2. Find all critical numbers for the function: f(x) = (9 - x 2 ) 3/5. -3, 0, 3

3. Find all critical numbers for the function:

4. Find all critical numbers for the function: f(x) = (x + 2) 3 (x - 1) 4 -2, 1,

5. Find all extrema in the interval [0, 2π] for y = x + sin x. Minimum: (0, 0) Maximum: (2π, 2π)

6. Find the absolute maximum and absolute minimum of f on the interval (0, 3]. Maximum: None Minimum: (2, 3)

7. on the interval [ - 1, 2]. Find the absolute minimum and the absolute maximum on the interval [-1, 2]. Maximum at (0, 10); Minimum at (2, 2)

8. Find the minimum and maximum values of f(x) = x 2 - 2x + 1 on the interval [0, 3]. Minimum at (1, 0); Maximum at (3, 4)

9. Find the value of the derivative (if it exists) at the indicated extremum. 0

10. Determine from the graph whether f possesses extrema on the interval (a, b). Maximum at x = c no minimum

11. Explain why has a minimum on the interval [1, 2] but not on the interval [-1, 1]. is continuous on the closed interval [1, 2] and, therefore, the Extreme Value Theorem guarantees the existence of extrema on that interval. However, f(x) is not continuous on the interval [-1, 1] and

12. Decide whether Rolle’s Theorem can be applied to f(x) = sin x on the interval If Rolle’s Theorem can be applied, find all value(s), c, in the interval such that If Rolle’s Theorem cannot be applied, state why. Rolle’s Theorem applies;

13. Given find all c in the interval (2, 8) such that 4

14. Determine whether the Mean Value Theorem applies to f(x) = 3x - x 2 on the interval [2, 3]. If the Mean Value Theorem can be applied, find all value(s) of c in the interval such that If the Mean Value Theorem does not apply, state why. The Mean Value Theorem applies;

15. Determine whether the Mean Value Theorem applies to on the interval. If the Mean Value Theorem applies, find all value(s) of c in the interval such that If the Mean Value Theorem does not apply, state why. Mean Value Theorem applies;

16. Consider Find all value(s), c, in the interval [0, 1] such that the slope of the tangent line to the graph of f at c is parallel to the secant line through the points (0, f(0)) and (1, f(1)).

17. State why the Mean Value Theorem does not apply to the function on the interval [-3, 0]. f is not continuous at x = -1.

18. Find all open intervals on which is decreasing (- , -2), (-2, 1), and (1,  )

19. Find the open intervals on which f(x) = x 3 - 3x 2 is increasing or decreasing. Increasing (- , 0) and (2,  ); decreasing (0, 2)

20. Use the graph to identify the open intervals on which the function is increasing or decreasing Increasing (- , 0) and (2,  ); decreasing (0, 2)

21. Find the values of x that give relative extrema for the function f(x) = 3x 5 - 5x 3 Relative maximum: x = -1; Relative minimum: x = 1

22. Find all relative extrema of relative maximum

23. Use the first derivative test to investigate f(x) = x 3 + x for relative extrema. f has no relative extrema

24. Use the first derivative test to find the x-values that give relative extrema for f(x) = -x 4 + 2x 3. Relative maximum at

25. Show that f has no critical numbers. ≠ 0 for all x ≠ 1

26. A differentiable function f has only one critical number: x = -3. Identify the relative extrema of f at (-3, f(-3)) if and Relative maximum

27. Find all intervals on which the graph of the function is concave upward: (- , 0) and (0,  )

28. Find all intervals for which the graph of the function y = 8x 3 - 2x 4 is concave downward (- , 0) and (2,  )

29. Find the intervals on which the graph of the function f(x) = x 4 - 4x 3 + 2 is concave upward or downward. Then find all points of inflection for the function. Concave upward: (- , 0), (2,  ) Concave downward: (0, 2) Points of inflection: (0, 2) and (2, -14)

