determine the critical numbers  f ’(x) = 3 x 2 + 6 x – 9  3 x 2 + 6 x – 9 = 0  3( x + 3)( x – 1) = 0  x = -3 or x = 1 (critical numbers) 2. Locate the critical numbers on a number line to identify “intervals”.

10. 3 intervals, (- , -3), (-3, 1), (1,  ) f(x) = x 3 + 3 x 2 – 9 x + 4 f ’(x) = 3 x 2 + 6 x – 9

11. Increasing and Decreasing Functions Example: f(x) = x 3 + 3 x 2 – 9 x + 4 1. Determine the critical numbers  f ’(x) = 3 x 2 + 6 x – 9  3 x 2 + 6 x – 9 = 0 x = -3 or x = 1 2. Locate the critical numbers on a number line to identify “intervals”. 3. Choose a value of x in each interval to determine if f ’(x) > 0 or < 0 in that interval. 4. Use the test above to determine if f is increasing or decreasing in that interval.

12. f ’(-4) = 3(-4) 2 + 6(-4) – 9 = 15 f ’(x) > 0. f(x) is increasing f ’(0) = -9 f ’ (x) < 0. f(x) is decreasing f ’(2) = 15 f ’ (x) > 0. f(x) is increasing f(x) = x 3 + 3 x 2 – 9 x + 4 f ’(x) = 3 x 2 + 6 x – 9

13. f(x) = x 3 + 3 x 2 – 9 x + 4 f ’(x) = 3 x 2 + 6 x – 9

14. Now You Try A manufacturer sells video games with the following cost and revenue functions, where x is the number of games sold. Determine the interval(s) on which the profit function is increasing. Remember: P ( x ) = R ( x ) – C ( x )

15. Relative Extrema RELATIVE MAXIMUM OR MINIMUM Let c be a number in the domain of a function f. Then f(c) is a relative maximum for f if there exists an open interval ( a, b ) containing c such that f ( x )  f ( c ) for all x in ( a, b ) f(c) is a relative minimum for f if there exists an open interval ( a, b ) containing c such that f ( x )  f ( c ) for all x in ( a, b ) A function has a relative extremum at c if it has either a relative maximum or relative minimum there.

16. Relative Extrema If a function f has a relative extremum at c, then c is a critical number or c is an endpoint of the domain. relative maximum relative minimum Increasing Decreasing

17. Relative Extrema Critical point ( f ’(x) = 0), but NOT a relative extremum Increasing

18. Relative Extrema - FIRST DERIVATIVE TEST Let c be a critical number for a function f. Then, the critical point ( c, f(c) ) is  A relative maximum if f ’(x) > 0 to the left of c and f ’(x) < 0 to the right of c  A relative minimum if f ’(x) 0 to the right of c  Not a relative extrema if f ’(x) has the same sign on both sides of c.

19. Relative Extrema relative maximum relative minimum f ’(x) > 0 f ’(x) < 0

20. Relative Extrema Critical point ( f ’(x) = 0), but NOT a relative extremum f ’(x) > 0

22. Now You Try The total profit P(x) (in thousands of dollars) from the sale of x units of a certain prescription drug is given by P(x) = ln(- x 3 + 3 x 2 + 72 x + 1) for x in [0, 10]. a. Find the number of units that should be sold in order to maximize the total profit. b. What is the maximum profit?

23. Try Another The number of passenger cars imported into the United States (in millions) in year x can be approximated by f ( x ) = –.051 x 3 + 1.43 x 2 – 12.55 x + 39.66 (0  x  13), where x = 5 corresponds to 1995. Find all relative extrema of this function and interpret your answers.

24. 13.3 Higher Derivatives If a function f has a derivative f ’, then the derivative of f ’ is the second derivative of f, written f ’’. f ’’’ is the third derivative of f, and so on.

25. Higher Derivatives NOTATION FOR HIGHER DERIVATIVES The second derivative of y = f(x) can be written using any of the following notations: For n  4, the n th derivative is written f (n) (x). The second derivative f ’’(x) is the rate of change of the first derivative f ’(x).

26. Quantity of Labor Law of Diminishing Marginal Returns Quantity of Labor Short-Run Production Relationships P(L) P’(L) Total Product Marginal Product Total product is increasing, marginal product > 0

27. Law of Diminishing Marginal Returns Short-Run Production Relationships Quantity of Labor Total Product Marginal Product P(L) P’(L)

28. Law of Diminishing Marginal Returns Short-Run Production Relationships Quantity of Labor Total Product Marginal Product Increasing Marginal Returns P(L) P’(L)

29. Quantity of Labor Diminishing Marginal Returns Law of Diminishing Marginal Returns Quantity of Labor Short-Run Production Relationships P(L) P’(L) Total Product Marginal Product

30. P(L) P’(L) Quantity of Labor Negative Marginal Returns Law of Diminishing Marginal Returns Quantity of Labor Short-Run Production Relationships Total Product Marginal Product

31. P’(L) L Marginal Product Second Derivative

32. Now You Try Find f ’’(x). Then find f ’’(0) and f ’’(2).

33. Concavity of a Graph First derivative shows... a. where a function f is increasing or decreasing and, b. where the extrema occur. Second derivative gives the rate of change of the first derivative, or  how fast the function is increasing or decreasing Rate of change of the derivative ( f ’’(x) ) affects the shape of the graph.  Concave upward  Concave downward

34. P(L) P’(L) Quantity of Labor Law of Diminishing Marginal Returns Quantity of Labor Short-Run Production Relationships Total Product Marginal Product Concave upward Concave downward

35. P(L) P’(L) Quantity of Labor Law of Diminishing Marginal Returns Quantity of Labor Short-Run Production Relationships Total Product Marginal Product Concave upward Concave downward

36. Concavity of a Graph TEST FOR CONCAVITY Let f be a function with derivatives f ’ and f ’’ existing for all points in an interval ( a, b ). 1.If f ’’(x) > 0 for all x in ( a, b ), then f is concave upward on ( a, b ). 2.If f ’’(x) < 0 for all x in ( a, b ), then f is concave downward on ( a, b ). Point of Inflection: Point where a graph changes concavity ( f ’’(x) = 0 ).

