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1969 AP Calculus AB: Section I
90 Minutes—No Calculator
Note: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e).
1.
2.
3.
Which of the following defines a function f for which f (− x) = − f ( x) ?
(A)
f ( x) = x 2
(B)
f ( x) = sin x
(D)
f ( x) = log x
(E)
f ( x) = e x
ln ( x − 2 ) < 0 if and only if
(A)
x<3
(B)
0< x<3
(D)
x>2
(E)
x>3
⎧
2x + 5 − x + 7
, for x ≠ 2,
⎪ f ( x) =
If ⎨
x−2
⎪ f (2) = k
⎩
(A) 0
4.
8
∫0
dx
1+ x
(A) 1
5.
f ( x) = cos x
(C)
2< x<3
and if f is continuous at x = 2 , then k =
(B)
1
6
(C)
1
3
(D) 1
(E)
7
5
(B)
3
2
(C)
2
(D) 4
(E) 6
(D) 4
(E) not defined
=
If 3 x 2 + 2 xy + y 2 = 2, then the value of
(A) –2
(C)
(B) 0
dy
at x = 1 is
dx
(C)
2
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1
1969 AP Calculus AB: Section I
8
6.
(A) 0
(E)
7.
1
2
For what value of k will x +
(C)
1
(D) The limit does not exist.
k
have a relative maximum at x = −2?
x
(B) –2
(C)
2
(D) 4
(E) None of these
If p ( x) = ( x + 2 )( x + k ) and if the remainder is 12 when p( x) is divided by x − 1, then k =
(A) 2
9.
(B)
It cannot be determined from the information given.
(A) –4
8.
8
⎛1
⎞
⎛1⎞
8⎜ + h ⎟ − 8⎜ ⎟
2
⎠
⎝2⎠ ?
What is lim ⎝
h →0
h
(B) 3
(C)
6
(D) 11
(E) 13
When the area in square units of an expanding circle is increasing twice as fast as its radius in
linear units, the radius is
(A)
1
4π
(B)
1
4
(C)
1
π
(D) 1
(E)
π
(E)
ln x
10. The set of all points (et , t ) , where t is a real number, is the graph of y =
(A)
1
ex
(B)
1
ex
(C)
1
xex
(D)
1
ln x
1⎞
⎛
11. The point on the curve x 2 + 2 y = 0 that is nearest the point ⎜ 0, − ⎟ occurs where y is
2⎠
⎝
1
1
(B) 0
(C) −
(D) −1
(E) none of the above
(A)
2
2
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2
1969 AP Calculus AB: Section I
12. If f ( x) =
(A)
4
and g ( x) = 2 x, then the solution set of f ( g ( x) ) = g ( f ( x) ) is
x −1
⎧1 ⎫
⎨ ⎬
⎩3⎭
(B)
{2}
(C)
{3}
(D)
{−1, 2}
(E)
⎧1 ⎫
⎨ , 2⎬
⎩3 ⎭
13. The region bounded by the x-axis and the part of the graph of y = cos x between x = −
π
and
2
π
π
is separated into two regions by the line x = k . If the area of the region for − ≤ x ≤ k is
2
2
π
three times the area of the region for k ≤ x ≤ , then k =
2
x=
⎛1⎞
(A) arcsin ⎜ ⎟
⎝4⎠
(D)
π
4
(B)
⎛1⎞
arcsin ⎜ ⎟
⎝3⎠
(E)
π
3
(C)
π
6
14. If the function f is defined by f ( x) = x5 − 1, then f −1 , the inverse function of f , is defined by
f −1 ( x) =
(A)
(D)
1
5
x +1
5
x −1
(B)
(E)
1
5
x +1
5
x +1
(C)
5
x −1
15. If f ′( x) and g ′( x) exist and f ′( x) > g ′( x) for all real x, then the graph of y = f ( x) and the graph
of y = g ( x)
(A) intersect exactly once.
(B) intersect no more than once.
(C) do not intersect.
(D) could intersect more than once.
(E) have a common tangent at each point of intersection.
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3
1969 AP Calculus AB: Section I
16. If y is a function of x such that y′ > 0 for all x and y′′ < 0 for all x, which of the following could
be part of the graph of y = f ( x) ?
17. The graph of y = 5 x 4 − x5 has a point of inflection at
(A) (0, 0) only
(B)
(3,162) only
(D) (0,0) and (3,162 )
(E)
(0, 0) and (4, 256)
(C)
(4, 256) only
18. If f ( x) = 2 + x − 3 for all x, then the value of the derivative f ′( x) at x = 3 is
(A)
−1
(B) 0
(C)
1
(D) 2
(E) nonexistent
19. A point moves on the x-axis in such a way that its velocity at time t ( t > 0 ) is given by v =
ln t
.
t
At what value of t does v attain its maximum?
(A) 1
(E)
(B)
1
e2
(C)
e
(D)
3
e2
There is no maximum value for v.
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4
1969 AP Calculus AB: Section I
20. An equation for a tangent to the graph of y = arcsin
(A)
x − 2y = 0
(B)
x− y =0
(C)
x
at the origin is
2
x=0
y=0
(D)
π x − 2y = 0
(E)
21. At x = 0 , which of the following is true of the function f defined by f ( x) = x 2 + e −2 x ?
(A) f is increasing.
(B)
f is decreasing.
(C)
f is discontinuous.
(D) f has a relative minimum.
(E)
22.
f has a relative maximum.
(
)
d
ln e 2 x =
dx
(A)
1
e
2x
(B)
2
e2 x
(C)
2x
(D) 1
(E) 2
23. The area of the region bounded by the curve y = e2x , the x-axis, the y-axis, and the line x = 2 is
equal to
(A)
e4
−e
2
(B)
e4
−1
2
(D)
2e4 − e
(E)
2e4 − 2
24. If sin x = e y , 0 < x < π, what is
(A)
− tan x
(B)
− cot x
(C)
e4 1
−
2 2
(E)
csc x
dy
in terms of x ?
dx
(C)
cot x
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(D)
tan x
5
1969 AP Calculus AB: Section I
25. A region in the plane is bounded by the graph of y =
x = 2m, m > 0 . The area of this region
1
, the x-axis, the line x = m, and the line
x
(A) is independent of m .
(B)
increases as m increases.
(C)
decreases as m increases.
1
1
; increases as m increases when m > .
2
2
1
1
increases as m increases when m < ; decreases as m increases when m > .
2
2
(D) decreases as m increases when m <
(E)
26.
1
∫0
x 2 − 2 x + 1 dx is
(A)
−1
(B)
−
1
2
1
2
(D) 1
(E) none of the above
(C)
27. If
dy
= tan x , then y =
dx
(A)
1
tan 2 x + C
2
(B)
sec 2 x + C
(D)
ln cos x + C
(E)
sec x tan x + C
(C)
ln sec x + C
(E)
3 3
28. The function defined by f ( x) = 3 cos x + 3sin x has an amplitude of
(A)
3− 3
(B)
3
(C)
2 3
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(D)
3+ 3
6
1969 AP Calculus AB: Section I
29.
∫π 4
cos x
dx =
sin x
(A)
ln 2
π 2
(B)
ln
π
4
(C)
ln 3
(D)
3
2
ln
(E)
ln e
30. If a function f is continuous for all x and if f has a relative maximum at (−1, 4) and a relative
minimum at (3, − 2) , which of the following statements must be true?
(A) The graph of f has a point of inflection somewhere between x = −1 and x = 3.
(B) f ′(−1) = 0
(C) The graph of f has a horizontal asymptote.
(D) The graph of f has a horizontal tangent line at x = 3 .
(E)
The graph of f intersects both axes.
31. If f ′( x) = − f ( x) and f (1) = 1, then f ( x) =
(A)
1 −2 x + 2
e
2
(B)
e − x −1
(C)
e1− x
(D)
e− x
(E)
−e x
32. If a, b, c, d , and e are real numbers and a ≠ 0 , then the polynomial equation
ax 7 + bx5 + cx3 + dx + e = 0 has
(A)
(B)
(C)
(D)
(E)
only one real root.
at least one real root.
an odd number of nonreal roots.
no real roots.
no positive real roots.
33. What is the average (mean) value of 3t 3 − t 2 over the interval −1 ≤ t ≤ 2 ?
(A)
11
4
(B)
7
2
(C) 8
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(D)
33
4
(E)
16
7
1969 AP Calculus AB: Section I
34. Which of the following is an equation of a curve that intersects at right angles every curve of the
1
family y = + k (where k takes all real values)?
x
1
1
(D) y = x3
(E) y = ln x
(C) y = − x3
(A) y = − x
(B) y = − x 2
3
3
35. At t = 0 a particle starts at rest and moves along a line in such a way that at time t its acceleration
is 24t 2 feet per second per second. Through how many feet does the particle move during the first
2 seconds?
(A) 32
(B) 48
(C) 64
(D) 96
(E)
192
36. The approximate value of y = 4 + sin x at x = 0.12 , obtained from the tangent to the graph at
x = 0, is
(A) 2.00
(B) 2.03
(C) 2.06
(D) 2.12
(E)
2.24
37. Which is the best of the following polynomial approximations to cos 2 x near x = 0 ?
(A) 1 +
38.
x2
∫ ex
3
x
2
(B) 1 + x
(C) 1 −
x2
2
(D) 1 − 2x 2
(E)
1 − 2x + x 2
(C)
−
(E)
sec2 e
dx =
3
(A)
3
1
− ln e x + C
3
(B)
(D)
3
1
ln e x + C
3
(E)
ex
−
+C
3
x3
3e
x3
1
3e x
3
+C
+C
1
dy
39. If y = tan u , u = v − , and v = ln x , what is the value of
at x = e ?
v
dx
(A) 0
(B)
1
e
(C) 1
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(D)
2
e
8
1969 AP Calculus AB: Section I
40. If n is a non-negative integer, then
1
∫0 x
(A) no n
(D) nonzero n, only
n
1
0
(1 − x )n dx
for
(B) n even, only
(E) all n
⎧⎪ f ( x) = 8 − x 2 for − 2 ≤ x ≤ 2,
41. If ⎨
then
2
elsewhere ,
⎪⎩ f ( x) = x
(A) 0 and 8
dx = ∫
(B) 8 and 16
(C) n odd, only
3
∫ −1 f ( x) dx is a number between
(C) 16 and 24
(D) 24 and 32
(E)
32 and 40
42. What are all values of k for which the graph of y = x3 − 3 x 2 + k will have three distinct
x-intercepts?
(A) All k > 0
43.
(B) All k < 4
(C)
k = 0, 4
(D) 0 < k < 4
(E)
All k
∫ sin ( 2 x + 3) dx =
(A)
1
cos ( 2 x + 3) + C
2
(B)
cos ( 2 x + 3) + C
(D)
1
− cos ( 2 x + 3) + C
2
(E)
1
− cos ( 2 x + 3) + C
5
(C)
44. The fundamental period of the function defined by f ( x) = 3 − 2 cos 2
(A) 1
(B) 2
(C) 3
− cos ( 2 x + 3) + C
πx
is
3
(D) 5
(E)
6
(C)
3x 2 g x3
d
d
d2
2
45. If
( f ( x) ) = g ( x) and ( g ( x) ) = f ( x ) , then 2 f ( x3 ) =
dx
dx
dx
(
( )
(A)
f x6
(D)
9 x 4 f x 6 + 6 x g x3
( )
( )
)
( )
(B)
g x3
(E)
f x 6 + g x3
( )
( ) ( )
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9
1969 Calculus AB Solutions
1.
B
Sine is the only odd function listed. sin(− x) = − sin( x) .
2.
C
ln t < 0 for 0 < t < 1 ⇒ ln ( x − 2 ) < 0 for 2 < x < 3 .
3.
B
Need to have lim f ( x) = f (2) = k .
x →2
k = lim
x →2
2x + 5 − x + 7
2x + 5 − x + 7 2x + 5 + x + 7
= lim
⋅
x→2
x−2
x−2
2x + 5 + x + 7
2x + 5 − ( x + 7)
1
1
1
⋅
= lim
=
x →2
x−2
2 x + 5 + x + 7 x→2 2 x + 5 + x + 7 6
= lim
8
dx
= 2 1+ x
1+ x
8
= 2 ( 3 − 1) = 4
4.
D
∫0
5.
E
Using implicit differentiation, 6 x + 2 xy′ + 2 y + 2 y ⋅ y′ = 0 . Therefore y′ =
0
−2 y − 6 x
.
2x + 2 y
When x = 1 , 3 + 2 y + y 2 = 2 ⇒ 0 = y 2 + 2 y + 1 = ( y + 1) 2 ⇒ y = −1
dy
Therefore 2 x + 2 y = 0 and so
is not defined at x = 1 .
dx
6.
B
This is the derivative of f ( x) = 8 x8 at x =
1
2
7
1
⎛1⎞
⎛1⎞
f ′ ⎜ ⎟ = 64 ⎜ ⎟ =
2
⎝2⎠
⎝2⎠
k
k
, we need 0 = f ′(−2) = 1 − and so k = 4. Since f ′′(−2) < 0 for k = 4, f
x
4
does have a relative maximum at x = −2 .
With f ( x) = x +
7.
D
8.
B
p ( x) = q ( x)( x − 1) + 12 for some polynomial q ( x) and so 12 = p (1) = (1 + 2 )(1 + k ) ⇒ k = 3
9.
C
A = π r2,
So, 2
dA
dr
dA
dr
and from the given information in the problem
= 2π r ⋅
=2 .
dt
dt
dt
dt
dr
dr
1
= 2π r ⋅ ⇒ r =
dt
dt
π
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160
1969 Calculus AB Solutions
10. E
x = e y ⇒ y = ln x
11. B
⎛
x2 ⎞
1⎞
⎛
Let L be the distance from ⎜ x , − ⎟ and ⎜ 0, − ⎟ .
⎜
⎟
2 ⎠
2⎠
⎝
⎝
2
⎛ x2 1 ⎞
L = ( x − 0) + ⎜ − ⎟
⎜ 2 2⎟
⎝
⎠
⎛ x2 1 ⎞
dL
2L ⋅
= 2x + 2 ⎜ − ⎟ ( x)
⎜ 2 2⎟
dx
⎝
⎠
2
2
⎛ x2 1 ⎞
2x + 2 ⎜ − ⎟ ( x)
2
⎜ 2 2⎟
2 x + x3 − x x3 + x x x + 1
dL
⎝
⎠
=
=
=
=
2L
2L
2L
2L
dx
(
)
dL
dL
< 0 for all x < 0 and
> 0 for all x > 0 , so the minimum distance occurs at x = 0 .
dx
dx
The nearest point is the origin.
12. A
13. C
4
1
⎛ 4 ⎞
= 2⎜
⎟ ⇒ x − 1 = 4 x − 2; x =
2x −1
3
⎝ x −1 ⎠
π
⎛ π⎞
⎛
⎞
cos x dx ; sin k − sin ⎜ − ⎟ = 3 ⎜ sin − sin k ⎟
2
⎝ 2⎠
⎝
⎠
π
sin k + 1 = 3 − 3sin k ; 4sin k = 2 ⇒ k =
6
k
π 2
∫ −π 2 cos x dx = 3∫ k
14. E
y = x5 − 1 has an inverse x = y 5 − 1 ⇒ y = 5 x + 1
15. B
The graphs do not need to intersect (eg. f ( x) = −e − x and g ( x) = e − x ) . The graphs could
intersect (e.g. f ( x) = 2 x and g ( x) = x ). However, if they do intersect, they will intersect no
more than once because f ( x) grows faster than g ( x) .
16. B
y′ > 0 ⇒ y is increasing; y′′ < 0 ⇒ the graph is concave down . Only B meets these
conditions.
17. B
y′ = 20 x3 − 5 x 4 , y′′ = 60 x 2 − 20 x3 = 20 x 2 ( 3 − x ) . The only sign change in y′′ is at x = 3 .
The only point of inflection is (3,162).
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1969 Calculus AB Solutions
18. E
There is no derivative at the vertex which is located at x = 3 .
19. C
dv 1 − ln t
dv
= 2 > 0 for 0 < t < e and
< 0 for t > e , thus v has its maximum at t = e .
dt
dt
t
20. A
y (0) = 0 and y′(0) =
y=
1
2
x2
1−
4
x =0
1
=
4 − x2
x =0
=
1
. The tangent line is
2
1
x ⇒ x − 2y = 0 .
2
21. B
f ′ ( x ) = 2 x − 2e−2 x , f ′ ( 0 ) = −2 , so f is decreasing
22. E
ln e2 x = 2 x ⇒
2 2x
e
0
(
2
1 2x
e
2
23. C
∫
24. C
y = ln sin x , y′ =
25. A
26. C
27. C
2m 1
∫m
x
dx =
)
d
d
ln e2 x = ( 2 x ) = 2
dx
dx
dx = ln x
0
=
(
)
1 4
e −1
2
cos x
= cot x
sin x
2m
m
= ln ( 2m ) − ln ( m ) = ln 2 so the area is independent of m.
1 1
1
2
x
−
1
=
(
)
∫0
0
0
0 2
2
Alternatively, the graph of the region is a right triangle with vertices at (0,0), (0,1), and (1,0).
1
The area is .
2
1
x 2 − 2 x + 1 dx = ∫
1
1
x − 1 dx = ∫ − ( x − 1) dx = −
sin x
∫ tan x dx = ∫ cos x dx = − ln
cos x + C = ln sec x + C
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1969 Calculus AB Solutions
28. C
3 cos x + 3sin x can be thought of as the expansion of sin ( x + y ) . Since
3 and 3 are too
large for values of sin y and cos y , multiply and divide by the result of the Pythagorean
Theorem used on those values, i.e. 2 3 . Then
⎛ 3
⎞
⎛1
⎞
3
3
3 cos x + 3sin x = 2 3 ⎜⎜
cos x +
sin x ⎟⎟ = 2 3 ⎜⎜ cos x +
sin x ⎟⎟
2
2 3
⎝2 3
⎠
⎝2
⎠
= 2 3 ( sin y cos x + cos y sin x ) = 2 3 sin ( y + x )
⎛1⎞
where y = sin −1 ⎜ ⎟ . The amplitude is 2 3 .
⎝2⎠
Alternatively, the function f ( x) is periodic with period 2π . f ′( x) = − 3 sin x + 3cos x = 0
π 4π
⎛π⎞
when tan x = 3 . The solutions over one period are x = ,
. Then f ⎜ ⎟ = 2 3 and
3 3
⎝3⎠
⎛ 4π ⎞
f ⎜ ⎟ = −2 3 . So the amplitude is 2 3 .
⎝ 3 ⎠
π2
cos x
dx = ln ( sin x )
sin x
π2
1
= ln 2
2
29. A
∫π 4
30. E
Because f is continuous for all x, the Intermediate Value Theorem implies that the graph of f
must intersect the x-axis. The graph must also intersect the y-axis since f is defined for all x,
in particular, at x = 0.
π4
= ln1 − ln
31. C
dy
= − y ⇒ y = ce − x and 1 = ce−1 ⇒ c = e ; y = e ⋅ e− x = e1− x
dx
32. B
If a < 0 then lim y = ∞ and lim y = −∞ which would mean that there is at least one root.
x→−∞
x→∞
If a > 0 then lim y = −∞ and lim y = ∞ which would mean that there is at least one root.
x→−∞
x→∞
In both cases the equation has at least one root.
33. A
1 2 3 2
1⎛ 3
1 ⎞
3t − t dt = ⎜ t 4 − t 3 ⎟
∫
−
1
3
3⎝ 4
3 ⎠
34. D
y′ = −
1
x
2
1 ⎛⎛
8 ⎞ ⎛ 3 1 ⎞ ⎞ 11
= ⎜ ⎜12 − ⎟ − ⎜ + ⎟ ⎟ =
−1 3 ⎝ ⎝
3 ⎠ ⎝ 4 3 ⎠⎠ 4
2
, so the desired curve satisfies y′ = x 2 ⇒ y =
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1 3
x +C
3
163
1969 Calculus AB Solutions
35. A
a ( t ) = 24t 2 , v(t ) = 8t 3 + C and v(0) = 0 ⇒ C = 0. The particle is always moving to the
right, so distance =
36. B
∫0
8t 3dt = 2t 4
y = 4 + sin x , y (0) = 2, y′(0) =
L( x ) = 2 +
37. D
2
2
0
= 32 .
cos 0
1
= . The linear approximation to y is
2 4 + sin 0 4
1
1
x . L(1.2) = 2 + (1.2) = 2.03
4
4
All options have the same value at x = 0 . We want the one that has the same first and second
derivatives at x = 0 as y = cos 2 x : y′(0) = −2sin 2 x
For y = 1 − 2 x 2 , y′(0) = −4 x
x2
x =0
x =0
= 0 and y′′(0) = −4 cos 2 x
1 − x3
1 3
1
+C
e (−3 x 2 dx) = − e − x + C = −
3
∫
3
3
3e− x
∫ ex
39. D
x = e ⇒ v = 1, u = 0, y = 0;
40. E
One solution technique is to evaluate each integral and note that the value is
dx = −
(
dy dy du dv
=
⋅ ⋅ = sec 2 u
dx du dv dx
Another technique is to use the substitution u = 1 − x ;
) ⎛⎜⎝1 + v12 ⎞⎟⎠ ⎜⎝⎛ 1x ⎟⎠⎞ = (1)( 2) ( e−1 ) = 2e
1
∫ 0 (1 − x )
Integrals do not depend on the variable that is used and so
3
= −4.
= 0 and y′′(0) = −4 and no other option works.
38. C
3
x =0
f ( x ) dx = ∫
2
(
)
3
1 ⎞
⎛
8 − x 2 dx + ∫ x 2 dx = ⎜ 8 x − x3 ⎟
2
3 ⎠
⎝
2
n
1 n
u
0
∫
1 3
x
3
1
for each.
n +1
0
1
1
0
dx = ∫ u n ( − du ) = ∫ u n du .
du is the same as
3
1 n
x
0
∫
41. D
∫ −1
42. D
y = x3 − 3 x 2 + k , y′ = 3 x 2 − 6 x = 3 x( x − 2) . So f has a relative maximum at (0, k ) and a
relative minimum at (2, k − 4) . There will be 3 distinct x-intercepts if the maximum and
minimum are on the opposite sides of the x-axis. We want k − 4 < 0 < k ⇒ 0 < k < 4 .
43. D
∫ sin ( 2 x + 3) dx = − 2 cos ( 2 x + 3) + C
−1
−1
+
2
= 27 1
dx .
3
1
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164
1969 Calculus AB Solutions
44. C
Since cos 2 A = 2 cos 2 A − 1 , we have 3 − 2 cos 2
2π
=3
⎛ 2π ⎞
⎜ ⎟
⎝ 3 ⎠
expression has period
45. D
πx
2π x
= 3 − (1 + cos
) and the latter
3
3
Let y = f ( x3 ) . We want y′′ where f ′( x) = g ( x) and f ′′( x) = g ′( x) = f ( x 2 )
y = f ( x3 )
y′ = f ′( x3 ) ⋅ 3x 2
(
)
y′′ = 3 x 2 f ′′( x3 ) ⋅ 3x 2 + f ′( x3 ) ⋅ 6 x
= 9 x 4 f ′′( x3 ) + 6 x f ′( x3 ) = 9 x 4 f (( x3 ) 2 ) + 6 x g ( x3 ) = 9 x 4 f ( x6 ) + 6 x g ( x3 )
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165
1973 AP Calculus AB: Section I
90 Minutes—No Calculator
Note: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e).
1.
2.
∫ (x
3
)
− 3 x dx =
(A)
3x 2 − 3 + C
(D)
x4
− 3x + C
4
5 x 2 + 15 x + 25
(D) 225
(B)
5 x3 + 15 x 2 + 20 x + 25
(E)
5
(C) 1125
1
e
(B)
2
2
e
2
(C)
4
e
2
(D)
1
e
4
(E)
4
e4
If f ( x) = x + sin x , then f ′( x) =
(D)
sin x − x cos x
(B)
1 − cos x
(E)
sin x + x cos x
(C)
cos x
y =1
If f ( x) = e x , which of the following lines is an asymptote to the graph of f ?
(A)
6.
x4
− 3x 2 + C
3
( )
(A) 1 + cos x
5.
(E)
x 4 3x 2
−
+C
4
2
(C)
The slope of the line tangent to the graph of y = ln x 2 at x = e 2 is
(A)
4.
4 x4 − 6 x2 + C
If f ( x) = x3 + 3 x 2 + 4 x + 5 and g ( x) = 5, then g ( f ( x) ) =
(A)
3.
(B)
y=0
If f ( x) =
(A) –1
(B)
x=0
(C)
y=x
(D)
y = −x
(E)
0
(D)
1
2
(E) 1
x −1
for all x ≠ −1, then f ′(1) =
x +1
(B)
−
1
2
(C)
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20
1973 AP Calculus AB: Section I
7.
8.
Which of the following equations has a graph that is symmetric with respect to the origin?
y=
(D)
y = ( x − 1) + 1
3
(B)
y = − x5 + 3 x
(E)
y = x2 + 1 − 1
(
)
(C)
y = x4 − 2 x2 + 6
2
A particle moves in a straight line with velocity v(t ) = t 2 . How far does the particle move between
times t = 1 and t = 2?
(A)
9.
x +1
x
(A)
1
3
(B)
If y = cos 2 3 x , then
7
3
(C)
3
(D) 7
(E) 8
dy
=
dx
(A)
−6sin 3 x cos 3 x
(B)
−2 cos 3x
(D)
6 cos 3x
(E)
2sin 3 x cos 3 x
x 4 x5
−
attains its maximum value at x =
10. The derivative of f ( x) =
3
5
4
(A) –1
(B) 0
(C) 1
(D)
3
(C)
2 cos 3x
(E)
5
3
11. If the line 3x − 4 y = 0 is tangent in the first quadrant to the curve y = x3 + k , then k is
(A)
1
2
(B)
1
4
(C)
0
(D)
−
1
8
(E)
−
1
2
12. If f ( x) = 2 x3 + Ax 2 + Bx − 5 and if f (2) = 3 and f (−2) = −37 , what is the value of A + B ?
(A) –6
(E)
(B) –3
(C)
–1
(D) 2
It cannot be determined from the information given.
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21
1973 AP Calculus AB: Section I
13. The acceleration α of a body moving in a straight line is given in terms of time t by α = 8 − 6t . If
the velocity of the body is 25 at t = 1 and if s (t ) is the distance of the body from the origin at time
t, what is s (4) − s (2) ?
(A) 20
(B) 24
14. If f ( x) = x
1
3
( x − 2)
2
3
(C)
28
(D) 32
(E) 42
for all x, then the domain of f ′ is
(A)
{x
x ≠ 0}
(B)
{x
x > 0}
(D)
{x
x ≠ 0 and x ≠ 2}
(E)
{x
x is a real number}
(C)
{x
0 ≤ x ≤ 2}
x
2
15. The area of the region bounded by the lines x = 0, x = 2, and y = 0 and the curve y = e is
(A)
e −1
2
(B)
e −1
(C)
2 ( e − 1)
(D)
2e − 1
(E)
2e
2t
3000e 5
16. The number of bacteria in a culture is growing at a rate of
per unit of time t. At t = 0 , the
number of bacteria present was 7,500. Find the number present at t = 5 .
(A) 1, 200e 2
(B)
3, 000e 2
(C)
7,500e 2
(D)
7,500e5
(E)
15, 000 7
e
7
17. What is the area of the region completely bounded by the curve y = − x 2 + x + 6 and the line
y =4?
(A)
18.
3
2
(B)
7
3
(C)
9
2
(D)
31
6
(E)
33
2
d
( arcsin 2 x ) =
dx
(A)
(D)
−1
2 1 − 4x 2
2
1 − 4x 2
(B)
(E)
−2
4 x2 −1
(C)
1
2 1 − 4x 2
2
4 x2 −1
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22
1973 AP Calculus AB: Section I
19. Suppose that f is a function that is defined for all real numbers. Which of the following conditions
assures that f has an inverse function?
(A) The function f is periodic.
(B)
The graph of f is symmetric with respect to the y-axis.
(C)
The graph of f is concave up.
(D) The function f is a strictly increasing function.
(E)
The function f is continuous.
20. If F and f are continuous functions such that F ′( x) = f ( x) for all x, then
21.
(A)
F ′(a ) − F ′(b)
(B)
F ′(b) − F ′(a )
(C)
F (a) − F (b)
(D)
F (b) − F (a)
(E)
none of the above
1
∫ 0 ( x + 1) e
(A)
x2 +2 x
e3
2
b
∫a
f ( x) dx is
dx =
(B)
e3 − 1
2
(C)
e4 − e
2
(D)
e3 − 1
(E)
e4 − e
22. Given the function defined by f ( x) = 3 x5 − 20 x3 , find all values of x for which the graph of f is
concave up.
(A)
x>0
(B)
− 2 < x < 0 or x > 2
(C)
−2 < x < 0 or x > 2
(D)
x> 2
(E)
−2 < x < 2
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23
1973 AP Calculus AB: Section I
23.
1 ⎛ 2+h⎞
ln ⎜
⎟ is
h→0 h ⎝ 2 ⎠
lim
(A)
e2
(B) 1
(C)
1
2
(D) 0
(E) nonexistent
24. Let f ( x) = cos ( arctan x ) . What is the range of f ?
25.
(A)
⎧
π
⎨x − < x <
2
⎩
(D)
{x
π 4
∫0
(A)
π⎫
⎬
2⎭
− 1 < x < 1}
(B)
{x
0 < x ≤ 1}
(E)
{x
− 1 ≤ x ≤ 1}
(C)
1
3
(C)
{x
0 ≤ x ≤ 1}
(E)
π
+1
4
tan 2 x dx =
π
−1
4
(B) 1 −
π
4
2 −1
(D)
26. The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant
when the surface area S becomes 100π square inches, what is the rate of increase, in cubic inches
4
⎛
⎞
per second, in the volume V ? ⎜ S = 4π r 2 and V = π r 3 ⎟
3
⎝
⎠
(A) 10π
27.
2x
12
∫0
(B) 12π
1− x
(A) 1 −
2
3
2
(C)
22.5 π
(D)
25 π
(E)
30 π
(C)
π
6
(D)
π
−1
6
(E)
2− 3
dx =
(B)
1 3
ln
2 4
28. A point moves in a straight line so that its distance at time t from a fixed point of the line is
8t − 3t 2 . What is the total distance covered by the point between t = 1 and t = 2?
(A) 1
(B)
4
3
(C)
5
3
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(D) 2
(E) 5
24
1973 AP Calculus AB: Section I
1
. The maximum value attained by f is
2
29. Let f ( x) = sin x −
1
2
(A)
30.
(B) 1
∫1
x−4
(A)
−
2
x2
3
2
(D)
π
2
(E)
3π
2
(C)
ln 2
(D) 2
(E)
ln 2 + 2
(C)
8
(D) 16
(E) 32
)
5
(C) 5x − + C
x
dx =
1
2
(B)
( )
ln 2 − 2
a
, then a =
4
31. If log a 2a =
(A) 2
32.
(C)
(B) 4
5
∫ 1 + x 2 dx =
−10 x
(
(A)
(1 + x2 )
+C
(B)
5
ln 1 + x 2 + C
2x
(D)
5arctan x + C
(E)
5ln 1 + x 2 + C
2
(
)
33. Suppose that f is an odd function; i.e., f (− x) = − f ( x) for all x. Suppose that f ′ ( x0 ) exists.
Which of the following must necessarily be equal to f ′ ( − x0 ) ?
(A)
f ′ ( x0 )
(B)
− f ′ ( x0 )
(C)
1
f ′ ( x0 )
(D)
−1
f ′ ( x0 )
(E)
None of the above
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25
1973 AP Calculus AB: Section I
x over the interval 0 ≤ x ≤ 2 is
34. The average value of
1
2
3
(A)
(B)
1
2
2
(C)
2
2
3
(D) 1
35. The region in the first quadrant bounded by the graph of y = sec x, x =
(E)
4
2
3
π
, and the axes is rotated
4
about the x-axis. What is the volume of the solid generated?
π2
4
(A)
36. If y = enx , then
dx n
π −1
(C)
π
(D)
2π
(E)
8π
3
n !e nx
(C)
n e nx
(D)
nn e x
(E)
n !e x
3 + e4 x
(D)
4 + e4 x
(E)
2 x2 + 4
(E)
–5
)
(E)
( 4,8)
(C)
cos 2 ( xy )
=
(B)
dy
= 4 y and if y = 4 when x = 0, then y =
dx
4e4 x
(A)
38. If
dny
n n enx
(A)
37. If
(B)
2
∫1
(A)
(B)
e4 x
(C)
f ( x − c ) dx = 5 where c is a constant, then
5+c
(B) 5
(C)
2−c
∫ 1−c f ( x ) dx =
5−c
(D) c − 5
39. The point on the curve 2 y = x 2 nearest to ( 4,1) is
(A)
( 0, 0 )
40. If tan( xy ) = x , then
(B)
( 2, 2 )
(C)
(
)
2,1
(D)
(2
2, 4
dy
=
dx
(A)
1 − y tan( xy ) sec( xy )
x tan( xy ) sec( xy )
(D)
cos 2 ( xy )
x
(B)
sec 2 ( xy ) − y
x
(E)
cos 2 ( xy ) − y
x
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26
1973 AP Calculus AB: Section I
⎧ x + 1 for x < 0,
41. Given f ( x) = ⎨
⎩cos π x for x ≥ 0,
(A)
1 1
+
2 π
(B)
−
1
∫ −1 f ( x) dx =
1
2
(C)
1 1
−
2 π
(D)
1
2
(E)
1
− +π
2
42. Calculate the approximate area of the shaded region in the figure by the trapezoidal rule, using
4
5
divisions at x = and x = .
3
3
(A)
50
27
(B)
251
108
(C)
7
3
(D)
127
54
(E)
77
27
(C)
−
⎛ x⎞
43. If the solutions of f ( x) = 0 are –1 and 2, then the solutions of f ⎜ ⎟ = 0 are
⎝2⎠
(A)
−1 and 2
(D)
−
1
and 1
2
44. For small values of h, the function
4
1
5
and
2
2
(B)
−
(E)
−2 and 4
3
3
and
2
2
16 + h is best approximated by which of the following?
(A)
4+
h
32
(B)
2+
h
32
(D)
4−
h
32
(E)
2−
h
32
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(C)
h
32
27
1973 AP Calculus AB: Section I
45. If f is a continuous function on [ a, b ] , which of the following is necessarily true?
(A)
f ′ exists on ( a , b ) .
(B)
If f ( x0 ) is a maximum of f, then f ′ ( x0 ) = 0 .
(C)
⎛
⎞
lim f ( x) = f ⎜ lim x ⎟ for x0 ∈ ( a , b )
x→ x0
⎝ x→ x0 ⎠
(D)
f ′( x) = 0 for some x ∈ [ a , b ]
(E)
The graph of f ′ is a straight line.
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1973 Calculus AB Solutions
1 4 3 2
x − x +C
4
2
1.
E
∫ (x
2.
E
g ( x) = 5 ⇒ g ( f ( x) ) = 5
3.
B
y = ln x 2 ; y′ =
4.
A
f ( x) = x + sin x ;
5.
A
lim e x = 0 ⇒ y = 0 is a horizontal asymptote
6.
D
7.
B
3
− 3x) dx =
2x
x
2
=
2
2
. At x = e 2 , y′ = 2 .
x
e
f ′( x) = 1 − cos x
x→−∞
f ′( x) =
(1)( x + 1) − ( x − 1)(1)
( x + 1)
2
, f ′(1) =
2 1
=
4 2
Replace x with (− x) and see if the result is the opposite of the original. This is true for B.
−(− x)5 + 3(− x) = x5 − 3 x = −(− x5 + 3 x) .
8.
B
9.
A
10. C
11. B
Distance = ∫
2
1
2
1
t 2 dx = ∫ t 2 dt = t 3
1
3
1
7
= (23 − 13 ) =
1 3
3
d
d
( cos 3x ) = 2 cos 3x ⋅ ( − sin 3x ) ⋅ ( 3x ) = 2 cos 3x ⋅ ( − sin 3x ) ⋅ ( 3)
dx
dx
y′ = −6sin 3x cos 3x
y′ = 2 cos 3x ⋅
x 4 x5
4 x3
− ; f ′( x) =
− x 4 ; f ′′ ( x ) = 4 x 2 − 4 x3 = 4 x 2 (1 − x )
3
5
3
f ′′ > 0 for x < 1 and f ′′ < 0 for x > 1 ⇒ f ′ has its maximum at x = 1 .
f ( x) =
Curve and line have the same slope when 3x 2 =
⎛1 3
tangency is ⎜ ,
⎝ 2 8
12. C
2
3
1
⇒ x = . Using the line, the point of
4
2
3
3 ⎛1⎞
1
⎞
⎟ . Since the point is also on the curve, = ⎜ ⎟ + k ⇒ k = .
8 ⎝2⎠
4
⎠
Substitute the points into the equation and solve the resulting linear system.
3 = 16 + 4 A + 2 B − 5 and − 37 = −16 + 4 A − 2 B − 5 ; A = −3, B = 2 ⇒ A + B = −1 .
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1973 Calculus AB Solutions
13. D
v(t ) = 8t − 3t 2 + C and v(1) = 25 ⇒ C = 20 so v(t ) = 8t − 3t 2 + 20 .
4
4
2
2
s (4) − s (2) = ∫ v ( t ) dt = (4t 2 − t 3 + 20t )
14. D
f ( x) = x1 3 ( x − 2 )
= 32
23
2
1
1
( x − 2 )−1 3 + ( x − 2 )2 3 ⋅ x −2 3 = x −2 3 ( x − 2 )−1 3 ( 3x − 2 )
3
3
3
′
f is not defined at x = 0 and at x = 2 .
f ′ ( x ) = x1 3 ⋅
2
x
2
e
x 2
2
2e
0
= 2 ( e − 1)
15. C
Area = ∫
16. C
t
t
dN
= 3000e 5 , N = 7500e 5 + C and N (0) = 7500 ⇒ C = 0
dt
0
dx =
2
N
17.
C
2
t
5
= 7500e
2
, N ( 5 ) = 7500e2
Determine where the curves intersect. − x 2 + x + 6 = 4 ⇒ x 2 − x − 2 = 0
( x − 2)( x + 1) = 0 ⇒ x = −1, x = 2 . Between these two x values the parabola lies above the
line y = 4.
2
1
⎛ 1
⎞ 2 9
Area = ∫ (− x 2 + x + 6) − 4 dx = ⎜ − x3 + x 2 + 2 x ⎟ =
−1
2
⎝ 3
⎠ −1 2
(
)
18. D
d
1
d
2
2
⋅ ( 2x) =
=
( arcsin 2 x ) =
2
2
dx
1 − 4 x2
1 − ( 2 x ) dx
1− ( 2x)
19. D
If f is strictly increasing then it must be one to one and therefore have an inverse.
20. D
By the Fundamental Theorem of Calculus,
1
2
b
∫a
f ( x) dx = F (b) − F ( a) where F ′( x) = f ( x) .
(
1 1 x2 +2 x
1 2
e
(2 x + 2) dx ) = e x + 2 x
(
∫
2 0
2
)
1
1 3 0
e3 − 1
e −e =
2
2
(
)
21. B
x
∫ 0 ( x + 1) e
22. B
f ( x) = 3 x5 − 20 x3 ; f ′( x) = 15 x 4 − 60 x 2 ; f ′′( x) = 60 x3 − 120 x = 60 x x 2 − 2
+2 x
dx =
0
=
(
)
The graph of f is concave up where f ′′ > 0 : f ′′ > 0 for x > 2 and for − 2 < x < 0 .
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1973 Calculus AB Solutions
23. C
ln ( 2 + h ) − ln 2
1
1
= f ′ ( 2 ) where f ( x ) = ln x ; f ′ ( x ) = ⇒ f ′ ( 2 ) =
h →0
h
x
2
24. B
f ( x) = cos ( arctan x ) ; −
lim
π
π
< arctan x < and the cosine in this domain takes on all values in
2
2
the interval (0,1].
π
4
0
∫
26. E
dV
dr
dr
= 4πr 2 ⋅ = S ⋅ = 100π ( 0.3) = 30π
dt
dt
dt
27. E
28. C
tan x dx = ∫
π
4
0
25. B
2
1
2x
∫0 2
1 − x2
(sec 2 x − 1) dx = (tan x − x)
dx = − ∫
1
0
)
( −2 x dx ) = − 2 (
v ( t ) = 8 − 6t changes sign at t =
)
= 2− 3
4
⎛4⎞
⎛4⎞ 5
. Distance = x(1) − x ⎜ ⎟ + x(2) − x ⎜ ⎟ = .
3
⎝3⎠
⎝3⎠ 3
2
1
30. B
π
4
0
Alternative Solution: Distance = ∫
29. C
= 1−
1
1 2
1 − x2 2
1
−
2 1 − x2 2
(
π4
0
v ( t ) dt = ∫
2
1
8 − 6t dt =
5
3
3
1 1
1
3
is .
−1 ≤ sin x ≤ 1 ⇒ − ≤ sin x − ≤ ; The maximum for sin x −
2
2 2
2
2
2
∫1
x−4
x2
2 ⎛1
4⎞
⎞
⎛
dx = ∫ ⎜ − 4 x −2 ⎟ dx = ⎜ ln x + ⎟
1 ⎝x
x⎠
⎠
⎝
( )
a
2
1
= ( ln 2 + 2 ) − ( ln1 + 4 ) = ln 2 − 2
1
a
1
= ⇒ log a 2 = ⇒ 2 = a 4 ; a = 16
4
4
31. D
log a 2
32. D
∫ 1 + x2 dx = 5 ∫ 1 + x 2 dx = 5 tan
33. A
f (− x) = − f ( x) ⇒ f ′(− x) ⋅ (−1) = − f ′( x) ⇒ f ′(− x) = − f ′( x) thus f ′(− x0 ) = − f ′( x0 ) .
5
1
−1
( x) + C
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174
1973 Calculus AB Solutions
3 2
34. C
35. C
1 2
1 2
x dx = ⋅ x 2
∫
2 0
2 3
Washers:
∑πr
Volume = π∫
36. A
37. A
38. B
39.
B
40. E
41. D
π
4
0
2
0
3
1
2
= ⋅ 22 =
2
3
3
∆x where r = y = sec x .
sec 2 x dx = π tan x
π4
0
= π(tan
π
− tan 0) = π
4
y = e nx , y′ = ne nx , y′′ = n 2 enx ," , y ( n) = n n e nx
dy
= 4 y , y (0) = 4 . This is exponential growth. The general solution is y = Ce 4 x . Since
dx
y (0) = 4 , C = 4 and so the solution is y = 4e4 x .
Let z = x − c . Then 5 = ∫
2
1
f ( x − c ) dx = ∫
2−c
1−c
f ( z ) dz
1 2
x ) on
2
2 ⎛1
⎞
the curve to the point (4,1) . The distance L satisfies the equation L2 = ( x − 4 ) + ⎜ x 2 − 1⎟ .
⎝2
⎠
Determine where L is a maximum by examining critical points. Differentiating with respect
dL
dL
⎛1
⎞
to x, 2 L ⋅
changes sign from positive to negative at
= 2( x − 4) + 2 ⎜ x 2 − 1⎟ x = x3 − 8 .
dx
dx
⎝2
⎠
x = 2 only. The point on the curve has coordinates (2, 2) .
Use the distance formula to determine the distance, L, from any point ( x, y ) = ( x ,
2
sec ( xy ) ⋅ ( xy′ + y ) = 1, xy′ sec2 ( xy ) + y sec2 ( xy ) = 1 , y′ =
1
∫ −1
0
1
−1
0
f ( x) dx = ∫ ( x + 1) dx + ∫ cos(π x) dx =
=
1
( x + 1) 2
2
0
1 − y sec2 ( xy )
cos 2 ( xy ) − y
=
x
x sec2 ( xy )
1
+ sin(π x)
−1 π
1
0
1 1
1
+ ( sin π − sin 0 ) =
2 π
2
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175
1973 Calculus AB Solutions
42. D
2
2
⎞ 127
1
1 1⎛
⎛4⎞
⎛5⎞
∆x = ; T = ⋅ ⎜12 + 2 ⎜ ⎟ + 2 ⎜ ⎟ + 22 ⎟ =
⎟ 54
3
2 3 ⎜⎝
⎝3⎠
⎝3⎠
⎠
43. E
Solve
x
x
= −1 and
= 2; x = −2, 4
2
2
3
44. B
45. C
1 −
1
Use the linearization of f ( x) = x at x = 16 . f ′( x) = x 4 , f ′(16) =
4
32
1
h
L( x) = 2 + ( x − 16); f (16 + h) ≈ L(16 + h) = 2 +
32
32
4
This uses the definition of continuity of f at x = x0 .
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176
1985 AP Calculus AB: Section I
90 Minutes—No Calculator
Notes: (1) In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e).
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f (x) is a real number.
1.
2
∫1
x −3 dx =
−
(A)
2.
7
8
5.
(C)
15
64
(D)
3
8
(E)
15
16
4
If y =
(A)
4.
3
4
If f ( x) = ( 2 x + 1) , then the 4th derivative of f ( x) at x = 0 is
(A) 0
3.
−
(B)
If
(B) 24
3
4+ x
2
( 4 + x2 )
2
48
(D) 240
(E) 384
dy
=
dx
, then
−6 x
(C)
(B)
3x
( 4 + x2 )
2
(C)
6x
( 4 + x2 )
2
(D)
−3
( 4 + x2 )
2
(E)
3
2x
dy
= cos ( 2 x ) , then y =
dx
(A)
1
− cos ( 2 x ) + C
2
1
(B) − cos 2 ( 2 x ) + C
2
(D)
1 2
sin ( 2 x ) + C
2
(E)
lim
n→∞
4n 2
n 2 + 10, 000n
(A) 0
(C)
1
sin ( 2 x ) + C
2
1
− sin ( 2 x ) + C
2
is
(B)
1
2,500
(C)
1
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(D) 4
(E) nonexistent
38
1985 AP Calculus AB: Section I
6.
If f ( x) = x, then f ′(5) =
(A) 0
7.
ln 3 + ln1
1
(D) 5
(E)
25
2
ln 8
ln 2
(B)
(C)
4
∫1
et dt
(D)
4
∫1
ln x dx
(E)
4
∫1
1
dt
t
⎛ x⎞
The slope of the line tangent to the graph of y = ln ⎜ ⎟ at x = 4 is
⎝2⎠
1
8
(A)
9.
(C)
Which of the following is equal to ln 4 ?
(A)
8.
1
5
(B)
If
1
∫ −1
(A)
1
4
(B)
2
0
e − x dx = k , then
∫ −1
−2k
−k
(B)
( x −1) , then
2
10. If y = 10
(A)
( ln10 )10(
(D)
2 x ( ln10 )10
(C)
1
2
(C)
−
(D) 1
(E) 4
2
e − x dx =
k
2
(D)
k
2
(E)
2k
dy
=
dx
)
(B)
( 2 x )10(
( x −1)
(E)
x 2 ( ln10 )10
x 2 −1
)
x 2 −1
(C)
(
)
( x −2)
x 2 − 1 10
2
( x −1)
2
2
11. The position of a particle moving along a straight line at any time t is given by
s (t ) = t 2 + 4t + 4 . What is the acceleration of the particle when t = 4 ?
(A) 0
(B) 2
(
(C)
)
4
(D) 8
(E) 12
( )
12. If f ( g ( x) ) = ln x 2 + 4 , f ( x) = ln x 2 , and g ( x) > 0 for all real x, then g (x) =
(A)
1
2
x +4
(B)
1
2
x +4
(C)
x2 + 4
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(D)
x2 + 4
(E)
x+2
39
1985 AP Calculus AB: Section I
13. If x 2 + xy + y 3 = 0 , then, in terms of x and y,
(A)
−
2x + y
x + 3y
2
(B)
x + 3y2
−
2x + y
(C)
dy
=
dx
−2 x
1+ 3y
(D)
2
14. The velocity of a particle moving on a line at time t is v
meters did the particle travel from t = 0 to t = 4?
(A) 32
(B) 40
(C)
1
= 3t 2
64
−2 x
x + 3y
3
2
+ 5t
2
−
2x + y
x + 3 y2 −1
meters per second. How many
(D) 80
(
(E)
(E) 184
)
15. The domain of the function defined by f ( x) = ln x 2 − 4 is the set of all real numbers x such that
(A)
x <2
x ≤2
(B)
x >2
(C)
(D)
x ≥2
(E)
x is a real number
16. The function defined by f ( x) = x3 − 3 x 2 for all real numbers x has a relative maximum at x =
(A)
17.
1
∫ 0 xe
−2
−x
(B) 0
(C)
1
(D) 2
(E) 4
(C)
1 − 2e −1
(D) 1
(E)
dx =
(A) 1 − 2e
(B)
−1
2e − 1
18. If y = cos 2 x − sin 2 x , then y′ =
(A) −1
(B)
(C) −2sin ( 2x )
0
−2 ( cos x + sin x )
(D)
(E) 2 ( cos x − sin x )
19. If f ( x1 ) + f ( x2 ) = f ( x1 + x2 ) for all real numbers x1 and x2 , which of the following could
define f ?
(A) f ( x) = x + 1
(B)
f ( x) = 2 x
(C)
f ( x) =
1
x
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(D) f ( x) = e x
(E)
f ( x) = x 2
40
1985 AP Calculus AB: Section I
20. If y = arctan ( cos x ) , then
dy
=
dx
− sin x
(A)
(B) − ( arcsec ( cos x ) ) sin x
2
2
1 + cos x
1
(D)
( arccos x )
2
(E)
+1
22.
1 + cos 2 x
1 − x2
is { x : x > 1} , what is the range of f ?
{ x : −∞ < x < −1}
(B)
{ x : −∞ < x < 0}
(D)
{ x : −1 < x < ∞}
(E)
{ x : 0 < x < ∞}
(C)
{ x : −∞ < x < 1}
x2 −1
dx =
x +1
1
2
(A)
23.
1
(A)
∫1
( arcsec ( cos x ) )2
1
21. If the domain of the function f given by f ( x) =
2
(C)
(B) 1
5
2
(C)
2
(D)
(E)
(C)
0
(D) 2
(E) 6
(C)
0
(D) 4
(E) 12
ln 3
d ⎛ 1 1
⎞
− + x 2 ⎟ at x = −1 is
⎜
3
dx ⎝ x
x
⎠
−6
(A)
24. If
∫ −2 ( x
(A)
2
(B)
7
−4
)
+ k dx = 16, then k =
−12
(B)
−4
25. If f ( x) = e x , which of the following is equal to f ′(e)?
(A)
lim
e x+h
h →0 h
(B) lim
(D)
e x+h − 1
h →0
h
(E) lim
lim
e x + h − ee
h →0
h
ee + h − e
h →0
h
(C) lim
ee + h − ee
h →0
h
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41
1985 AP Calculus AB: Section I
26. The graph of y 2 = x 2 + 9 is symmetric to which of the following?
I.
II.
III.
The x-axis
The y-axis
The origin
(A) I only
27.
3
∫0
(B) II only
(C)
III only
(D)
