1. AP Exam Review Competition First to finish calls “15 seconds”.First to finish calls “15 seconds”. Answers must be written down & then revealed.Answers must be written down & then revealed. Add 1 pt to your score for each correct response.Add 1 pt to your score for each correct response. Come back later for review & practice!Come back later for review & practice!

2. 1.Find the limit. Recall key step: divide all terms by the highest power x 3 Answer: *green terms  0

3. 2.Find the derivative. Answer:

4. 3. Evaluate: Answer:

5. 4. Fill in the blanks: Since polynomial functions are continuous over the reals and for f(x) = x 3 -1, we know f(0) = -1 and f(2) = 7, there exists a value c in the interval ________such that f(c) = 5 by the ___________________ Theorem. Answer: (0, 2) Intermediate Value

6. 5. Given f(x) and g(x) are diff. over R and g(x) = f -1 (x). If f(5)=7, f ’(5)=2, f(9)=5 and f ’(9)=6, find g’(5). Answer: 1/6 Slopes on inverses are reciprocals at corresponding pts. Since (9,5) is on f, then (5, 9) is on g... so we simply take the reciprocal of f’(9) to get g’(5).

7. 6. Evaluate: Answer: Fundamental Thrm of Calculus 7. Evaluate f ’ (x): 8. Name the theorem used in problem 7 above. Answer: Answer:

8. 9. Evaluate: Answer: 9 (Two triangles with b = 3 and h = 3.) (2,0) (5,3) (-1,3)

9. 10. Differentiate with respect to t (time): PV = c where c is a constant Answer:

10. 11. For s(t) = t 2 + 1, what is the average velocity over the time interval (0,4) seconds if distance is given in ft? 12. For s(t) = t 2 + 1 above, what is the instantaneous velocity at t = 4? Answer: Answer:

11. 13. Evaluate. (You must have both correct!) Answer: tan -1 x + C and sin -1 x + C

12. 14. Find the derivative. Recall key step: apply the quotient rule apply the quotient rule Answer:

13. 15. Given the graph of f ‘ (x) shown, give the x-coordinate(s) where f(x) has local minima. Answer: 0 and 3 0 and 3 (where slopes change from neg to pos) 24 3 2 1 -2 -3 f ' (x) 0 3 2 1 (a,b) (c,d)

14. 16. Given the graph of f ‘ (x) shown, give the x-coordinates where f(x) has points of inflection. Answer: a and c a and c (where f ‘ changes from incr  decr, the concavity will change) 24 3 2 1 -2 -3 f ' (x) 0 3 2 1 (a,b) (c,d)

15. 17. If f (x) = g ( h (x) ), then f ’(x) = __?__ Answer: f ’(x) = g’(h(x)) ● h’(x) *Derivative of a composite function requires the chain rule

16. 18.By the 2 nd Derivative Test, if f ”(x) is continuous, f ’(2) =0, and f ”(2) > 0, then (2, f(2) ) is a ____ ____. Answer: local minimum. *horiz. tangent in a concave up interval  local min

17. 19. Find f ’(x) if: Answer: (by FTC 2)

18. 20. Evaluate: Answer:

19. 21. If f (x) = sin 2 (3x), find f ’(x). Answer: f ’(x) = 2sin(3x) ● cos(3x) ● 3 *Power rule and two chain rules on the inside functions.

20. 22.Name the type of discontinuity at x = 3 for Answer: jump discontinuity (3, 1) (3, -1)

21. 23. Find f ’(x) if: Answer: (by FTC 2)

22. 24. Evaluate: Answer:

23. 25. Find the average value of f(x) over the interval (2,7) given: Answer: Recall this is the height of a rectangle that has the same area as the area under the curve.

24. 26. Find the critical #s of f(x) if: Answer: -5, -1, 0, 1 *Crit # are where f ’(x) = 0  -1, 0, 1 or f ’(x) is undefined  -5

25. 27. If oil leaks from a tank at a rate of r(t) gallons per minute what does represent? Answer: the net number of gallons that leaked from the tank in the first five minutes.