30. Find all points of inflection of the graph of the function

31. Find all points of inflection of the graph of the function f(x) = 2x(x - 4) 3. (4, 0), (2, -32)

32. Let and let f(x) have critical numbers -2, 0, and 2. Use the Second Derivative Test to determine which critical numbers, if any, give a relative maximum. 0

33. Let f(x) = x 3 - x 2 + 3. Use the Second Derivative Test to determine which critical numbers, if any, give relative extrema. x = 0, relative maximum; relative minimum

34. Let f(x) be a polynomial function such that f(4) = -1, If x 4, then The point (4, -1) is a ___________ of the graph of f. [A] Critical number [B] Relative minimum [C] Point of inflection [D] Relative maximum [E] None of these

35. Give the sign of the second derivative of f at the indicated point. Positive

36. The graph of a polynomial function, f, is given. On the same coordinate axes sketch f’ and f’’

37. Find the horizontal asymptote fory = 0

38. Find the horizontal asymptote for y = 3

39. Find the horizontal asymptote for y = 2

40. Find the horizontal asymptote for y = ±5

41. Find the horizontal asymptote for y = ±6

42. Find the horizontal asymptote for y = -1

43. Find the horizontal asymptote for y = 0

44. Which of the following is the correct sketch of the graph of the function

45. Which of the following is the correct sketch of the graph of the function y = x 3 – 12x + 20

46. Which of the following is the correct sketch of the graph of the function

48. Sketch the graph of y = x 3 - 3x + 1

49. Sketch the graph of

50. Sketch the graph of f(x) = x sin 2x

51. Sketch the graph of f(x) = x 5 + 2

52. Sketch the graph of f(x) = 2x 4 - 8x 2

53. The product of two positive numbers is 675. Minimize the sum of the first and three times the second. 45 and 15

54. A rancher has 300 feet of fencing to enclose a pasture bordered on one side by a river. The river side of the pasture needs no fence. Find the dimensions of the pasture that will produce a pasture with a maximum area 75 feet by 150 feet

55. Find two positive numbers whose product is a maximum if the sum of the numbers is 10. 5, 5

56. The management of a large store wishes to add a fenced-in rectangular storage yard of 20,000 square feet, using the building as one side of the yard. Find the minimum amount of fencing that must be used to enclose the remaining 3 sides of the yard. 400 ft

57. A person has 400 feet of fencing to enclose two adjacent rectangular regions of the same size. What dimensions should each region be so that the enclosed area will be a maximum? ft. by 50 ft.

58. A dog owner has 25 feet of fencing to use for a rectangular dog run. If he puts the run next to the garage, he only needs to use the wire for three sides. What dimensions will yield the maximum area? x = 6.25 ft, y = 12.5 ft

59. An open box is to be made from a rectangular piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box of maximum volume if the material has dimensions 6 inches by 6 inches 4 in. by 4 in. by 1 in

60. An open box is to be made from a rectangular piece of cardboard, 7 inches by 3 inches, by cutting equal squares from each corner and turning up the sides. a. Write the volume, V, as a function of the edge of the square, x, cut from each corner. b. Use a graphing utility to graph the function, V. Then use the graph of the function to estimate the size of the square that should be cut from each corner and the volume of the largest such box. V = x(7 - 2x)(3 - 2x) b. 0.65 in. by 0.65 in.; V = 6.3 cubic inches

61. Find the point on the graph of y = x 3 closest to the point (2, 0). Find the x-value accurate to the nearest 0.1. (0.8, 0.512)

62. A manufacturer determines that x employees on a certain production line will produce y units per month where y = 75x 2 - 0.2x 4. To obtain maximum monthly production, how many employees should be assigned to the production line? 14

63. Use differentials to approximate1.850

64. Find dy for dx

65. The measurement of the edge of a piece of square floor tile is found to be 12 inches with a possible error of 0.02 inches. a. Use differentials to approximate the maximum possible error in the area of the tile. b. Use the answer from part a to estimate the relative error. c. Use the answer from part b to estimate the percentage error. a. 0.48 square inches b. c.