I and II only
(E)
I, II, and III
x − 1 dx =
(A) 0
(B)
3
2
(C)
2
(D)
5
2
(E) 6
28. If the position of a particle on the x-axis at time t is −5t 2 , then the average velocity of the particle
for 0 ≤ t ≤ 3 is
(A)
−45
(B)
−30
(C)
−15
(D)
−10
(E)
−5
29. Which of the following functions are continuous for all real numbers x ?
I.
II.
III.
y=
2
x3
y = ex
y = tan x
(A) None
30.
(B) I only
(C)
II only
(D) I and II
(E) I and III
∫ tan ( 2x ) dx =
(A)
−2 ln cos(2 x) + C
(B)
1
− ln cos(2 x) + C
2
(D)
2 ln cos(2 x) + C
(E)
1
sec(2 x) tan(2 x) + C
2
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(C)
1
ln cos(2 x) + C
2
42
1985 AP Calculus AB: Section I
1
31. The volume of a cone of radius r and height h is given by V = π r 2 h . If the radius and the height
3
1
both increase at a constant rate of centimeter per second, at what rate, in cubic centimeters per
2
second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters?
(A)
32.
∫
π
3
0
(A)
1
π
2
(B) 10 π
(C)
24 π
(D) 54 π
(E)
108 π
sin ( 3x ) dx =
−2
(B)
−
2
3
(C) 0
(D)
2
3
(E) 2
33. The graph of the derivative of f is shown in the figure above. Which of the following could be the
graph of f ?
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43
1985 AP Calculus AB: Section I
34. The area of the region in the first quadrant that is enclosed by the graphs of y = x3 + 8 and
y = x + 8 is
(A)
1
4
(B)
1
2
(C)
3
4
(D) 1
(E)
65
4
35. The figure above shows the graph of a sine function for one complete period. Which of the
following is an equation for the graph?
(A)
⎛π ⎞
y = 2sin ⎜ x ⎟
⎝2 ⎠
(B) y = sin ( π x )
(D)
y = 2sin ( π x )
(E) y = sin ( 2 x )
(C) y = 2sin ( 2 x )
36. If f is a continuous function defined for all real numbers x and if the maximum value of f ( x) is 5
and the minimum value of f ( x) is −7 , then which of the following must be true?
I.
The maximum value of f ( x ) is 5.
II.
The maximum value of f ( x) is 7.
III.
The minimum value of f ( x ) is 0.
(A) I only
37.
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III
(D) 1
∞
lim ( x csc x ) is
x →0
(A)
−∞
(B) –1
(C) 0
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(E)
44
1985 AP Calculus AB: Section I
38. Let f and g have continuous first and second derivatives everywhere. If f ( x ) ≤ g ( x ) for all real
x, which of the following must be true?
f ′( x) ≤ g ′( x) for all real x
f ′′( x) ≤ g ′′( x) for all real x
I.
II.
1
∫0
III.
f ( x) dx ≤
(A) None
39. If f ( x) =
(A)
(B)
(C)
(D)
(E)
f
f
f
f
f
1
∫ 0 g ( x) dx
(B) I only
(C)
III only
(D)
I and II only
(E) I, II, and III
ln x
, for all x > 0, which of the following is true?
x
is increasing for all x greater than 0.
is increasing for all x greater than 1.
is decreasing for all x between 0 and 1.
is decreasing for all x between 1 and e.
is decreasing for all x greater than e.
40. Let f be a continuous function on the closed interval [ 0, 2] . If 2 ≤ f ( x) ≤ 4, then the greatest
possible value of
(A) 0
2
∫0
f ( x) dx is
(B) 2
(C) 4
(D) 8
(E) 16
41. If lim f ( x) = L, where L is a real number, which of the following must be true?
x →a
(A)
f ′(a ) exists.
(B)
f ( x) is continuous at x = a.
(C)
f ( x) is defined at x = a.
(D)
f (a) = L
(E)
None of the above
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45
1985 AP Calculus AB: Section I
42.
d
dx
(A)
(D)
x
∫2
1 + t 2 dt =
x
1+ x
(B)
2
x
1 + x2
−
1
5
(E)
1 + x2 − 5
1
2 1 + x2
−
(C)
1 + x2
1
2 5
43. An equation of the line tangent to y = x3 + 3 x 2 + 2 at its point of inflection is
(A)
y = −6 x − 6
(B) y = −3x + 1
(D)
y = 3x − 1
(E) y = 4 x + 1
(C) y = 2 x + 10
44. The average value of f ( x) = x 2 x3 + 1 on the closed interval [ 0, 2] is
(A)
26
9
(B)
13
3
(C)
26
3
(D) 13
(E) 26
45. The region enclosed by the graph of y = x 2 , the line x = 2, and the x-axis is revolved about the
y -axis. The volume of the solid generated is
(A)
8π
(B)
32
π
5
(C)
16
π
3
(D)
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4π
(E)
8
π
3
46
1985 Calculus AB Solutions
2 −3
x dx
1
1
= − x −2
2
2
1⎛1 ⎞ 3
= − ⎜ − 1⎟ = .
1
2⎝4 ⎠ 8
1.
D
∫
2.
E
f ′( x) = 4(2 x + 1)3 ⋅ 2, f ′′(1) = 4 ⋅ 3(2 x + 1) 2 ⋅ 22 , f ′′′(1) = 4 ⋅ 3 ⋅ 2(2 x + 1)1 ⋅ 23 ,
f (4) (1) = 4!⋅ 24 = 384
3.
A
y = 3(4 + x 2 ) −1 so y′ = −3(4 + x 2 ) −2 (2 x) =
−6 x
(4 + x 2 ) 2
( 4 + x ) (0) − 3(2 x) = −6 x
Or using the quotient rule directly gives y′ =
(4 + x )
(4 + x )
2
2
1
C
∫ cos(2 x) dx = 2 ∫ cos(2 x) (2 dx) = 2 sin(2 x) + C
5.
D
lim
6.
C
f ′( x) = 1 ⇒ f ′(5) = 1
7.
E
∫1
8.
B
1
1
⎛ x⎞
y = ln ⎜ ⎟ = ln x − ln 2, y′ = , y′(4) =
4
x
⎝2⎠
9.
D
Since e − x is even,
n→∞
4
2
n + 10000n
2
2
1
4.
4n 2
2
4
=4
10000
1+
n
= lim
n→∞
1
4
dt = ln t 1 = ln 4 − ln1 = ln 4
t
2
2
−1)
0
∫ −1 e
− x2
(
dx =
1 1 − x2
1
e dx = k
∫
2 −1
2
)
2
d
( x 2 − 1) = 2 x ⋅10( x −1) ⋅ ln(10)
dx
10. D
y′ = 10( x
11. B
v(t ) = 2t + 4 ⇒ a(t ) = 2 ∴ a(4) = 2
12. C
f ( g ( x) ) = ln g ( x) 2 = ln x 2 + 4 ⇒ g ( x) = x 2 + 4
⋅ ln(10) ⋅
(
)
(
)
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183
1985 Calculus AB Solutions
13. A
2 x + x ⋅ y′ + y + 3 y 2 ⋅ y′ = 0 ⇒ y′ = −
4
∫0
2x + y
x + 3y2
v ( t ) dt = ∫
4
3⎞
5⎞
⎛ 1
⎛ 3
2
2
2
⎜ 3t + 5t ⎟ dt = ⎜ 2t + 2t 2 ⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
4
14. D
Since v(t ) ≥ 0, distance =
15. C
x2 − 4 > 0 ⇒ x > 2
16. B
f ′( x) = 3 x 2 − 6 x = 3 x( x − 2) changes sign from positive to negative only at x = 0.
17. C
0
0
= 80
Use the technique of antiderivatives by parts:
u=x
dv = e− x dx
du = dx
v = −e − x
(
− xe − x + ∫ e − x dx = − xe− x − e− x
) 0 = 1 − 2e − 1
1
18. C
y = cos 2 x − sin 2 x = cos 2 x , y′ = −2sin 2 x
19. B
Quick solution: lines through the origin have this property.
Or, f ( x 1 ) + f ( x 2 ) = 2 x 1 + 2 x 2 = 2( x 1 + x 2 ) = f ( x 1 + x 2 )
20. A
dy
1
d
− sin x
=
⋅ ( cos x ) =
2
dx 1 + cos x dx
1 + cos 2 x
21. B
x > 1 ⇒ x 2 > 1 ⇒ f ( x) < 0 for all x in the domain. lim f ( x) = 0 . lim f ( x) = −∞ . The only
x →∞
x →1
option that is consistent with these statements is (B).
2
2 ( x + 1)( x − 1)
2
x2 − 1
1
dx = ∫
dx = ∫ ( x − 1) dx = ( x − 1) 2
1
1
x +1
x +1
2
22. A
∫1
23. B
d −3
x − x −1 + x 2
dx
24. D
16 = ∫
(
2
−2
)
x =−1
( x 7 + k ) dx = ∫
(
= − 3 x −4 + x − 2 + 2 x
2
−2
x 7 dx + ∫
2
−2
)
x =−1
2
1
=
1
2
= − 3 + 1 − 2 = −4
k dx = 0 + ( 2 − (−2) ) k = 4k ⇒ k = 4
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184
1985 Calculus AB Solutions
25. E
26.
E
f ′(e) = lim
h →0
f (e + h ) − f (e )
ee + h − ee
= lim
h →0
h
h
I: Replace y with (− y ) : (− y ) 2 = x 2 + 9 ⇒ y 2 = x 2 + 9 , no change, so yes.
II: Replace x with (− x) : y 2 = (− x) 2 + 9 ⇒ y 2 = x 2 + 9 , no change, so yes.
III: Since there is symmetry with respect to both axes there is origin symmetry.
27. D
The graph is a V with vertex at x = 1 . The integral
gives the sum of the areas of the two triangles that the
V forms with the horizontal axis for x from 0 to 3.
These triangles have areas of 1/2 and 2 respectively.
28. C
Let x(t ) = −5t 2 be the position at time t. Average velocity =
29. D
The tangent function is not defined at x = π 2 so it cannot be continuous for all real
numbers. Option E is the only one that includes item III. In fact, the functions in I and II are a
power and an exponential function that are known to be continuous for all real numbers x.
30. B
∫ tan(2 x) dx = − 2 ∫
31. C
1
dV 1 ⎛
dr
dh ⎞ 1 ⎛
⎛1⎞
⎛ 1 ⎞⎞
V = π r 2h ,
= π ⎜ 2rh + r 2 ⎟ = π ⎜ 2(6)(9) ⎜ ⎟ + 62 ⎜ ⎟ ⎟ = 24π
3
dt 3 ⎝
dt
dt ⎠ 3 ⎝
⎝2⎠
⎝ 2 ⎠⎠
32. D
∫0
33. B
f ′ changes sign from positive to negative at x = –1 and therefore f changes from increasing
to decreasing at x = –1.
1
π3
x(3) − x(0) −45 − 0
=
= −15
3−0
3
−2sin(2 x)
1
dx = − ln cos(2 x) + C
cos(2 x)
2
1
sin(3x) dx = − cos(3x)
3
π3
=
0
−
1
2
( cos π − cos 0 ) =
3
3
Or f ′ changes sign from positive to negative at x = −1 and from negative to positive at
x = 1 . Therefore f has a local maximum at x = −1 and a local minimum at x = 1 .
34. A
⎛
3
3
∫ 0 ( ( x + 8) − ( x + 8) ) dx = ∫ 0 ( x − x ) dx = ⎜
1
1
1 2 1 4⎞
x − x ⎟
4 ⎠
⎝2
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1
0
=
1
4
185
1985 Calculus AB Solutions
35. D
The amplitude is 2 and the period is 2.
y = A sin Bx where A = amplitude = 2 and B =
36. B
2π
2π
=
=π
period 2
II is true since − 7 = 7 will be the maximum value of f ( x) . To see why I and III do not
⎧ 5 if
⎪
have to be true, consider the following: f ( x ) = ⎨− x if
⎪ −7 if
⎩
For f
37. D
38. C
x ≤ −5
−5 < x < 7
x≥7
( x ) , the maximum is 0 and the minimum is –7.
x
=1
x→0 sin x
lim x csc x = lim
x →0
To see why I and II do not have to be true consider f ( x) = sin x and g ( x) = 1 + e x . Then
f ( x) ≤ g ( x) but neither f ′( x) ≤ g ′( x) nor f ′′( x) < g ′′( x) is true for all real values of x.
III is true, since
f ( x) ≤ g ( x) ⇒ g ( x) − f ( x) ≥ 0 ⇒
39. E
f ′( x) =
2
1
1
1
∫ 0 ( g ( x) − f ( x) ) dx ≥ 0 ⇒ ∫ 0 f ( x) dx ≤ ∫ 0 g ( x) dx
1 1 1
1
⋅ − 2 ln x = 2 (1 − ln x) < 0 for x > e . Hence f is decreasing. for x > e .
x x x
x
2
40. D
f ( x) dx ≤ ∫ 4 dx = 8
∫0
41. E
Consider the function whose graph is the horizontal line y = 2 with a hole at x = a .
For this function lim f ( x) = 2 and none of the given statements are true.
0
x →a
42. C
This is a direct application of the Fundamental Theorem of Calculus: f ′( x) = 1 + x 2
43. B
y′ = 3 x 2 + 6 x , y′′ = 6 x + 6 = 0 for x = −1. y′(−1) = −3 . Only option B has a slope of –3.
44. A
1 2 2 3
x x +1
2 ∫0
(
)
1
2
(
) (
1 1 2
dx = ⋅ ∫ x3 + 1
2 3 0
1
2
) (
3 x 2 dx =
)
1 3
x +1
6
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3
2
⋅
2
3
2
0
=
26
9
186
1985 Calculus AB Solutions
45. A
Washers:
∑ π ( R 2 − r 2 ) ∆y
Volume = π ∫
4
0
(
where R = 2, r = x
)
4
1 ⎞
⎛
22 − x 2 dy = π ∫ (4 − y ) dy = π ⎜ 4 y − y 2 ⎟
0
2 ⎠
⎝
AP Calculus Multiple-Choice Question Collection
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4
0
= 8π
187
1988 AP Calculus AB: Section I
90 Minutes—No Calculator
Notes: (1) In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e).
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f (x) is a real number.
1.
2.
3.
If y = x 2e x , then
2 xe x
(B)
x x + 2e x
(D)
2x + ex
(E)
2x + e
What is the domain of the function f given by f ( x) =
(A)
{x :
(D)
{x :
x ≠ 3}
x ≥ 2 and x ≠ 3}
(B)
{x :
(E)
{x :
)
(C)
xe x ( x + 2 )
(C)
{x :
x2 − 4
?
x−3
x ≤ 2}
x ≥ 2}
x ≥ 2 and x ≠ 3}
A particle with velocity at any time t given by v(t ) = et moves in a straight line. How far does the
particle move from t = 0 to t = 2 ?
e2 − 1
The graph of y =
(A)
5.
(
(A)
(A)
4.
dy
=
dx
∫ sec
x<0
2
(B)
e −1
(C)
2e
(D)
e2
(E)
e3
3
(E)
x>2
(C)
cos 2 x + C
−5
is concave downward for all values of x such that
x−2
(B)
x<2
(C)
x<5
(D)
x>0
x dx =
(A)
tan x + C
(B)
csc 2 x + C
(D)
sec3 x
+C
3
(E)
2sec2 x tan x + C
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57
1988 AP Calculus AB: Section I
6.
7.
8.
If y =
ln x
dy
, then
=
x
dx
(A)
1
x
(B)
x dx
∫
3x 2 + 5
x
2
(C)
ln x − 1
x
(D)
2
1 − ln x
x
2
1 + ln x
(E)
x2
=
)
3
2
+C
(B)
1
3x 2 + 5
4
)
1
2
+C
(E)
3
3x 2 + 5
2
(
(A)
1
3x 2 + 5
9
(D)
1
3x 2 + 5
3
(
1
(
(
)
3
2
+C
)
1
2
+C
(
1
3x 2 + 5
(C)
12
)
1
2
+C
The graph of y = f ( x) is shown in the figure above. On which of the following intervals are
dy
d2y
> 0 and
<0?
dx
dx 2
I.
II.
III.
a< x<b
b<x<c
c<x<d
(A) I only
(B) II only
(C) III only
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(D) I and II
(E) II and III
58
1988 AP Calculus AB: Section I
9.
If x + 2 xy − y 2 = 2, then at the point (1,1) ,
3
2
(A)
10. If
k
∫0
(B)
1
2
dy
is
dx
−
(C)
0
(D)
(C)
3
(D) 9
3
2
(E) nonexistent
( 2kx − x2 ) dx = 18, then k =
(A) –9
(B) –3
(E) 18
11. An equation of the line tangent to the graph of f ( x) = x(1 − 2 x)3 at the point (1, − 1) is
(A)
y = −7 x + 6
(B)
y = −6 x + 5
(D)
y = 2x − 3
(E)
y = 7x − 8
(C)
2
2
(C)
y = −2 x + 1
(E)
3
⎛π⎞
12. If f ( x) = sin x , then f ′ ⎜ ⎟ =
⎝3⎠
(A)
−
1
2
(B)
1
2
(D)
13. If the function f has a continuous derivative on [ 0, c ] , then
(A) f (c) − f (0)
14.
∫
π
2
0
(B) f (c) − f (0)
(C)
f (c )
c
∫0
3
2
f ′( x) dx =
(D) f ( x) + c
(E) f ′′(c) − f ′′(0)
cos θ
dθ =
1 + sin θ
(A) −2
(D) 2
(
(
)
2 −1
)
2 −1
(B) −2 2
(E) 2
(
(C) 2 2
)
2 +1
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59
1988 AP Calculus AB: Section I
15. If f ( x) = 2 x , then f ′(2) =
(A)
1
4
(B)
1
2
(C)
2
2
(D) 1
2
(E)
16. A particle moves along the x-axis so that at any time t ≥ 0 its position is given by
x(t ) = t 3 − 3t 2 − 9t + 1 . For what values of t is the particle at rest?
(A) No values
17.
1
∫ 0 ( 3x − 2 )
(A) −
2
(B) 1 only
(C)
3 only
(D) 5 only
(E) 1 and 3
dx =
7
3
(B) −
7
9
1
9
(D) 1
(E) 3
⎛ x⎞
(C) − sin ⎜ ⎟
⎝2⎠
⎛x⎞
(D) − cos ⎜ ⎟
⎝2⎠
1
⎛ x⎞
(E) − cos ⎜ ⎟
2
⎝2⎠
(C) ln 2
(D) 2 ln 2
(E)
(C)
d2y
⎛x⎞
18. If y = 2 cos ⎜ ⎟ , then
=
dx 2
⎝2⎠
⎛ x⎞
(A) −8cos ⎜ ⎟
⎝2⎠
19.
3
∫2
(A)
x
2
x +1
⎛ x⎞
(B) −2 cos ⎜ ⎟
⎝2⎠
dx =
1 3
ln
2 2
(B)
1
ln 2
2
1
ln 5
2
20. Let f be a polynomial function with degree greater than 2. If a ≠ b and f (a) = f (b) = 1 , which
of the following must be true for at least one value of x between a and b?
I.
II.
III.
f ( x) = 0
f ′( x) = 0
f ′′( x) = 0
(A) None
(B) I only
(C) II only
(D) I and II only
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(E) I, II, and III
60
1988 AP Calculus AB: Section I
21. The area of the region enclosed by the graphs of y = x and y = x 2 − 3 x + 3 is
(A)
2
3
(B) 1
(C)
4
3
(C)
e
(D) 2
(E)
14
3
(E)
e2
⎛1⎞
22. If ln x − ln ⎜ ⎟ = 2, then x =
⎝ x⎠
(A)
1
e
1
e
(B)
2
(D)
2e
f ( x)
is
x→0 g ( x )
23. If f ′( x) = cos x and g ′( x) = 1 for all x, and if f (0) = g (0) = 0 , then lim
(A)
24.
π
2
(B) 1
(D)
−1
(E) nonexistent
( )
d ln x
x
=
dx
(A) x ln x
(B)
( ln x ) x
25. For all x > 1, if f ( x) = ∫
(A) 1
26.
(C) 0
∫
π
2
0
(A)
(B)
(C)
x
1
( )
2
( ln x ) xln x
x
(D)
( ln x ) ( xln x−1 )
( )
(E) 2 ( ln x ) x ln x
1
dt , then f ′( x) =
t
1
x
(C)
ln x − 1
(D) ln x
(E)
ex
(D) 1
(E)
π
−1
2
x cos x dx =
−
π
2
(B) –1
(C) 1 −
π
2
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61
1988 AP Calculus AB: Section I
⎧⎪ x 2 , x < 3
27. At x = 3 , the function given by f ( x ) = ⎨
is
⎪⎩6 x − 9, x ≥ 3
(A)
(B)
(C)
(D)
(E)
28.
4
∫1
(A)
undefined.
continuous but not differentiable.
differentiable but not continuous.
neither continuous nor differentiable.
both continuous and differentiable.
x − 3 dx =
−
3
2
(B)
3
2
(C)
5
2
(D)
9
2
(E) 5
tan 3( x + h ) − tan 3x
is
h →0
h
29. The lim
(A) 0
(B)
3sec 2 (3x)
(C)
sec2 (3 x)
(D)
3cot(3x)
(E) nonexistent
30. A region in the first quadrant is enclosed by the graphs of y = e 2 x , x = 1, and the coordinate axes.
If the region is rotated about the y -axis , the volume of the solid that is generated is represented by
which of the following integrals?
1
(A)
2π ∫ xe2 x dx
(B)
2π ∫ e2 x dx
(C)
π ∫ e 4 x dx
(D)
π∫
(E)
π e 2
ln y dy
4 ∫0
0
1
0
1
0
e
0
y ln y dy
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62
1988 AP Calculus AB: Section I
31. If f ( x) =
(A)
x
, then the inverse function, f −1 , is given by f −1 ( x) =
x +1
x −1
x
(B)
x +1
x
(C)
x
1− x
(D)
x
x +1
(E) x
32. Which of the following does NOT have a period of π ?
(A)
⎛1 ⎞
f ( x) = sin ⎜ x ⎟
⎝2 ⎠
(B)
f ( x) = sin x
(D)
f ( x) = tan x
(E)
f ( x) = tan 2 x
(C)
f ( x) = sin 2 x
33. The absolute maximum value of f ( x) = x3 − 3 x 2 + 12 on the closed interval [ −2, 4] occurs at x =
(A) 4
(B) 2
(C)
1
(D) 0
(E) –2
34. The area of the shaded region in the figure above is represented by which of the following
integrals?
c
(A)
∫ a ( f ( x)
(B)
∫b
(C)
∫ a ( g ( x) − f ( x) ) dx
(D)
∫ a ( f ( x) − g ( x) ) dx
(E)
c
− g ( x) ) dx
c
f ( x) dx − ∫ g ( x) dx
a
c
c
b
c
∫ a ( g ( x) − f ( x) ) dx + ∫ b ( f ( x) − g ( x) ) dx
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63
1988 AP Calculus AB: Section I
35.
π⎞
⎛
4 cos ⎜ x + ⎟ =
3⎠
⎝
(A)
2 3 cos x − 2sin x
(B)
2 cos x − 2 3 sin x
(D)
2 3 cos x + 2sin x
(E)
4 cos x + 2
(C)
2 cos x + 2 3 sin x
36. What is the average value of y for the part of the curve y = 3 x − x 2 which is in the first quadrant ?
(A) –6
(B) –2
(C)
3
2
(D)
9
4
(E)
9
2
37. If f ( x) = e x sin x , then the number of zeros of f on the closed interval [ 0, 2π] is
(A) 0
(B) 1
38. For x > 0,
1
(A)
x3
+C
(B)
( ) +C
(E)
2
10
∫1
f ( x) dx = 4 and
(A) –3
2
(D) 3
(E) 4
⎛ 1 x du ⎞
⎜ ∫1
⎟ dx =
u ⎠
⎝x
ln x 2
(D)
39. If
∫
(C)
3
∫ 10
f ( x) dx = 7, then
(B) 0
8
x4
−
2
x2
+C
(C)
ln ( ln x ) + C
(E)
11
( ln x )2 + C
2
3
∫1
f ( x) dx =
(C) 3
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(D) 10
64
1988 AP Calculus AB: Section I
z
y
x
40. The sides of the rectangle above increase in such a way that
when x = 4 and y = 3 , what is the value of
(A)
1
3
(B) 1
dz
dx
dy
= 1 and
= 3 . At the instant
dt
dt
dt
dx
?
dt
(C) 2
(D)
5
(E)
5
41. If lim f ( x) = 7 , which of the following must be true?
x→3
I.
II.
III.
f is continuous at x = 3 .
f is differentiable at x = 3 .
f (3) = 7
(A) None
(B) II only
(D) I and III only
(E) I, II, and III
(C) III only
42. The graph of which of the following equations has y = 1 as an asymptote?
(A)
y = ln x
(B)
y = sin x
(C)
y=
x
x +1
(D)
y=
x2
x −1
(E)
y = e− x
43. The volume of the solid obtained by revolving the region enclosed by the ellipse x 2 + 9 y 2 = 9
about the x-axis is
(A)
2π
(B)
4π
(C)
6π
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(D) 9π
(E)
12π
65
1988 AP Calculus AB: Section I
44. Let f and g be odd functions. If p, r, and s are nonzero functions defined as follows, which must
be odd?
I.
II.
III.
p ( x) = f ( g ( x) )
r ( x) = f ( x) + g ( x)
s ( x) = f ( x) g ( x)
(A) I only
(B) II only
(D) II and III only
(E) I, II, and III
(C) I and II only
45. The volume of a cylindrical tin can with a top and a bottom is to be 16π cubic inches. If a
minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the
can?
(A)
3
2 2
(B)
2 2
(C)
3
2 4
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(D) 4
(E)
8
66
1988 Calculus AB Solutions
1.
C
dy
d
d
= x 2 ⋅ (e x ) + e x ⋅ ( x 2 ) = x 2e x + 2 xe x = xe x ( x + 2)
dx
dx
dx
2.
D
x 2 − 4 ≥ 0 and x ≠ 3 ⇒ x ≥ 2 and x ≠ 3
3.
A
Distance = ∫
4.
E
Students should know what the graph looks like without a calculator and choose option E.
2
0
2
v ( t ) dt = ∫ et dt = et
0
−1
Or y = −5 ( x − 2 ) ; y ′ = 5 ( x − 2 )
5.
A
∫ sec
2
−2
2
0
= e 2 − e0 = e 2 − 1
−3
; y ′′ = −10 ( x − 2 ) . y ′′ < 0 for x > 2 .
x dx = ∫ d ( tan x ) = tan x + C
x⋅
⎛1⎞
d
d
(ln x) − ln x ⋅ ( x) x ⋅ ⎜ ⎟ − ln x ⋅ (1)
1 − ln x
⎝ x⎠
dx
dx
=
=
x2
x2
x2
D
dy
=
dx
7.
D
∫ x(3x
8.
B
dy
d2y
> 0 ⇒ y is increasing;
< 0 ⇒ graph is concave down . This is only on b < x < c .
dx
dx 2
9.
E
1 + ( 2 x ⋅ y ′ + 2 y ) − 2 y ⋅ y ′ = 0; y ′ =
6.
2
+ 5)
−
1
2
1
1
1
−
1
1
1
dx = ∫ (3x 2 + 5) 2 ( 6 x dx ) = ⋅ 2(3x 2 + 5) 2 + C = (3x 2 + 5) 2 + C
6
6
3
1+ 2y
. This cannot be evaluated at (1,1) and so y ′ does
2 y − 2x
not exist at (1,1) .
10. C
1 ⎞
⎛
18 = ⎜ kx 2 − x3 ⎟
3 ⎠
⎝
k
0
=
2 3
k ⇒ k 3 = 27, so k = 3
3
11. A
f ′( x) = x ⋅ 3(1 − 2 x) 2 (−2) + (1 − 2 x)3 ; f ′(1) = −7 . Only option A has a slope of –7.
12. B
⎛π⎞
⎛π⎞ 1
f ′ ⎜ ⎟ = cos ⎜ ⎟ =
⎝3⎠
⎝3⎠ 2
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194
1988 Calculus AB Solutions
13. A
By the Fundamental Theorem of Calculus
π
2
0
−1 2
∫0
π
12 2
14. D
∫ (1 + sin θ )
15. B
f ( x) = 2 x = 2 ⋅ x ; f ′( x) = 2 ⋅
16. C
c
( cos θ d θ ) = 2 (1 + sin θ )
1
2 x
0
f ′( x) dx = f ( x)
=2
(
c
0
= f (c) − f (0)
)
2 −1
; f ′(2) = 2 ⋅
1
2 2
(
=
1
2
)
At rest when 0 = v(t ) = x′(t ) = 3t 2 − 6t − 9 = 3 t 2 − 2t − 3 = 3(t − 3)(t + 1)
t = −1, 3 and t ≥ 0 ⇒ t = 3
1
( 3x − 2 ) dx =
2
1 1
1 1
( 3x − 2 )2 ( 3 dx ) = ⋅ ( 3x − 2 )3
∫
0
3
3 3
1
1
(1 − ( −8) ) = 1
9
17. D
∫0
18. E
⎛
⎛
1
⎛ x⎞ 1⎞
⎛ x⎞
⎛ x ⎞ ⎛ 1 ⎞⎞
⎛ x⎞
y ′ = 2 ⋅ ⎜ − sin ⎜ ⎟ ⋅ ⎟ = − sin ⎜ ⎟ ; y ′′ = − ⎜ cos ⎜ ⎟ ⋅ ⎜ ⎟ ⎟ = − cos ⎜ ⎟
2
⎝ 2⎠ 2⎠
⎝ 2⎠
⎝ 2 ⎠ ⎝ 2 ⎠⎠
⎝2⎠
⎝
⎝
3
x
∫ 2 x2 + 1
20. C
Consider the cases:
I. false if f ( x ) = 1
dx =
(
) 2 = 12 ( ln10 − ln 5) = 12 ln 2
1 3 2 x dx 1
= ln x 2 + 1
2 ∫ 2 x2 + 1 2
19. B
0
=
3
II. This is true by the Mean Value Theorem
III. false if the graph of f is a parabola with vertex at x =
a+b
.
2
Only II must be true.
21. C
x = x 2 − 3x + 3 at x = 1 and at x = 3.
Area =
22. C
∫1 ( (
3
))
4
⎛ 1
⎞
− x 2 + 4 x − 3) dx = ⎜ − x3 + 2 x 2 − 3 x ⎟ =
(
1
⎝ 3
⎠1 3
x − x 2 − 3 x + 3 dx = ∫
2 = ln x − ln
3
3
1
= ln x + ln x ⇒ ln x = 1 ⇒ x = e
x
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1988 Calculus AB Solutions
23. B
By L’Hôpital’s rule (which is no longer part of the AB Course Description),
f ( x)
f ′( x) f ′(0) cos 0 1
= lim
=
=
= =1
lim
x→0 g ( x )
x→0 g ′( x )
g ′(0)
1
1
Alternatively, f ′( x) = cos x and f (0) = 0 ⇒ f ( x) = sin x . Also g ′( x) = 1 and
f ( x)
sin x
g (0) = 0 ⇒ g ( x) = x . Hence lim
= lim
= 1.
x→0 g ( x )
x →0 x
24. C
Let y = x ln x and take the ln of each side. ln y = ln x ln x = ln x ⋅ ln x . Take the derivative of
y′
1
1
each side with respect to x.
= 2 ln x ⋅ ⇒ y ′ = 2 ln x ⋅ ⋅ x ln x
y
x
x
25. B
Use the Fundamental Theorem of Calculus. f ′( x) =
26. E
Use the technique of antiderivatives by parts: Let u = x and dv = cos x dx .
∫
27. E
π
2
0
(
x cos x dx = x sin x − ∫ sin x dx
)
π
2
0
1
x
= ( x sin x + cos x )
π
2
0
=
π
−1
2
The function is continuous at x = 3 since lim− f ( x) = lim+ f ( x) = 9 = f (3) . Also, the
x→3
x→3
derivative as you approach x = 3 from the left is 6 and the derivative as you approach x = 3
from the right is also 6. These two facts imply that f is differentiable at x = 3. The function
is clearly continuous and differentiable at all other values of x.
28. C
The graph is a V with vertex at x = 3 . The
integral gives the sum of the areas of the two
triangles that the V forms with the horizontal
axis for x from 1 to 4. These triangles have
areas of 2 and 0.5 respectively.
29. B
This limit gives the derivative of the function f ( x) = tan(3x) . f ′( x) = 3sec 2 (3 x)
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1988 Calculus AB Solutions
30. A
Shells (which is no longer part of the AB Course
Description)
∑ 2πrh∆x , where r = x, h = e2 x
1
Volume = 2π ∫ xe 2 x dx
0
31. C
32. A
33. A
Let y = f ( x) and solve for x.
x
y
x
; xy + y = x ; x( y − 1) = − y ; x =
⇒ f −1 ( x) =
y=
x +1
1− y
1− x
⎛ x⎞
The period for sin ⎜ ⎟ is
⎝2⎠
2π
= 4π .
1
2
Check the critical points and the endpoints.
f ′( x) = 3 x 2 − 6 x = 3 x( x − 2) so the critical points are 0 and 2.
x
−2 0 2 4
f ( x ) −8 12 8 28
Absolute maximum is at x = 4.
34. D
35. B
The interval is x = a to x = c. The height of a rectangular slice is the top curve, f ( x) , minus
the bottom curve, g ( x) . The area of the rectangular slice is therefore ( f ( x) − g ( x))∆x . Set
up a Riemann sum and take the limit as ∆x goes to 0 to get a definite integral.
π⎞
⎛
⎛
⎛π⎞
⎛ π ⎞⎞
4 cos ⎜ x + ⎟ = 4 ⎜ cos x ⋅ cos ⎜ ⎟ − sin x ⋅ sin ⎜ ⎟ ⎟
3⎠
⎝
⎝3⎠
⎝ 3 ⎠⎠
⎝
⎛
1
3⎞
= 4 ⎜⎜ cos x ⋅ − sin x ⋅
⎟ = 2 cos x − 2 3 sin x
2
2 ⎟⎠
⎝
36. C
3x − x 2 = x ( 3 − x ) > 0 for 0 < x < 3
Average value =
(
)
1 3
1⎛ 3
1 ⎞
3 x − x 2 dx = ⎜ x 2 − x3 ⎟
∫
3 0
3⎝ 2
3 ⎠
3
0
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=
3
2
197
1988 Calculus AB Solutions
37. D
Since e x > 0 for all x, the zeros of f ( x) are the zeros of sin x , so x = 0, π , 2π .
38. E
∫ ⎜⎝ x ∫1
⎛1
( ln x )2
2
10
x
du ⎞
1
⎟ dx = ∫ ln x dx =
u ⎠
x
⎛ dx ⎞
⎟ .This is ∫ u du with u = ln x , so the value is
x ⎠
∫ ln x ⎜⎝
+C
f ( x) dx = − ∫
3
3
40. B
x 2 + y 2 = z 2 , take the derivative of both sides with respect to t. 2 x ⋅
10
Divide by 2 and substitute: 4 ⋅
41. A
42. C
43. B
1
f ( x) dx − ∫
3
f ( x) dx = 4 − (−7) = 11
dx
dy
dz
+ 2y ⋅
= 2z ⋅
dt
dt
dt
dx
1 dx
dx
+ 3⋅
= 5 ⋅1 ⇒
=1
3 dt
dt
dt
The statement makes no claim as to the behavior of f at x = 3 , only the value of f for input
arbitrarily close to x = 3 . None of the statements are true.
x
x
1
x
= lim
= lim
= 1.
1 x→∞
1
x→∞ x + 1 x→∞ x
1+
+
x x
x
None of the other functions have a limit of 1 as x → ∞
lim
The cross-sections are disks with radius r = y where y =
Volume = π ∫
44. C
f ( x) dx = ∫
10
∫3
f ( x) dx ;
∫1
10
39. E
3
−3
y 2 dx = 2π ∫
3
0
(
)
1
9 − x2 .
3
1
2π ⎛
1 3⎞
9 − x 2 dx =
⎜ 9x − x ⎟
9
9 ⎝
3 ⎠
3
0
= 4π
For I: p (− x) = f ( g (− x) ) = f ( − g ( x) ) = − f ( g ( x) ) = − p( x) ⇒ p is odd.
For II: r (− x) = f (− x) + g (− x) = − f ( x) − g ( x) = − ( f ( x) + g ( x) ) = − r ( x) ⇒ r is odd.
For III: s (− x) = f (− x) ⋅ g (− x) = ( − f ( x) ) ⋅ ( − g ( x) ) = f ( x) ⋅ g ( x) = s( x) ⇒ s is not odd.
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1988 Calculus AB Solutions
45. D
(
Volume = π r 2 h = 16π ⇒ h = 16r −2 . A = 2π rh + 2π r 2 = 2π 16 r −1 + r 2
(
)
(
)
)
dA
dA
dA
= 2π −16 r −2 + 2r = 4π r −2 −8 + r 3 ;
< 0 for 0 < r < 2 and
> 0 for r > 2
dr
dr
dr
The minimum surface area of the can is when r = 2 ⇒ h = 4 .
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1993 AP Calculus AB: Section I
90 Minutes—Scientific Calculator
Notes: (1) The exact numerical value of the correct answer does not always appear among the choices
given. When this happens, select from among the choices the number that best approximates
the exact numerical value.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f (x) is a real number.
1.
If f ( x) =
3
x2
, then f ′(4) =
(A) –6
2.
3.
(B) –3
(C)
3
(D) 6
(E) 8
Which of the following represents the area of the shaded region in the figure above?
d
(A)
∫c
(D)
(b − a ) [ f (b) − f (a) ]
lim
f ( y )dy
3n3 − 5n
n→∞ n3
− 2n 2 + 1
(A) –5
b
(B)
∫ a ( d − f ( x) ) dx
(E)
(d − c) [ f (b) − f (a) ]
(C)
1
(C)
f ′(b) − f ′(a)
is
(B) –2
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(D) 3
(E) nonexistent
78
1993 AP Calculus AB: Section I
4.
If x3 + 3 xy + 2 y 3 = 17 , then in terms of x and y,
(A)
(B)
−
(C)
−
(D)
−
(E)
5.
−
dy
=
dx
x2 + y
x + 2 y2
x2 + y
x + y2
x2 + y
x + 2y
x2 + y
2 y2
− x2
1+ 2 y2
If the function f is continuous for all real numbers and if f ( x) =
then f (−2) =
(A) –4
6.
(C) –1
The area of the region enclosed by the curve y =
(A)
7.
(B) –2
5
36
(B)
ln
2
3
(C)
ln
(D) 0
(D)
x + 13 y = 66
4
3
(D) ln
2
3
2
(E)
ln 6
2x + 3
at the point (1,5 ) is
3x − 2
(B) 13x + y = 18
(E)
(E)
1
, the x-axis, and the lines x = 3 and x = 4 is
x −1
An equation of the line tangent to the graph of y =
(A) 13 x − y = 8
x2 − 4
when x ≠ −2 ,
x+2
(C)
x − 13 y = 64
−2 x + 3 y = 13
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79
1993 AP Calculus AB: Section I
8.
If y = tan x − cot x, then
(A) sec x csc x
9.
dy
=
dx
(B) sec x − csc x
(C) sec x + csc x
(D) sec2 x − csc2 x
(E) sec2 x + csc2 x
If h is the function given by h( x) = f ( g ( x)), where f ( x) = 3 x 2 − 1 and g ( x) = x , then h( x) =
(A)
3x3 − x
(B)
3x 2 − 1
(C)
3x 2 x − 1
(D) 3 x − 1
(E)
3x 2 − 1
(D) 1
(E)
2
10. If f ( x) = ( x − 1) 2 sin x, then f ′(0) =
(A) –2
(B) –1
(C) 0
11. The acceleration of a particle moving along the x-axis at time t is given by a (t ) = 6t − 2 . If the
velocity is 25 when t = 3 and the position is 10 when t = 1 , then the position x(t ) =
(A)
9t 2 + 1
(B)
3t 2 − 2t + 4
(C)
t 3 − t 2 + 4t + 6
(D)
t 3 − t 2 + 9t − 20
(E)
36t 3 − 4t 2 − 77t + 55
12. If f and g are continuous functions, and if f ( x) ≥ 0 for all real numbers x , which of the
following must be true?
I.
II.
III.
b
∫a
b
b
f ( x) g ( x)dx = ⎛⎜ ∫ f ( x)dx ⎞⎟ ⎛⎜ ∫ g ( x)dx ⎞⎟
⎝ a
⎠⎝ a
⎠
b
b
∫ a ( f ( x) + g ( x) ) dx = ∫ a
b
∫a
f ( x) dx =
(A) I only
b
∫a
b
f ( x)dx + ∫ g ( x)dx
a
f ( x)dx
(B) II only
(C) III only
(D) II and III only
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(E) I, II, and III
80
1993 AP Calculus AB: Section I
13. The fundamental period of 2 cos(3x) is
(A)
14.
∫
2π
3
3x 2
x3 + 1
(B)
2π
(C)
6π
(D) 2
(E)
3
dx =
(A)
2 x3 + 1 + C
(B)
3 3
x +1 + C
2
(C)
x3 + 1 + C
(D)
ln x3 + 1 + C
(E)
ln( x3 + 1) + C
15. For what value of x does the function f ( x) = ( x − 2)( x − 3) 2 have a relative maximum?
(A) –3
(B)
−
7
3
(C)
−
5
2
(D)
16. The slope of the line normal to the graph of y = 2 ln(sec x) at x =
(A)
(E)
5
2
π
is
4
−2
(B) −
1
2
(C)
1
2
(D)
2
(E)
7
3
nonexistent
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1993 AP Calculus AB: Section I
17.
∫ (x
2
+ 1) 2 dx =
(A)
( x 2 + 1)3
+C
3
(B)
( x 2 + 1)3
+C
6x
(C)
⎛ x3
⎞
⎜⎜ + x ⎟⎟ + C
⎝ 3
⎠
(D)
2 x( x 2 + 1)3
+C
3
(E)
x5 2 x3
+
+ x+C
5
3
2
π
3π
⎛ x⎞
that satisfies the
18. If f ( x) = sin ⎜ ⎟ , then there exists a number c in the interval < x <
2
2
⎝2⎠
conclusion of the Mean Value Theorem. Which of the following could be c ?
(A)
2π
3
(B)
3π
4
(C)
⎪⎧ x3
19. Let f be the function defined by f ( x) = ⎨
⎪⎩ x
about f is true?
(A)
f is an odd function.
(B)
f is discontinuous at x = 0 .
(C)
f has a relative maximum.
(D)
f ′(0) = 0
(E)
f ′( x) > 0 for x ≠ 0
5π
6
(D)
π
(E)
3π
2
for x ≤ 0,
Which of the following statements
for x > 0.
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82
1993 AP Calculus AB: Section I
1
= ( x + 1) 3
20. Let R be the region in the first quadrant enclosed by the graph of y
, the line x = 7 ,
the x-axis, and the y-axis. The volume of the solid generated when R is revolved about the y -axis
is given by
(A) π ∫
7
0
(D) 2π ∫
2
( x + 1) 3 dx
2
0
(B) 2π ∫
1
x( x + 1) 3 dx
(B)
ln( x 2 − 2 x + 2)
(C)
ln
(D)
arcsec( x − 1)
(E)
arctan( x − 1)
1
x2
−
1
x3
have a point of inflection?
(C) 2
x2 − 2x + 2
−( x 2 − 2 x + 2) −2
0
2
( x + 1) 3 dx
0
1
(A)
(C) π ∫
2
7
(B) 1
22. An antiderivative for
0
1
x( x + 1) 3 dx
(E) π ∫ ( y 3 − 1) 2 dy
21. At what value of x does the graph of y =
(A) 0
7
(D) 3
(E) At no value of x
is
x−2
x +1
23. How many critical points does the function f ( x) = ( x + 2)5 ( x − 3) 4 have?
(A) One
(B) Two
(C) Three
(D) Five
(E)
Nine
(E)
–2
2
24. If f ( x) = ( x 2 − 2 x − 1) 3 , then f ′(0) is
(A)
4
3
(B) 0
(C)
−
2
3
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(D)
−
4
3
83
1993 AP Calculus AB: Section I
25.
( )
d x
2 =
dx
2 x−1
(A)
(B)
(2 x −1 ) x
(C)
(2 x ) ln 2
(D) (2 x−1 ) ln 2
(E)
2x
ln 2
26. A particle moves along a line so that at time t, where 0 ≤ t ≤ π , its position is given by
t2
s (t ) = −4 cos t − + 10 . What is the velocity of the particle when its acceleration is zero?
2
(A) –5.19
(B) 0.74
(C) 1.32
(D) 2.55
(E)
8.13
(D) 46.000
(E)
136.364
(D) 1
(E)
nonexistent
27. The function f given by f ( x) = x3 + 12 x − 24 is
(A) increasing for x < −2, decreasing for −2 < x < 2, increasing for x > 2
(B) decreasing for x < 0, increasing for x > 0
(C) increasing for all x
(D) decreasing for all x
(E) decreasing for x < −2, increasing for −2 < x < 2, decreasing for x > 2
28.
500
∫1
(13x − 11x ) dx + ∫ 2500 (11x − 13x ) dx =
(A) 0.000
29.
lim
θ→0
1 − cos θ
2sin 2 θ
(A) 0
(B) 14.946
(C) 34.415
is
(B)
1
8
(C)
1
4
30. The region enclosed by the x-axis, the line x = 3 , and the curve y = x is rotated about the
x-axis. What is the volume of the solid generated?
(A)
3π
(B)
2 3π
(C)
9
π
2
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(D) 9 π
(E)
36 3
π
5
84
1993 AP Calculus AB: Section I
2
31. If f ( x) = e3ln( x ) , then f ′( x) =
(A)
32.
2
)
dx
3
∫0
4 − x2
(A)
33. If
e3ln( x
(B)
3
x
e3ln( x
2
2
)
(C)
6(ln x) e3ln( x
(C)
π
6
2
)
(D) 5x 4
(E)
6x5
(D)
1
ln 2
2
(E)
− ln 2
(D)
1
3
(E)
2
3
=
π
3
(B)
π
4
dy
= 2 y 2 and if y = −1 when x = 1, then when x = 2, y =
dx
(A)
−
2
3
(B)
−
1
3
(C) 0
34. The top of a 25-foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute.
When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance
between the bottom of the ladder and the wall?
(A) −
7
feet per minute
8
(B) −
7
feet per minute
24
(C)
7
feet per minute
24
(D)
7
feet per minute
8
(E)
21
feet per minute
25
35. If the graph of y =
then a + c =
(A) –5
ax + b
has a horizontal asymptote y = 2 and a vertical asymptote x = −3 ,
x+c
(B) –1
(C) 0
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(D) 1
(E)
5
85
1993 AP Calculus AB: Section I
36. If the definite integral
2 x2
∫0 e
dx is first approximated by using two inscribed rectangles of equal
width and then approximated by using the trapezoidal rule with n = 2 , the difference between the
two approximations is
(A) 53.60
(B) 30.51
(C) 27.80
(D) 26.80
(E)
12.78
37. If f is a differentiable function, then f ′(a ) is given by which of the following?
I.
II.
III.
lim
f ( a + h) − f ( a )
h
lim
f ( x) − f (a)
x−a
lim
f ( x + h) − f ( x )
h
h →0
x →a
x →a
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
38. If the second derivative of f is given by f ′′( x) = 2 x − cos x , which of the following could be f ( x) ?
(A)
x3
+ cos x − x + 1
3
(B)
x3
− cos x − x + 1
3
(C)
x3 + cos x − x + 1
(D)
x 2 − sin x + 1
(E)
x 2 + sin x + 1
39. The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in
the area of the circle is numerically equal to the rate of increase in its circumference. At this
instant, the radius of the circle is
(A)
1
π
(B)
1
2
(C)
2
π
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(D) 1
(E)
2
86
1993 AP Calculus AB: Section I
40. The graph of y = f ( x) is shown in the figure above. Which of the following could be the graph
of y = f
41.
( x )?
d x
cos(2π u ) du is
dx ∫ 0
(A) 0
(B)
1
sin x
2π
(C)
1
cos(2πx)
2π
(D) cos(2πx)
(E) 2π cos(2πx)
42. A puppy weighs 2.0 pounds at birth and 3.5 pounds two months later. If the weight of the puppy
during its first 6 months is increasing at a rate proportional to its weight, then how much will the
puppy weigh when it is 3 months old?
(A) 4.2 pounds
(B) 4.6 pounds (C) 4.8 pounds
AP Calculus Multiple-Choice Question Collection
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(D) 5.6 pounds
(E) 6.5 pounds
87
1993 AP Calculus AB: Section I
43.
∫ x f ( x) dx =
(A)
x f ( x) − ∫ x f ′( x) dx
(B)
x2
x2
f ( x) − ∫
f ′( x)dx
2
2
(C)
x f ( x) −
(D)
x f ( x) − ∫ f ′( x) dx
(E)
x2
2
∫
x2
f ( x) + C
2
f ( x) dx
44. What is the minimum value of f ( x) = x ln x ?
(A)
−e
(B) −1
(C) −
1
e
(D) 0
(E)
f ( x) has no minimum value.
45. If Newton’s method is used to approximate the real root of x3 + x − 1 = 0 , then a first
approximation x1 = 1 would lead to a third approximation of x3 =
(A) 0.682
(B) 0.686