26. 28. Evaluate: Answer:

27. 29. Evaluate: Answer:

28. 30. If f(x) is differentiable over the reals & f ” (x) = (x – 1)(x – 2), over which interval(s) is f(x) concave down? Answer: (1, 2) f ” < 0  concave down f ” < 0  concave down (-∞,1) (1, 2) (2, ∞) (-∞,1) (1, 2) (2, ∞) f ”(x)>0 f ”(x) 0 f ”(x)>0 f ”(x) 0

29. 31. If f is continuous at (c, f(c)), which of the following could be FALSE? A. B. C.D. Answer: C (e.g., a corner is continuous, but not differentiable) A, B & D are the very def of continuous

30. 32. A particle moves along the x-axis so that its position at any time t  0 is given by The particle is at rest when t = ? Answer: when t=3 and t=5 seconds When v(t) = 0 When v(t) = 0

31. 33. P(t) = 520e 570t is the model for the number of fruit flies at time t hours in a biology experiment. What do you know about the population at t = 0 hours? Answer: 520 fruit flies For P(t) = Ce kt, C is the initial pop

32. 34. Evaluate both: Answer:

33. 35. Given the graph of f (x) shown, find the interval(s) where f ”(x) < 0. Answer: (-∞, c) (-∞, c) (where f is concave down) 24 3 2 1 -2 -3 f (x) 0 3 2 1 (a,b) (e,f) (c,d)

34. 36. Given the graph of f (x) shown give the interval(s) where f ’(x) < 0. Answer: (a, e) (a, e) (where f is decreasing) 24 3 2 1 -2 -3 f (x) 0 3 2 1 (a,b) (e,f) (c,d)

35. 37. Given the graph of f(x), evaluate evaluate Answer: – ½ – ½ Sum of 2 Δs ½ + -1 f(x) ½ –1

36. 38. Evaluate: Answer: 0 (it is an odd function)

37. 39. Give the third part of the definition of continuity: “f is continuous at c if: “f is continuous at c if:i.ii. iii. ??? Answer:

38. 40. Find the derivative (and factor the GCF). Answer:

39. 41. Evaluate: Answer:

40. 42. Find the equation of the tangent to y = x 3 – 1 at x =1. Answer:

41. 43. Evaluate: Answer:

42. 44. If f is differentiable over (1, 3), f(1)=4, and f(3)= 8, what can you conclude by the Mean Value Theorem? Answer: Since f is differentiable over (1,3) and the slope of the secant between endpoints is, there exists a value, there exists a value c in (1,3) such that f ’(c) = 2.

43. 45. Evaluate: Answer: When degrees are equal the horiz asymptote is at ratio of leading coefficients.

44. 46. Evaluate: Answer: - ∞ Next to vertical asymp, think of the signs of num. over denom.

45. 47. Find the slope of the normal to y = x 3 – 1 at x =1. Answer:

46. 48. Evaluate: Answer:

47. 49. If f is differentiable over (0,4) and f(1)=7 and f(3)=5, then we know there exists a c in ___?___ such that f(c)=6 by the _________?_________. Answers: (1,3) Intermediate Value Theorem

48. 50. If f is differentiable over (0,4) and we know f(2) = 7 and f ’(2)=3, what is the best approximation we can give for f(2.1)? Answers: 7.3 by Linear Approx. Tangent line is: y – 7 = 3(x – 2) *Find (2.1, ?) on tangent as a close approximation since the tangent lies close to the f(x) curve.

49. 51. Evaluate: Answer: 12 Shortcut: this is def of derivative for Shortcut: this is def of derivative for f ’(2) where f(x) = x 3, so use f ’(x)= 3x 2 f ’(2) where f(x) = x 3, so use f ’(x)= 3x 2 Or Alg: Or Alg:

50. 52. Evaluate: Answer:

51. 53.Find the limit. Answer:

52. 54. Evaluate each: Answer: -1 and 7/3 For pos exponents the ratio of is -1 (while the 7 and 3 become neglible). (while the 7 and 3 become neglible). For neg exponents, the power terms become neglible and the ratio is 7/3 become neglible and the ratio is 7/3

53. 55. Evaluate: Answer:

