AP Exam

1
AP Exam Review Competition

• Everyone must WRITE his own work.
• Be ready to hold up your circled answer (Set pen
  down but do not reveal until directed.)
• The class period with highest average by May 8
  wins brownies ice cream on May 10.

2
Sample Find the derivative.
Recall key step apply the quotient
rule
3
1.Find the limit.
Recall key step divide all terms by the highest
power x3
4
2.Find the derivative.
5
3. Evaluate
6
4. Fill in the blanks Since polynomial functions
are continuous over the reals and for f(x) x3
-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
7
5.Find the limit.
8
6. Evaluate
7. Evaluate f (x)
8. Name the theorem used in problem 7 above.
Answer Fundamental Thrm of Calculus
9
9. Evaluate
10
10. Differentiate with respect to t (time) PV
c where c is a constant
11
11. For s(t) t2 1, what is the average
velocity over the time interval (0,4) seconds if
distance is given in ft?
12. For s(t) t2 1 above, what is the
instantaneous velocity at t 4?
12
13. Evaluate. (You must have both correct!)
Answer tan-1x C and sin-1x C
13
14. Differentiate implicitly with respect to
x x2 xy y2 9
14
15. Given the graph of f (x) shown, give the
x-coordinate(s) where f(x) has local
minima.
Answer 0 and 3 (where slopes change
from neg to pos)
15
16. Given the graph of f (x) shown, give the
x-coordinates where f(x) has points of
inflection.
Answer a and c (where f changes from
incr ?? decr, the concavity will change)
16
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
17
18.By the 2nd Derivative Test, if f (x) is
continuous, f (2) 0, and f (2) gt 0, then (2,
f(2) ) is a ____ ____.
Answer local minimum. horiz. tangent in a
concave up interval ? local min
18
19. Find f (x) if
19
20. Evaluate
20
21. If f (x) sin2(3x), find f (x).
Answer f (x) 2sin(3x) ? cos(3x) ? 3 Power
rule and two chain rules on the inside functions.
21
22.Name the type of discontinuity at x 3 for
22
23. Find f (x) if
23
24. Evaluate
24
25. Find the average value of f(x) over the
interval (2,7) given
25
26. Find the critical s of f(x) if
26
27. If oil leaks from a tank at a rate of r(t)
gallons per minute what does represent?
Answer the total number of gallons that leaked
from the tank in the first five minutes.
27
28. Evaluate
28
29. Evaluate
29
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 lt 0 ?? concave down
(-8 ,1) (1, 2) (2, 8) f (x)gt0
f (x) lt0 f (x) gt0
30
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
31
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 ?
32
33. P(t) 520e570t 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 In P(t) Cekt, C is
the initial pop
33
34. Evaluate both
34
35. Given the graph of f (x) shown, find the
interval(s) where f (x) lt 0.
Answer (-8, c) (where f is concave down)
35
36. Given the graph of f (x) shown give the
interval(s) where f (x) lt 0.
Answer (a, e) (where f is decreasing)
36
37. Given the graph of f(x), evaluate
Answer ½ Sum of 2 ?s ½ -1
f(x)
37
38. Evaluate
38
39. Give the third part of the definition of
continuity f is continuous at c
if i. ii. iii. ???
39
40. Find the derivative (and factor the GCF).
40
41. Evaluate
41
42. Find the equation of the tangent to y x3
1 at x 1.
42
43. Evaluate
43
44. If f is differentiable over 1, 3, f(1)4,
and f(3) 8, what can you conclude by the Mean
Value Theorem?
44
45. Evaluate
45
46. Evaluate
46
47. Find the slope of the normal to y x3 1
at x 1.
Answer
47
48. Evaluate
48
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
49
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.
50
51. Evaluate
51
52. Evaluate
52
53. 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).
53
54. Evaluate each
54
55. Evaluate
55
56. Evaluate
56
57. Evaluate
57

• 58.
• Which of the following are true about f? (may be
  one or more answers)
• f has a limit at x 3
• f is continuous at x3
• f is differentiable at x3

Answer I only
58
59. If f(x) is differentiable over R and f (x)
x2(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 (8, 0) (0, 1) (1, 2) (2,
8 ) f (x)gt0 f (x)gt0 f (x)lt0 f (x)gt0
- - - - -
59
60. If f(x) is differentiable over R and f
(x) x2(x 1)(x 2), what term describes the
point (0, f(0)) on the graph of f?
Answer Inflection Pt Has a horiz tangent
there, but graph is increasing on both sides.
(2nd deriv will change signs there.)
60
61. Use the graph of f (x) to give the
x-coordinates where the tangent to f(x) will be
horizontal.
Answer x0, 2 and 4 Where f (x) 0
61
62. Use the graph of f (x) to give the
interval(s) where f(x) will be concave down.
Answer (-8, 0) U (1.1, 3.2) Where f is
decreasing, f will be neg
62
63. If f(x) ln e?, then f (x) ?
Answer 0 ln e? ? and deriv of a constant is
zero
63
64. Evaluate
64
65. Evaluate
65
66. If f(x) (x 5)(x - 1)3, then f (x) ?
66
67. Evaluate
Hint Convert from complex to simple fraction
67
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).
68
69. Find a general solution if
69
70. Find a specific solution if f(-3) -2
70
71. For the diff EQ below, if given f(0) 3,
then find f(1).
71
72. Find the volume if the region bounded by y
1-x2 and y0 is revolved about the x-axis.
72
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.
73
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.
74
75-76. Let R be the region bounded by y 1-x2
and y0. 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
75
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.
76
78. Solve the integral by substitution with u
cos 2x.
77
79. Given y 5x k is a tangent to f(x) x3
2x in quadrant I. Find k.
78
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 8 and - 8 (in
either order)
79
81. Differentiate the formula for surface area of
a sphere implicitly with respect to t (time)
A 4?r2
80
82. If f (x) gt 0 f (x) lt 0 over a, b, which
graph could represent the shape of f(x) on this
interval?
Answer
81
83. Name the type of discontinuity for where
x 1.
Answer Removable Discontinuity Others
infinite discontinuity at vert. asymp
jump discontinuity where y bumps up
82
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
83
85. Find two derivatives
84
86. Find
85
87. Given f(x) g ( h(x) ) g(x) x3 and
h(x) 5x. Find f (2).
Answer g(x) 3x2 and h(x)
5 So by the CHAIN RULE f(2) g ( h(2) ) ?
h(2) g (10) ?5 300 ?5 1500
86
88. For Find c d given that f(x) is
differentiable over (0, 8).
87
89.Find the derivative.
88
90. Evaluate
89
91. Evaluate
90
92. For Find c d given that f(x) is
differentiable over (0, 6).
91
93. If the rate at which ballots are collected
(per hour) is given by r(t) t2 2t, how many
ballots are collected in the first three hours?
92
94. The rate at which ballots are collected (per
hour) is given by r(t) t2 2t. Using correct
units, how is this rate changing at t 1 hour?
93
95. The rate at which ballots are collected (per
hour) is given by r(t) t2 2t and b(t)
represents the total ballots collected at any
time t. Describe the concavity of b(t) for all
times, t gt 0.
Answer since the deriv of the rate function,
r(t) 2t 2 is greater than zero for all t gt0,
the function b(t) is concave up for all t gt 0.
94
96. Evaluate
95
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