(C) 0.694
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(D) 0.750
(E)
1.637
88
1993 Calculus AB Solutions
1
1
1.
C
2.
B
3
3
3
f ′( x) = x 2 ; f ′(4) = ⋅ 4 2 = ⋅ 2 = 3
2
2
2
Summing pieces of the form: (vertical) ⋅ (small width) , vertical = ( d − f ( x) ) , width = ∆x
Area =
b
∫ a ( d − f ( x) ) dx
3n3 − 5n
3−
5
n2
3.
D
Divide each term by n3 . lim
4.
A
3x 2 + 3 ( y + x ⋅ y′ ) + 6 y 2 ⋅ y′ = 0; y′(3 x + 6 y 2 ) = −(3 x 2 + 3 y )
n→∞ n3
y′ = −
3x2 + 3 y
3x + 6 y 2
=−
− 2n 2 + 1
= lim
n→∞
2 1
1− + 3
n n
=3
x2 + y
x + 2 y2
5.
A
x2 − 4
( x + 2)( x − 2)
lim
= lim
= lim ( x − 2) = −4. For continuity f (−2) must be –4.
x→−2 x + 2
x→−2
x→−2
x+2
6.
D
Area = ∫
7.
B
y′ =
8.
E
y′ = sec 2 x + csc2 x
9.
E
h( x ) = f
4
3
1
dx = ( ln x − 1
x −1
2 ⋅ (3 x − 2) − (2 x + 3) ⋅ 3
(3x − 2) 2
( x )=3
x
2
) 3 = ln 3 − ln 2 = ln 32
4
; y′(1) = −13 . Tangent line: y − 5 = −13( x − 1) ⇒ 13x + y = 18
− 1 = 3x2 − 1
10. D
f ′( x) = 2( x − 1) ⋅ sin x + ( x − 1) 2 cos x ; f ′(0) = (−2) ⋅ 0 + 1⋅1 = 1
11. C
a (t ) = 6t − 2; v(t ) = 3t 2 − 2t + C and v(3) = 25 ⇒ 25 = 27 − 6 + C ; v(t ) = 3t 2 − 2t + 4
x(t ) = t 3 − t 2 + 4t + K ; Since x(1) = 10, K = 6; x(t ) = t 3 − t 2 + 4t + 6 .
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1993 Calculus AB Solutions
12. B
The only one that is true is II. The others can easily been seen as false by examples. For
example, let f ( x) = 1 and g ( x) = 1 with a = 0 and b = 2. Then I gives 2 = 4 and III gives
2 = 2 , both false statements.
2π 2π
=
B
3
13. A
period =
14. A
Let u = x3 + 1. Then
15. D
f ′( x) = ( x − 3) 2 + 2( x − 2)( x − 3) = ( x − 3)(3 x − 7); f ′( x) changes from positive to negative at
7
x= .
3
16. B
17. E
18. D
19. E
3x 2
∫
3
x +1
dx = ∫ u −1/ 2 du = 2u1/ 2 + C = 2 x3 + 1 + C
sec x tan x
= 2 tan x; y′(π 4) = 2 tan(π 4) = 2 . The slope of the normal line
sec x
1
1
−
=−
y′(π 4)
2
y′ = 2
Expand the integrand.
∫ (x
2
+ 1) 2 dx = ∫ ( x 4 + 2 x 2 + 1) dx =
1 5 2 3
x + x + x+C
5
3
⎛ 3π ⎞
⎛π⎞
⎛ 3π ⎞
⎛ π⎞
f ⎜ ⎟ − f ⎜ ⎟ sin ⎜ ⎟ − sin ⎜ ⎟
2
⎝2⎠ =
⎝ 4 ⎠
⎝4⎠ = 0 .
Want c so that f ′(c) = ⎝ ⎠
3π π
π
π
−
2 2
1
⎛c⎞
f ′(c) = cos ⎜ ⎟ = 0 ⇒ c = π
2
⎝2⎠
The only one that is true is E. A consideration of the graph of y = f ( x) , which is a standard
cubic to the left of 0 and a line with slope 1 to the right of 0, shows the other options to be
false.
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207
1993 Calculus AB Solutions
20. B
Use Cylindrical Shells which is no part of the AP Course Description. The volume of each
7
1
3
shell is of the form (2π rh) ∆x with r = x and h = y. Volume = 2π ∫ x ( x + 1) dx .
0
21. C
y = x −2 − x −3 ; y′ = −2 x −3 + 3 x −4 ; y′′ = 6 x −4 − 12 x −5 = 6 x −5 ( x − 2) . The only domain value at
which there is a sign change in y′′ is x = 2 . Inflection point at x = 2 .
22. E
∫
23. C
A quick way to do this problem is to use the effect of the multiplicity of the zeros of f on the
graph of y = f ( x) . There is point of inflection and a horizontal tangent at x = −2 . There is a
horizontal tangent and turning point at x = 3 . There is a horizontal tangent on the interval
(−2,3) . Thus, there must be 3 critical points. Also, f ′( x) = ( x − 3)3 ( x + 2) 4 (9 x − 7) .
1
2
x − 2x + 2
dx = ∫
(
1
2
( x − 2 x + 1) + 1
)
−
1
3
1
dx = ∫
( x − 1)
2
+1
dx = tan −1 ( x − 1) + C
24. A
2
f ′( x) = x 2 − 2 x − 1
3
25. C
d x
(2 ) = 2 x ⋅ ln 2
dx
26. D
v(t ) = 4sin t − t ; a (t ) = 4 cos t − 1 = 0 at t = cos −1 (1 4) = 1.31812; v(1.31812) = 2.55487
27. C
f ′( x) = 3 x 2 + 12 > 0 . Thus f is increasing for all x.
28. B
∫1
500
(13x − 11x ) dx + ∫
500
2
( 2x − 2) ,
2
4
f ′(0) = ⋅ (−1) ⋅ (−2) =
3
3
(11x − 13x ) dx = ∫
⎛ 13x 11x ⎞
2
= ∫ (13x − 11x ) dx = ⎜
−
⎜ ln13 ln11 ⎟⎟
1
⎝
⎠
2
1
=
500
1
(13x − 11x ) dx − ∫
500
2
(13x − 11x ) dx
132 − 13 112 − 11
−
= 14.946
ln13
ln11
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208
1993 Calculus AB Solutions
29. C
Use L’Hôpital’s Rule (which is no longer part of the AB Course Description).
lim
θ→0
1 − cos θ
2
2sin θ
sin θ
1
1
= lim
=
θ→0 4sin θ cos θ θ→0 4 cos θ
4
= lim
A way to do this without L’Hôpital’s rule is the following
lim
θ→0
30. C
1 − cos θ
2sin 2 θ
= lim
1 − cos θ
θ→0 2(1 − cos 2 θ)
1 − cos θ
1
1
= lim
=
θ→0 2(1 − cos θ)(1 + cos θ) θ→0 2(1 + cos θ)
4
= lim
Each slice is a disk whose volume is given by π r 2 ∆x , where r = x .
3
3
0
0
Volume = π∫ ( x ) 2 dx = π ∫ x dx =
2
π 2
x
2
f ( x) = e3ln( x ) = eln( x ) = x6 ; f ′( x) = 6 x5
32. A
∫
33. B
0
=
9
π.
2
6
31. E
∫0
3
⎛u⎞
= sin −1 ⎜ ⎟ + C , a > 0
⎝a⎠
a2 − u 2
du
3
⎛ x⎞
= sin −1 ⎜ ⎟
2
⎝2⎠
4− x
dx
3
0
⎛ 3⎞
π
−1
= sin −1 ⎜⎜
⎟⎟ − sin (0) =
3
⎝ 2 ⎠
1
−1
= 2x + C ; y =
. Substitute the point (1, −1)
y
2x + C
−1
1
1
⇒ C = −1, so y =
. When x = 2, y = − .
to find the value of C. Then −1 =
2+C
1− 2x
3
Separate the variables. y −2 dy = 2dx ; −
34. D
Let x and y represent the horizontal and vertical sides of the triangle formed by the ladder, the
wall, and the ground.
dx
dy
dx
dx 7
x 2 + y 2 = 25; 2 x + 2 y
= 0; 2(24) + 2(7)(−3) = 0;
= .
dt
dt
dt
dt 8
35. E
For there to be a vertical asymptote at x = −3 , the value of c must be 3. For y = 2 to be a
horizontal asymptote, the value of a must be 2. Thus a + c = 5 .
36. D
Rectangle approximation = e0 + e1 = 1 + e
(
)
Trapezoid approximation. = 1 + 2e + e 4 / 2 .
Difference = (e 4 − 1) / 2 = 26.799 .
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209
1993 Calculus AB Solutions
37. C
I and II both give the derivative at a. In III the denominator is fixed. This is not the derivative
of f at x = a . This gives the slope of the secant line from ( a , f (a ) ) to ( a + h , f (a + h) ) .
38. A
1
f ′( x) = x 2 − sin x + C , f ( x) = x3 + cos x + Cx + K . Option A is the only one with this form.
3
39. D
A = π r 2 and C =2π r ;
40. C
The graph of y = f
x > 0 , x and x
dA
dr
dC
dr
dA dC
= 2π r
and
= 2π . For
=
, r = 1.
dt
dt
dt
dt
dt
dt
( x ) is symmetric to the y-axis. This leaves only options C and E. For
are the same, so the graphs of f ( x ) and f ( x ) must be the same. This is
option C.
41. D
Answer follows from the Fundamental Theorem of Calculus.
t
42. B
⎛ 3.5 ⎞ 2
This is an example of exponential growth. We know from pre-calculus that w = 2 ⎜
⎟ is
⎝ 2 ⎠
an exponential function that meets the two given conditions. When t = 3 , w = 4.630 . Using
calculus the student may translate the statement “increasing at a rate proportional to its
weight” to mean exponential growth and write the equation w = 2e kt . Using the given
conditions, 3.5 = 2e
43. B
t⋅
ln(1.75)
; w = 2e
; ln(1.75) = 2k ; k =
2
ln(1.75)
2
.
When t = 3 , w = 4.630 .
Use the technique of antiderivative by parts, which is no longer in the AB Course
Description. The formula is ∫ u dv = uv − ∫ v du . Let u = f ( x) and dv = x dx. This leads to
1
∫ x f ( x) dx = 2 x
44. C
2k
2
f ( x) −
1
x 2 f ′( x) dx .
2∫
1
f ′( x) = ln x + x ⋅ ; f ′( x) changes sign from negative to positive only at x = e −1 .
x
1
f (e−1 ) = −e−1 = − .
e
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210
1993 Calculus AB Solutions
45. B
Let f ( x) = x3 + x − 1 . Then Newton’s method (which is no longer part of the AP Course
Description) gives
xn+1 = xn −
x2 = 1 −
f ( xn )
x 3 + x −1
= xn − n 2 n
f ′( xn )
3xn + 1
1+1−1 3
=
3 +1
4
3
⎛3⎞ 3
+ −1
3 ⎜⎝ 4 ⎟⎠ 4
59
x3 = −
=
= 0.686
2
4
86
⎛3⎞
3⎜ ⎟ +1
⎝4⎠
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211
1997 AP Calculus AB:
Section I, Part A
50 Minutes—No Calculator
Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f (x) is a real number.
1.
2.
2
3
(A)
(B)
(C)
(D)
(E)
2
4
6
36
42
∫ 1 (4 x
If f ( x) = x 2 x − 3, then f ′( x) =
3x − 3
(A)
2x − 3
x
(B)
2x − 3
1
(C)
2x − 3
−x + 3
(D)
2x − 3
5x − 6
(E)
3.
If
2 2x − 3
b
∫a
(A)
4.
− 6 x) dx =
f ( x) dx = a + 2b, then
a + 2b + 5
(B)
b
∫ a ( f ( x) + 5) dx =
5b − 5a
(C)
7b − 4 a
(D)
7b − 5a
(C)
–1
(D) –3
(E)
7b − 6 a
1
If f ( x) = − x3 + x + , then f ′(−1) =
x
(A) 3
(B) 1
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(E) –5
100
1997 AP Calculus AB:
Section I, Part A
5.
The graph of y = 3 x 4 − 16 x3 + 24 x 2 + 48 is concave down for
(A)
x<0
(B)
x>0
(C)
x < −2 or x > −
(D)
x<
(E)
2
<x<2
3
2
3
2
or x > 2
3
t
6.
1 2
e dt =
2∫
(A)
7.
−t
e +C
(B)
e
−
t
2
+C
(C)
t
2
e
+C
(D)
t
2
2e
+C
(E)
et + C
d
cos 2 ( x3 ) =
dx
(A)
6 x 2 sin( x3 ) cos( x3 )
(B)
6 x 2 cos( x3 )
(C)
sin 2 ( x3 )
(D) −6 x 2 sin( x3 ) cos( x3 )
(E) −2sin( x3 ) cos( x3 )
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101
1997 AP Calculus AB:
Section I, Part A
Questions 8-9 refer to the following situation.
A bug begins to crawl up a vertical wire at time t = 0 . The velocity v of the bug at time t,
0 ≤ t ≤ 8 , is given by the function whose graph is shown above.
8.
At what value of t does the bug change direction?
(A) 2
9.
(B) 4
(C)
6
(D) 7
(E) 8
What is the total distance the bug traveled from t = 0 to t = 8 ?
(A) 14
(B) 13
(C)
11
(D) 8
10. An equation of the line tangent to the graph of y = cos(2 x) at x =
(A)
π⎞
⎛
y −1 = − ⎜ x − ⎟
4⎠
⎝
(B)
π⎞
⎛
y − 1 = −2 ⎜ x − ⎟
4⎠
⎝
(C)
π⎞
⎛
y = 2⎜ x − ⎟
4⎠
⎝
(D)
π⎞
⎛
y = −⎜ x − ⎟
4⎠
⎝
(E)
π⎞
⎛
y = −2 ⎜ x − ⎟
4⎠
⎝
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(E) 6
π
is
4
102
1997 AP Calculus AB:
Section I, Part A
11. The graph of the derivative of f is shown in the figure above. Which of the following could be the
graph of f ?
12. At what point on the graph of y =
(A)
⎛1 1⎞
⎜ ,− ⎟
⎝2 2⎠
⎛1 1⎞
(B) ⎜ , ⎟
⎝ 2 8⎠
1 2
x is the tangent line parallel to the line 2 x − 4 y = 3 ?
2
(C)
1⎞
⎛
⎜1, − ⎟
4⎠
⎝
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(D)
⎛ 1⎞
⎜1, ⎟
⎝ 2⎠
(E)
( 2, 2 )
103
1997 AP Calculus AB:
Section I, Part A
13. Let f be a function defined for all real numbers x. If f ′( x) =
4 − x2
x−2
, then f is decreasing on the
interval
(A)
( −∞, 2 )
(B)
( −∞, ∞ )
(C)
( −2, 4 )
(D)
( −2, ∞ )
(E)
( 2, ∞ )
14. Let f be a differentiable function such that f (3) = 2 and f ′(3) = 5 . If the tangent line to the graph
of f at x = 3 is used to find an approximation to a zero of f, that approximation is
(A) 0.4
(B) 0.5
(C)
2.6
(D) 3.4
(E) 5.5
15. The graph of the function f is shown in the figure above. Which of the following statements about
f is true?
(A)
(B)
(C)
(D)
(E)
lim f ( x) = lim f ( x)
x →a
x→b
lim f ( x) = 2
x →a
lim f ( x) = 2
x→b
lim f ( x) = 1
x→b
lim f ( x) does not exist.
x →a
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104
1997 AP Calculus AB:
Section I, Part A
16. The area of the region enclosed by the graph of y = x 2 + 1 and the line y = 5 is
(A)
14
3
(B)
16
3
17. If x 2 + y 2 = 25 , what is the value of
(A)
18.
∫
π
4
0
−
25
27
e tan x
cos 2 x
(B)
−
7
27
(C)
d2y
dx 2
28
3
(D)
32
3
(E)
8π
at the point ( 4,3) ?
(C)
7
27
(D)
3
4
(E)
25
27
(C)
e −1
(D)
e
(E)
e +1
dx is
(A) 0
(B) 1
19. If f ( x) = ln x 2 − 1 , then f ′( x) =
(A)
(B)
(C)
(D)
(E)
2x
x2 − 1
2x
2
x −1
2 x
x2 − 1
2x
2
x −1
1
2
x −1
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105
1997 AP Calculus AB:
Section I, Part A
20. The average value of cos x on the interval [ −3,5] is
21.
(A)
sin 5 − sin 3
8
(B)
sin 5 − sin 3
2
(C)
sin 3 − sin 5
2
(D)
sin 3 + sin 5
2
(E)
sin 3 + sin 5
8
x
is
x→1 ln x
lim
(A) 0
(B)
1
e
(C)
1
(D)
e
(E) nonexistent
22. What are all values of x for which the function f defined by f ( x) = ( x 2 − 3)e − x is increasing?
(A)
There are no such values of x .
(B)
x < −1 and x > 3
(C)
−3 < x < 1
(D)
−1 < x < 3
(E)
All values of x
23. If the region enclosed by the y-axis, the line y = 2 , and the curve y = x is revolved about the
y-axis, the volume of the solid generated is
(A)
32π
5
(B)
16π
3
(C)
16π
5
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(D)
8π
3
(E)
π
106
1997 AP Calculus AB:
Section I, Part A
24. The expression
25.
1
1 ⎛ 1
2
3
50 ⎞
+
+
+ ⋅⋅⋅ +
⎜⎜
⎟ is a Riemann sum approximation for
50 ⎝ 50
50
50
50 ⎟⎠
x
dx
50
(A)
∫0
(B)
∫0
(C)
1 1 x
dx
50 ∫ 0 50
(D)
1 1
x dx
50 ∫ 0
(E)
1 50
x dx
50 ∫ 0
1
x dx
∫ x sin(2 x) dx =
(A)
1
x
− cos(2 x) + sin(2 x) + C
2
4
(B)
1
x
− cos(2 x) − sin(2 x) + C
2
4
(C)
1
x
cos(2 x) − sin(2 x) + C
2
4
(D)
−2 x cos(2 x) + sin(2 x) + C
(E)
−2 x cos(2 x) − 4sin(2 x) + C
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107
1997 AP Calculus AB:
Section I, Part B
40 Minutes—Graphing Calculator Required
Notes: (1) The exact numerical value of the correct answer does not always appear among the choices
given. When this happens, select from among the choices the number that best approximates
the exact numerical value.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f (x) is a real number.
e2 x
76. If f ( x) =
, then f ′( x) =
2x
(A) 1
(B)
(C)
(D)
(E)
e 2 x (1 − 2 x)
2 x2
e 2x
e 2 x (2 x + 1)
x2
e 2 x (2 x − 1)
2 x2
77. The graph of the function y = x3 + 6 x 2 + 7 x − 2 cos x changes concavity at x =
(A) –1.58
(B) –1.63
(C)
–1.67
78. The graph of f is shown in the figure above. If
3
∫1
(D) –1.89
(E) –2.33
f ( x) dx = 2.3 and F ′( x) = f ( x), then
F (3) − F (0) =
(A) 0.3
(B) 1.3
(C)
3.3
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(D) 4.3
(E) 5.3
108
1997 AP Calculus AB:
Section I, Part B
79. Let f be a function such that lim
h →0
I.
f (2 + h) − f (2)
= 5 . Which of the following must be true?
h
f is continuous at x = 2.
II. f is differentiable at x = 2.
III. The derivative of f is continuous at x = 2 .
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) II and III only
2
80. Let f be the function given by f ( x) = 2e 4 x . For what value of x is the slope of the line tangent to
the graph of f at ( x, f ( x) ) equal to 3?
(A) 0.168
(B) 0.276
(C)
0.318
(D) 0.342
(E) 0.551
81. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of
the crossing and watches an eastbound train traveling at 60 meters per second. At how many
meters per second is the train moving away from the observer 4 seconds after it passes through the
intersection?
(A) 57.60
(B) 57.88
(C)
59.20
(D) 60.00
(E) 67.40
82. If y = 2 x − 8 , what is the minimum value of the product xy ?
(A) –16
(B) –8
(C)
–4
(D) 0
(E) 2
83. What is the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x,
and the y-axis?
(A) 0.127
(B) 0.385
(C)
0.400
(D) 0.600
(E) 0.947
84. The base of a solid S is the region enclosed by the graph of y = ln x , the line x = e, and the
x-axis. If the cross sections of S perpendicular to the x-axis are squares, then the volume of S is
(A)
1
2
(B)
2
3
(C)
1
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(D) 2
(E)
1 3
(e − 1)
3
109
1997 AP Calculus AB:
Section I, Part B
85. If the derivative of f is given by f ′( x) = e x − 3 x 2 , at which of the following values of x does f
have a relative maximum value?
(A) –0.46
(B) 0.20
(C)
0.91
(D) 0.95
(E) 3.73
86. Let f ( x) = x . If the rate of change of f at x = c is twice its rate of change at x = 1 , then c =
(A)
1
4
(B) 1
(C)
4
(D)
1
2
(E)
1
2 2
87. At time t ≥ 0 , the acceleration of a particle moving on the x-axis is a (t ) = t + sin t . At t = 0 , the
velocity of the particle is –2. For what value t will the velocity of the particle be zero?
(A) 1.02
(B) 1.48
(C) 1.85
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(D) 2.81
(E) 3.14
110
1997 AP Calculus AB:
Section I, Part B
88. Let f ( x) = ∫
x
a
h(t ) dt , where h has the graph shown above. Which of the following could be the
graph of f ?
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111
1997 AP Calculus AB:
Section I, Part B
x
0 0.5 1.0 1.5 2.0
f ( x) 3 3
5
8 13
89. A table of values for a continuous function f is shown above. If four equal subintervals of [ 0, 2]
are used, which of the following is the trapezoidal approximation of
(A) 8
(B) 12
(C)
16
(D) 24
2
∫0
f ( x) dx ?
(E) 32
90. Which of the following are antiderivatives of f ( x) = sin x cos x ?
I.
F ( x) =
sin 2 x
2
II.
F ( x) =
cos 2 x
2
III.
F ( x) =
− cos(2 x)
4
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
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112
1997 Calculus AB Solutions: Part A
1.
C
2
∫1
(4 x3 − 6 x) dx = ( x 4 − 3 x 2 )
f ( x) =
1
x(2 x − 3) 2 ;
2
1
= (16 − 12) − (1 − 3) = 6
1
f ′( x) = (2 x − 3) 2
+ x(2 x − 3)
−
1
2
= (2 x − 3)
−
1
2 (3 x − 3) =
(3x − 3)
2x − 3
2.
A
3.
C
∫a
4.
D
1
1
1
= −3 + 1 − 1 = −3
f ( x) = − x3 + x + ; f ′( x) = −3 x 2 + 1 − 2 ; f ′(−1) = −3(−1) 2 + 1 −
x
x
(−1) 2
5.
E
b
b
b
a
a
( f ( x) + 5) dx = ∫ f ( x)dx + 5∫ 1 dx = a + 2b + 5(b − a) = 7b − 4a
y = 3 x 4 − 16 x3 + 24 x 2 + 48; y′ = 12 x3 − 48 x 2 + 48 x; y′′ = 36 x 2 − 96 x + 48 = 12(3x − 2)( x − 2)
2
2
y′′ < 0 for
< x < 2, therefore the graph is concave down for < x < 2
3
3
t
t
6.
C
1 2
e dt = e 2 + C
2∫
7.
D
d
⎛d
⎞
⎛d
⎞
cos 2 ( x3 ) = 2 cos( x3 ) ⎜ (cos( x3 ) ⎟ = 2 cos( x3 )(− sin( x3 ) ⎜ ( x3 ) ⎟
dx
⎝ dx
⎠
⎝ dx
⎠
= 2 cos( x3 )(− sin( x3 )(3 x 2 )
8.
C
The bug change direction when v changes sign. This happens at t = 6 .
9.
B
Let A1 be the area between the graph and t-axis for 0 ≤ t ≤ 6 , and let A 2 be the area between
the graph and the t-axis for 6 ≤ t ≤ 8 Then A1 = 12 and A 2 = 1 . The total distance is
A1 + A 2 = 13 .
10. E
π⎞
⎛π⎞
⎛π⎞
⎛
y = cos(2 x); y′ = −2sin(2 x); y′ ⎜ ⎟ = −2 and y ⎜ ⎟ = 0; y = −2 ⎜ x − ⎟
4⎠
⎝4⎠
⎝4⎠
⎝
11. E
Since f ′ is positive for −2 < x < 2 and negative for x < −2 and for x > 2, we are looking
for a graph that is increasing for −2 < x < 2 and decreasing otherwise. Only option E.
12. B
y=
1 2
1
1
⎛1 1⎞
x ; y′ = x; We want y′ =
⇒ x = . So the point is ⎜ , ⎟ .
2
2
2
⎝ 2 8⎠
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217
1997 Calculus AB Solutions: Part A
4 − x2
; f is decreasing when f ′ < 0 . Since the numerator is non-negative, this is
x−2
only when the denominator is negative. Only when x < 2 .
13. A
f ′( x) =
14. C
f ( x) ≈ L( x) = 2 + 5( x − 3); L( x) = 0 if 0 = 5 x − 13 ⇒ x = 2.6
15. B
Statement B is true because lim− f ( x) = 2 = lim+ f ( x) . Also, lim f ( x) does not exist
x →a
x→b
x→a
because the left- and right-sided limits are not equal, so neither (A), (C), nor (D) are true.
16. D
17. A
18. C
19. D
20. E
1 3
2
∫ −2 (5 − ( x + 1))dx = 2(4 x − 3 x )
2
The area of the region is given by
2
⎛ 8 ⎞ 32
= 2⎜8 − ⎟ =
0
⎝ 3⎠ 3
4
x 2 + y 2 = 25; 2 x + 2 y ⋅ y′ = 0; x + y ⋅ y′ = 0; y′(4,3) = − ;
3
25
⎛ 4⎞ ⎛ 4⎞
x + y ⋅ y′ = 0 ⇒ 1+y ⋅ y′′ + y′ ⋅ y′ = 0; 1 + (3) y′′ + ⎜ − ⎟ ⋅ ⎜ − ⎟ = 0; y′′ = −
27
⎝ 3⎠ ⎝ 3⎠
∫
π
4
0
∫
π
4
0
e tan x
2
cos x
e tan x
2
cos x
dx is of the form
dx = e tan x
f ( x) = ln x 2 − 1 ;
π
4
0
∫e
u
du where u = tan x. .
= e1 − e0 = e − 1
f ′( x) =
1
d 2
2x
( x − 1) = 2
x − 1 dx
x −1
2
⋅
1 5
1
1
cos x dx = (sin 5 − sin(−3)) = (sin 5 + sin 3) ; Note: Since the sine is an odd function,
∫
8 −3
8
8
sin(−3) = − sin(3) .
x
is nonexistent since lim ln x = 0 and lim x ≠ 0 .
x→1 ln x
x→1
x→1
21. E
lim
22. D
f ( x) = ( x 2 − 3)e − x ; f ′( x) = e− x (− x 2 + 2 x + 3) = −e− x ( x − 3)( x + 1); f ′( x) > 0 for − 1 < x < 3
23. A
2
2
0
0
Disks where r = x . V = π ∫ x 2 dy = π ∫ y 4 dy =
π 5
y
5
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2
0
=
32π
5
218
1997 Calculus AB Solutions: Part A
24. B
Let [ 0,1] be divided into 50 subintervals. ∆x =
Using f ( x) = x , the right Riemann sum
1
1
2
3
; x1 = , x2 = , x3 = , ⋅⋅⋅, x50 = 1
50
50
50
50
50
∑ f ( xi )∆x
is an approximation for
i =1
25. A
1
∫0
x dx .
Use the technique of antiderivatives by parts, which was removed from the AB Course
Description in 1998.
u=x
du = dx
dv = sin 2 x dx
1
v = − cos 2 x
2
1
1
1
1
∫ x sin(2 x) dx = − 2 x cos(2 x) + ∫ 2 cos(2 x) dx = − 2 x cos(2 x) + 4 sin(2 x) + C
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219
1997 Calculus AB Solutions: Part B
76. E
f ( x) =
e2 x
2e2 x ⋅ 2 x − 2e2 x e 2 x (2 x − 1)
; f ′( x) =
=
2x
4x2
2 x2
77. D
y = x3 + 6 x 2 + 7 x − 2 cos x . Look at the graph of y′′ = 6 x + 12 + 2 cos x in the window
[–3,–1] since that domain contains all the option values. y′′ changes sign at x = −1.89 .
78. D
F (3) − F (0) = ∫ f ( x)dx = ∫ f ( x)dx + ∫ f ( x)dx = 2 + 2.3 = 4.3
3
1
3
0
0
1
( Count squares for
1
∫ 0 f ( x)dx )
79. C
The stem of the questions means f ′(2) = 5 . Thus f is differentiable at x = 2 and therefore
continuous at x = 2. We know nothing of the continuity of f ′ . I and II only.
80. A
f ( x) = 2e4 x ; f ′( x) = 16 xe4 x ; We want 16 xe4 x = 3. Graph the derivative function and the
function y = 3, then find the intersection to get x = 0.168 .
81. A
Let x be the distance of the train from the
dx
crossing. Then
= 60 .
dt
dS
dx
dS x dx
S 2 = x 2 + 702 ⇒ 2 S
= 2x ⇒
=
.
dt
dt
dt S dt
After 4 seconds, x = 240 and so S = 250 .
dS 240
=
(60) = 57.6
Therefore
dt 250
82. B
P ( x) = 2 x 2 − 8 x; P′( x) = 4 x − 8; P′ changes from negative to positive at x = 2. P (2) = −8
83. C
cos x = x at x = 0.739085. Store this in A.
84. C
Cross sections are squares with sides of length y.
2
2
e
e
1
1
2
Volume = ∫ y 2 dx = ∫ ln x dx = (x ln x − x)
85. C
86. A
A
∫ 0 (cos x − x) dx = 0.400
e
1
= (e ln e − e) − (0 − 1) = 1
Look at the graph of f ′ and locate where the y changes from positive to negative. x = 0.91
f ( x) = x ; f ′( x) =
1
2 x
;
1
2 c
= 2⋅
1
2 1
⇒ c=
1
4
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220
1997 Calculus AB Solutions: Part B
87. B
1
a(t ) = t + sin t and v(0) = −2 ⇒ v(t ) = t 2 − cos t − 1; v(t ) = 0 at t = 1.48
2
88. E
f ( x) = ∫ h( x)dx ⇒ f (a) = 0, therefore only (A) or (E) are possible. But f ′( x) = h( x) and
x
a
therefore f is differentiable at x = b. This is true for the graph in option (E) but not in option
(A) where there appears to be a corner in the graph at x = b. Also, Since h is increasing at
first, the graph of f must start out concave up. This is also true in (E) but not (A).
89. B
90. D
1 1
T = ⋅ (3 + 2 ⋅ 3 + 2 ⋅ 5 + 2 ⋅ 8 + 13) = 12
2 2
1
F ( x) = sin 2 x
2
1
F ( x) = cos 2 x
2
1
F ( x) = − cos(2 x)
4
F ′( x) = sin x cos x
Yes
F ′( x) = − cos x sin x
No
1
F ′( x) = sin(2 x) = sin x cos x
2
Yes
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221
1998 AP Calculus AB:
Section I, Part A
55 Minutes—No Calculator
Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f (x) is a real number.
1.
1
What is the x-coordinate of the point of inflection on the graph of y = x3 + 5 x 2 + 24 ?
3
(A) 5
2.
(B)
0
(C)
−
10
3
(D) –5
(E)
−10
The graph of a piecewise-linear function f , for −1 ≤ x ≤ 4 , is shown above. What is the value of
4
∫ −1 f ( x) dx ?
(A) 1
3.
2
∫1
(A)
1
x2
(B) 2.5
(C)
4
(D) 5.5
(E) 8
7
24
(C)
1
2
(D) 1
(E)
dx =
−
1
2
(B)
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2 ln 2
125
1998 AP Calculus AB:
Section I, Part A
4.
5.
If f is continuous for a ≤ x ≤ b and differentiable for a < x < b , which of the following could be
false?
f (b) − f (a)
for some c such that a < c < b.
b−a
(A)
f ′(c) =
(B)
f ′(c) = 0 for some c such that a < c < b.
(C)
f has a minimum value on a ≤ x ≤ b.
(D)
f has a maximum value on a ≤ x ≤ b.
(E)
∫a
b
f ( x) dx exists.
x
∫ 0 sin t dt =
(A) sin x
6.
If x 2 + xy = 10, then when x = 2,
(A)
7.
e
∫1
(A)
8.
(B) − cos x
−
7
2
(B) –2
(C) cos x
(D) cos x − 1
(E) 1 − cos x
dy
=
dx
(C)
2
7
(C)
e2
1
−e+
2
2
(D)
3
2
(E)
7
2
(E)
e2 3
−
2 2
(E)
1
⎛ x2 − 1 ⎞
⎜⎜
⎟⎟ dx =
x
⎝
⎠
1
e−
e
(B)
2
e −e
2
e −2
(D)
Let f and g be differentiable functions with the following properties:
(i)
(ii)
g ( x) > 0 for all x
f (0) = 1
If h( x) = f ( x) g ( x) and h′( x) = f ( x) g ′( x), then f ( x) =
(A)
f ′( x)
(B)
g ( x)
(C)
ex
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(D)
0
126
1998 AP Calculus AB:
Section I, Part A
9.
The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown
above. Of the following, which best approximates the total number of barrels of oil that passed
through the pipeline that day?
(A)
500
(B)
600
(C)
2, 400
(D)
3, 000
(E)
10. What is the instantaneous rate of change at x = 2 of the function f given by f ( x) =
(A)
−2
(B)
1
6
11. If f is a linear function and 0 < a < b, then
(A)
0
(B) 1
1
2
(C)
(C)
b
∫a
(D) 2
4,800
x2 − 2
?
x −1
(E) 6
f ′′( x) dx =
ab
2
(D)
b−a
(E)
b2 − a 2
2
⎪⎧ ln x for 0 < x ≤ 2
then lim f ( x) is
12. If f ( x) = ⎨ 2
x →2
⎪⎩ x ln 2 for 2 < x ≤ 4,
(A)
ln 2
(B)
ln 8
(C)
ln16
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(D) 4
(E) nonexistent
127
1998 AP Calculus AB:
Section I, Part A
13. The graph of the function f shown in the figure above has a vertical tangent at the point ( 2, 0 ) and
horizontal tangents at the points (1, − 1) and ( 3,1) . For what values of x, −2 < x < 4 , is f not
differentiable?
(A) 0 only
(B) 0 and 2 only
(C) 1 and 3 only
(D) 0, 1, and 3 only
(E) 0, 1, 2, and 3
14. A particle moves along the x-axis so that its position at time t is given by x(t ) = t 2 − 6t + 5 . For
what value of t is the velocity of the particle zero?
(A) 1
15. If F ( x) = ∫
(A)
(B) 2
x
0
(C)
3
(D) 4
(E) 5
(C)
2
(D) 3
(E) 18
t 3 + 1 dt , then F ′(2) =
−3
(B)
−2
( )
16. If f ( x) = sin e − x , then f ′(x) =
(A)
− cos(e − x )
(B)
cos(e − x ) + e − x
(C)
cos(e − x ) − e − x
(D)
e − x cos(e− x )
(E)
−e − x cos(e− x )
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1998 AP Calculus AB:
Section I, Part A
17. The graph of a twice-differentiable function f is shown in the figure above. Which of the
following is true?
(A)
f (1) < f ′ (1) < f ′′ (1)
(B)
f (1) < f ′′ (1) < f ′ (1)
(C)
f ′ (1) < f (1) < f ′′ (1)
(D)
f ′′ (1) < f (1) < f ′ (1)
(E)
f ′′ (1) < f ′ (1) < f (1)
18. An equation of the line tangent to the graph of y = x + cos x at the point ( 0,1) is
(A)
y = 2x +1
y = x +1
(B)
(C)
y=x
(D)
y = x −1
(E)
y=0
19. If f ′′( x) = x ( x + 1)( x − 2 ) , then the graph of f has inflection points when x =
2
(A) –1 only (B) 2 only
(C) –1 and 0 only
20. What are all values of k for which
(A) –3
21. If
(B)
0
k
∫ −3 x
2
(D) –1 and 2 only (E) –1, 0, and 2 only
dx = 0 ?
(C)
3
(D)
–3 and 3
(E) –3, 0, and 3
dy
= ky and k is a nonzero constant, then y could be
dt
(A)
2e kty
(B)
2e kt
(C)
e kt + 3
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(D)
kty + 5
(E)
1 2 1
ky +
2
2
129
1998 AP Calculus AB:
Section I, Part A
22. The function f is given by f ( x) = x 4 + x 2 − 2 . On which of the following intervals is f
increasing?
(A)
⎛ 1
⎞
, ∞⎟
⎜−
2
⎝
⎠
(B)
1 ⎞
⎛ 1
,
⎜−
⎟
2
2⎠
⎝
(C)
( 0, ∞ )
(D)
( −∞, 0 )
(E)
1 ⎞
⎛
⎜ −∞, −
⎟
2⎠
⎝
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1998 AP Calculus AB:
Section I, Part A
23. The graph of f is shown in the figure above. Which of the following could be the graph of the
derivative of f ?
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131
1998 AP Calculus AB:
Section I, Part A
24. The maximum acceleration attained on the interval 0 ≤ t ≤ 3 by the particle whose velocity is given
by v(t ) = t 3 − 3t 2 + 12t + 4 is
(A) 9
(B) 12
(C)
14
(D) 21
(E) 40
25. What is the area of the region between the graphs of y = x 2 and y = − x from x = 0 to x = 2?
(A)
2
3
(B)
8
3
(C)
4
(D)
x
0
1
2
f ( x)
1
k
2
14
3
(E)
16
3
26. The function f is continuous on the closed interval [ 0, 2] and has values that are given in the table
above. The equation f ( x) =
(A)
(B)
0
1
must have at least two solutions in the interval [ 0, 2] if k =
2
1
2
(C)
1
(D) 2
(E) 3
27. What is the average value of y = x 2 x3 + 1 on the interval [ 0, 2] ?
(A)
26
9
(B)
52
9
(C)
26
3
(D)
52
3
(E) 24
(C)
4
(D)
4 3
(E) 8
⎛π⎞
28. If f ( x) = tan(2 x), then f ′ ⎜ ⎟ =
⎝6⎠
(A)
3
(B)
2 3
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1998 AP Calculus AB:
Section I, Part B
50 Minutes—Graphing Calculator Required
Notes: (1) The exact numerical value of the correct answer does not always appear among the choices
given. When this happens, select from among the choices the number that best approximates
the exact numerical value.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f (x) is a real number.
76.
The graph of a function f is shown above. Which of the following statements about f is false?
(A)
f is continuous at x = a .
(B)
f has a relative maximum at x = a .
(C)
x = a is in the domain of f.
(D)
(E)
lim f ( x) is equal to lim− f ( x) .
x →a +
x →a
lim f ( x) exists .
x →a
77. Let f be the function given by f ( x) = 3e 2 x and let g be the function given by g ( x) = 6 x3 . At what
value of x do the graphs of f and g have parallel tangent lines?
(A)
(B)
(C)
(D)
(E)
−0.701
−0.567
−0.391
−0.302
−0.258
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1998 AP Calculus AB:
Section I, Part B
78. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the
circumference C, what is the rate of change of the area of the circle, in square centimeters per
second?
(A)
− ( 0.2 ) π C
(B)
− ( 0.1) C
(C)
−
( 0.1) C
2π
(D)
( 0.1)2 C
(E)
( 0.1)2 π C
79. The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions
f, g, or h have a relative maximum on the open interval a < x < b ?
(A)
(B)
(C)
(D)
(E)
f only
g only
h only
f and g only
f, g, and h
80. The first derivative of the function f is given by f ′( x) =
does f have on the open interval ( 0,10 ) ?
(A)
(B)
(C)
(D)
(E)
cos 2 x 1
− . How many critical values
x
5
One
Three
Four
Five
Seven
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1998 AP Calculus AB:
Section I, Part B
81. Let f be the function given by f ( x) = x . Which of the following statements about f are true?
f is continuous at x = 0 .
f is differentiable at x = 0 .
f has an absolute minimum at x = 0 .
I.
II.
III.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
82. If f is a continuous function and if F ′( x) = f ( x) for all real numbers x, then
(A)
2 F (3) − 2 F (1)
(B)
1
1
F (3) − F (1)
2
2
(C)
2 F (6) − 2 F (2)
(D)
F (6) − F (2)
(E)
1
1
F (6) − F (2)
2
2
83. If a ≠ 0, then lim
x →a
(A)
1
a2
x2 − a2
x4 − a4
(B)
3
∫ 1 f ( 2 x ) dx =
is
1
2a 2
(C)
1
6a 2
(D)
0
(E) nonexistent
dy
= ky , where k is a constant and t is measured in
dt
years. If the population doubles every 10 years, then the value of k is
84. Population y grows according to the equation
(A) 0.069
(B) 0.200
(C)
0.301
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(D) 3.322
(E) 5.000
135
1998 AP Calculus AB:
Section I, Part B
2
x
f ( x)
5
7
8
10 30 40 20
85. The function f is continuous on the closed interval [ 2,8] and has values that are given in the table
above. Using the subintervals [ 2,5] , [5, 7 ] , and [ 7,8] , what is the trapezoidal approximation of
8
∫ 2 f ( x) dx ?
(A) 110
(B) 130
(C)
160
(D) 190
(E) 210
86. The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line
x + 2 y = 8 , as shown in the figure above. If cross sections of the solid perpendicular to the x-axis
are semicircles, what is the volume of the solid?
(A) 12.566
(B) 14.661
(C)
16.755
(D) 67.021
(E) 134.041
87. Which of the following is an equation of the line tangent to the graph of f ( x) = x 4 + 2 x 2 at the
point where f ′( x) = 1?
(A)
(B)
(C)
(D)
(E)
y = 8x − 5
y = x+7
y = x + 0.763
y = x − 0.122
y = x − 2.146
88. Let F ( x) be an antiderivative of
(A) 0.048
(B) 0.144
( ln x )3 . If
x
(C)
F (1) = 0, then F (9) =
5.827
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(D) 23.308
(E) 1,640.250
136
1998 AP Calculus AB:
Section I, Part B
89. If g is a differentiable function such that g ( x) < 0 for all real numbers x and if
(
)
f ′( x) = x 2 − 4 g ( x) , which of the following is true?
(A)
(B)
(C)
(D)
(E)
f has a relative maximum at x = −2 and a relative minimum at x = 2 .
f has a relative minimum at x = −2 and a relative maximum at x = 2 .
f has relative minima at x = −2 and at x = 2 .
f has relative maxima at x = −2 and at x = 2 .
It cannot be determined if f has any relative extrema.
90. If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is
decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of
the triangle?
(A)
(B)
(C)
(D)
(E)
A is always increasing.
A is always decreasing.
A is decreasing only when b < h .
A is decreasing only when b > h .
A remains constant.
91. Let f be a function that is differentiable on the open interval (1,10 ) . If f (2) = −5, f (5) = 5, and
f (9) = −5 , which of the following must be true?
I.
II.
III.
(A)
(B)
(C)
(D)
(E)
f has at least 2 zeros.
The graph of f has at least one horizontal tangent.
For some c, 2 < c < 5, f (c) = 3 .
None
I only
I and II only
I and III only
I, II, and III
92. If 0 ≤ k <
π
π
and the area under the curve y = cos x from x = k to x = is 0.1, then k =
2
2
(A) 1.471
(B) 1.414
(C)
1.277
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(D) 1.120
(E) 0.436
137
1998 Calculus AB Solutions: Part A
1.
D
y′ = x 2 + 10 x ; y′′ = 2 x + 10; y′′ changes sign at x = −5
2.
B
∫ −1 f ( x)dx = ∫ −1 f ( x)dx + ∫ 2
4
2
4
f ( x)dx
= Area of trapezoid(1) – Area of trapezoid(2) = 4 − 1.5 = 2.5
2
1
2
dx = ∫ x −2 dx = − x −1
2
1
2
3.
C
∫1
4.
B
This would be false if f was a linear function with non-zero slope.
5.
E
∫ 0 sin t dt = − cos t 0 = − cos x − (− cos 0) = − cos x + 1 = 1 − cos x
6.
A
Substitute x = 2 into the equation to find y = 3. Taking the derivative implicitly gives
d 2
x + xy = 2 x + xy′ + y = 0 . Substitute for x and y and solve for y′ .
dx
7
4 + 2 y′ + 3 = 0; y′ = −
2
x
2
1
x
x
(
e
1
=
)
e
e
x2 − 1
1
3
⎛1
⎞
⎛1
⎞ ⎛1
⎞ 1
dx = ∫ x − dx = ⎜ x 2 − ln x ⎟ = ⎜ e 2 − 1⎟ − ⎜ − 0 ⎟ = e 2 −
1
2
x
x
⎝2
⎠1 ⎝2
⎠ ⎝2
⎠ 2
7.
E
∫1
8.
E
h( x) = f ( x) g ( x) so, h′( x) = f ′( x) g ( x) + f ( x) g ′( x) . It is given that h′( x) = f ( x) g ′( x) .
Thus, f ′( x) g ( x) = 0 . Since g ( x) > 0 for all x, f ′( x) = 0 . This means that f is constant. It
is given that f (0) = 1 , therefore f ( x) = 1 .
9.
D
Let r (t ) be the rate of oil flow as given by the graph, where t is measured in hours. The total
number of barrels is given by
24
∫0
r (t )dt . This can be approximated by counting the squares
below the curve and above the horizontal axis. There are approximately five squares with
area 600 barrels. Thus the total is about 3, 000 barrels.
10. D
11. A
f ′( x) =
( x − 1)(2 x) − ( x 2 − 2)(1)
( x − 1) 2
; f ′(2) =
(2 − 1)(4) − (4 − 2)(1)
(2 − 1) 2
=2
Since f is linear, its second derivative is zero. The integral gives the area of a rectangle with