54. 56. Evaluate: Answer: 4 Shortcut: this is the def of derivative for Shortcut: this is the def of derivative for f(x) = x 4 when finding f ’(1), so use f ’(x)= 4x 3 f(x) = x 4 when finding f ’(1), so use f ’(x)= 4x 3 Or Alg:

55. 57. Evaluate: Answer:

56. 58. Which of the following are true about f? (may be one or more answers) I.f has a limit at x =3 II.f is continuous at x=3 III.f is differentiable at x=3 Answer: I only

57. 59. If f(x) is differentiable over R and f ’(x) = x 2 (x – 1)(x – 2), at what values of x, does f have local minima? Answer: at x = 2 Where f ’ changes from neg to pos ( ∞, 0) (0, 1) (1, 2) (2, ∞ ) f ’(x)>0 f ’(x)>0 f ’(x) 0 + - - + - - + + - + + + + - - + - - + + - + + +

58. 60. If f(x) is differentiable over R and f ’(x) = x 2 (x – 1)(x – 2), what term describes the point (0, f(0)) on the graph of f? Answer: Inflection Pt Inflection Pt Has a horiz tangent there, but graph is increasing on both sides. (2 nd deriv will change signs there.) f ’ (x) f(x)

59. 61. Use the graph of f ’(x) to give the x-coordinates where the tangent to f(x) will be horizontal. Answer: x=0, 2 and 4 Where f ’(x) = 0 f ’ (x) 123 4 5 f(x) 1 2 3 4 5

60. 62. Use the graph of f ’(x) to give the interval(s) where f(x) will be concave down. f ’ (x) 123 4 5 (1.1,0.2) (3.2,-0.6) Answer: (-∞, 0) U (1.1, 3.2) Where f ’ is decreasing, f ” will be neg

61. 63. If f(x) = ln e , then f ’(x) = ? Answer: 0 ln e  =  and deriv of a constant is zero

62. 64. Evaluate: Answer: - cos x

63. 65. Evaluate: Answer: - ∞ Only two possible answers ∞ or - ∞ Here: Here:

64. 66. If f(x) = (x + 5)(x - 1) 3, then f ’(x) = ? Answer: Product rule w/ power rule f ’(x) = (x + 5) ● 3(x - 1) 2 + (x - 1) 3 (1) = (x – 1) 2 [3(x+5) + (x-1)] = (x – 1) 2 [3(x+5) + (x-1)] = 2(x – 1) 2 (2x +7) = 2(x – 1) 2 (2x +7)

65. 67. Evaluate: Hint: Convert from complex to simple fraction Answer:

66. 68. Given f(x) and g(x) are diff. over R and g(x) = f - 1 (x). If f(1)=6, f ’(1)=4, f(7)=2 and f ’(7)=3, find g’(2). Answer: 1/3 Slopes on inverses are reciprocals at corresponding pts. Since (7,2) is on f, then (2, 7) is on g... so we simply take the reciprocal of f ’(7) to get g ’(2).

67. 69. Find a general solution if: Answer:

68. 70. Find a particular solution if f(-3) = -2 if f(-3) = -2 Answer:

69. 71. For the diff EQ below, if given f(0) = 3, then find f(1). Answer:

70. 72. Find the volume if the region bounded by y = 1-x 2 and y=0 is revolved about the x-axis. Answer:

71. 73. Let R be a region in quadrant I bounded by f(x) and g(x) as shown. Set up an integral to find the volume if R is revolved about the x-axis. Answer: (c,d) f(x) g(x) (a,b)

72. 74. Let R be a region in quadrant I bounded by f(x) and g(x) as shown. Set up an integral to find the volume if cross-sections perpendicular to the x-axis are squares. (c,d) f(x) g(x) (a,b) Answer:

73. 75-76. Let R be the region bounded by y = 1-x 2 and y=0. On base R, cross- sections perpendicular to the y-axis are semi-circles. Find the volume. 1 pt for correct set-up of integral & limits 1 pt for correct answer for volume Answer:

74. 77. The radius of a circular water spill is increasing at a rate of 3cm/sec. Find the rate at which the Area of the spill is increasing when the radius is 10cm. Answer:

75. 78. Solve the integral by substitution with u = cos 2x. Answer:

