CALCULUS 2 – BC PRACTICE TEST #2
Name
DAY 1 – SECTION A – NO CALCULATOR
1.
2.
Period
5
5
⌠ ƒ(x) dx = 2. What is ⌡
⌠ 2 ƒ(x) – 1 dx ?
Let ƒ(x) be a function such that ⌡
1
1
A) 0 B) 1 C) 2 D) 3 E) 4
π
6
What is ⌠
⌡ cos x(sin2 x + 1)dx
0
A)
13
24
B) 0
C)
13 3
24
3e3t
A) 3e3t B)
cos t
D)
3.
dy
If x = sin t and y = e3t, what is
?
dx
4.
lim
5.
Where does the function 2x2 + sin 2x have a point of inflection?
π
π
π
π
A) x =
B) x = –
C) x = 0 D) x =
E) x = –
4
4
3
3
6.
A function ƒ(x) is equal to
7.
4
⌠ x(5x + 3) dx ? A) 7.5 B) 12
What is ⌡
1
8.
What is lim
9.
What is the particular solution to the differential equation
sin 2 (3x)
is
x→0
x2
A) 0
B) 1
C) 3 D) 9
–13
24
E)
–13 3
24
C) 3e3t cos t D) cos t E) e3t cos t
E) undefined
x2 – 4
for all x > 0 except x = 2. In order for the function to be continuous at
x–2
x = 2, what must the value of ƒ(2) be?
A) –4 B) –2 C) 0 D) 2 E) 4
3 x 4 − 3x
x→∞ 2x 2 + cos x
A) ∞ B) undefined
x2
x2
– 1 B) cos y = 3 –
2
2
2
2
x
x
D) sin y =
– 2 E) sin y = 2 –
2
2
A) cos y =
C) 64
C)
D) 70.4 E) 76
3
2
D) 0
E) 6
dy
x
=
that passes through the point (2,0) ?
dx
sin y
C) cos y = x2 – 3
10. Which of the following improper integrals diverge?
π/2
1
∞
2
sin
x
⌠ 1
A) ⌠
dx
B)
dx
C)
xe− x dx dx D) All of the above E) None of the above
⎮ 1– cos x
⎮
∫
⌡
⌡ 1–x
4
0
0
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CALCULUS 2 – BC PRACTICE TEST #2
Name
DAY 1 – SECTION A – NO CALCULATOR
Period
5
1
14
23
C)
D)
E)
3
3
3
6
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⎛π
⎞
sin ⎜ + h ⎟ − 1
⎝2
⎠
12. What is lim
h→0
h
⎛π⎞
A) ƒ ' ⎜ ⎟ , where ƒ(x) = cos x
B) ƒ ' (1), where ƒ(x) = sin x C) ƒ ' (1), where ƒ(x) = cos x
⎝2⎠
⎛π⎞
⎛π⎞
D) ƒ ' ⎜ ⎟ , where ƒ(x) = sin 2x E) ƒ ' ⎜ ⎟ , where ƒ(x) = sin x
2
⎝ ⎠
⎝2⎠
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2
13. What is the area under the curve y = x e(x ) between x = 0 and x = 2?
e4 1
e4
A)
–
B)
C) e4 – 1 D) 4e4 E) 4e4 – 4
2 2
2
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11. If ƒ(x) = x 2x – 1 , what is ƒ ' (5)? A) 3
B)
∞
14. Which of the following guarantee that
∑ f (n) converges?
n=0
1
1
(I) lim f (x) = 0 (II) ƒ(x) < 2 (III) 2 < ƒ(x) when x ≥ 1.
x→∞
x
x
A) (I) only B) (II) only C) (I) and (II) only D) (I) and (III) only E) (I), (II), and (III)
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2x
15. What is the absolute maximum value of the function y = 2
?
x + 16
1
1
A) 4 B) –
C) 0 D)
E) –4
4
4
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16. Let f(x) be a function with a continuous derivative on the interval (0, 5) such that
ƒ ' (0) = 3, ƒ ' (1) = 2, ƒ ' (2) = –3, ƒ ' (3) = –4, ƒ ' (4) = 1. Which of the following must be true about ƒ(x)?