zero height and width (b − a) . This area is zero.
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1998 Calculus AB Solutions: Part A
12. E
lim f ( x) = ln 2 ≠ 4 ln 2 = lim+ f ( x) . Therefore the limit does not exist.
x →2−
x →2
13. B
At x = 0 and x = 2 only. The graph has a non-vertical tangent line at every other point in the
interval and so has a derivative at each of these other x’s.
14. C
v(t ) = 2t − 6; v(t ) = 0 for t = 3
15
By the Fundamental Theorem of Calculus, F ′( x) = x3 + 1, thus F ′(2) = 23 + 1 = 9 = 3 .
D
16. E
f ′( x) = cos(e− x ) ⋅
d −x
d
⎛
⎞
(e ) = cos(e− x ) ⎜ e− x ⋅ (− x) ⎟ = −e− x cos(e− x )
dx
dx
⎝
⎠
17. D
From the graph f (1) = 0 . Since f ′(1) represents the slope of the graph at x = 1 , f ′(1) > 0 .
Also, since f ′′(1) represents the concavity of the graph at x = 1 , f ′′(1) <0 .
18. B
y′ = 1 − sin x so y′(0) = 1 and the line with slope 1 containing the point (0,1) is y = x + 1 .
19. C
Points of inflection occur where f ′′ changes sign. This is only at x = 0 and x = −1 . There is
no sign change at x = 2.
20. A
∫ −3 x
21. B
The solution to this differential equation is known to be of the form y = y (0) ⋅ ekt . Option (B)
is the only one of this form. If you do not remember the form of the solution, then separate
the variables and antidifferentiate.
dy
= k dt ; ln y = kt + c1; y = ekt +c1 = ekt ec1 ; y = cekt .
y
22. C
f is increasing on any interval where f ′( x) > 0 . f ′( x) = 4 x3 + 2 x = 2 x(2 x 2 + 1) > 0 .
k
2
1
dx = x3
3
k
−3
=
(
) (
)
1 3
1
k − (−3)3 = k 3 + 27 = 0 only when k = −3.
3
3
Since ( x 2 + 1) > 0 for all x, f ′( x) > 0 whenever x > 0 .
23. A
The graph shows that f is increasing on an interval (a, c) and decreasing on the interval
(c, b) , where a < c < b . This means the graph of the derivative of f is positive on the
interval (a, c) and negative on the interval (c, b) , so the answer is (A) or (E). The derivative
is not (E), however, since then the graph of f would be concave down for the entire interval.
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1998 Calculus AB Solutions: Part A
24. D
25. D
26. A
The maximum acceleration will occur when its derivative changes from positive to negative
or at an endpoint of the interval. a (t ) = v′(t ) = 3t 2 − 6t + 12 = 3(t 2 − 2t + 4) which is always
positive. Thus the acceleration is always increasing. The maximum must occur at t = 3
where a(3) = 21
The area is given by
∫
2 2
x
0
1 ⎞
⎛1
− (− x) dx = ⎜ x3 + x 2 ⎟
2 ⎠
⎝3
2
0
Any value of k less than 1/2 will require the function to assume the value of 1/2 at least twice
because of the Intermediate Value Theorem on the intervals [0,1] and [1,2]. Hence k = 0 is
the only option.
1
27. A
28. E
8
14
= +2= .
3
3
3 2
1 2 2 3
1 2
1 1 2
⎛1
⎞
x x + 1 dx = ∫ ( x3 + 1) 2 ⎜ ⋅ 3 x 2 ⎟ dx = ⋅ ⋅ ( x3 + 1) 2
∫
0
0
2
2
2 3 3
⎝3
⎠
f ′( x) = sec2 (2 x) ⋅
0
3
3
1 ⎛ 2 2 ⎞ 26
⎜
= 9 −1 ⎟ =
⎟ 9
9⎜
⎝
⎠
d
⎛π⎞
⎛π⎞
(2 x) = 2sec 2 (2 x); f ′ ⎜ ⎟ = 2sec 2 ⎜ ⎟ = 2(4) = 8
dx
⎝6⎠
⎝3⎠
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1998 Calculus AB Solutions: Part B
76. A
From the graph it is clear that f is not continuous at x = a . All others are true.
77. C
Parallel tangents will occur when the slopes of f and g are equal. f ′( x) = 6e 2 x and
g ′( x) = 18 x 2 . The graphs of these derivatives reveal that they are equal only at x = −0.391 .
dA
dr
dr
dA
= 2πr . However, C = 2πr and
= −0.1 . Thus
= −0.1C .
dt
dt
dt
dt
78. B
A = πr 2 ⇒
79. A
The graph of the derivative would have to change from positive to negative. This is only true
for the graph of f ′ .
80. B
Look at the graph of f ′( x) on the interval (0,10) and count the number of x-intercepts in the
interval.
81. D
Only II is false since the graph of the absolute value function has a sharp corner at x = 0.
82. E
Since F is an antiderivative of f,
83. B
84. A
lim
x →a
x2 − a2
4
x −a
4
= lim
x→a ( x
3
∫1
f (2 x) dx =
x2 − a2
2
2
2
2
− a )( x + a )
= lim
x→a ( x
1
1
3
F (2 x) 1 = ( F (6) − F (2) )
2
2
1
2
2
+a )
=
1
2a 2
A known solution to this differential equation is y (t ) = y (0)e k t . Use the fact that the
population is 2 y (0) when t = 10. Then
2 y (0) = y (0)ek (10) ⇒ e10 k = 2 ⇒ k = (0.1) ln 2 = 0.069
1
1
1
⋅ 3 ( f (2) + f (5) ) + ⋅ 2 ( f (5) + f (7) ) + ⋅1( f (7) + f (8) )
2
2
2
85. C
There are 3 trapezoids.
86. C
Each cross section is a semicircle with a diameter of y. The volume would be given by
8
∫0
87. D
2
2
1 ⎛ y⎞
π 8 ⎛8− x ⎞
π ⎜ ⎟ dx = ∫ ⎜
⎟ dx = 16.755
2 ⎝2⎠
8 0 ⎝ 2 ⎠
Find the x for which f ′( x) = 1 . f ′( x) = 4 x3 + 4 x = 1 only for x = 0.237 . Then
f (0.237) = 0.115 . So the equation is y − 0.115 = x − 0.237 . This is equivalent to option (D).
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231
1998 Calculus AB Solutions: Part B
88. C
F (9) − F (1) = ∫
9
1
(ln t )3
dt = 5.827 using a calculator. Since F (1) = 0 , F (9) = 5.827.
t
Or solve the differential equation with an initial condition by finding an antiderivative for
(ln x)3
1
. This is of the form u 3du where u = ln x . Hence F ( x) = (ln x) 4 + C and since
x
4
1
F (1) = 0 , C = 0. Therefore F (9) = (ln 9)4 = 5.827
4
89. B
The graph of y = x 2 − 4 is a parabola that changes from positive to negative at x = −2 and
from negative to positive at x = 2 . Since g is always negative, f ′ changes sign opposite to
the way y = x 2 − 4 does. Thus f has a relative minimum at x = −2 and a relative maximum
at x = 2 .
90. D
91. E
1
The area of a triangle is given by A = bh . Taking the derivative with respect to t of both
2
dA 1 ⎛ db
dh ⎞
sides of the equation yields
= ⎜ ⋅ h + b ⋅ ⎟ . Substitute the given rates to get
dt 2 ⎝ dt
dt ⎠
dA
3
dA 1
< 0 . This is true
= (3h − 3b) = (h − b) . The area will be decreasing whenever
dt
2
dt 2
whenever b > h .
I. True. Apply the Intermediate Value Theorem to each of the intervals [2,5] and [5,9] .
II. True. Apply the Mean Value Theorem to the interval [2,9] .
III. True. Apply the Intermediate Value Theorem to the interval [2,5].
92. D
∫
π
2 cos x dx
k
⎛π⎞
= 0.1 ⇒ sin ⎜ ⎟ − sin k = 0.1 ⇒ sin k = 0.9 . Therefore k = sin −1 (0.9) = 1.120 .
⎝2⎠
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AP Calculus 2003
Calculators ARE NOT Permitted On This Portion Of The Exam
28 Questions - 55 Minutes
1) Give f(g(1)), given that
a) -8/9
b) 7/3
c) 2
d) 4/3
e) -2/9
2) Find the slope of the tangent line to the graph of f at x = 4, given that
a) -8
b) -10
c) -9
d) -5
e) -7
3) Determine
a) ∞
b) 0
c) ½
d) 3/10
e) 1
4) Let f(x) = x 3
A region is bounded between the graphs of y = -1 and y = f(x) for x between -1 and 0, and
between the graphs of y = 1 and y = f(x) for x between 0 and 1. Give an integral that
corresponds to the area of this region.
a)
b)
c)
d)
e)
5) Given that
Determine the change in y with respect to x.
a)
b)
c)
d)
e)
6) Compute the derivative of -4 sec (x) + 2 csc (x)
a)
b)
c)
d)
e)
7) Compute
a) - π
b) 3/2 π
c) ½ π
d) π
e) 0
8) Determine
a)
b)
c)
d)
e)
9) Give the equation of the normal line to the graph of
at the point ( 0 , 2 ).
a)
b)
c)
d)
e)
10) Determine the concavity of the graph of
at x = π.
a) 8
b) -10
c) 4
d) -8
e) -6
11) Compute
a)
b)
c)
d)
e)
12) Give the value of x where the function
has a local maximum.
a) 4
b) -2
c) 2
d) -4
e) 3
13) The slope of the tangent line to the graph of
at x = 0 is 4. Give the value of c.
a) -2
b) 4
c) 8
d) -4
e) -8
14) Compute
a)
b)
c)
d)
e)
15) What is the average value of the function
x = -3 to x = -1?
a) 7/3
b) -4
c) 5
d) 14/3
e) 3
16) Compute
a) 1
b) ¼ π
c) π
g (x) = (2x + 3) 2 on the interval from
d) 2
e) -1
17) Find the instantaneous rate of change of
at t = 0.
a) -3
b) -3/4
c) 0
d) -4
e) -5/4
18) Compute
a)
b)
c)
d)
e)
19) A solid is generated by rotating the region enclosed by the graph of
the lines x = 1, x = 2, and y = 1, about the x-axis. Which of the following integrals gives
the volume of the solid?
a)
b)
c)
d)
e)
20) Compute
a) ∞
b) 0
c) -5/2
d) -2
e) undefined
21) Given y > 0 and
If the point
is on the graph relating x and y, then what is y when x = 0?
a) 3
b) 2
c) 1
d) 6
e) 10
22) Determine
a) ½ π
b) 1/3 π
c) π
d) 1/6 π
e) ¼ π
23) Determine
a)
b)
c)
d)
e)
24) A particle's acceleration for t > 0 is given by a(t) = 12 t + 4. The particle's initial
position is 2 and its velocity at t = 1 is 5. What is the position of the particle at t = 2?
a) 10
b) 12
c) 16
d) 4
e) 20
25) Determine
a) -1
b) 1
c) 0
d) 2/3
e) -2/3
26) Determine the derivative of
at x =
/b.
a)
b)
c)
d)
e)
27) Compute the derivative of
a)
b)
c)
d)
e)
28) Determine
a)
b)
c)
d)
e)
Calculators ARE Permitted On This Portion Of The Exam
17 Questions - 50 Minutes
1) Give a value of c that satisfies the conclusion of the Mean Value Theorem for
Derivatives for the function
on the interval [1,3].
a) 9/4
b) 3/2
c) ½
d) 2
e) 5/4
2) The function
is invertible. Give the derivative of f -1 at x = 2.
a)
b)
c)
d)
e)
3) The derivative of f is graphed below.
Give a value of x where f has a local maximum.
a) -4
b) -1
c) -5/2
d) There is no such value of x
e) 1
4) Let
Which of the following is (are) true?
1) f is continuous at x = -2.
2) f is differentiable at x = 1.
3) f has a local minimum at x = 0.
4) f has an absolute maximum at x = -2.
a) 2 and 4
b) 3 only
c) 2 only
d) 1 and 3
e) 1 and 4
5) Given
Determine
a) 10
b) -3
c) There is not enough infomration
d) -6
e) 5
6) Give the approximate location of a local maximum for the function
a)
b)
c)
d)
e)
7) Give the approximate average value of the function f(x) = 4 x ln (2x) over the interval
[1,4].
a) 19.71
b) 12.54
c) 16.71
d) 18.02182670
e) 18.71
8) The region enclosed by the graphs of
is rotated around the y-axis to generate a solid. What is the volume of the solid?
a) 0.8380
b) 0.7855
c) 1.676
d) 1.047
e) 2.356
9) What is the approximate instantaneous rate of change of the function
at t = /7?
a) -.9009
b) -7.207
c) 3.3473
d) 0.4341
e) -1.030
10) What is the error when the integral
is approximated by the Trapezoidal rule with n = 3?
a) 0.011
b) 0.032
c) 0.109
d) 0.059
e) 0.051
11) The amount of money in a bank account is increasing at the rate of
dollars per year, where t is measured in years. If t = 0 corresponds to the year 2005, then
what is the approximate total amount of increase from 2005 to 2007.
a) $18,350
b) $4,500
c) $21,250
d) $32,560
e) $16,250
12) A particle moves with acceleration
and its initial velocity is 0. For how many values of t does the particle change direction?
a) 3
b) 2
c) 1
d) 0
e) 4
13) At what approximate rate (in cubic meters per minute) is the volume of a sphere
changing at the instant when the surface area is 5 square meters and the radius is
increasing at the rate of 1/3 meters per minute?
a) 5.271
b) 1.700
c) 1.667
d) 1.080
e) 2.714
14) A rectangle has one side on the x-axis and the upper two vertices on the graph of
Give a decimal approximation to the maximum possible area for this rectangle.
a) 1.649
b) 1
c) -1
d) 0.5458
e) 0.6065
15) A rough approximation for ln(5) is 1.609. Use this approximation and differentials to
approximate ln(128/25).
a) 1.633
b) 1.621
c) 1.632
d) 1.585
e) 1.597
16) The function
is differentiable everywhere. What is n?
a) -9
b) 13
c) -17
d) -11
e) -14
17) Which of the following functions has a vertical asymptote at x = -1 and a horizontal
asymptote at y = 2?
a)
b)
c)
d)
e)
Answers
Part A
1) d)
2) e)
3) b)
4) b)
5) e)
6) a)
7) c)
8) c)
9) b)
10) d)
11) c)
12) c)
13) c)
14) e)
15) a)
16) d)
17) a)
18) b)
19) a)
20) d)
21) b)
22) b)
23) a)
24) c)
25) d)
26) b)
27) b)
28) c)
Part B
1) d)
2) e)
3) b)
4) b)
5) e)
6) a)
7) c)
8) c)
9) b)
10) d)
11) c)
12) c)
13) c)
14) e)
15) a)
16) d)
17) a)
Advanced Placement
Program
AP® Calculus AB
Practice Exam
The questions contained in this AP® Calculus AB Practice Exam are written to the content specifications
of AP Exams for this subject. Taking this practice exam should provide students with an idea of their
general areas of strengths and weaknesses in preparing for the actual AP Exam. Because this AP
Calculus AB Practice Exam has never been administered as an operational AP Exam, statistical data are
not available for calculating potential raw scores or conversions into AP grades.
This AP Calculus AB Practice Exam is provided by the College Board for AP Exam preparation. Teachers
are permitted to download the materials and make copies to use with their students in a classroom setting
only. To maintain the security of this exam, teachers should collect all materials after their administration
and keep them in a secure location. Teachers may not redistribute the files electronically for any reason.
© 2008 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central,
SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may
be trademarks of their respective owners. Visit the College Board on the Web: www.collegeboard.com.
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
CALCULUS AB
SECTION I, Part A
Time—55 minutes
Number of questions—28
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining
the form of the choices, decide which is the best of the choices given and place the letter of your choice in the
corresponding box on the student answer sheet. Do not spend too much time on any one problem.
In this exam:
(1)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x ) is a real number.
(2)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
GO ON TO THE NEXT PAGE.
-2-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
Ú cos (3x ) dx =
1.
(A) -3sin (3x ) + C
1
(B) - sin (3 x ) + C
3
(C)
1
sin (3 x ) + C
3
(D) sin (3x ) + C
(E) 3sin (3x ) + C
2.
2 x6 + 6 x3
is
xÆ0 4 x5 + 3x3
lim
(A) 0
(B)
1
2
(C) 1
(D) 2
(E) nonexistent
GO ON TO THE NEXT PAGE.
-3-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
Ï x 2 - 3 x + 9 for x £ 2
f ( x) = Ì
for x > 2
Ó kx + 1
3. The function f is defined above. For what value of k, if any, is f continuous at x = 2 ?
(A) 1
(B) 2
(C) 3
(D) 7
(E) No value of k will make f continuous at x = 2.
4. If f ( x ) = cos3 (4 x ) , then f ¢( x ) =
(A) 3cos2 (4x )
(B) -12 cos2 ( 4 x ) sin (4 x )
(C) -3cos2 ( 4 x ) sin (4 x )
(D) 12 cos2 ( 4 x ) sin (4 x )
(E) - 4 sin 3 ( 4x )
GO ON TO THE NEXT PAGE.
-4-
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5. The function f given by f ( x ) = 2 x 3 - 3 x 2 - 12 x has a relative minimum at x =
(A) -1
(B) 0
(C) 2
(D)
3 - 105
4
(E)
3 + 105
4
6. Let f be the function given by f ( x ) = (2 x - 1) ( x + 1) . Which of the following is an equation for the line
tangent to the graph of f at the point where x = 1 ?
5
(A) y = 21x + 2
(B) y = 21x - 19
(C) y = 11x - 9
(D) y = 10 x + 2
(E) y = 10 x - 8
GO ON TO THE NEXT PAGE.
-5-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
Û e x dx =
Ù
ı x
7.
(A) 2e
(B)
1
e
2
(C) e
x
x
x
+C
+C
(D) 2 x e
(E)
+C
x
+C
1e x
+C
2 x
x
0
2
4
6
f ( x)
4
k
8
12
8. The function f is continuous on the closed interval [0, 6] and has the values given in the table above.
The trapezoidal approximation for
6
Ú0 f ( x ) dx found with 3 subintervals of equal length is 52. What
is the value of k ?
(A) 2
(B) 6
(C) 7
(D) 10
(E) 14
GO ON TO THE NEXT PAGE.
-6-
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9. A particle moves along the x-axis so that at any time t > 0, its velocity is given by v(t ) = 4 - 6t 2 . If the
particle is at position x = 7 at time t = 1, what is the position of the particle at time t = 2 ?
(A) -10
(B) -5
(C) -3
(D) 3
(E) 17
ax 2 + 12
. The figure above shows a portion of the graph of f. Which of the
x2 + b
following could be the values of the constants a and b ?
10. The function f is given by f ( x ) =
(A) a = -3, b = 2
(B) a = 2, b = -3
(C) a = 2, b = -2
(D) a = 3, b = - 4
(E) a = 3, b = 4
GO ON TO THE NEXT PAGE.
-7-
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11. What is the slope of the line tangent to the graph of y =
(A) -
1
e
12. If f ¢( x ) =
(A) 2
(B) -
3
4e
(C) -
1
4e
e- x
at x = 1 ?
x +1
1
4e
(D)
(E)
1
e
2
and f ( e ) = 5, then f (e ) =
x
(B) ln 25
(C) 5 +
2
2
- 2
e e
(D) 6
(E) 25
GO ON TO THE NEXT PAGE.
-8-
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Ú (x
13.
3
)
+1
2
dx =
(A)
1 7
x + x+C
7
(B)
1 7 1 4
x + x + x +C
7
2
(
)
(C) 6 x 2 x 3 + 1 + C
(D)
(
)
3
1 3
x +1 +C
3
( x3 + 1)
3
(E)
14.
9x2
+C
e( 2 + h ) - e 2
=
h
hÆ0
lim
(A) 0
(B) 1
(C) 2e
(D) e2
(E) 2e2
GO ON TO THE NEXT PAGE.
-9-
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15. The slope field for a certain differential equation is shown above. Which of the following could be a solution to
the differential equation with the initial condition y(0 ) = 1 ?
(A) y = cos x
(B) y = 1 - x 2
(C) y = e x
(D) y = 1 - x 2
(E) y =
1
1 + x2
GO ON TO THE NEXT PAGE.
-10-
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16. If f ¢( x ) = x - 2 , which of the following could be the graph of y = f ( x ) ?
(A)
(B)
(C)
(D)
(E)
GO ON TO THE NEXT PAGE.
-11-
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17. What is the area of the region enclosed by the graphs of f ( x ) = x - 2 x 2 and g( x ) = -5 x ?
(A)
7
3
(B)
16
3
(C)
20
3
(D) 9
(E) 36
18. For the function f, f ¢( x ) = 2 x + 1 and f (1) = 4. What is the approximation for f (1.2 ) found by using the line
tangent to the graph of f at x = 1 ?
(A) 0.6
(B) 3.4
(C) 4.2
(D) 4.6
(E) 4.64
GO ON TO THE NEXT PAGE.
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19. Let f be the function given by f ( x ) = x 3 - 6 x 2 . The graph of f is concave up when
(A) x > 2
(B) x < 2
(C) 0 < x < 4
(D) x < 0 or x > 4 only
(E) x > 6 only
20. If g( x ) = x 2 - 3 x + 4 and f ( x ) = g ¢( x ) , then
(A) -
14
3
(B) -2
(C) 2
3
Ú1 f ( x ) dx =
(D) 4
(E)
14
3
GO ON TO THE NEXT PAGE.
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21. The graph of f ¢, the derivative of the function f, is shown above for 0 £ x £ 10. The areas of the regions
between the graph of f ¢ and the x-axis are 20, 6, and 4, respectively. If f (0 ) = 2, what is the maximum value
of f on the closed interval 0 £ x £ 10 ?
(A) 16
(B) 20
(C) 22
(D) 30
(E) 32
22. If f ¢( x ) = ( x - 2 )( x - 3) ( x - 4 ) , then f has which of the following relative extrema?
2
3
I. A relative maximum at x = 2
II. A relative minimum at x = 3
III. A relative maximum at x = 4
(A) I only
(B) III only
(C) I and III only
(D) II and III only
(E) I, II, and III
GO ON TO THE NEXT PAGE.
-14-
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23. The graph of the even function y = f ( x ) consists of 4 line segments, as shown above. Which of the following
statements about f is false?
(A) lim ( f ( x ) - f (0 )) = 0
x Æ0
(B) lim
f ( x ) - f (0 )
=0
x
(C) lim
f ( x ) - f (- x )
=0
2x
(D) lim
f ( x ) - f (2 )
=1
x-2
(E) lim
f ( x ) - f (3 )
does not exist.
x-3
xÆ0
xÆ0
xÆ2
x Æ3
GO ON TO THE NEXT PAGE.
-15-
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24. The radius of a circle is increasing. At a certain instant, the rate of increase in the area of the circle is numerically
equal to twice the rate of increase in its circumference. What is the radius of the circle at that instant?
(A)
1
2
(B) 1
(C)
(D) 2
2
25. If x 2 y - 3 x = y3 - 3, then at the point ( -1, 2 ) ,
(A) -
7
11
(B) -
7
13
(C) -
1
2
(E) 4
dy
=
dx
(D) -
3
14
(E) 7
GO ON TO THE NEXT PAGE.
-16-
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26. For x > 0, f is a function such that f ¢( x ) =
ln x
1 - ln x
. Which of the following is true?
and f ¢¢( x ) =
x
x2
(A) f is decreasing for x > 1, and the graph of f is concave down for x > e.
(B) f is decreasing for x > 1, and the graph of f is concave up for x > e.
(C) f is increasing for x > 1, and the graph of f is concave down for x > e.
(D) f is increasing for x > 1, and the graph of f is concave up for x > e.
(E) f is increasing for 0 < x < e, and the graph of f is concave down for 0 < x < e3 2 .
27. If f is the function given by f ( x ) =
(A) 0
(B)
7
2 12
(C)
2x
Ú4
2
t 2 - t dt, then f ¢(2 ) =
(D)
12
(E) 2 12
GO ON TO THE NEXT PAGE.
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28. If y = sin -1 (5 x ) , then
(A)
1
1 + 25x 2
(B)
5
1 + 25x 2
(C)
(D)
(E)
dy
=
dx
-5
1 - 25x 2
1
1 - 25x 2
5
1 - 25x 2
END OF PART A OF SECTION I
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WORK ON PART A ONLY.
DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.
-18-
B
B
B
B
B
B
B
B
B
CALCULUS AB
SECTION I, Part B
Time—50 minutes
Number of questions—17
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON
THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining
the form of the choices, decide which is the best of the choices given and place the letter of your choice in the
corresponding box on the student answer sheet. Do not spend too much time on any one problem.
In this exam:
(1)
The exact numerical value of the correct answer does not always appear among the choices given. When this
happens, select from among the choices the number that best approximates the exact numerical value.
(2)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x ) is a real number.
(3)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
GO ON TO THE NEXT PAGE.
-19-
B
B
B
B
B
B
B
B
B
76. A particle moves along the x-axis so that at any time t ≥ 0 its velocity is given by v(t ) = t 2 ln (t + 2 ) . What is
the acceleration of the particle at time t = 6 ?
(A) 1.500
77. If
(B) 20.453
3
5
(C) 29.453
(D) 74.860
(E) 133.417
5
Ú0 f ( x ) dx = 6 and Ú3 f ( x ) dx = 4, then Ú0 (3 + 2 f ( x )) dx =
(A) 10
(B) 20
(C) 23
(D) 35
(E) 50
GO ON TO THE NEXT PAGE.
-20-
B
B
B
B
B
B
B
B
B
78. For t ≥ 0 hours, H is a differentiable function of t that gives the temperature, in degrees Celsius, at an Arctic
weather station. Which of the following is the best interpretation of H ¢(24 ) ?
(A) The change in temperature during the first day
(B) The change in temperature during the 24th hour
(C) The average rate at which the temperature changed during the 24th hour
(D) The rate at which the temperature is changing during the first day
(E) The rate at which the temperature is changing at the end of the 24th hour
79. A spherical tank contains 81.637 gallons of water at time t = 0 minutes. For the next 6 minutes, water flows out
of the tank at the rate of 9sin ( t + 1 ) gallons per minute. How many gallons of water are in the tank at the end
of the 6 minutes?
(A) 36.606
(B) 45.031
(C) 68.858
(D) 77.355
(E) 126.668
GO ON TO THE NEXT PAGE.
-21-
B
B
B
B
B
B
B
B
B
80. A left Riemann sum, a right Riemann sum, and a trapezoidal sum are used to approximate the value of
1
Ú0 f ( x ) dx, each using the same number of subintervals. The graph of the function f is shown in the figure
above. Which of the sums give an underestimate of the value of
1
Ú0 f ( x ) dx ?
I. Left sum
II. Right sum
III. Trapezoidal sum
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
GO ON TO THE NEXT PAGE.
-22-
B
B
B
B
B
B
B
B
B
81. The first derivative of the function f is given by f ¢( x ) = x - 4e - sin(2 x ) . How many points of inflection does the
graph of f have on the interval 0 < x < 2 p ?
(A) Three
(B) Four
(C) Five
(D) Six
(E) Seven
82. If f is a continuous function on the closed interval [a, b], which of the following must be true?
(A) There is a number c in the open interval (a, b ) such that f (c ) = 0.
(B) There is a number c in the open interval (a, b ) such that f ( a ) < f (c ) < f (b ) .
(C) There is a number c in the closed interval [a, b] such that f (c ) ≥ f ( x ) for all x in [a, b].
(D) There is a number c in the open interval (a, b ) such that f ¢(c ) = 0.
(E) There is a number c in the open interval (a, b ) such that f ¢(c ) =
f (b ) - f ( a )
.
b-a
GO ON TO THE NEXT PAGE.
-23-
B
B
B
B
B
B
B
x
2.5
2.8
3.0
3.1
f ( x)
31.25
39.20
45
48.05
B
B
83. The function f is differentiable and has values as shown in the table above. Both f and f ¢ are strictly increasing
on the interval 0 £ x £ 5. Which of the following could be the value of f ¢(3) ?
(A) 20
(B) 27.5
(C) 29
(D) 30
(E) 30.5
84. The graph of f ¢, the derivative of the function f, is shown above. On which of the following intervals is f
decreasing?
(A) [2, 4] only
(B) [3, 5] only
(C) [0, 1] and [3, 5]
(D) [2, 4] and [6, 7]
(E)
[0, 2] and [4, 6]
GO ON TO THE NEXT PAGE.
-24-
B
B
B
B
B
B
B
B
B
x2
x2
and y = for 1 £ x £ 4, as shown in
10
10
the figure above. For this loudspeaker, the cross sections perpendicular to the x-axis are squares. What is the
volume of the loudspeaker, in cubic units?
85. The base of a loudspeaker is determined by the two curves y =
(A) 2.046
(B) 4.092
(C) 4.200
(D) 8.184
(E) 25.711
GO ON TO THE NEXT PAGE.
-25-
B
B
B
B
x
f ( x)
B
B
B
3
4
5
6
7
20
17
12
16
20
B
B
86. The function f is continuous and differentiable on the closed interval [3, 7]. The table above gives selected
values of f on this interval. Which of the following statements must be true?
I. The minimum value of f on [3, 7] is 12.
II. There exists c, for 3 < c < 7, such that f ¢(c ) = 0.
III. f ¢( x ) > 0 for 5 < x < 7.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
GO ON TO THE NEXT PAGE.
-26-
B
B
B
B
B
B
B
B
B
87. The figure above shows the graph of f ¢, the derivative of the function f, on the open interval -7 < x < 7. If
f ¢ has four zeros on -7 < x < 7, how many relative maxima does f have on -7 < x < 7 ?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
88. The rate at which water is sprayed on a field of vegetables is given by R (t ) = 2 1 + 5t 3 , where t is in minutes
and R (t ) is in gallons per minute. During the time interval 0 £ t £ 4, what is the average rate of water flow, in
gallons per minute?
(A) 8.458
(B) 13.395
(C) 14.691
(D) 18.916
(E) 35.833
GO ON TO THE NEXT PAGE.
-27-
B
B
B
B
B
B
B
x
f ( x)
f ¢( x )
g( x )
g ¢( x )
1
3
–2
–3
4
B
B
89. The table above gives values of the differentiable functions f and g and their derivatives at x = 1. If
h( x ) = (2 f ( x ) + 3) (1 + g ( x )) , then h ¢(1) =
(A) -28
(B) -16
(C) 40
(D) 44
(E) 47
90. The functions f and g are differentiable. For all x, f ( g( x )) = x and g( f ( x )) = x.
If f (3) = 8 and f ¢(3) = 9, what are the values of g(8) and g¢(8) ?
(A) g(8) =
1
1
and g ¢(8) = 3
9
(B) g(8) =
1
1
and g ¢(8) =
3
9
(C) g(8) = 3 and g ¢(8) = -9
(D) g(8) = 3 and g ¢(8) = (E) g(8) = 3 and g ¢(8) =
1
9
1
9
GO ON TO THE NEXT PAGE.
-28-
B
B
B
B
B
B
B
B
B
91. A particle moves along the x-axis so that its velocity at any time t ≥ 0 is given by v(t ) = 5te - t - 1. At t = 0,
the particle is at position x = 1. What is the total distance traveled by the particle from t = 0 to t = 4 ?
(A) 0.366
(B) 0.542
(C) 1.542
(D) 1.821
(E) 2.821
( )
92. Let f be the function with first derivative defined by f ¢( x ) = sin x 3 for 0 £ x £ 2. At what value of x does f
attain its maximum value on the closed interval 0 £ x £ 2 ?
(A) 0
(B) 1.162
(C) 1.465
(D) 1.845
(E) 2
END OF SECTION I
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WORK ON PART B ONLY.
DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO.
________________________________________________
-29-
AP® Calculus
Instructions for Section II Free-Response Questions
Write clearly and legibly. Cross out any errors you make; erased or crossed-out work will not be graded.
Manage your time carefully. During the timed portion for Part A, work only on the questions in Part A. You are
permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the
value of a definite integral. However, you must clearly indicate the setup of your question, namely the equation,
function, or integral you are using. If you use other built-in features or programs, you must show the mathematical
steps necessary to produce your results. During the timed portion for Part B, you may continue to work on the
questions in Part A without the use of a calculator.
For each part of Section II, you may wish to look over the questions before starting to work on them. It is not
expected that everyone will be able to complete all parts of all questions.
•
Show all of your work. Clearly label any functions, graphs, tables, or other objects that you use. Your work
will be graded on the correctness and completeness of your methods as well as your answers. Answers
without supporting work may not receive credit. Justifications require that you give mathematical
(noncalculator) reasons.
•
Your work must be expressed in standard mathematical notation rather than calculator syntax. For example,
5 2
Ú1 x
dx may not be written as fnInt(X2, X, 1, 5).
•
Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If you use decimal
approximations in calculations, your work will be graded on accuracy. Unless otherwise specified, your final
answers should be accurate to three places after the decimal point.
•
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for
which f ( x ) is a real number.
-31-
CALCULUS AB
SECTION II, Part A
Time—45 minutes
Number of problems—3
A graphing calculator is required for some problems or parts of problems.
Ê pt2 ˆ
gallons per hour
1. The rate at which raw sewage enters a treatment tank is given by E (t ) = 850 + 715cos Á
Ë 9 ˜¯
for 0 £ t £ 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per hour.
The treatment tank is empty at time t = 0.
(a) How many gallons of sewage enter the treatment tank during the time interval 0 £ t £ 4 ? Round your
answer to the nearest gallon.
(b) For 0 £ t £ 4, at what time t is the amount of sewage in the treatment tank greatest? To the nearest gallon,
what is the maximum amount of sewage in the tank? Justify your answers.
(c) For 0 £ t £ 4, the cost of treating the raw sewage that enters the tank at time t is (0.15 - 0.02t ) dollars
per gallon. To the nearest dollar, what is the total cost of treating all the sewage that enters the tank during
the time interval 0 £ t £ 4 ?
GO ON TO THE NEXT PAGE.
-32-
2. Let R and S in the figure above be defined as follows: R is the region in the first and second quadrants
bounded by the graphs of y = 3 - x 2 and y = 2 x. S is the shaded region in the first quadrant bounded by
the two graphs, the x-axis, and the y-axis.
(a) Find the area of S.
(b) Find the volume of the solid generated when R is rotated about the horizontal line y = -1.
(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an
isosceles right triangle with one leg across the base of the solid. Write, but do not evaluate, an integral
expression that gives the volume of the solid.
t (minutes)
0
4
8
12
16
H (t ) (∞C )
65
68
73
80
90
3. The temperature, in degrees Celsius (∞C ) , of an oven being heated is modeled by an increasing differentiable
function H of time t, where t is measured in minutes. The table above gives the temperature as recorded every
4 minutes over a 16-minute period.
(a) Use the data in the table to estimate the instantaneous rate at which the temperature of the oven is changing
at time t = 10. Show the computations that lead to your answer. Indicate units of measure.
(b) Write an integral expression in terms of H for the average temperature of the oven between time t = 0 and
time t = 16. Estimate the average temperature of the oven using a left Riemann sum with four subintervals
of equal length. Show the computations that lead to your answer.
(c) Is your approximation in part (b) an underestimate or an overestimate of the average temperature? Give a
reason for your answer.
(d) Are the data in the table consistent with or do they contradict the claim that the temperature of the oven is
increasing at an increasing rate? Give a reason for your answer.
END OF PART A OF SECTION II
-33-
CALCULUS AB
SECTION II, Part B
Time—45 minutes
Number of problems—3
No calculator is allowed for these problems.
4. Let f be the function given by f ( x ) = (ln x )(sin x ) . The figure above shows the graph of f for 0 < x £ 2 p .
The function g is defined by g( x ) =
x
Ú1
f (t ) dt for 0 < x £ 2 p .
(a) Find g(1) and g¢(1) .
(b) On what intervals, if any, is g increasing? Justify your answer.
(c) For 0 < x £ 2 p , find the value of x at which g has an absolute minimum. Justify your answer.
(d) For 0 < x < 2 p , is there a value of x at which the graph of g is tangent to the x-axis? Explain why
or why not.
GO ON TO THE NEXT PAGE.
-34-
5. Consider the differential equation
dy
x
= , where y π 0.
dx
y
(a) The slope field for the given differential equation is shown below. Sketch the solution curve that passes
through the point (3, -1) , and sketch the solution curve that passes through the point (1, 2 ) .
(Note: The points (3, -1) and (1, 2 ) are indicated in the figure.)
(b) Write an equation for the line tangent to the solution curve that passes through the point (1, 2 ) .
(c) Find the particular solution y = f ( x ) to the differential equation with the initial condition f (3) = -1,
and state its domain.
6. Let g( x ) = xe - x + be - x , where b is a positive constant.
(a) Find lim g ( x ) .
x Æ•
(b) For what positive value of b does g have an absolute maximum at x =
2
? Justify your answer.
3
(c) Find all values of b, if any, for which the graph of g has a point of inflection on the interval 0 < x < •.
Justify your answer.
STOP
END OF EXAM
-35-
AP® Calculus AB
Multiple-Choice Answer Key
No.
1
Correct
Answer
C
No.
76
Correct
Answer
C
2
D
77
D
3
C
78
E
4
B
79
A
5
C
80
D
6
B
81
B
7
A
82
C
8
D
83
D
9
C
84
E
10
D
85
D
11
B
86
B
12
D
87
A
13
B
88
C
14
D
89
D
15
E
90
E
16
E
91
D
17
D
92
C
18
D
19
A
20
C
21
C
22
A
23
B
24
D
25
A
26
C
27
E
28
E
-37-
AP® Calculus AB
Free-Response Scoring Guidelines
Question 1
Ê pt 2 ˆ
The rate at which raw sewage enters a treatment tank is given by E (t ) = 850 + 715cos Á
gallons
Ë 9 ˜¯
per hour for 0 £ t £ 4 hours. Treated sewage is removed from the tank at the constant rate of 645