76. 79. Given y = 5x + k is a tangent to f(x) = x 3 + 2x in quadrant I. Find k. Answer: We see the slope of y = 5x + k is 5. So at what pt on f is the slope equal to 5? f ’(x) = 3x 2 + 2 So at what pt on f is the slope equal to 5? f ’(x) = 3x 2 + 2 5 = 3x 2 + 2 when x =  1 Quad I is given so f(1) = 3 is pt of tangency (1, 3) lies both on f(x) and on the tangent (1, 3) lies both on f(x) and on the tangent So plug into y = 5x + k  3 = 5(1) + k K = -2

77. 80. If (a,b) is a cusp on f(x), what do you know about the values of the left and right hand derivatives at x = a? Answer: the two slopes must go to ∞ and - ∞ (in either order)

78. 81. Differentiate the formula for surface area of a sphere implicitly with respect to t (time): A = 4  r 2 Answer:

79. 82. If f ’(x) > 0 & f ”(x) 0 & f ”(x) < 0 over [a, b], which graph could represent the shape of f(x) on this interval? Answer: A. B. C. E. D.

80. 83. Name the type of discontinuity for where x = 1. Answer: Removable Discontinuity Removable DiscontinuityOthers: “infinite discontinuity” at vert. asymp “infinite discontinuity” at vert. asymp “jump discontinuity” where y bumps up “jump discontinuity” where y bumps up

81. 84. Give an example of any function that is continuous, but not differentiable, at a specific x-coordinate. Explain your choice. Sample Answer: f(x) = | x | is continuous at x = 0, but the left & right slopes do not agree, so it has no derivative at this “corner”

82. 85. Find two derivatives: Answers:

83. 86. Find Answer: this is the definition of the derivative of the square root function!

84. 87. Given f(x) = g ( h(x) ); g(x) = x 3 and h(x) = 5x. g(x) = x 3 and h(x) = 5x. Find f ’ (2). Answer: g’(x) = 3x 2 and h’(x) = 5 g’(x) = 3x 2 and h’(x) = 5 So by the CHAIN RULE: f’(2) = g’ ( h(2) ) ● h’(2) f’(2) = g’ ( h(2) ) ● h’(2) = g’ (10) ● 5 = 300 ● 5 = 1500 = g’ (10) ● 5 = 300 ● 5 = 1500

85. 88. For Find c & d given that f(x) is differentiable over (0, 8). Answer: Cont  equal values when x =4 c(4) = 4 2 + d Diff  equal derivs when x = 4 c = 2x  c = 2(4) Thus c = 8 and by back-sub, d = 16 Thus c = 8 and by back-sub, d = 16

86. 89.Find the derivative. Answer:

87. 90. Evaluate: Answer:

88. 91. Evaluate: Answer: 0 (it is an odd function)

89. 92. For Find c & d given that f(x) is differentiable over (0, 6). Answer: Cont  equal values when x =3 c(3) = 3 2 + d Diff  equal derivs when x = 3 c = 2x  c = 2(3) = 6 Thus c = 6 and by back-sub, d = 9 Thus c = 6 and by back-sub, d = 9

90. 93. If the rate at which ballots are collected (per hour) is given by r(t) = t 2 + 2t, how many ballots are collected in the first three hours? Answer:

91. 94. The rate at which ballots are collected (per hour) is given by r(t) = t 2 + 2t. Using correct units, how is this rate changing at t = 1 hour? Answer:

92. 95. The rate at which ballots are collected (per hour) is given by r(t) = t 2 + 2t and b(x) represents the total # ballots collected at any time t. Describe the concavity of b(t) for all times, t > 0.

93. 96. Evaluate: Answer:

94. 97. Differentiate implicitly with respect to x: x 2 + xy + y 2 = 9 Answer:

95. THE END Congratulations! Repeat as needed until you remember your basics amd build your confidence.Repeat as needed until you remember your basics amd build your confidence. Look at the links on our class website to find more review lessons, videos and practice as needed.Look at the links on our class website to find more review lessons, videos and practice as needed.links Repetition and practice are keys to be successful on the AP Exam!Repetition and practice are keys to be successful on the AP Exam!