(I) ƒ(x) has a critical point between x = 1 and x = 2
(II) ƒ(x) has a critical point between x = 0 and x = 1
(III) ƒ(x) has a critical point between x = 2 and x = 3
A) (I) only B) (III) only C) (I) and (II) only D) (I) and (III) only E) (I), (II), and (III)
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17. Let ƒ be a continuous function on the interval [–1,3]. If ƒ(–1) = 9 and ƒ(3) = 1, then the Mean Value
Theorem guarantees that
A) ƒ ' (0) = 0 B) ƒ ' (c) = –2 for some c between –1 and 3
C) ƒ ' (c) = 2 for some c between –1 and 3 D) ƒ = 5 for some c between –1 and 3
E) ƒ < 0 for all c between –1 and 3
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18. A young tree is measured every week. Each week, it is taller than it was the week before, but it has grown
a lesser amount. If h(t) is the height of the tree, which of the following is positive?
(I) h(t) (II) h ' (t) (III) h '' (t)
A) (II) only B) (I) and (II) only C) (I) and (III) only D) (II) and (III) only E) (I), (II), and (III)
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CALCULUS 2 – BC PRACTICE TEST #2
Name
DAY 1 – SECTION A – NO CALCULATOR
Period
6x
19. On what interval is the function y = 2
increasing?
x +9
A) (– , 0) B) (–3, 0) C) (0, ) D) (0,3) E) (–3,3)
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x
20. Let ƒ(x) =
. Which of the following are true?
x+1
(I) ƒ(x) has exactly one local maximum
(II) ƒ(x) has a point of inflection at x = 0
(III) ƒ(x) has a vertical asymptote at x = – 1
A) (I) only B) (III) only C) (I) and (III) only D) (II) and (III) only E) (I), (II), and (III)
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∞
(x − 2)n
21. What is the radius of convergence of the series ∑
?
A) 0 B) 1 C) 2 D) 3 E)
n!
n=0
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6
22. What is the volume of the solid obtained by rotating the region between y =
and y = 4 – x around the
x+1
3
3
⌠
⎛ 6 ⎞2
⌠⎛ 6 ⎞
x-axis? A) π⎮ (4 – x)2 – ⎜
⎟ dx B) π⎮⎜x + 1⎟ 2 – (4 – x)2 dx
x
+
1
⎝
⎠
⎠
⌡
⌡⎝
1
1
3
2
2
2
6
6
⎛
⎞
⎛
⎞
⌠
⎛ 6 ⎞
C) π ∫ (4 − x)2 − ⎜
(4 – x) – ⎜
dx
E) 2π ⎮(4 – x) – ⎜
dx D) 2π ⌠
⎟
⎟ dx
⌡
⎟
⎝x + 1⎠
⎝x + 1⎠
⎝ x + 1⎠
⌡
1
1
1
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23. This is a slope field for which of the following differential equations?
4
2
–2
1
–1
–2
–2
–4
dy
dy
dy x
dy x2 E) dy = 2
= xy2 B)
= xy C)
=
D)
= y
dx x y
dx
dx
dx y
dx
________________________________________________________________________
A)
CALCULUS 2 – BC PRACTICE TEST #2
Name
DAY 1 – SECTION A – NO CALCULATOR
Period
This graph of ƒ(x) is for problems 24 and 25.
t
24. Let g(t) = ⌠
A) 0 B) 6 C) 12 D) 18 E) unknown
⌡ ƒ(x) dx. What is g(6)?
0
-------------------------------------------------------------------------------------------------------25. Where is ƒ ' (x) = 0?
(I) When x = 1
(II) When x is between 4 and 5
(III) When x = 2
A) (I) only B) (II) only C) (I) and (II) only D) (II) and (III) only E) (I), (II), and (III)
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9x + 10
26. What is ⌠
dx?
⎮ 2
⌡ 2x – x – 6
ln|2x + 3|
ln|2x + 3|
A) 4 ln|x – 2| +
+ C B) 4 ln|9x + 10| +
+C
2
2
C) 3 ln|x – 2| + ln|2x + 3| + C D) 3 ln|9x + 10| + 2 ln|2x + 3| + C
E) 4 ln|x – 2| + 2 ln|2x + 3| + C
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27. A particle is moving along the x–axis. Its position at time t > 0 is e2–t. What is its acceleration when t = 2?
A) e B) 1 C) 0 D) –1 E) –e
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28. A particle, initially at position (1,2), moves with velocity (t, sin(π t)). Where is the object when t = 2?
1 ⎞
1 ⎞
1 ⎞
⎛
⎛
⎛
A) ⎜1,2 +
B) ⎜3,2 +
C) ⎜3,2 – ⎟ D) (1,2) E) (3,2)
⎟
⎟
π ⎠
π ⎠
π ⎠
⎝
⎝
⎝