gallons per hour. The treatment tank is empty at time t = 0.
(a) How many gallons of sewage enter the treatment tank during the time interval 0 £ t £ 4 ? Round
your answer to the nearest gallon.
(b) For 0 £ t £ 4, at what time t is the amount of sewage in the treatment tank greatest? To the nearest
gallon, what is the maximum amount of sewage in the tank? Justify your answers.
(c) For 0 £ t £ 4, the cost of treating the raw sewage that enters the tank at time t is (0.15 - 0.02t )
dollars per gallon. To the nearest dollar, what is the total cost of treating all the sewage that enters
the tank during the time interval 0 £ t £ 4 ?
(a)
4
Ú0 E (t ) dt ª 3981 gallons
2:
(b) Let S (t ) be the amount of sewage in the treatment tank at
time t. Then S ¢(t ) = E (t ) - 645 and S ¢(t ) = 0 when
E (t ) = 645. On the interval 0 £ t £ 4, E (t ) = 645
when t = 2.309 and t = 3.559.
t (hours)
0
1 : integral
1 : answer
Ï 1 : sets E (t ) = 645
Ô 1 : identifies t = 2.309 as
Ô
4: Ì
a candidate
Ô 1 : amount of sewage at t = 2.309
Ô
Ó 1 : conclusion
amount of sewage in treatment tank
0
2.309
E (t ) dt - 645 (2.309) = 1637.178
3.559
E (t ) dt - 645 (3.559) = 1228.520
2.309
Ú0
3.559
Ú0
4
{
3981.022 - 645(4) = 1401.022
The amount of sewage in the treatment tank is greatest at
t = 2.309 hours. At that time, the amount of sewage in
the tank, rounded to the nearest gallon, is 1637 gallons.
(c) Total cost =
4
Ú0 (0.15 - 0.02t ) E (t ) dt = 474.320
The total cost of treating the sewage that enters the tank
during the time interval 0 £ t £ 4, to the nearest dollar,
is $474.
-38-
Ï 1 : integrand
Ô
3 : Ì 1 : limits
ÔÓ 1 : answer
AP® Calculus AB
Free-Response Scoring Guidelines
Question 2
Let R and S in the figure above be defined as follows: R is the region in the first and second quadrants
bounded by the graphs of y = 3 - x 2 and y = 2 x. S is the shaded region in the first quadrant bounded
by the two graphs, the x-axis, and the y-axis.
(a) Find the area of S.
(b) Find the volume of the solid generated when R is rotated about the horizontal line y = -1.
(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is
an isosceles right triangle with one leg across the base of the solid. Write, but do not evaluate, an
integral expression that gives the volume of the solid.
3 - x 2 = 2 x when x = -1.63658 and x = 1
Let a = -1.63658
(a) Area of S =
1 x
Ú0 2
dx +
= 2.240
3-x
ı ((
Ù
(b) Volume = p Û
1
a
2
2
Ú1 (3 - x ) dx
3
Ï 1 : integrands
Ô
3 : Ì 1 : limits
ÔÓ 1 : answer
) - (2 x + 1) ) dx
2
+1
2
Ï 2 : integrand
Ô
4 : Ì 1 : limits and constant
ÔÓ 1 : answer
= 63.106 or 63.107
(c) Volume =
(
1 1
3 - x2 - 2 x
2 Úa
)
2
dx
2:
-39-
{
1 : integrand
1 : limits and constant
AP® Calculus AB
Free-Response Scoring Guidelines
Question 3
t (minutes)
0
4
8
12
16
H (t ) (∞C)
65
68
73
80
90
The temperature, in degrees Celsius (∞C) , of an oven being heated is modeled by an increasing
differentiable function H of time t, where t is measured in minutes. The table above gives the
temperature as recorded every 4 minutes over a 16-minute period.
(a) Use the data in the table to estimate the instantaneous rate at which the temperature of the oven is
changing at time t = 10. Show the computations that lead to your answer. Indicate units of measure.
(b) Write an integral expression in terms of H for the average temperature of the oven between time
t = 0 and time t = 16. Estimate the average temperature of the oven using a left Riemann sum with
four subintervals of equal length. Show the computations that lead to your answer.
(c) Is your approximation in part (b) an underestimate or an overestimate of the average temperature?
Give a reason for your answer.
(d) Are the data in the table consistent with or do they contradict the claim that the temperature of the
oven is increasing at an increasing rate? Give a reason for your answer.
(a) H ¢ (10 ) ª
H (12 ) - H (8) 80 - 73 7
=
= ∞C min
12 - 8
4
4
(b) Average temperature is
16
Ú0
1
16
16
Ú0
{
1 : difference quotient
1 : answer with units
Ï 1 : 1 16 H (t ) dt
Ô
16 Ú0
3: Ì
1 : left Riemann sum
Ô
Ó 1 : answer
H (t ) dt
H (t ) dt ª 4 ◊ (65 + 68 + 73 + 80 )
Average temperature ª
2:
4 ◊ 286
= 71.5∞C
16
(c) The left Riemann sum approximation is an underestimate of the
integral because the graph of H is increasing. Dividing by 16
will not change the inequality, so 71.5∞C is an underestimate of
the average temperature.
1 : answer with reason
(d) If a continuous function is increasing at an increasing rate, then
the slopes of the secant lines of the graph of the function are
increasing. The slopes of the secant lines for the four intervals in
3 5 7
10
, respectively.
the table are , , , and
4 4 4
4
Since the slopes are increasing, the data are consistent with
the claim.
OR
By the Mean Value Theorem, the slopes are also the values
of H ¢(ck ) for some times c1 < c2 < c3 < c4 , respectively.
Since these derivative values are positive and increasing, the
data are consistent with the claim.
Ï 1 : considers slopes of
Ô
four secant lines
Ô
3 : Ì 1 : explanation
Ô 1 : conclusion consistent
Ô
with explanation
Ó
-40-
AP® Calculus AB
Free-Response Scoring Guidelines
Question 4
Let f be the function given by f ( x ) = (ln x )(sin x ) . The figure above shows the graph of f for
0 < x £ 2 p . The function g is defined by g( x ) =
x
Ú1
f (t ) dt for 0 < x £ 2 p .
(a) Find g(1) and g¢(1) .
(b) On what intervals, if any, is g increasing? Justify your answer.
(c) For 0 < x £ 2 p , find the value of x at which g has an absolute minimum. Justify your answer.
(d) For 0 < x < 2 p , is there a value of x at which the graph of g is tangent to the x-axis? Explain why
or why not.
(a) g(1) =
1
Ú1 f (t ) dt = 0 and g ¢(1) =
Ï 1 : g(1)
2: Ì
Ó 1 : g ¢(1)
f (1) = 0
(b) Since g ¢( x ) = f ( x ) , g is increasing on the interval
1 £ x £ p because f ( x ) > 0 for 1 < x < p .
(c) For 0 < x < 2 p , g ¢( x ) = f ( x ) = 0 when x = 1, p .
g ¢ = f changes from negative to positive only at
x = 1. The absolute minimum must occur at x = 1
or at the right endpoint. Since g(1) = 0 and
g (2 p ) =
2p
Ú1
f (t ) dt =
p
Ú1
f (t ) dt +
2p
Úp
f (t ) dt < 0
by comparison of the two areas, the absolute
minimum occurs at x = 2 p .
(d) Yes, the graph of g is tangent to the x-axis at x = 1
since g(1) = 0 and g ¢(1) = 0.
-41-
2:
{
1 : interval
1 : reason
Ï 1 : identifies 1 and 2 p as candidates
Ô
- or Ô
indicates that the graph of g
Ô
3: Ì
decreases, increases, then decreases
Ô
Ô 1 : justifies g(2 p ) < g(1)
Ô 1 : answer
Ó
2:
{
1 : answer of “yes” with x = 1
1 : explanation
AP® Calculus AB
Free-Response Scoring Guidelines
Question 5
Consider the differential equation
dy
x
= , where y π 0.
dx
y
(a) The slope field for the given differential equation is shown below.
Sketch the solution curve that passes through the point (3, -1) , and
sketch the solution curve that passes through the point (1, 2 ) .
(Note: The points (3, -1) and (1, 2 ) are indicated in the figure.)
(b) Write an equation for the line tangent to the solution curve that
passes through the point (1, 2 ) .
(c) Find the particular solution y = f ( x ) to the differential equation
with the initial condition f (3) = -1, and state its domain.
Ï 1 : solution curve through (3, -1)
2: Ì
Ó 1 : solution curve through (1, 2 )
(a)
Curves must go through the indicated
points, follow the given slope lines, and
extend to the boundary of the slope field
or the x-axis.
(b)
dy
1
=
dx (1, 2) 2
1 : equation of tangent line
An equation for the line tangent to the solution
1
curve is y - 2 = ( x - 1) .
2
(c) y dy = x dx
1 2 1 2
y = x +A
2
2
y2 = x 2 + C
C = -8
Since the particular solution goes through (3, -1) ,
y must be negative.
y = - x 2 - 8 for x > 8
-42-
Ï
Ï 1 : separates variables
Ô 1 : antiderivatives
Ô
Ô
Ô
Ô 5 : Ì 1 : constant of integration
Ô 1 : uses initial condition
ÔÔ
Ô
6: Ì
Ó 1 : solves for y
Ô
Ô Note: max 2 5 [1-1-0-0-0] if no
Ô
constant of integration
Ô
ÓÔ 1 : domain
AP® Calculus AB
Free-Response Scoring Guidelines
Question 6
Let g( x ) = xe - x + be - x , where b is a positive constant.
(a) Find lim g( x ) .
xƕ
(b) For what positive value of b does g have an absolute maximum at x =
2
? Justify your answer.
3
(c) Find all values of b, if any, for which the graph of g has a point of inflection on the interval
0 < x < •. Justify your answer.
(a)
lim g( x ) = 0
1 : answer
xƕ
Ï 2 : g ¢( x )
ÔÔ
2
4 : Ì 1 : solves g ¢
= 0 for b
3
Ô
ÔÓ 1 : justification
(b) g ¢( x ) = e - x - xe - x - be - x = (1 - x - b ) e - x
g¢
( 23 ) = ( 13 - b) e
When b =
-2 3
=0fib=
( )
()
1
3
1
2
- x e- x .
, g ¢( x ) =
3
3
2
2
, g ¢( x ) > 0 and for x > , g ¢( x ) < 0.
3
3
1
Therefore, when b = , g has an absolute maximum
3
2
at x = .
3
For x <
(c) g ¢¢( x ) = -e - x - (1 - x - b ) e - x = ( x - 2 + b ) e - x
If 0 < b < 2, then g ¢¢( x ) will change sign at x = 2 - b > 0.
Therefore, the graph of g will have a point of inflection
on the interval 0 < x < • when 0 < b < 2.
-43-
Ï 2 : g ¢¢( x )
Ô
4 : Ì 1 : interval for b
ÔÓ 1 : justification
®
AP Calculus AB Exam
SECTION I: Multiple Choice
2012
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.
At a Glance
Total Time
1 hour, 45 minutes
Number of Questions
45
Percent of Total Score
50%
Writing Instrument
Pencil required
Part A
Instructions
Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For
Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet.
For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer
sheet. The survey questions are numbers 93 through 96.
Indicate all of your answers to the multiple-choice questions on the answer sheet. No
credit will be given for anything written in this exam booklet, but you may use the booklet
for notes or scratch work. After you have decided which of the suggested answers is best,
completely fill in the corresponding circle on the answer sheet. Give only one answer to
each question. If you change an answer, be sure that the previous mark is erased
completely. Here is a sample question and answer.
Number of Questions
28
Time
55 minutes
Electronic Device
None allowed
Part B
Number of Questions
17
Time
50 minutes
Electronic Device
Graphing calculator
required
Use your time effectively, working as quickly as you can without losing accuracy. Do not
spend too much time on any one question. Go on to other questions and come back to
the ones you have not answered if you have time. It is not expected that everyone will
know the answers to all of the multiple-choice questions.
Your total score on the multiple-choice section is based only on the number of questions
answered correctly. Points are not deducted for incorrect answers or unanswered
questions.
Form I
Form Code 4IBP4-Q-S
Minimum 20% post-consumer waste
66
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
CALCULUS AB
SECTION I, Part A
Time— 55 minutes
Number of questions—28
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
In this exam:
(1)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x ) is a real number.
(2)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-3-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
1. If y = x sin x, then
dy
=
dx
(A) sin x + cos x
(B) sin x + x cos x
(C) sin x - x cos x
(D) x (sin x + cos x )
(E) x (sin x - cos x )
2. Let f be the function given by f ( x ) = 300 x - x 3 . On which of the following intervals is the
function f increasing?
(A) ( - •, -10] and [10, • )
(B) [ -10, 10]
(C) [0, 10] only
(D) [0, 10 3 ] only
(E)
[0, •)
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-4-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
Ú sec x tan x dx =
3.
(A) sec x + C
(B) tan x + C
(C)
sec 2 x
+C
2
(D)
tan 2 x
+C
2
(E)
sec 2 x tan 2 x
+C
2
4. If f ( x ) = 7 x - 3 + ln x, then f ¢(1) =
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-5-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
5. The graph of the function f is shown above. Which of the following statements is false?
(A) lim f ( x ) exists.
xÆ2
(B) lim f ( x ) exists.
x Æ3
(C) lim f ( x ) exists.
xÆ4
(D) lim f ( x ) exists.
xÆ5
(E) The function f is continuous at x = 3.
6. A particle moves along the x-axis. The velocity of the particle at time t is 6t - t 2 . What is the total distance
traveled by the particle from time t = 0 to t = 3 ?
(A) 3
(B) 6
(C) 9
(D) 18
(E) 27
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-6-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
(
)
(
)
5
7. If y = x 3 - cos x , then y ¢ =
(A) 5 x 3 - cos x
4
(
)
5 (3 x 2 + sin x )
4
(B) 5 3 x 2 + sin x
(C)
(
)
(
) ⴢ (3x 2 + sin x )
(D) 5 3 x 2 + sin x
(E) 5 x 3 - cos x
4
ⴢ (6 x + cos x )
4
t (hours)
4
7
12
15
R (t )
(liters/hour)
6.5
6.2
5.9
5.6
8. A tank contains 50 liters of oil at time t = 4 hours. Oil is being pumped into the tank at a rate R(t ) , where R(t )
is measured in liters per hour, and t is measured in hours. Selected values of R(t ) are given in the table above.
Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the
number of liters of oil that are in the tank at time t = 15 hours?
(A) 64.9
(B) 68.2
(C) 114.9
(D) 116.6
Unauthorized copying or reuse of
any part of this page is illegal.
(E) 118.2
GO ON TO THE NEXT PAGE.
-7-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
ÏÔ (2 x + 1)( x - 2 )
f (x) = Ì
x-2
ÔÓ k
for x π 2
for x = 2
9. Let f be the function defined above. For what value of k is f continuous at x = 2 ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5
10. What is the area of the region in the first quadrant bounded by the graph of y = e x
(A) 2e - 2
(B) 2e
(C)
e
-1
2
(D)
Unauthorized copying or reuse of
any part of this page is illegal.
e -1
2
2
and the line x = 2 ?
(E) e - 1
GO ON TO THE NEXT PAGE.
-8-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
11. Let f be the function defined by f ( x ) =
x - 2 for all x. Which of the following statements is true?
(A) f is continuous but not differentiable at x = 2.
(B) f is differentiable at x = 2.
(C) f is not continuous at x = 2.
(D) lim f ( x ) π 0
xÆ2
(E) x = 2 is a vertical asymptote of the graph of f.
4
e
12. Using the substitution u = x , Û
Ù
ı1
(A) 2
16 u
Ú1
e du
(B) 2
4 u
Ú1
e du
x
x
dx is equal to which of the following?
(C) 2
2 u
Ú1
e du
Unauthorized copying or reuse of
any part of this page is illegal.
(D)
1 2 u
e du
2 Ú1
(E)
4 u
Ú1 e
du
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-9-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
13. The function f is defined by f ( x ) =
(A) 2
(B) 6
(C) 8
{
2
for x < 3
What is the value of
x - 1 for x ≥ 3.
(D) 10
5
Ú1
f ( x ) dx ?
(E) 12
14. If f ( x ) = x 2 - 4 and g( x ) = 3 x - 2, then the derivative of f ( g( x )) at x = 3 is
(A)
7
5
(B)
14
5
(C)
18
5
(D)
15
21
Unauthorized copying or reuse of
any part of this page is illegal.
(E)
30
21
GO ON TO THE NEXT PAGE.
-10-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
15. The graph of a differentiable function f is shown above. If h( x ) =
x
Ú0 f (t ) dt, which of the following is true?
(A) h(6 ) < h ¢(6 ) < h ¢¢(6 )
(B) h(6 ) < h ¢¢(6 ) < h ¢(6 )
(C) h ¢(6 ) < h(6 ) < h ¢¢(6 )
(D) h ¢¢(6 ) < h(6 ) < h ¢(6 )
(E) h ¢¢(6 ) < h ¢(6 ) < h(6 )
Unauthorized copying or reuse of
any part of this page is illegal.
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-11-
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16. A particle moves along the x-axis with its position at time t given by x (t ) = (t - a )(t - b ) , where a and b are
constants and a π b. For which of the following values of t is the particle at rest?
(A) t = ab
(B) t =
a+b
2
(C) t = a + b
(D) t = 2 ( a + b )
(E) t = a and t = b
Unauthorized copying or reuse of
any part of this page is illegal.
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-12-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
17. The figure above shows the graph of f. If f ( x ) =
of y = g( x ) ?
x
Ú2 g(t ) dt, which of the following could be the graph
(A)
(B)
(C)
(D)
(E)
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any part of this page is illegal.
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-13-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
lim
18.
hÆ0
(A) 0
ln (4 + h ) - ln ( 4 )
is
h
(B)
1
4
(C) 1
(D) e
19. The function f is defined by f ( x ) =
line tangent to f at ( x, y ) has slope
(E) nonexistent
x
. What points ( x, y ) on the graph of f have the property that the
x+2
1
?
2
(A) (0,0 ) only
(B)
( 21 , 15 ) only
(C) (0,0 ) and ( -4,2 )
( 23 )
(D) (0,0 ) and 4,
(E) There are no such points.
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any part of this page is illegal.
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-14-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
20. Let f ( x ) = (2 x + 1) and let g be the inverse function of f. Given that f (0 ) = 1, what is the value of g¢(1) ?
3
(A) -
2
27
(B)
1
54
(C)
1
27
(D)
1
6
(E) 6
21. The line y = 5 is a horizontal asymptote to the graph of which of the following functions?
(A) y =
sin (5 x )
x
(B) y = 5 x
(C) y =
1
x-5
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(D) y =
5x
1- x
(E) y =
20 x 2 - x
1 + 4 x2
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
22. Let f be the function defined by f ( x ) =
ln x
. What is the absolute maximum value of f ?
x
(A) 1
(B)
1
e
(C) 0
(D) -e
(E) f does not have an absolute maximum value.
23. If P (t ) is the size of a population at time t, which of the following differential equations describes linear growth
in the size of the population?
(A)
dP
= 200
dt
(B)
dP
= 200t
dt
(C)
dP
= 100t 2
dt
(D)
dP
= 200 P
dt
(E)
dP
= 100 P 2
dt
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
24. Let g be the function given by g( x ) = x 2e kx , where k is a constant. For what value of k does g have a critical
2
point at x = ?
3
(A) -3
(B) -
3
2
(C) -
1
3
(D) 0
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(E) There is no such k.
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
25. Which of the following is the solution to the differential equation
condition y( p ) = 1 ?
dy
= 2sin x with the initial
dx
(A) y = 2 cos x + 3
(B) y = 2 cos x - 1
(C) y = -2 cos x + 3
(D) y = -2 cos x + 1
(E) y = -2 cos x - 1
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
26. Let g be a function with first derivative given by g ¢( x ) =
interval 0 < x < 2 ?
x -t2
Ú0 e
dt. Which of the following must be true on the
(A) g is increasing, and the graph of g is concave up.
(B) g is increasing, and the graph of g is concave down.
(C) g is decreasing, and the graph of g is concave up.
(D) g is decreasing, and the graph of g is concave down.
(E) g is decreasing, and the graph of g has a point of inflection on 0 < x < 2.
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
27. If ( x + 2 y ) ◊
(A) -
10
3
dy
d2y
at the point (3, 0 ) ?
= 2 x - y, what is the value of
dx
dx 2
(B) 0
(C) 2
(D)
10
3
(E) Undefined
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
28. For t ≥ 0, the position of a particle moving along the x-axis is given by x (t ) = sin t - cos t. What is the
acceleration of the particle at the point where the velocity is first equal to 0 ?
(A) - 2
(B) -1
(C) 0
(D) 1
(E)
2
END OF PART A OF SECTION I
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WORK ON PART A ONLY.
DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.
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-21-
B
B
B
B
B
B
B
B
B
CALCULUS AB
SECTION I, Part B
Time— 50 minutes
Number of questions—17
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON
THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO
QUESTIONS NUMBERED 76–92.
YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET.
In this exam:
(1)
The exact numerical value of the correct answer does not always appear among the choices given. When this
happens, select from among the choices the number that best approximates the exact numerical value.
(2)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x ) is a real number.
(3)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
Unauthorized copying or reuse of
any part of this page is illegal.
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-24-
B
B
B
B
B
B
B
B
B
76. The graph of the function f is shown in the figure above. For which of the following values of x is f ¢( x )
positive and increasing?
(A) a
(B) b
(C) c
(D) d
(E) e
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-25-
B
B
B
B
B
B
B
B
B
77. Let f be a function that is continuous on the closed interval [2, 4] with f (2 ) = 10 and f ( 4 ) = 20. Which of the
following is guaranteed by the Intermediate Value Theorem?
(A) f ( x ) = 13 has at least one solution in the open interval (2, 4 ) .
(B) f (3) = 15
(C) f attains a maximum on the open interval (2, 4 ) .
(D) f ¢( x ) = 5 has at least one solution in the open interval (2, 4 ) .
(E) f ¢( x ) > 0 for all x in the open interval (2, 4 ) .
78. The graph of y = e tan x - 2 crosses the x-axis at one point in the interval [ 0, 1]. What is the slope of the graph
at this point?
(A) 0.606
(B) 2
(C) 2.242
(D) 2.961
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(E) 3.747
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-26-
B
B
B
B
B
B
B
B
B
79. A particle moves along the x-axis. The velocity of the particle at time t is given by v(t ) , and the acceleration of
the particle at time t is given by a(t ) . Which of the following gives the average velocity of the particle from
time t = 0 to time t = 8 ?
(A)
a(8) - a(0 )
8
(B)
1 8
v(t ) dt
8 Ú0
(C)
1 8
v(t ) dt
8 Ú0
(D)
1 8
v(t ) dt
2 Ú0
(E)
v(0 ) + v(8)
2
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-27-
B
B
B
B
B
B
B
B
B
80. The graph of f ¢, the derivative of the function f, is shown above. Which of the following statements must be
true?
I. f has a relative minimum at x = -3.
II. The graph of f has a point of inflection at x = -2.
III. The graph of f is concave down for 0 < x < 4.
(A) I only
(B) II only
(C) III only
(D) I and II only
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-28-
B
B
B
B
B
(
B
B
B
B
)
81. Water is pumped into a tank at a rate of r (t ) = 30 1 - e - 0.16t gallons per minute, where t is the number of
minutes since the pump was turned on. If the tank contained 800 gallons of water when the pump was turned
on, how much water, to the nearest gallon, is in the tank after 20 minutes?
(A) 380 gallons
(B)
420 gallons
(C)
829 gallons
(D) 1220 gallons
(E) 1376 gallons
82. If f ¢ ( x ) = x 4 + 1 + x 3 - 3 x, then f has a local maximum at x =
(A) -2.314
(B) -1.332
(C) 0.350
(D) 0.829
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(E) 1.234
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-29-
B
B
B
B
B
B
B
B
B
83. The graph above gives the velocity, v, in ft/sec, of a car for 0 £ t £ 8, where t is the time in seconds. Of the
following, which is the best estimate of the distance traveled by the car from t = 0 until the car comes to a
complete stop?
(A) 21 ft
(B) 26 ft
(C) 180 ft
(D) 210 ft
(E) 260 ft
84. For -1.5 < x < 1.5, let f be a function with first derivative given by f ¢( x ) = e
following are all intervals on which the graph of f is concave down?
( x4 - 2 x2 +1) - 2. Which of the
(A) ( - 0.418, 0.418) only
(B)
( -1, 1)
(C)
( -1.354, - 0.409) and (0.409, 1.354)
(D) ( -1.5, -1) and (0, 1)
(E)
( -1.5, -1.354) , ( - 0.409, 0 ) , and (1.354, 1.5)
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-30-
B
B
B
B
B
B
B
B
B
85. The graph of f ¢, the derivative of f, is shown in the figure above. The function f has a local maximum at x =
(A) -3
(B) -1
(C) 1
(D) 3
(E) 4
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-31-
B
B
B
B
86. If f ¢ ( x ) > 0 for all real numbers x and
B
B
B
B
B
7
Ú4 f (t ) dt = 0, which of the following could be a table of values for the
function f ?
(A)
x
4
5
7
(B) x
4
5
7
f (x)
-4
-3
0
f (x)
-4
-2
5
4
5
7
f (x)
-4
6
3
(D) x
4
5
7
f (x)
0
0
0
(E) x
4
5
7
f (x)
0
4
6
(C) x
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-32-
B
B
B
B
B
B
B
B
B
87. The graph of f ¢¢, the second derivative of f, is shown above for -2 £ x £ 4. What are all intervals on which
the graph of the function f is concave down?
(A) -1 < x < 1
(B)
0< x<2
(C)
1 < x < 3 only
(D) -2 < x < -1 only
(E) -2 < x < -1 and 1 < x < 3
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-33-
B
B
B
B
B
B
B
B
B
88. A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate
of 4 feet per second. If the height of the streetlight is 15 feet, what is the rate at which the person’s shadow is
lengthening?
(A) 1.5 ft/sec
(B) 2.667 ft/sec
(C) 3.75 ft/sec
(D) 6 ft/sec
89. A particle moves along a line so that its acceleration for t ≥ 0 is given by a (t ) =
velocity at t = 0 is 5, what is the velocity of the particle at t = 3 ?
(A) 0.713
(B) 1.134
(C) 6.134
(D) 6.710
Unauthorized copying or reuse of
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(E) 10 ft/sec
t+3
t3 + 1
. If the particle’s
(E) 11.710
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-34-
B
B
B
90. Let f be a function such that
24
f (t ) dt = 5
24
f (t ) dt = 20
12
f (t ) dt = 5
12
f (t ) dt = 20
(A)
Ú12
(B)
Ú12
(C)
Ú6
(D)
Ú6
(E)
B
12
Ú6
B
B
B
B
B
f (2 x ) dx = 10. Which of the following must be true?
6
Ú3 f (t ) dt = 5
x
–2
0
3
5
6
f ¢( x )
3
1
4
7
5
91. Let f be a polynomial function with values of f ¢( x ) at selected values of x given in the table above. Which of
the following must be true for -2 < x < 6 ?
(A) The graph of f is concave up.
(B) The graph of f has at least two points of inflection.
(C) f is increasing.
(D) f has no critical points.
(E) f has at least two relative extrema.
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-35-
B
B
B
B
B
B
B
B
B
92. Let R be the region in the first quadrant bounded below by the graph of y = x 2 and above by the graph of
y = x . R is the base of a solid whose cross sections perpendicular to the x-axis are squares. What is the
volume of the solid?
(A) 0.129
(B) 0.300
(C) 0.333
(D) 0.700
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(E) 1.271
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-36-
Answer Key for AP Calculus AB
Practice Exam, Section I
Multiple-Choice Questions
Question #
Key
1
B
2
B
3
A
4
E
5
C
6
D
7
E
8
C
9
E
10
A
11
A
12
C
13
D
14
A
15
A
16
B
17
A
18
B
19
C
20
D
21
E
22
B
23
24
25
26
27
28
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
A
A
E
A
A
A
E
A
D
B
E
D
C
D
D
C
B
E
B
E
B
B
A
®
AP Calculus AB Exam
2013
SECTION I: Multiple Choice
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.
At a Glance
Total Time
1 hour, 45 minutes
Number of Questions
45
Percent of Total Score
50%
Writing Instrument
Pencil required
Part A
Instructions
Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For
Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet.
For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer
sheet. The survey questions are numbers 93 through 96.
Indicate all of your answers to the multiple-choice questions on the answer sheet. No
credit will be given for anything written in this exam booklet, but you may use the booklet
for notes or scratch work. After you have decided which of the suggested answers is best,
completely fill in the corresponding circle on the answer sheet. Give only one answer to
each question. If you change an answer, be sure that the previous mark is erased
completely. Here is a sample question and answer.
Number of Questions
28
Time
55 minutes
Electronic Device
None allowed
Part B
Number of Questions
17
Time
50 minutes
Electronic Device
Graphing calculator
required
Use your time effectively, working as quickly as you can without losing accuracy. Do not
spend too much time on any one question. Go on to other questions and come back to
the ones you have not answered if you have time. It is not expected that everyone will
know the answers to all of the multiple-choice questions.
Your total score on the multiple-choice section is based only on the number of questions
answered correctly. Points are not deducted for incorrect answers or unanswered
questions.
Form I
Form Code 4JBP4-Q-S
66
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
CALCULUS AB
SECTION I, Part A
Time—55 minutes
Number of questions—28
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
In this exam:
(1)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
is a real number.
(2)
The inverse of a trigonometric function f may be indicated using the inverse function notation
prefix “arc” (e.g.,
or with the
).
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-3-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
is
1.
(A)
(B) 0
(D)
(E) nonexistent
, then
2. If
(A) 10
(C) 1
(B) 9
(C) 7
(D) 5
(E) 3
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-4-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
3. Which of the following definite integrals has the same value as
?
(A)
(B)
(C)
(D)
(E)
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-5-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
4. Which of the following is an equation of the line tangent to the graph of
at the point
?
(A)
(B)
(C)
(D)
(E)
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-6-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
5. If g is the function given by
, on which of the following intervals
is g decreasing?
(A)
(B)
(C)
(D)
(E)
6.
(A)
(B)
(C)
(D)
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(E)
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-7-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
7. Let f be the function given by
of f at
?
(A)
(B)
. What is the instantaneous rate of change
(C)
(D) 6
(E) 17
at time t is shown above
8. A particle moves along a straight line. The graph of the particle’s velocity
, where j, k, l, and m are constants. The graph intersects the horizontal axis at
,
,
for
and
and has horizontal tangents at
and
. For what values of t is the speed of the particle
decreasing?
(A)
(B)
(C)
(D)
(E)
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-8-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
. For which of the following values of x is f not
9. Let f be the function given by
continuous?
and
(A)
(B)
,
only
, and 2
(C)
only
(D)
and 2 only
(E) 2 only
for time
10. A particle moves along the x-axis with velocity given by
?
at time
, what is the position of the particle at time
position
(A) 13
(B) 15
(C) 16
(D) 17
. If the particle is at
(E) 25
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11. Let f be the function defined by
concave down?
graph of
(A)
. On which of the following intervals is the
only
(B)
(C)
(D)
(E)
only
only
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
12. For which of the following does
?
I.
II.
III.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
13. Let f be a differentiable function such that
possible value for
?
(A)
(B)
(C) 0
for all x. Of the following, which is not a
and
(D) 1
(E) 2
14. Let f be the function given above. What are all values of a and b for which f is differentiable at
(A)
and
(B)
and
(C)
and b is any real number
(D)
?
, where b is any real number
(E) There are no such values of a and b.
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15. The table above gives values for the functions f and g and their derivatives at
given by
(A)
16. If
(A)
0. What is the value of
, where
(B)
, then
(B)
(C) 2
at
(D) 3
. Let k be the function
?
(E) 8
is
(C)
(D)
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(E)
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
, which of the following could be the value of a ?
17. If
(A)
(B)
(D)
(E) 2
, then
18. If
(A)
(C)
(B)
(C)
(D)
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
19. The figure above shows the graph of the function g and the line tangent to the graph of g at
. What is the value of
the function given by
(A)
(B)
(C)
(D)
. Let h be
?
(E)
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20. For
(A)
(B)
(C)
(D)
(E)
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
21. The graph of a function f is shown above. What is the value of
(A) 6
(B) 8
(C) 10
(D) 14
?
(E) 18
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
22. The function f is continuous for all real numbers, and the average rate of change of f on the closed interval
is
, there is no value of c such that
. For
. Of the following, which must
be true?
(A)
(B)
does not exist.
(C)
(D)
for all x in the open interval
.
(E) f is not differentiable on the open interval
23. Let f be the function defined by
?
of f, what is the value of
(A)
(B)
.
for all x and the point
. If
(C)
(D) 3
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is on the graph
(E)
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-18-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
24. The function g is given by
interval
?
(A)
(B)
. What is the absolute minimum value of g on the closed
(C) 0
(D) 2
(E) 6
25. Which of the following is the solution to the differential equation
condition
with the initial
?
(A)
(B)
(C)
(D)
(E)
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-19-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
?
26. Which of the following is an antiderivative of
(A)
(B)
(C)
(D)
(E)
27. For time
, the height h of an object suspended from a spring is given by
the average height of the object from
(A) 16
(B)
(C)
to
. What is
?
(D)
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(E)
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-20-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
28. The function f is defined by
for
. What is the x-coordinate of the point of
inflection where the graph of f changes from concave down to concave up?