37. P(L) P’(L) Quantity of Labor Law of Diminishing Marginal Returns Quantity of Labor Short-Run Production Relationships Total Product Marginal Product Concave upward Concave downward Point of Inflection m = 0

38. Concavity of a Graph TEST FOR CONCAVITY Let f be a function with derivatives f ’ and f ’’ existing for all points in an interval ( a, b ). 1.If f ’’(x) > 0 for all x in ( a, b ), then f is concave upward on ( a, b ). 2.If f ’’(x) < 0 for all x in ( a, b ), then f is concave downward on ( a, b ). Find all intervals where f(x) = x 4 – 8 x 3 + 18 x 2 is concave upward or downward, and find all inflection points. Point of Inflection: Point where a graph changes concavity ( f ’’(x) = 0 ).

39. Concavity of a Graph 3 intervals: (- , 1), (1, 3), (3,  ) Test each interval for concavity: f concave upward on (- , 1) f concave downward on (1, 3) f concave upward on (3,  )

40. Concavity of a Graph Inflection points at x = 1 and x = 3 Points of inflection: (1, 11) and (3, 27)

41. Concavity of a Graph Concave upward f ’’(x) > 0

42. Concavity of a Graph Concave downward f ’’(x) < 0

43. Concavity of a Graph Concave upward f ’’(x) > 0

44. Point of Diminishing Returns Law of Diminishing Returns – The principle that as additional units of a variable resource (labor) are added to a fixed resource (capital), the marginal product will eventually decrease. Point of Diminishing Returns – The point on the production function at which the marginal product begins to decrease as additional units of a variable resource are added to a fixed resource.  Occurs at the point of inflection where concavity changes from upward to downward.

45. P(L) P’(L) Quantity of Labor Law of Diminishing Marginal Returns Quantity of Labor Short-Run Production Relationships Total Product Marginal Product Point of Diminishing Marginal Returns

46. Point of Diminishing Returns - example An efficiency study of the morning shift (8:00 – noon) at a factory indicates that an average worker who starts at 8:00 A.M. will have produced Q(t) = -t 3 + 9 t 2 + 12 t units t hours later. At what time during the shift is the worker performing most efficiently (point of diminishing returns)? Solution: The worker’s rate of production is the derivative of Q(t) R(t) = Q’(t) = - 3 t 2 + 18 t + 12 The point of diminishing returns occurs at a point of inflection where Q’’(t) = 0 for 0  t  4 Q’’(t) = -6 t + 18

47. Point of Diminishing Returns - example Q’’(t) = -6 t + 18 = 0 when t = 3,  is positive for 0 < t < 3 (concave upward),  and negative for 3 < t < 4 (concave downward). Therefore, maximum efficiency occurs at t = 3, or at 11:00.

48. Point of Diminishing Returns - example Q(t) = -t 3 + 9 t 2 + 12 t

49. Point of Diminishing Returns - example Q(t) = -t 3 + 9 t 2 + 12 t Point of inflection (3.00, 90)

50. Point of Diminishing Returns - example Q’(t) = - 3 t 2 + 18 t + 12 Q’’(t) = 0

52. Concavity of a Graph SECOND DERIVATIVE TEST FOR RELATIVE EXTREMA Let f ’’ exist on some open interval containing c, and let f ‘(c) = 0 1. If f ‘’(c) > 0, then f has a relative minimum at c. f(c) is concave upward on a < c < b. c, f ( c )

53. Concavity of a Graph SECOND DERIVATIVE TEST FOR RELATIVE EXTREMA Let f ’’ exist on some open interval containing c, and let f ‘(c) = 0 1. If f ‘’(c) > 0, then f has a relative minimum at c. 2. If f ‘’(c) < 0, then f has a relative maximum at c. f(c) is concave downward on a < c < b. c, f ( c )

54. Concavity of a Graph SECOND DERIVATIVE TEST FOR RELATIVE EXTREMA Let f ’’ exist on some open interval containing c, and let f ‘(c) = 0 1. If f ‘’(c) > 0, then f has a relative minimum at c. 2. If f ‘’(c) < 0, then f has a relative maximum at c. 3. If f ‘’(c) = 0, then the test gives no information about extrema; use the first derivative test.

55. A Previous Exercise The number of passenger cars imported into the United States (in millions) in year x can be approximated by f ( x ) = –.051 x 3 + 1.43 x 2 – 12.55 x + 39.66 (0  x  13), where x = 5 corresponds to 1995. Find all relative extrema of this function and interpret your answers.

56. A Previous Exercise There is a relative minimum at x = 7.0 There is a relative maximum at x = 11.7

57. A Previous Exercise f ( x ) = –.051 x 3 + 1.43 x 2 – 12.55 x + 39.66

58. A Previous Exercise To determine points of inflection : Set f “( x ) equal to zero and solve for x Calculate f (9.3) Point of inflection : (9.3, 5.6)

59. A Previous Exercise f ( x ) = –.051 x 3 + 1.43 x 2 – 12.55 x + 39.66

60. Now You Try Find any critical numbers for f and then use the second derivative test to decide whether the critical numbers are relative maxima or relative minima.

61. Chapter 13 End 