(A)
(B)
(C)
(D)
(E)
END OF PART A OF SECTION I
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WORK ON PART A ONLY.
DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.
Unauthorized copying or reuse of
any part of this page is illegal.
-21-
B
B
B
B
B
B
B
B
B
CALCULUS AB
SECTION I, Part B
Time—50 minutes
Number of questions—17
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON
THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO
QUESTIONS NUMBERED 76–92.
YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET.
In this exam:
(1)
The exact numerical value of the correct answer does not always appear among the choices given. When this
happens, select from among the choices the number that best approximates the exact numerical value.
(2)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
is a real number.
(3)
The inverse of a trigonometric function f may be indicated using the inverse function notation
prefix “arc” (e.g.,
or with the
).
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-24-
B
B
B
B
B
B
B
B
B
76. The graph of the function f shown above consists of two line segments and a semicircle. Let g be defined by
. What is the value of
(A) 0
(B)
(C)
?
(D)
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(E)
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-25-
B
B
B
B
B
B
B
B
B
77. The volume of a sphere is decreasing at a constant rate of 3 cubic centimeters per second. At the instant when the
radius of the sphere is decreasing at a rate of 0.25 centimeter per second, what is the radius of the sphere?
(The volume V of a sphere with radius r is
(A) 0.141 cm
(B) 0.244 cm
.)
(C) 0.250 cm
(D) 0.489 cm
78. Let f and g be continuous functions such that
, and
?
. What is the value of
(A) 3
(B) 7
(C) 11
(D) 15
(E) 0.977 cm
(E) 19
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-26-
B
B
B
79. The figure above shows the graph of
could be the graph of f ?
B
B
B
, the derivative of the function f . If
(A)
(B)
(C)
(D)
B
B
B
, which of the following
(E)
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-27-
B
B
B
B
B
B
B
B
B
80. For time
, the position of a particle traveling along a line is given by a differentiable function s. If s is
and s is decreasing for
, which of the following is the total distance the particle
increasing for
?
travels for
(A)
(B)
(C)
(D)
(E)
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-28-
B
B
B
B
B
B
B
B
B
. If the initial
81. A cup of tea is cooling in a room that has a constant temperature of 70 degrees Fahrenheit
minutes, is
and the temperature of the tea changes at the rate
temperature of the tea, at time
degrees Fahrenheit per minute, what is the temperature, to the nearest degree, of the tea
after 4 minutes?
(A)
(B)
(C)
(D)
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(E)
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-29-
B
B
B
B
B
B
82. The derivative of the function f is given by
B
. On the interval
B
B
, at which of
the following values of x does f have a relative maximum?
(A)
and 0
(B)
and 1.075
(C)
, 0.542, and 1.396
(D)
and 1.396 only
(E)
0.542 only
0
0
0.5
4
1
10
1.5
18
2
28
2.5
40
3
54
83. The table above gives selected values for a continuous function f . If f is increasing over the closed interval
, which of the following could be the value of
(A) 50
(B) 62
(C) 77
?
(D) 100
(E) 154
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-30-
B
B
B
B
B
B
B
B
B
84. The graph of a function f is shown in the figure above. Which of the following statements is true?
(A)
(B) f is continuous at
.
(C)
(D)
(E)
does not exist.
its position is given by
85. A particle moves along the x-axis so that at time
velocity of the particle at the first instance the particle is at the origin?
(A)
(B)
(C)
(D) 0
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. What is the
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-31-
B
B
B
B
for all x and
86. If
(A)
B
B
B
B
B
for all x, which of the following could be a table of values for f ?
(B)
(D)
(C)
4
(E)
4
4
4
4
0
3
0
4
0
5
0
5
0
6
1
1
1
4
1
6
1
7
1
7
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-32-
B
B
B
B
B
B
87. Let f be the function with first derivative given by
extrema does f have on the open interval
(A) Two
(B) Three
(C) Four
B
B
B
. How many relative
?
(D) Five
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(E) Six
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-33-
B
B
B
B
B
B
B
B
B
88. The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true?
(A)
(B)
(C)
(D)
(E)
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B
B
B
B
B
B
B
89. What is the volume of the solid generated when the region bounded by the graph of
and
is revolved about the y-axis?
(A) 3.464
(B) 4.500
(C) 7.854
(D) 10.883
B
B
and the lines
(E) 14.137
90. The population P of a city grows according to the differential equation
, where k is a constant and t
is measured in years. If the population of the city doubles every 12 years, what is the value of k ?
(A) 0.058
(B) 0.061
(C) 0.167
(D) 0.693
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(E) 8.318
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-35-
B
B
B
B
91. The function f is continuous and
(A)
(B) 3
(C) 6
B
(B) f is not bounded on
(D) 12
B
B
?
(E) 24
. If f does not attain a maximum value on
,
.
.
(C) f does not attain a minimum value on
.
(D) The graph of f has a vertical asymptote in the interval
(E) The equation
B
. What is the value of
92. The function f is defined for all x in the closed interval
which of the following must be true?
(A) f is not continuous on
B
.
does not have a solution in the interval
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.
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-36-
Answer Key for AP Calculus AB
Practice Exam, Section I
Question 1: D
Question 24: A
Question 2: B
Question 25: C
Question 3: B
Question 26: E
Question 4: E
Question 27: D
Question 5: D
Question 28: B
Question 6: B
Question 76: D
Question 7: C
Question 77: E
Question 8: C
Question 78: A
Question 9: D
Question 79: C
Question 10: A
Question 80: E
Question 11: D
Question 81: A
Question 12: E
Question 82: E
Question 13: E
Question 83: B
Question 14: A
Question 84: D
Question 15: C
Question 85: C
Question 16: D
Question 86: E
Question 17: B
Question 87: E
Question 18: B
Question 88: D
Question 19: B
Question 89: E
Question 20: D
Question 90: A
Question 21: A
Question 91: B
Question 22: E
Question 92: A
Question 23: B
®
AP Calculus AB Exam
2014
SECTION I: Multiple Choice
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.
At a Glance
Total Time
1 hour, 45 minutes
Number of Questions
45
Percent of Total Score
50%
Writing Instrument
Pencil required
Part A
Instructions
Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For
Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet.
For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer
sheet. The survey questions are numbers 93 through 96.
Indicate all of your answers to the multiple-choice questions on the answer sheet. No
credit will be given for anything written in this exam booklet, but you may use the booklet
for notes or scratch work. After you have decided which of the suggested answers is best,
completely fill in the corresponding circle on the answer sheet. Give only one answer to
each question. If you change an answer, be sure that the previous mark is erased
completely. Here is a sample question and answer.
Number of Questions
28
Time
55 minutes
Electronic Device
None allowed
Part B
Number of Questions
17
Time
50 minutes
Electronic Device
Graphing calculator
required
Use your time effectively, working as quickly as you can without losing accuracy. Do not
spend too much time on any one question. Go on to other questions and come back to
the ones you have not answered if you have time. It is not expected that everyone will
know the answers to all of the multiple-choice questions.
Your total score on the multiple-choice section is based only on the number of questions
answered correctly. Points are not deducted for incorrect answers or unanswered
questions.
Form I
Form Code 4JBP6-S
66
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
CALCULUS AB
SECTION I, Part A
Time— 55 minutes
Number of questions—28
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
In this exam:
(1)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x) is a real number.
(2)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-3-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
Ú2 (3t
x
1.
2
)
- 1 dt =
(A) x 3 - x - 6
(B) x 3 - x
(C) 3 x 2 - 12
(D) 3 x 2 - 1
(E) 6 x - 12
2. What is the slope of the line tangent to the graph of y = ln (2 x ) at the point where x = 4 ?
(A)
1
8
(B)
1
4
(C)
1
2
(D)
3
4
(E) 4
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-4-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
3. If f ( x ) = 4 x -2 +
(A) -62
4.
2
1 2
x + 4, then f ¢(2 ) =
4
(B) -58
(C) -3
(D) 0
(E) 1
dx
Ú1 2 x + 1 =
(A) 2 ln 2
(B)
1
ln 2
2
(C) 2 (ln 5 - ln 3)
Unauthorized copying or reuse of
any part of this page is illegal.
(D) ln 5 - ln 3
(E)
1
(ln 5 - ln 3)
2
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-5-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
5. The figure above shows the graph of the function f . Which of the following statements are true?
I.
lim f ( x ) = f (2 )
x Æ 2-
II. lim f ( x ) = lim f ( x )
x Æ 6-
III.
x Æ6+
lim f ( x ) = f (6 )
x Æ6
(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III
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-6-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
( ( )) =
d
sin3 x 2
dx
6.
( )
3sin 2 ( x 2 )
6 x sin 2 ( x 2 )
3sin 2 ( x 2 ) cos ( x 2 )
6 x sin 2 ( x 2 ) cos ( x 2 )
(A) cos3 x 2
(B)
(C)
(D)
(E)
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-7-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
x3
is
x Æ • e3 x
7.
lim
(A) 0
(B)
2
9
(C)
2
3
8. Using the substitution u = sin (2 x ) ,
(A) -2
1
Ú1 2 u
5
1 1 5
u du
2 Ú1 2
(C)
1 3 2 5
u du
2 Ú0
(D)
1 0
u 5 du
2Ú 3 2
0
Ú 3 2u
5
p 2
Úp
6
(E) infinite
sin 5 (2 x ) cos (2 x ) dx is equivalent to
du
(B)
(E) 2
(D) 1
du
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-8-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
9. The function f has a first derivative given by f ¢( x ) = x ( x - 3) ( x + 1) . At what values of x does f have
a relative maximum?
2
(A) -1 only
(C) -1 and 0 only
(B) 0 only
(D) -1 and 3 only
Ï x 2 - 7 x + 10
Ô
f ( x ) = Ì b ( x - 2)
ÔÓ b
(E) -1, 0, and 3
for x π 2
for x = 2
10. Let f be the function defined above. For what value of b is f continuous at x = 2 ?
(A) -3
(B)
2
(C) 3
(D) 5
(E) There is no such value of b.
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-9-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
11. For 0 £ x £ 6, the graph of f ¢, the derivative of f, is piecewise linear as shown above. If f (0 ) = 1, what is the
maximum value of f on the interval?
(A) 1
(B) 1.5
(C) 2
(D) 4
(E) 6
12. Let f be the function given by f ( x ) = 9 x. If four subintervals of equal length are used, what is the value of the
right Riemann sum approximation for
(A) 20
(B) 40
(C) 60
2
Ú0 f ( x ) dx ?
(D) 80
(E) 120
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-10-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
d Ê x +1 ˆ
=
dx ÁË x 2 + 1 ˜¯
13.
(A)
(B)
x2 + 2 x - 1
( x2 + 1)
2
- x2 - 2 x + 1
x2 + 1
(C)
- x2 - 2 x + 1
(D)
3x2 + 2 x + 1
(E)
( x2 + 1)
( x2 + 1)
2
2
1
2x
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-11-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
14. The velocity of a particle moving along the x-axis is given by v(t ) = sin (2t ) at time t. If the particle is at x = 4
p
when t = 0, what is the position of the particle when t = ?
2
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
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-12-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
15. The function y = g( x ) is differentiable and increasing for all real numbers. On what intervals is the function
(
)
y = g x 3 - 6 x 2 increasing?
(A) ( - •, 0] and [ 4, • ) only
(B) [0, 4] only
(C) [2, • ) only
(D) [6, • ) only
(E)
16.
( -•, • )
lim
x Æ 3-
(A) -3
x-3
is
x-3
(B) -1
(C) 1
(D) 3
(E) nonexistent
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-13-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
17. If f ( x ) = ae - ax for a > 0, then f ¢( x ) =
(A) e - ax
(B) ae - ax
(C) a 2 e - ax
(D) - ae - ax
(E) - a 2e - ax
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-14-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
18. A student attempted to solve the differential equation
dy
= xy with initial condition y = 2 when x = 0. In
dx
which step, if any, does an error first appear?
1
Step 1: Û
Ù dy =
ıy
x2
+C
2
Step 2: ln y =
Step 3: y = e x
Ú x dx
2
2
+C
Step 4: Since y = 2 when x = 0, 2 = e0 + C.
Step 5: y = e x
2
2
+1
(A) Step 2
(B) Step 3
(C) Step 4
(D) Step 5
(E) There is no error in the solution.
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-15-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
19. For what values of x does the graph of y = 3 x 5 + 10x 4 have a point of inflection?
(A) x = -
8
only
3
(B) x = -2 only
(C) x = 0 only
(D) x = 0 and x = -
8
3
(E) x = 0 and x = -2
20.
ln ( x + 3) - ln (5)
is
x-2
xÆ2
lim
(A) 0
(B)
1
5
(C)
1
2
(D) 1
(E) nonexistent
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-16-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
21. Functions w, x, and y are differentiable with respect to time and are related by the equation w = x 2 y. If x is
decreasing at a constant rate of 1 unit per minute and y is increasing at a constant rate of 4 units per minute, at
what rate is w changing with respect to time when x = 6 and y = 20 ?
(A) -384
(B) -240
(C) -96
(D) 276
(E) 384
22. Let f be the function defined by f ( x ) = 2 x 3 - 3 x 2 - 12 x + 18. On which of the following intervals is the graph
of f both decreasing and concave up?
(A)
( -•, -1)
(B)
(-1, 21 )
(C)
( -1, 2)
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(D)
( 12 , 2)
(E) (2, • )
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-17-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
when x < -1
Ï3 x + 5
f (x) = Ì 2
Ó - x + 3 when x ≥ -1
23. If f is the function defined above, then f ¢( -1) is
(A) -3
(B) -2
(C) 2
24. Let f be the function defined by f ( x ) =
(D) 3
(E) nonexistent
(3 x + 8)(5 - 4 x )
. Which of the following is a horizontal asymptote to
(2 x + 1)2
the graph of f ?
(A) y = - 6
(B) y = -3
(C) y = -
1
2
(D) y = 0
(E) y =
3
2
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-18-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
25. If y = x 2 - 2 x and u = 2 x + 1, then
(A)
(
(2 x + 1)
26. For x > 0,
(A)
)
2 x2 + x - 1
d
dx
(B) 6 x 2 - 3 x - 2
2
Ú1
x
dy
=
du
(C) 4x
(D) x - 1
(E)
1
x -1
1
dt =
1 + t2
1
2 x (1 + x )
(B)
1
2 x (1 + x )
(C)
1
1+ x
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any part of this page is illegal.
(D)
x
1+ x
(E)
1
1+ x
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-19-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
27. A particle moves on the x-axis so that at any time t, 0 £ t £ 1, its position is given by x (t ) = sin (2 p t ) + 2 p t.
For what value of t is the particle at rest?
(A) 0
(B)
1
8
(C)
1
4
(D)
1
2
(E) 1
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-20-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
28. Shown above is a slope field for which of the following differential equations?
(A)
dy
= xy - x
dx
(B)
dy
= xy + x
dx
(C)
dy
= y - x2
dx
(D)
dy
= ( y - 1) x 2
dx
(E)
dy
3
= ( y - 1)
dx
END OF PART A OF SECTION I
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WORK ON PART A ONLY.
DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.
Unauthorized copying or reuse of
any part of this page is illegal.
-21-
B
B
B
B
B
B
B
B
B
CALCULUS AB
SECTION I, Part B
Time— 50 minutes
Number of questions—17
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON
THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO
QUESTIONS NUMBERED 76–92.
YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET.
In this exam:
(1)
The exact numerical value of the correct answer does not always appear among the choices given. When this
happens, select from among the choices the number that best approximates the exact numerical value.
(2)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x) is a real number.
(3)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-24-
B
B
B
B
B
B
B
B
B
76. A particle moves along a straight line so that at time t > 0 the position of the particle is given by s(t ) , the
velocity is given by v(t ) , and the acceleration is given by a(t ) . Which of the following expressions gives the
average velocity of the particle on the interval [2, 8] ?
(A)
1 8
a(t ) dt
6 Ú2
(B)
1 8
s(t ) dt
6 Ú2
(C)
s(8) - s(2 )
6
(D)
v(8) - v(2 )
6
(E) v(8) - v(2 )
Ê 1 ˆ
77. If sin Á 2
is an antiderivative for f ( x ) , then
Ë x + 1 ˜¯
(A) - 0.281
(B) - 0.102
2
Ú1
f ( x ) dx =
(C) 0.102
(D) 0.260
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(E) 0.282
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-25-
B
B
B
B
B
B
B
B
B
78. The function f is differentiable and increasing for all real numbers x, and the graph of f has exactly one point of
inflection. Of the following, which could be the graph of f ¢, the derivative of f ?
(A)
(B)
(C)
(D)
(E)
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-26-
B
B
B
B
B
B
B
B
B
79. A vase has the shape obtained by revolving the curve y = 2 + sin x from x = 0 to x = 5 about the x-axis,
where x and y are measured in inches. What is the volume, in cubic inches, of the vase?
(A) 10.716
(B) 25.501
(C) 33.666
(D) 71.113
x
f (x)
1
2.4
3
3.6
5
5.4
(E) 80.115
80. The table above gives selected values of a function f. The function is twice differentiable with f ¢¢( x ) > 0.
Which of the following could be the value of f ¢(3) ?
(A) 0.6
(B) 0.7
(C) 0.9
(D) 1.2
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(E) 1.5
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-27-
B
B
B
B
B
B
B
B
B
81. At time t = 0 years, a forest preserve has a population of 1500 deer. If the rate of growth of the population is
modeled by R(t ) = 2000e0.23t deer per year, what is the population at time t = 3 ?
(A) 3987
(B) 5487
(C) 8641
(D) 10,141
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(E) 12,628
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-28-
B
B
B
B
B
B
B
B
B
82. The figure above shows the graph of f ¢, the derivative of function f, for -6 < x < 8. Of the following,
which best describes the graph of f on the same interval?
(A) 1 relative minimum, 1 relative maximum, and 3 points of inflection
(B) 1 relative minimum, 1 relative maximum, and 4 points of inflection
(C) 2 relative minima, 1 relative maximum, and 2 points of inflection
(D) 2 relative minima, 1 relative maximum, and 4 points of inflection
(E) 2 relative minima, 2 relative maxima, and 3 points of inflection
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-29-
B
B
B
B
83. Let f and g be continuous functions such that
1
f ( x ) - 3g ( x )) dx ?
(
ı 2
the value of Û
Ù
B
6
B
6
B
B
B
0
Ú0 f ( x ) dx = 9, Ú3 f ( x ) dx = 5, and Ú3 g( x ) dx = -7. What is
3
0
(A) -23
(B) -19
(C) -
17
2
(D) 19
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(E) 23
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-30-
B
B
B
B
B
B
B
B
B
84. The regions A, B, and C in the figure above are bounded by the graph of the function f and the x-axis. The area
of region A is 14, the area of region B is 16, and the area of region C is 50. What is the average value of f on
the interval [ 0, 8] ?
(A) 6
(B) 10
(C)
40
3
(D)
80
3
(E) 48
85. A particle moves along the x-axis so that its velocity at time t ≥ 0 is given by v(t ) =
distance traveled by the particle from t = 0 to t = 2 ?
(A) 0.214
(B) 0.320
(C) 0.600
(D) 0.927
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t2 - 1
. What is the total
t2 + 1
(E) 1.600
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-31-
B
B
B
B
B
(
B
B
B
B
)
86. Line  is tangent to the graph of y = e x at the point k , e k . What is the positive value of k for which the
y-intercept of  is
1
?
2
(A) 0.405
(B) 0.768
(C) 1.500
(D) 1.560
(E) There is no such value of k.
87. A differentiable function f has the property that f ¢( x ) £ 3 for 1 £ x £ 8 and f (5) = 6. Which of the following
could be true?
I. f (2 ) = 0
II. f (6 ) = -2
III. f (7) = 13
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) II and III only
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-32-
B
B
B
B
B
B
B
B
B
88. The graph of the differentiable function f is shown in the figure above. Let h be the function defined
by h( x ) =
x
Ú0 f (t ) dt. Which of the following correctly orders h(2),
h¢(2 ) , and h¢¢(2 ) ?
(A) h(2 ) < h ¢(2 ) < h¢¢(2 )
(B) h ¢(2 ) < h(2 ) < h¢¢(2 )
(C) h ¢(2 ) < h ¢¢(2 ) < h(2 )
(D) h ¢¢(2 ) < h(2 ) < h ¢(2 )
(E) h ¢¢(2 ) < h ¢(2 ) < h(2 )
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-33-
B
B
B
B
B
B
B
B
B
89. What is the area of the region enclosed by the graphs of y = e x - 2, y = sin x, and x = 0 ?
(A) 0.239
(B) 0.506
(C) 0.745
(D) 2.340
(E) 3.472
90. A particle moves along a line so that its velocity is given by v(t ) = -t 3 + 2t 2 + 2 - t for t ≥ 0. For what values
of t is the speed of the particle increasing?
(A) (0, 0.177) and (1.256, • )
(B) (0, 1.256 ) only
(C) (0, 2.057) only
(D) (0.177, 1.256 ) only
(E)
(0.177, 1.256) and (2.057, • )
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-34-
B
B
B
B
B
B
B
B
B
91. Let F be a function defined for all real numbers x such that F ¢( x ) > 0 and F ¢¢( x ) > 0. Which of the following
could be a table of values for F ?
(A)
x
1
2
3
4
F(x)
–3
–4
–6
–9
(B)
x
1
2
3
4
F(x)
–3
–1
3
19
(C)
x
1
2
3
4
F(x)
–3
0
3
6
(D)
x
1
2
3
4
F(x)
–3
5
11
13
(E)
x
1
2
3
4
F(x)
–3
–4
–3
0
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-35-
B
B
B
B
B
B
x
f (x)
g( x )
f ¢( x )
–4
0
–9
5
–2
4
–7
4
0
6
–4
2
2
7
–3
1
4
10
–2
3
B
B
B
92. The table above gives values of the differentiable functions f and g, and f ¢, the derivative of f, at selected
values of x. If g( x ) = f -1 ( x ) , what is the value of g¢( 4 ) ?
(A) -
1
3
(B
-
1
4
(C) -
3
100
(D
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1
4
(E)
1
3
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-36-
Question 1: A
Question 24: B
Question 2: B
Question 25: D
Question 3: D
Question 26: A
Question 4: E
Question 27: D
Question 5: C
Question 28: A
Question 6: E
Question 76: C
Question 7: A
Question 77: A
Question 8: D
Question 78: A
Question 9: A
Question 79: E
Question 10: E
Question 80: B
Question 11: D
Question 81: D
Question 12: C
Question 82: A
Question 13: C
Question 83: B
Question 14: D
Question 84: A
Question 15: A
Question 85: D
Question 16: B
Question 86: B
Question 17: E
Question 87: C
Question 18: B
Question 88: E
Question 19: B
Question 89: C
Question 20: B
Question 90: E
Question 21: C
Question 91: B
Question 22: D
Question 92: D
Question 23: E
®
AP Calculus AB Exam
2015
SECTION I: Multiple Choice
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.
At a Glance
Total Time
1 hour, 45 minutes
Number of Questions
45
Percent of Total Score
50%
Writing Instrument
Pencil required
Part A
Instructions
Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For
Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet.
For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer
sheet. The survey questions are numbers 93 through 96.
Indicate all of your answers to the multiple-choice questions on the answer sheet. No
credit will be given for anything written in this exam booklet, but you may use the booklet
for notes or scratch work. After you have decided which of the suggested answers is best,
completely fill in the corresponding circle on the answer sheet. Give only one answer to
each question. If you change an answer, be sure that the previous mark is erased
completely. Here is a sample question and answer.
Number of Questions
28
Time
55 minutes
Electronic Device
None allowed
Part B
Number of Questions
17
Time
50 minutes
Electronic Device
Graphing calculator
required
Use your time effectively, working as quickly as you can without losing accuracy. Do not
spend too much time on any one question. Go on to other questions and come back to
the ones you have not answered if you have time. It is not expected that everyone will
know the answers to all of the multiple-choice questions.
Your total score on the multiple-choice section is based only on the number of questions
answered correctly. Points are not deducted for incorrect answers or unanswered
questions.
Form I
Form Code 4KBP6-S
66
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
CALCULUS AB
SECTION I, Part A
Time— 55 minutes
Number of questions—28
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
In this exam:
(1)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x) is a real number.
(2)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-3-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
(
)
1
Û
dx =
Ù 5e2 x +
x
ı
1.
(A)
5 2x
2
e + 2 +C
2
x
(B)
5 2x
e + ln x + C
2
(C) 5e 2 x +
2
+C
x2
(D) 5e 2 x + ln x + C
(E) 10e 2 x -
1
+C
x2
2. If f ( x ) = x +
(A)
1
16
3
, then f ¢( 4) =
x
(B)
5
16
(C) 1
(D)
7
2
(E)
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any part of this page is illegal.
49
4
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-4-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
2 3
Ú x ( x + 5)
6
3.
(
)
dx =
(A)
1 3
x +5
3
(B)
1 3 1 4
x
x + 5x
3
4
(C)
1 3
x +5
7
(D)
3 2 3
x x +5
7
(E)
1 3
x +5
21
6
+C
(
(
)
7
(
(
)
)
6
+C
+C
)
7
7
+C
+C
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-5-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
x
0
25
30
50
f (x)
4
6
8
12
4. The values of a continuous function f for selected values of x are given in the table above. What is the value of
the left Riemann sum approximation to
(A) 290
(B) 360
(C) 380
50
Ú0
f ( x ) dx using the subintervals [0, 25], [25, 30], and [30, 50] ?
(D) 390
(E) 430
ÏÔ x 2 sin ( p x )
for x < 2
f (x) = Ì
2
ÔÓ x + cx - 18 for x ≥ 2
5. Let f be the function defined above, where c is a constant. For what value of c, if any, is f continuous
at x = 2 ?
(A) 2
(B) 7
(C) 9
(D) 4p - 4
(E) There is no such value of c.
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-6-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
6. Which of the following is an antiderivative of 3sec 2 x + 2 ?
(A) 3 tan x
(B) 3tan x + 2 x
(C) 3sec x + 2 x
(D) sec3 x + 2 x
(E) 6sec 2 x tan x
7. If f ( x ) = x 2 - 4 and g is a differentiable function of x, what is the derivative of f ( g ( x )) ?
(A) 2 g ( x )
(B) 2 g ¢ ( x )
(C) 2 xg ¢ ( x )
(D) 2 g ( x ) g ¢ ( x )
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(E) 2 g ( x ) - 4
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-7-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
(
)
dy
= y 2 4 - y 2 . If y = g( x ) is the solution to the
dx
differential equation with the initial condition g( -2 ) = -1, then lim g( x ) is
8. Shown above is a slope field for the differential equation
x Æ•
(A) - •
(B) -2
(C) 0
(D) 2
(E) 3
9. If f ¢¢ ( x ) = x ( x + 2 ) , then the graph of f is concave up for
2
(A) x < 0
(B) x > 0
(C) -2 < x < 0
(D) x < -2 and x > 0
(E) all real numbers
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-8-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
10. If y = sin x cos x, then at x =
(A) -
11.
1
2
(B) -
1
4
p dy
,
=
3 dx
(C)
1
4
(D)
1
2
(E) 1
x2 - 9
is
xÆ-3 x 2 - 2x - 15
lim
(A) 0
(B)
3
5
(C)
3
4
(D) 1
(E) nonexistent
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-9-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
p
12. What is the average rate of change of y = cos ( 2 x ) on the interval È 0, ˘ ?
ÎÍ 2 ˙˚
(A) -
4
p
(B) -1
13. If y3 + y = x 2 , then
(A) 0
(B)
(C) 0
(D)
2
2
(E)
4
p
dy
=
dx
x
2
(C)
2x
3y 2
(D) 2x - 3y 2
Unauthorized copying or reuse of
any part of this page is illegal.
(E)
2x
1 + 3y 2
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-10-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
14. The graph of y = f ( x ) on the closed interval [0, 4] is shown above. Which of the following could be the graph
of y = f ¢( x ) ?
(A)
(B)
(C)
(D)
(E)
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-11-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
if x < 1
Ï 3x - 2
f (x) = Ì
Ó ln (3 x - 2 ) if x ≥ 1
15. Let f be the function defined above. Which of the following statements about f are true?
I. lim f ( x ) = lim f ( x )
xÆ1-
xÆ1+
II. lim f ¢( x ) = lim f ¢( x )
xÆ1-
xÆ1+
III. f is differentiable at x = 1.
(A) None
(B) I only
(C) II only
(D) II and III only
(E) I, II, and III
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-12-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
16. The function f is defined by f ( x ) = 2x 3 - 4x 2 + 1. The application of the Mean Value Theorem to f on the
interval 1 £ x £ 3 guarantees the existence of a value c, where 1 < c < 3, such that f ¢( c ) =
(A) 0
(B) 9
(C) 10
(D) 14
(E) 16
17. The velocity v, in meters per second, of a certain type of wave is given by v(h ) = 3 h , where h is the depth, in
meters, of the water through which the wave moves. What is the rate of change, in meters per second per meter,
of the velocity of the wave with respect to the depth of the water, when the depth is 2 meters?
(A) -
3
4 2
(B) -
3
8 2
(C)
3
2 2
(D)
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3
2
(E) 4 2
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-13-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
18. If
dy
= -10e - t
dt
2
(A) 20 e - 6
and y (0 ) = 20, what is the value of y (6 ) ?
(B) 20 e -3
(C) 20 e -2
(D) 10 e -3
(E) 5e -3
19. Let f be the function with derivative defined by f ¢( x ) = x 3 - 4x. At which of the following values of x does
the graph of f have a point of inflection?
(A) 0
(B)
2
3
(C)
2
3
(D)
4
3
(E) 2
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-14-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
20. Let f be the function given by f ( x ) =
( x - 4 )(2 x - 3)
. If the line y = b is a horizontal asymptote to the graph
( x - 1)2
of f, then b =
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
21. The base of a solid is the region bounded by the x-axis and the graph of y = 1 - x 2 . For the solid, each cross
section perpendicular to the x-axis is a square. What is the volume of the solid?
(A)
2
3
(B)
4
3
(C) 2
(D)
2p
3
(E)
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4p
3
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-15-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
kx
, where k is a constant. For what values of k, if any,
x +1
is f strictly decreasing on the interval ( -1, 1) ?
22. Let f be the function given by f ( x ) =
2
(A) k < 0
(B) k = 0
(C) k > 0
(D) k > 1 only
(E) There are no such values of k.
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-16-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
23. Which of the following is an equation for the line tangent to the graph of y = 3 where x = -1 ?
x
Ú-1 e
-t3
dt at the point
(A) y - 3 = -3e ( x + 1)
(B) y - 3 = -e ( x + 1)
(C) y - 3 = 0
(D) y - 3 =
1
( x + 1)
e
(E) y - 3 = 3e ( x + 1)
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-17-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
24. Which of the following is the solution to the differential equation
y( 0 ) = 3 ?
dy
= 5y 2 with the initial condition
dx
(A) y = 9e 5 x
(B) y =
1 5x
e
9
(C) y = e 5x + 9
(D) y =
3
1 - 15 x
(E) y =
3
1 + 15 x
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-18-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
sin
25.
lim
hÆ0
(A) 0
( p3 + h) - sin ( p3 ) is
h
(B)
1
2
(C) 1
(D)
3
2
(E) nonexistent
26. An object moves along a straight line so that at any time t ≥ 0 its velocity is given by v(t ) = 2 cos (3t ) . What is
the distance traveled by the object from t = 0 to the first time that it stops?
(A) 0
(B)
p
6
(C)
2
3
(D)
p
3
(E)
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4
3
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-19-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
x
f (x)
f ¢( x )
0
49
0
1
2
–8
2
–1
–80
27. The table above gives selected values for a differentiable and decreasing function f and its derivative. If f -1 is
( )
the inverse function of f, what is the value of f -1 ¢ (2 ) ?
(A) -80
(B) -
1
8
(C) -
1
80
(D)
1
80
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any part of this page is illegal.
(E)
1
8
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-20-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
28. The top of a 15-foot-long ladder rests against a vertical wall with the bottom of the ladder on level ground, as
shown above. The ladder is sliding down the wall at a constant rate of 2 feet per second. At what rate, in radians
per second, is the acute angle between the bottom of the ladder and the ground changing at the instant the bottom
of the ladder is 9 feet from the base of the wall?
(A) -
2
9
(B) -
1
6
(C) -
2
25
(D)
2
25
(E)
1
9
END OF PART A OF SECTION I
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WORK ON PART A ONLY.
DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.
Unauthorized copying or reuse of
any part of this page is illegal.
-21-
B
B
B
B
B
B
B
B
B
CALCULUS AB
SECTION I, Part B
Time— 50 minutes
Number of questions—17
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON
THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO
QUESTIONS NUMBERED 76–92.
YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET.
In this exam:
(1)
The exact numerical value of the correct answer does not always appear among the choices given. When this
happens, select from among the choices the number that best approximates the exact numerical value.
(2)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x) is a real number.
(3)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-24-
B
B
B
B
B
B
B
B
B
76. The function P (t ) models the population of the world, in billions of people, where t is the number of years
since January 1, 2010. Which of the following is the best interpretation of the statement P¢(1) = 0.076 ?
(A) On February 1, 2010, the population of the world was increasing at a rate of 0.076 billion people per year.
(B) On January 1, 2011, the population of the world was increasing at a rate of 0.076 billion people per year.
(C) On January 1, 2011, the population of the world was 0.076 billion people.
(D) From January 1, 2010 to January 1, 2011, the population of the world was increasing at an average rate of
0.076 billion people per year.
(E) When the population of the world was 1 billion people, the population of the world was increasing at a rate
of 0.076 billion people per year.
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-25-
B
B
B
B
B
B
B
x
0
2
4
6
8
10
f (x)
5
7
8
0
–15
–20
B
B
77. Let f be a differentiable function with selected values given in the table above. What is the average rate of
change of f over the closed interval 0 £ x £ 10 ?
(A) -6
(B) -
5
2
(C) -2
(D) -
2
5
(E)
2
5
78. The rate at which motor oil is leaking from an automobile is modeled by the function L defined by
L (t ) = 1 + sin t 2 for time t ≥ 0. L (t ) is measured in liters per hour, and t is measured in hours. How
( )
much oil leaks out of the automobile during the first half hour?
(A) 1.998 liters
(B) 1.247 liters
(C) 0.969 liters
(D) 0.541 liters
(E) 0.531 liters
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-26-
B
B
B
B
x
0
B
f (x)
3
f ¢( x )
4
B
g( x )
2
B
B
B
g ¢( x )
p
79. The table above gives values of the differentiable functions f and g and their derivatives at x = 0.
f (x)
, what is the value of h ¢( 0 ) ?
If h( x ) =
g( x )
(A)
8 - 3p
4
(B)
3p - 8
4
(C)
4
p
(D)
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2 - 3p
2
(E)
8 + 3p
4
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-27-
B
B
B
B
B
B
B
B
B
80. The figure above shows the graph of f ¢, the derivative of a function f, for 0 £ x £ 2. What is the value of x
at which the absolute minimum of f occurs?
(A) 0
(B)
1
2
(C) 1
(D)
3
2
(E) 2
81. What is the area of the region enclosed by the graphs of y = 4 x - x 2 and y =
(A) 1.707
(B) 2.829
(C) 5.389
(D) 8.886
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x
?
2
(E) 21.447
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-28-
B
B
B
B
B
B
B
B
B
82. The graph of f ¢, the derivative of f, is shown above. The line tangent to the graph of f ¢ at x = 0 is vertical,
and f ¢ is not differentiable at x = 2. Which of the following statements is true?
(A) f ¢ does not exist at x = 2.
(B) f is decreasing on the interval (2, 4 ) .
(C) The graph of f has a point of inflection at x = 2.
(D) The graph of f has a point of inflection at x = 0.
(E) f has a local maximum at x = 0.
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-29-
B
B
B
B
B
B
B
B
B
83. A particle moves along the x-axis so that its position at time t > 0 is given by x (t ) and
dx
= -10t 4 + 9t 2 + 8t. The acceleration of the particle is zero when t =
dt
(A) 0.387
(B) 0.831
(C) 1.243
(D) 1.647
84. The function f is continuous on the closed interval [1, 7]. If
then
7
Ú1
(E) 8.094
f ( x ) dx = 42 and
3
Ú7 f ( x ) dx = -32,
3
Ú1 2 f ( x ) dx =
(A) -148
(B) 10
(C) 12
(D) 20
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(E) 148
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-30-
B
B
B
B
B
B
B
B
B
85. Let y = f ( x) define a twice-differentiable function and let y = t ( x) be the line tangent to the graph of f at
x = 2. If t ( x ) ≥ f ( x) for all real x, which of the following must be true?
(A) f (2) ≥ 0
(B) f ¢(2) ≥ 0
(C) f ¢(2) £ 0
(D) f ¢¢(2) ≥ 0
(E) f ¢¢(2) £ 0
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-31-
B
B
B
B
B
B
B
B
B
86. The vertical line x = 2 is an asymptote for the graph of the function f. Which of the following statements must
be false?
(A) lim f ( x ) = 0
xÆ2
(B) lim f ( x ) = - •
xÆ2
(C) lim f ( x ) = •
xÆ2
(D) lim f ( x ) = 2
xÆ •
(E) lim f ( x ) = •
xÆ •
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-32-
B
B
B
B
B
B
B
B
87. The graph of the piecewise linear function f is shown above. Let h be the function given by h( x ) =
B
x
Ú-1 f (t ) dt.
On which of the following intervals is h increasing?
(A) [ -1, 3]
(B) [0, 5]
(C) [2, 5] only
(D) [2, 9]
(E)
[3, 9]
only
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-33-
B
B
B
B
B
B
B
B
B
( )
88. The first derivative of the function f is given by f ¢ ( x ) = sin x 2 . At which of the following values of x does
f have a local minimum?
(A) 2.507
(B) 2.171
(C) 1.772
(D) 1.253
(E) 0
89. If lim f ( x ) = f (a ) , then which of the following statements about f must be true?
xÆa
(A) f is continuous at x = a.
(B) f is differentiable at x = a.
(C) For all values of x, f ( x ) = f (a ) .
(D) The line y = f (a ) is tangent to the graph of f at x = a.
(E) The line x = a is a vertical asymptote of the graph of f.
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-34-
B
B
B
B
B
B
B
B
B
90. The temperature F, in degrees Fahrenheit (∞F ) , of a cup of coffee t minutes after it is poured is given by
F (t ) = 72 + 118e -0.093t . To the nearest degree, what is the average temperature of the coffee between t = 0 and
t = 10 minutes?
(A)
93∞F
(B) 119∞F
(C) 146∞F
(D) 149∞F
(E) 154∞F
( )
91. If f ¢( x ) = cos x 2 and f (3) = 7, then f (2 ) =
(A) 0.241
(B) 5.831
(C) 6.416
(D) 6.759
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(E) 7.241
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-35-
B
B
B
B
B
B
B
B
B
92. The graph of the function h is shown in the figure above. Of the following, which has the greatest value?
(A) Average value of h over [ -3,2]
(B) Average rate of change of h over [ -3,2]
2
(C)
Ú-3 h ( x ) dx
(D)
Ú-3 h ( x ) dx
0
(E) h ¢ (0 )
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-36-
Answer Key for AP Calculus AB
Practice Exam, Section I
Question 24: D
Question 2: A
Question 25: B
Question 3: E
Question 26: C
Question 4: A
Question 27: B
Question 5: B
Question 28: A
Question 6: B
Question 76: B
Question 7: D
Question 77: B
Question 8: C
Question 78: D
Question 9: B
Question 79: A
Question 10: A
Question 80: E
Question 11: C
Question 81: B
Question 12: A
Question 82: C
Question 13: E
Question 83: B
Question 14: D
Question 84: D
Question 15: C
Question 85: E
Question 16: C
Question 86: A
Question 17: C
Question 87: E
Question 18: B
Question 88: A
Question 19: C
Question 89: A
Question 20: C
Question 90: D
Question 21: B
Question 91: D
Question 22: A
Question 92: B
Question 23: B
®
AP Calculus AB Exam
2016
SECTION I: Multiple Choice
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.
At a Glance
Total Time
1 hour, 45 minutes
Number of Questions
45
Percent of Total Score
50%
Writing Instrument
Pencil required
Part A
Instructions
Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For
Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet.
For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer
sheet. The survey questions are numbers 93 through 96.
Indicate all of your answers to the multiple-choice questions on the answer sheet. No
credit will be given for anything written in this exam booklet, but you may use the booklet
for notes or scratch work. After you have decided which of the suggested answers is best,
completely fill in the corresponding circle on the answer sheet. Give only one answer to
each question. If you change an answer, be sure that the previous mark is erased
completely. Here is a sample question and answer.
Number of Questions
28
Time
55 minutes
Electronic Device
None allowed
Part B
Number of Questions
17
Time
50 minutes
Electronic Device
Graphing calculator
required
Use your time effectively, working as quickly as you can without losing accuracy. Do not
spend too much time on any one question. Go on to other questions and come back to
the ones you have not answered if you have time. It is not expected that everyone will
know the answers to all of the multiple-choice questions.
Your total score on the multiple-choice section is based only on the number of questions
answered correctly. Points are not deducted for incorrect answers or unanswered
questions.
Form I
Form Code 4LBP6-S
66
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
CALCULUS AB
SECTION I, Part A
Time— 55 minutes
Number of questions—28
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
In this exam:
(1)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x) is a real number.
(2)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
Unauthorized copying or reuse of
any part of this page is illegal.
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-3-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
dy
=
dx
1. If y = cos 2 x, then
(A) -2sin 2x
(B) - sin 2x
2
3
Ú x ( x - 1)
10
2.
(A)
ˆ
x3 Ê x 4
- x˜
3 ÁË 4
¯
( x3 - 1)
(C) sin 2x
(D) 2sin 2x
(E) 2sin x
dx =
10
+C
11
(B)
(C)
+C
11
(
)
x 2 x3 - 1
11
11
( x3 - 1)
+C
11
(D)
(E)
+C
33
(
)
x3 x3 - 1
33
11
+C
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-4-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
9x4 + 1
is
x Æ• 4 x 2 + 3
lim
3.
(A)
1
3
4. If y =
(B)
3
4
(C)
3
2
(D)
9
4
(E) infinite
( )
dy
x 5
, then
=
x +1
dx
(A) 5 (1 + x )
4
(B)
x4
( x + 1)4
(C)
5x 4
( x + 1)4
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any part of this page is illegal.
(D)
5x 4
( x + 1)6
(E)
5 x 4 (2 x + 1)
( x + 1)6
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-5-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
t
(minutes)
r (t )
(gallons per minute)
0
4
7
9
9
6
4
3
5. Water is flowing into a tank at the rate r (t ) , where r (t ) is measured in gallons per minute and t is measured
in minutes. The tank contains 15 gallons of water at time t = 0. Values of r (t ) for selected values of t are
given in the table above. Using a trapezoidal sum with the three intervals indicated by the table, what is the
approximation of the number of gallons of water in the tank at time t = 9 ?
(A) 52
(B) 57
(C) 67
(D) 77
(E) 79
6. The slope of the line tangent to the graph of y = ln (1 - x ) at x = -1 is
(A) -1
(B) -
1
2
(C)
1
2
(D) ln 2
Unauthorized copying or reuse of
any part of this page is illegal.
(E) 1
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-6-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
7. For which of the following pairs of functions f and g is lim
x Æ•
f (x)
infinite?
g( x )
(A) f ( x ) = x 2 + 2 x and g( x ) = x 2 + ln x
(B) f ( x ) = 3 x 3 and g( x ) = x 4
(C) f ( x ) = 3 x and g( x ) = x 3
(D) f ( x ) = 3e x + x 3 and g( x ) = 2e x + x 2
(E) f ( x ) = ln (3 x ) and g( x ) = ln ( 2 x )
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-7-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
Ú
8.
4
0
x
x +9
2
(A) -2
dx =
(B) -
2
15
(C) 1
(D) 2
(E) 5
9. Let f be the function with derivative given by f ¢( x ) =
-2 x
(1 + x 2 )
2
. On what interval is f decreasing?
(A) [ 0, •) only
(B)
( - •, 0] only
1 1 ˘
,
(C) È only
ÍÎ 3 3 ˙˚
(D) ( - •, •)
(E) There is no such interval.
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-8-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
10.
Ú (e
x
)
+ e dx =
(A) e x + C
(B) 2e x + C
(C) e x + e + C
Unauthorized copying or reuse of
any part of this page is illegal.
(D) e x +1 + ex + C
(E) e x + ex + C
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-9-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
11. The graph of the function f is shown in the figure above. Which of the following could be the graph of f ¢, the
derivative of f ?
(A)
(B)
(C)
(D)
(E)
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-10-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
12. If 0 < c < 1, what is the area of the region enclosed by the graphs of y = 0, y =
(A) ln (1 - c )
(B) ln
(
( 1c )
(C) ln c
(D)
1
-1
c2
1
, x = c, and x = 1 ?
x
(E) 1 -
1
c2
)
d
tan -1 x + 2 x =
dx
13.
(A) (B)
(C)
1
1
+
2
x
sin x
1
1- x
2
1
1- x
2
- 43 x
+
1
x
(D)
1
- 43 x
2
1+ x
(E)
1
1
+
2
x
1+ x
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-11-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
14. If y = f ( x ) is a solution to the differential equation
2
dy
= e x with the initial condition f (0 ) = 2, which of the
dx
following is true?
(A) f ( x ) = 1 + e x
(B) f ( x ) = 2 xe x
2
2
x t2
Ú1 e
dt
(D) f ( x ) = 2 +
Ú0 e
(E) f ( x ) = 2 +
Ú2 e
(C) f ( x ) =
x t2
x t2
dt
dt
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-12-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
15. A function f (t ) gives the rate of evaporation of water, in liters per hour, from a pond, where t is measured in
hours since 12 noon. Which of the following gives the meaning of
10
Ú4
f (t ) dt in the context described?
(A) The total volume of water, in liters, that evaporated from the pond during the first 10 hours after 12 noon
(B) The total volume of water, in liters, that evaporated from the pond between 4 P.M. and 10 P.M.
(C) The net change in the rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M.
(D) The average rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M.
(E) The average rate of change in the rate of evaporation, in liters per hour per hour, from the pond between
4 P.M. and 10 P.M.
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-13-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
16. The first derivative of the function f is given by f ¢( x ) = 3 x 4 - 12 x 3 . What are the x-coordinates of the points
of inflection of the graph of f ?
(A) x = 3 only
(B) x = 4 only
(C) x = 0 and x = 2
(D) x = 0 and x = 3
(E) x = 0 and x = 4
17. Let f be the function defined by f ( x ) =
(A) -
1
24
(B)
5
24
(C)
1
. What is the average value of f on the interval [ 4, 6] ?
x
1 3
ln
2 2
(D) ln
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any part of this page is illegal.
3
2
(E)
1
ln 2
2
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-14-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
Ê x, 1 ˆ , and Ê 3, 1 ˆ are the vertices of a rectangle, where x ≥ 3, as shown in the
ÁË
˜
ÁË
˜
x2 ¯
x2 ¯
figure above. For what value of x does the rectangle have a maximum area?
18. The points (3, 0 ) ,
( x, 0 ) ,
(A) 3
(B) 4
(C) 6
(D) 9
(E) There is no such value of x.
19. What are all values of x for which
(A) -2 only
(B) 0 only
2 3
Úx t
dt is equal to 0 ?
(D) -2 and 2 only
(C) 2 only
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(E) -2, 0, and 2
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-15-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
20. Let h be the function defined by h( x ) =
to the graph of h at the point where x =
(A) y =
x
Úp
4
sin 2 t dt. Which of the following is an equation for the line tangent
p
?
4
1
2
(B) y = 2 x
p
4
(C) y = x (D) y =
(E) y =
(
1
p
x2
4
(
)
2
p
x2
4
)
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-16-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
x
f (x)
–1
0
3
5
–30
–2
10
18
21. The table above gives selected values for a twice-differentiable function f. Which of the following must be true?
(A) f has no critical points in the interval -1 < x < 5.
(B) f ¢( x ) = 8 for some value of x in the interval -1 < x < 5.
(C) f ¢( x ) > 0 for all values of x in the interval -1 < x < 5.
(D) f ¢¢( x ) < 0 for all values of x in the interval -1 < x < 5.
(E) The graph of f has no points of inflection in the interval -1 < x < 5.
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-17-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
22. A particle moves along the x-axis so that at time t ≥ 0, the acceleration of the particle is a (t ) = 15 t . The
position of the particle is 10 when t = 0, and the position of the particle is 20 when t = 1. What is the velocity
of the particle at time t = 0 ?
(A) -14
(B) 0
(C) 5
(D) 6
(E) 10
23. Which of the following is the solution to the differential equation
the point ( 0, 1) ?
(A) y = e x
dy
2 xy
whose graph contains
= 2
dx
x +1
2
(B) y = x 2 + 1
(
)
(C) y = ln x 2 + 1
(
)
(D) y = 1 + ln x 2 + 1
(
)
(E) y = 1 + 2 ln x 2 + 1
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-18-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
r3
, where r is the
3
radius of the base, in feet. The circumference of the base is increasing at a constant rate of 5p feet per hour.
When the circumference of the base is 8p feet, what is the rate of change of the volume of the pile, in cubic feet
per hour?
24. Sand is deposited into a pile with a circular base. The volume V of the pile is given by V =
(A)
25.
8
p
(B) 16
(C) 40
(D) 40p
(E) 80p
e -1- h - e -1
is
h
hÆ 0
lim
(A) -1
(B)
-1
e
(C) 0
(D)
1
e
(E) nonexistent
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-19-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
26. Let f be the function given by f ( x ) = x 3 + 5 x. For what value of x in the closed interval [1,3] does the
instantaneous rate of change of f equal the average rate of change of f on that interval?
7
3
(A)
13
3
(B)
(C)
27. If e xy - y 2 = e - 4, then at x =
(A)
e
4
(B)
e
2
(C)
5
6
(D)
(E)
19
3
dy
1
=
and y = 2,
dx
2
4e
8-e
(D)
4e
4-e
Unauthorized copying or reuse of
any part of this page is illegal.
(E)
8 - 4e
e
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-20-
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
28. Let f be the function defined by f ( x ) = x 3 + x 2 + x. Let g( x ) = f -1 ( x ) , where g(3) = 1. What is the value
of g¢(3) ?
(A)
1
39
(B)
1
34
(C)
1
6
(D)
1
3
(E) 39
END OF PART A OF SECTION I
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WORK ON PART A ONLY.
DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.
Unauthorized copying or reuse of
any part of this page is illegal.
-21-
B
B
B
B
B
B
B
B
B
CALCULUS AB
SECTION I, Part B
Time— 50 minutes
Number of questions—17
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON
THIS PART OF THE EXAM.
Directions: Solve each of the following problems, using the available space for scratch work. After examining the
form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer
sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one
problem.
BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO
QUESTIONS NUMBERED 76–92.
YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET.
In this exam:
(1)
The exact numerical value of the correct answer does not always appear among the choices given. When this
happens, select from among the choices the number that best approximates the exact numerical value.
(2)
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which
f ( x) is a real number.
(3)
The inverse of a trigonometric function f may be indicated using the inverse function notation f -1 or with the
prefix “arc” (e.g., sin -1 x = arcsin x ).
Unauthorized copying or reuse of
any part of this page is illegal.
GO ON TO THE NEXT PAGE.
-24-
B
B
B
B
B
B
B
B
B
76. The graph of a function f is shown above. Which of the following limits does not exist?
(A) lim f ( x )
x Æ1-
(B) lim f ( x )
x Æ1
(C) lim f ( x )
x Æ 3-
Unauthorized copying or reuse of
any part of this page is illegal.
(D) lim f ( x )
x Æ3
(E) lim f ( x )
xÆ5
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-25-
B
B
B
B
B
B
B
B
B
77. Let f be a function that is continuous on the closed interval [1, 3] with f (1) = 10 and f (3) = 18. Which of the
following statements must be true?
(A) 10 £ f (2 ) £ 18
(B) f is increasing on the interval [1, 3].
(C) f ( x ) = 17 has at least one solution in the interval [1, 3].
(D) f ¢( x ) = 8 has at least one solution in the interval (1, 3) .
(E)
3
Ú1 f ( x ) dx > 20
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any part of this page is illegal.
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-26-
B
B
B
B
B
B
B
B
B
78. Let R be the region bounded by the graphs of y = e x , y = e3 , and x = 0. Which of the following gives the
volume of the solid formed by revolving R about the line y = -1?
(A) p
Ú0 (e
3
3
- e x + 1 dx
(B) p
Ú0 (e
3
- e x - 1 dx
(C) p
Ú0 ÈÎÍ(e
3
- ex
)
(D) p
Ú0 ÈÍÎ(e
3
- ex
)
(E) p
Ú0 ÈÍÎ(e
3
+1
3
3
3
3
)
)
2
2
2
+ 1˘ dx
˚˙
2
- 1˘ dx
˙˚
) - (e x + 1)
2
2˘
˙˚
dx
79. The number of people who have entered a museum on a certain day is modeled by a function f (t ) ,
where t is measured in hours since the museum opened that day. The number of people who have left
(
)
the museum since it opened that same day is modeled by a function g(t ) . If f ¢(t ) = 380 1.02t and
Ê p (t - 4) ˆ
g ¢(t ) = 240 + 240sin Á
, at what time t, for 1 £ t £ 11, is the number of people in the
Ë 12 ˜¯
museum at a maximum?
(A) 1
(B) 7.888
(C) 9.446
(D) 10.974
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any part of this page is illegal.
(E) 11
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-27-
B
B
B
B
B
B
B
x
0
1
2
3
f (x)
5
2
3
6
f ¢( x )
–3
1
3
4
B
B
80. The derivative of the function f is continuous on the closed interval [ 0, 4]. Values of f and f ¢ for selected
values of x are given in the table above. If
(A) 0
(B) 3
(C) 5
4
Ú0 f ¢(t ) dt = 8, then
(D) 10
f ( 4) =
(E) 13
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-28-
B
B
B
B
B
B
B
B
B
81. A slope field for a differential equation is shown in the figure above. If y = f ( x ) is the particular solution to the
differential equation through the point ( -1, 2 ) and h( x ) = 3 x ⴢ f ( x ) , then h ¢( -1) =
(A) - 6
(B) -2
(C) 0
(D) 1
(E) 12
82. If f is a continuous function such that f ( 2 ) = 6, which of the following statements must be true?
(A) lim f ( 2 x ) = 3
x Æ1
(B) lim f (2 x ) = 12
xÆ2
(C) lim
x Æ2
f ( x ) - f ( 2)
=6
x-2
( )
(D) lim f x 2 = 36
x Æ2
(E) lim ( f ( x )) = 36
2
xÆ2
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-29-
B
B
B
B
B
B
B
83. A particle moves along a straight line with velocity given by v(t ) = 5 + et
acceleration of the particle at time t = 4 ?
(A) 0.422
(B) 0.698
(C) 1.265
84. A home uses fuel oil at the rate r (t ) = 10 + 8sin
(D) 8.794
3
B
B
for time t ≥ 0. What is the
(E) 28.381
( 60t ) gallons per day, where t is the number of days from
the beginning of the heating season. To the nearest gallon, what is the total amount of fuel oil used from t = 0
to t = 60 days?
(A) 7 gal
(B) 14 gal
(C) 600 gal
(D) 821 gal
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(E) 1004 gal
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-30-
B
B
B
B
B
B
B
B
B
85. The function f is defined on the open interval 0.4 < x < 2.4 and has first derivative f ¢ given by
( )
f ¢ ( x ) = sin x 2 . Which of the following statements are true?
I. f has a relative maximum on the interval 0.4 < x < 2.4.
II. f has a relative minimum on the interval 0.4 < x < 2.4.
III. The graph of f has two points of inflection on the interval 0.4 < x < 2.4.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
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-31-
B
B
B
B
B
B
B
B
B
86. The graph of the function f, which has a domain of [0, 7], is shown in the figure above. The graph consists of a
quarter circle of radius 3 and a segment with slope -1. Let b be a positive number such that
b
Ú0 f ( x ) dx = 0.
What is the value of b ?
(A) 3.760
(B) 5.548
(C) 5.659
(D) 6.760
(E) There is no such value of b.
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-32-
B
B
B
B
B
B
B
B
B
( )
87. The first derivative of the function g is given by g ¢ ( x ) = cos p x 2 for - 0.5 < x < 1.5. On which of the
following intervals is g decreasing?
(A) - 0.5 < x < 0
(B) 0 < x < 1
(C) 0.707 < x < 1.225
(D) 1.225 < x < 1.414
(E) 1.414 < x < 1.5
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-33-
B
B
B
B
B
B
B
B
B
88. The height above the ground of a passenger on a Ferris wheel t minutes after the ride begins is modeled by the
differentiable function H, where H (t ) is measured in meters. Which of the following is an interpretation of the
statement H ¢(7.5) = 15.708 ?
(A) The Ferris wheel is turning at a rate of 15.708 meters per minute when the passenger is 7.5 meters above the
ground.
(B) The Ferris wheel is turning at a rate of 15.708 meters per minute 7.5 minutes after the ride begins.
(C) The passenger’s height above the ground is increasing by 15.708 meters per minute when the passenger is
7.5 meters above the ground.
(D) The passenger’s height above the ground is increasing by 15.708 meters per minute 7.5 minutes after the
ride begins.
(E) The passenger is 15.708 meters above the ground 7.5 minutes after the ride begins.
89. A particle moves along a straight line for 6 seconds so that its velocity, in centimeters per second, is modeled by
the graph shown. During the time interval 0 £ t £ 6, what is the total distance the particle travels?
(A) 2 cm
(B) 3.5 cm
(C) 4 cm
(D) 6.5 cm
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(E) 8.5 cm
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-34-
B
B
B
B
B
B
B
B
B
90. Let f be a twice-differentiable function on the open interval (a, b ) . If f ¢( x ) > 0 on (a, b ) and f ¢¢( x ) < 0 on
(a, b ) , which of the following could be the graph of f ?
(A)
(B)
(C)
(D)
(E)
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-35-
B
B
B
B
B
B
B
B
B
91. The graphs of f and g are shown above. If h ( x ) = f ( x ) g( x ) , then h ¢(6 ) =
(A) -9
(B) -7
(C) 1
(D) 7
(E) 9
92. In the xy-plane, the graph of the twice-differentiable function y = f ( x ) is concave up on the open
interval (0, 2 ) and is tangent to the line y = 3 x - 2 at x = 1. Which of the following statements
must be true about the derivative of f ?
(A) f ¢( x ) £ 3 on the interval (0.9, 1) .
(B) f ¢( x ) ≥ 3 on the interval (0.9, 1) .
(C) f ¢( x ) < 0 on the interval (0.9, 1.1) .
(D) f ¢( x ) > 0 on the interval (0.9, 1.1) .
(E) f ¢( x ) is constant on the interval (0.9, 1.1) .
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-36-
Answer Key for AP Calculus AB
Practice Exam, Section I
Question 1: A
Question 24: C
Question 2: D
Question 25: B
Question 3: B
Question 26: B
Question 4: D
Question 27: C
Question 5: C
Question 28: C
Question 6: B
Question 76: D
Question 7: C
Question 77: C
Question 8: D
Question 78: E
Question 9: A
Question 79: B
Question 10: E
Question 80: E
Question 11: B
Question 81: E
Question 12: B
Question 82: E
Question 13: E
Question 83: C
Question 14: D
Question 84: D
Question 15: B
Question 85: D
Question 16: A
Question 86: D
Question 17: C
Question 87: C
Question 18: C
Question 88: D
Question 19: D
Question 89: D
Question 20: D
Question 90: C
Question 21: B
Question 91: A
Question 22: D
Question 92: A
Question 23: B
Answer Key for AP Calculus AB
Practice Exam, Section I
Question 1: D
Question 24: C
Question 2: B
Question 25: D
Question 3: C
Question 26: C
Question 4: B
Question 27: C
Question 5: A
Question 28: B
Question 6: B
Question 29: A
Question 7: D
Question 30: B
Question 8: A
Question 76: D
Question 9: D
Question 77: B
Question 10: A
Question 78: C
Question 11: A
Question 79: B
Question 12: D
Question 80: A
Question 13: D
Question 81: D
Question 14: B
Question 82: D
Question 15: C
Question 83: C
Question 16: A
Question 84: D
Question 17: A
Question 85: B
Question 18: D
Question 86: C
Question 19: C
Question 87: B
Question 20: C
Question 88: C
Question 21: A
Question 89: A
Question 22: B
Question 90: C
Question 23: A
2017 AP Calculus AB
Question Descriptors and Performance Data
Multiple-Choice Questions
Question
Learning
Objective
Essential
Knowledge
1
2.1C
2.1C4
2
3.3B(a)
3.3B3
3
2.1C
2.1C3
4
3.2B
3.2B2
5
2.1C
2.1C4
6
2.1A
2.1A1
7
3.2C
3.2C2
8
2.3B
2.3B1
9
3.2C
3.2C1
Mathematical Practice
for AP Calculus 1
Implementing algebraic/computational
processes
Implementing algebraic/computational
processes
Implementing algebraic/computational
processes
Connecting multiple representations
Implementing algebraic/computational
processes
Implementing algebraic/computational
processes
Reasoning with definitions and
theorems
Implementing algebraic/computational
processes
Connecting multiple representations
Implementing algebraic/computational
processes
Implementing algebraic/computational
processes
Mathematical Practice
for AP Calculus 2
Key
% Correct
Building notational fluency
D
91
Building notational fluency
B
63
Building notational fluency
C
80
Connecting concepts
B
68
Building notational fluency
A
76
Connecting concepts
B
57
Building notational fluency
D
69
Connecting concepts
A
68
Connecting concepts
D
84
Building notational fluency
A
42
Building notational fluency
A
61
10
3.3B(b)
3.3B5
11
2.1C
2.1C5
12
2.3D
2.3D1
Connecting concepts
Implementing
algebraic/computational
processes
D
80
13
2.3F
2.3F1
Connecting multiple representations
Connecting concepts
D
52
14
3.5A
3.5A1
Connecting concepts
Building notational fluency
B
18
15
1.1A(b)
1.1A3
Connecting multiple representations
Connecting concepts
C
41
Connecting concepts
A
60
Connecting concepts
A
42
Connecting concepts
D
68
Connecting concepts
C
57
Connecting concepts
C
24
Implementing algebraic/computational
processes
Implementing algebraic/computational
processes
Reasoning with definitions and
theorems
Implementing algebraic/computational
processes
Reasoning with definitions and
theorems
16
2.2A
2.2A1
17
2.3C
2.3C1
18
3.3A
3.3A2
19
2.3B
2.3B2
20
1.2A
1.2A3
21
2.1A
2.1A2
Connecting concepts
Building notational fluency
A
47
Connecting concepts
B
39
Building notational fluency
A
42
C
44
D
60
C
30
22
3.5A
3.5A2
Implementing algebraic/computational
processes
23
3.4D
3.4D2
Connecting concepts
24
3.4B
3.4B1
Connecting concepts
25
2.1C
2.1C3
Connecting multiple representations
26
1.1C
1.1C3
Implementing algebraic/computational
processes
Implementing
algebraic/computational
processes
Implementing
algebraic/computational
processes
Reasoning with definitions
and theorems
27
2.2A
2.2A3
Connecting multiple representations
Connecting concepts
C
61
Implementing algebraic/computational
processes
Connecting concepts
B
46
28
2.3C
2.3C2
29
1.1D
1.1D1
Building notational fluency
Connecting concepts
A
30
3.2A2
Implementing algebraic/computational
processes
Reasoning with definitions
and theorems
B
21
30
3.2A(a)
2017 AP Calculus AB
Question Descriptors and Performance Data
Question
Learning
Objective
Essential
Knowledge
76
2.2A
2.2A2
Mathematical Practice
for AP Calculus 1
Mathematical Practice
for AP Calculus 2
Key
% Correct
Connecting multiple representations
Connecting concepts
D
81
Reasoning with definitions
and theorems
B
55
77
3.3B(b)
3.3B2
Implementing algebraic/computational
processes
78
1.1D
1.1D1
Building notational fluency
Connecting concepts
C
63
79
2.3A
2.3A2
Connecting concepts
Building notational fluency
B
67
Implementing algebraic/computational
processes
Connecting concepts
A
65
Connecting multiple representations
Connecting concepts
D
49
Connecting concepts
D
42
Connecting concepts
C
72
80
2.2A
2.2A1
81
2.2A
2.2A3
Implementing algebraic/computational
processes
Implementing algebraic/computational
processes
82
3.4C
3.4C1
83
2.2A
2.2A1
84
3.4E
3.4E1
Connecting concepts
Building notational fluency
D
89
85
2.4A
2.4A1
Reasoning with definitions and
theorems
Connecting concepts
B
63
86
2.2A
2.2A1
Connecting concepts
Connecting multiple
representations
C
44
87
2.2A
2.2A2
Connecting multiple representations
Connecting concepts
B
44
Connecting concepts
C
61
Connecting concepts
A
37
Connecting concepts
C
54
88
2.3B
2.3B1
89
3.4B
3.4B1
90
1.2A
1.2A1
Implementing algebraic/computational
processes
Reasoning with definitions and
theorems
Reasoning with definitions and
theorems
Free-Response Questions
Question
Learning Objective
Essential Knowledge
1
1.2B|2.3C|3.4C
1.2B1|2.3C1|3.4C1
2
2.3D|3.2C|3.4A|3.4D
2.3D1|3.2C2|3.4A2|3.4D1
3
2.1A|2.2A|3.2C|3.3A
2.1A1|2.2A1|3.2C1|3.3A2,3.3A3
4
1.1C|2.1C|3.2B|3.3B(b)
1.1C3|2.1C2|3.2B2|3.3B2
5
3.3B(b)|3.4D
3.3B2,3.3B5|3.4D1,3.4D2
6
2.1C|2.3A|2.3C|3.1A|3.3B(b)|3.4A
2.1C2,2.1C4|2.3A1|2.3C2|3.1A2
|3.3B2|3.4A2
Mathematical Practice for AP Calculus
Reasoning with definitions and theorems|Connecting
concepts|Implementing algebraic/computational
processes|Building notational fluency|Communicating
Reasoning with definitions and theorems|Connecting
concepts|Implementing algebraic/computational
processes|Connecting multiple
representations|Building notational
fluency|Communicating
Reasoning with definitions and theorems|Connecting
concepts|Implementing algebraic/computational
processes|Connecting multiple
representations|Building notational
fluency|Communicating
Reasoning with definitions and theorems|Connecting
concepts|Implementing algebraic/computational
processes|Connecting multiple
representations|Building notational
fluency|Communicating
Reasoning with definitions and theorems|Connecting
concepts|Implementing algebraic/computational
processes|Connecting multiple
representations|Building notational
fluency|Communicating
Reasoning with definitions and theorems|Connecting
concepts|Implementing algebraic/computational
processes|Building notational fluency|Communicating
Mean
3.04
3.01
3.49
2.44
4.46
5.39
Sample Questions for Calculus AB: Section I
1.
What is 
(a) 1

Æ 
(  + ) -  (  ) 

2
2
(c) 0
(d) -1
(e) The limit does not exist.
(b)
2.
At which of the five points on the graph in the figure
at the right are
both negative?
(a)
(b)
(c)
( d)
(e)
3.
A
B
C
D
E
y
d2y
dy
and 2
dx
dx
A
B
D
C
O
E
x
The slope of the tangent to the curve y 3 x + y 2 x 2 = 6 at ( 2, 1) is
(a) - 3
2
(b) -1
(c) - 5
14
3
(d) 14
(e) 0
18
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Sample Questions for Calculus AB: Section I
4.
Let S be the region enclosed by the graphs of y = 2 x and y = 2 x 2 for 0 £ x £ 1.
What is the volume of the solid generated when S is revolved about the line y = 3 ?
 -   ) - ( -   ) ) 
ı ((
 Û (( -   ) - ( -   ) ) 
ı
(a)  Û

 


(b)

 


(

)
(c)  Ú    -    

5.
Û
(d)  Ù
ı

ÊÊ
 ˆ Ê

- Á ÁË
¯
Ë
Ë
Û
(e)  Ù
ı

ÊÊ
Á ÁË  Ë

ˆ ˆ

 ˜¯ ˜¯


ˆ
Ê  -  ˆ ˆ 
Ë
 ˜¯
 ¯ ˜¯
Which of the following statements about the function given by f ( x ) = x 4 - 2 x3 is
true?
(a) The function has no relative extremum.
(b) The graph of the function has one point of inflection and the function has two
relative extrema.
(c) The graph of the function has two points of inflection and the function has one
relative extremum.
(d) The graph of the function has two points of inflection and the function has two
relative extrema.
(e) The graph of the function has two points of inflection and the function has
three relative extrema.
6.
If f ( x ) = sin 2 (3 - x ) , then f ¢(0) =
(a)
(b)
(c)
(d)
(e)
7.
–2 cos 3
–2 sin 3 cos 3
6 cos 3
2 sin 3 cos 3
6 sin 3 cos 3
Which of the following is the solution to the differential equation
(a)
(b)
(c)
where for
for
for
(d)
for
(e)
for
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
19
Sample Questions for Calculus AB: Section I
8.
What is the average rate of change of the function f given by f ( x ) = x 4 - 5 x
on the closed interval [0, 3] ?
(a) 8.5
(b) 8.7
(c) 22
(d) 33
(e) 66
9.
The position of a particle moving along a line is given by
s (t ) = 2t 3 - 24t 2 + 90t + 7 for t ≥ 0. For what values of t is the speed of the
particle increasing?
(a)
(b)
(c)
( d)
(e)
10.
3 < t < 4 only
t > 4 only
t > 5 only
0 < t < 3 and t > 5
3 < t < 4 and t > 5
Ú ( x - 1)
(a)
(b)
(c)
(d)
(e)
x dx =
3
1
x +C
2
x
2 3 2 1 1 2
x
+ x
+C
3
2
1 2
x -x+C
2
2 5 2 2 3 2
x
- x
+C
5
3
1 2
x + 2 x3 2 - x + C
2
x2 - 4
?
xƕ 2 + x - 4 x2
11. What is lim
(a) -2
1
(b) 4
1
(c)
2
( d) 1
(e) The limit does not exist.
20
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Sample Questions for Calculus AB: Section I
y
O
x
12. The figure above shows the graph of y = 5 x - x 2 and the graph of the line
y = 2 x. What is the area of the shaded region?
25
6
9
(b)
2
(c) 9
(a)
(d) 27
2
45
(e)
2
13. If
which of the following is true?
(a)
and (b)
and (c)
and (d)
and (e)
and © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
21
Sample Questions for Calculus AB: Section I
14. Which of the following is a slope field for the differential equation
(a)
(b)
(c)
(d)
dy
x
= ?
dx
y
(e)
22
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Sample Questions for Calculus AB: Section I
Part B Sample Multiple-Choice Questions
A graphing calculator is required for some questions on this part of the exam.
Part B consists of 17 questions. Following are the directions for Section I, Part B, and a
representative set of 10 ­questions.
Directions: Solve each of the following problems, using the available space for scratch
work. After examining the form of the choices, decide which is the best of the choices
given and fill in the corresponding circle on the answer sheet. No credit will be given for
anything written in the exam book. Do not spend too much time on any one problem.
In this exam:
(1) The exact numerical value of the correct answer does not always appear among the
choices given. When this happens, select from among the choices the number that
best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of
all real numbers x for which f (x) is a real number.
(3) The inverse of a trigonometric function f may be indicated using the inverse
function notation f –1 or with the prefix “arc” (e.g., sin–1 x = arcsin x).
15. A particle travels along a straight line with a velocity of v(t ) = 3e(- t 2) sin ( 2t )
meters per second. What is the total distance, in meters, traveled by the particle
during the time interval 0 £ t £ 2 seconds?
( a)
(b)
(c)
(d)
(e)
0.835
1.850
2.055
2.261
7.025
16. A city is built around a circular lake that has a radius of 1 mile. The population
density of the city is f ( r ) people per square mile, where r is the distance from the
center of the lake, in miles. Which of the following expressions gives the number of
people who live within 1 mile of the lake?

(a)  Ú   (  ) 


(b)  Ú  ( +  (  )) 


(c)  Ú  ( +  (  )) 


(d)  Ú   (  ) 


(e)  Ú  ( +  (  )) 

© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
23
Sample Questions for Calculus AB: Section I
y
2
1
O
–1
1
2
3
4
x
17. The graph of a function f is shown above. If lim f ( x ) exists and f is not
xÆb
continuous at b, then b =
(a)
(b)
(c)
(d)
(e)
–1
0
1
2
3
x
1.1
1.2
1.3
1.4
f (x)
4.18
4.38
4.56
4.73
18. Let f be a function such that f ¢¢( x ) < 0 for all x in the closed interval [1, 2].
Selected values of f are shown in the table above. Which of the ­following must be
true about f ¢(1.2) ?
(a)
(b)
(c)
(d)
(e)
f ¢(1.2) < 0
0 < f ¢(1.2) < 1.6
1.6 < f ¢(1.2) < 1.8
1.8 < f ¢(1.2) < 2.0
f ¢(1.2) > 2.0
19. Two particles start at the origin and move along the x-axis. For 0 £ t £ 10, their
respective position functions are given by x1 = sin t and x2 = e -2t - 1. For how
many values of t do the particles have the same velocity?
(a)
(b)
(c)
(d)
(e)
24
None
One
Two
Three
Four
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Sample Questions for Calculus AB: Section I
y
(0, 2)
(−1, 0)
0
(2, 0)
x
(−2, −2)
Graph of f
20. The graph of the function f shown above consists of two line segments. If g is the
function defined by g ( x ) =
(a)
(b)
(c)
(d)
(e)
–2
–1
0
1
2
x
Ú0
f (t ) dt , then g ( -1) =
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
25
Sample Questions for Calculus AB: Section I
21. The graphs of five functions are shown below. Which function has a nonzero
average value over the closed interval [ -   ] 
(a)
(b)
(c)
(d)
(e)
and
22. A differentiable function f has the property that
using the local linear approximation for f at
is the estimate for
(a)
(b)
(c)
(d)
(e)
26
What
2.2
2.8
3.4
3.8
4.6
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Sample Questions for Calculus AB: Section I
23. Oil is leaking from a tanker at the rate of R(t ) = 2,000e - 0.2t gallons per hour,
where t is measured in hours. How much oil leaks out of the tanker from time
t = 0 to t = 10 ?
(a)
(b)
(c)
( d)
(e)
54 gallons
271 gallons
865 gallons
8,647 gallons
14,778 gallons
Ê  ˆ
24. If  ¢(  ) =  Á
and f (0) = 1, then f ( 2) =
Ë  ˜¯
(a) –1.819
(b) –0.843
(c) –0.819
(d) 0.157
(e) 1.157
Answers to Calculus AB Multiple-Choice Questions
Part A
1. a
2. b
3. c
4. a
5. c
6. b
†7. c
8. c
9. e
10. d
11. b
12. b
13. e
14. e
Part B
15.* d
16. d
17. b
18. d
19.* d
20. b
21. e
22. a
23.* d
24.* e
*Indicates a graphing calculator-active question.
†For resources on differential equations, see the Course Home Pages for Calculus AB and Calculus BC at
AP Central.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
27
SAMPLE QUESTIONS
AP Calculus AB and
AP Calculus BC Exam
®
®
Originally published in the Fall 2014
AP® Calculus AB and AP® Calculus
BC Curriculum Framework
Sample Questions
AP Calculus AB/BC Exam
AP Calculus AB Sample Exam Questions
Multiple Choice: Section I, Part A
A calculator may not be used on questions on this part of the exam.
1.
is
(A)
(B)
(C) 1
(D) nonexistent
Learning Objectives
Essential Knowledge
LO 1.1C: Determine limits of
functions.
EK 1.1C3: Limits of the indeterminate forms
LO 2.1C: Calculate derivatives.
EK 2.1C2: Specific rules can be used to calculate
derivatives for classes of functions, including
polynomial, rational, power, exponential,
logarithmic, trigonometric, and inverse
trigonometric.
© 2016 The College Board
and
may be evaluated using L’Hospital’s Rule.
Mathematical Practices
for AP Calculus
MPAC 1: Reasoning with
definitions and theorems
MPAC 3: Implementing
algebraic/computational
processes
Return to
Table of Contents
1
Sample Questions
2.
AP Calculus AB/BC Exam
is
(A) 1
(B) 3
(C) 9
(D) nonexistent
Learning Objectives
Essential Knowledge
LO 1.1C: Determine limits of
functions.
EK 1.1C2: The limit of a function may be found
by using algebraic manipulation, alternate forms
of trigonometric functions, or the squeeze
theorem.
LO 1.1A(b): Interpret limits
expressed symbolically.
© 2016 The College Board
EK 1.1A2: The concept of a limit can be extended
to include one-sided limits, limits at infinity, and
infinite limits.
Mathematical Practices
for AP Calculus
MPAC 3: Implementing
algebraic/computational
processes
MPAC 2: Connecting
concepts
Return to
Table of Contents
2
Sample Questions
AP Calculus AB/BC Exam
3. The graph of the piecewise-defined function f is shown in the figure above. The graph has a
vertical tangent line at
and horizontal tangent lines at
and
What are all
values of x,
at which f is continuous but not differentiable?
(A)
(B)
and
(C)
and
(D)
and
Learning Objectives
Essential Knowledge
Mathematical Practices
for AP Calculus
LO 2.2B: Recognize the
connection between
differentiability and continuity.
EK 2.2B1: A continuous function may fail to
be differentiable at a point in its domain.
MPAC 4: Connecting
multiple representations
LO 1.2A: Analyze functions
for intervals of continuity
or points of discontinuity.
EK 1.2A3: Types of discontinuities include
removable discontinuities, jump discontinuities,
and discontinuities due to vertical asymptotes.
© 2016 The College Board
MPAC 2: Connecting
concepts
Return to
Table of Contents
3
Sample Questions
AP Calculus AB/BC Exam
4. An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical
cubic meters per hour. At
shape. The volume of the sphere is decreasing at a constant rate of
what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment
and the
when the radius is 5 meters? (Note: For a sphere of radius r, the surface area is
)
volume is
(A)
(B)
(C)
(D)
Learning Objectives
Essential Knowledge
LO 2.3C: Solve problems
involving related rates,
optimization, rectilinear motion,
(BC) and planar motion.
EK 2.3C2: The derivative can be used to solve
related rates problems, that is, finding a rate at
which one quantity is changing by relating it to
other quantities whose rates of change are
known.
LO 2.1C: Calculate derivatives.
EK 2.1C5: The chain rule is the basis for implicit
differentiation.
© 2016 The College Board
Mathematical Practices
for AP Calculus
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/computational
processes
Return to
Table of Contents
4
Sample Questions
AP Calculus AB/BC Exam
5. Shown above is a slope field for which of the following differential equations?
(A)
(B)
(C)
(D)
Learning Objective
Essential Knowledge
LO 2.3F: Estimate solutions
to differential equations.
EK 2.3F1: Slope fields provide visual clues to the
behavior of solutions to first order differential
equations.
© 2016 The College Board
Mathematical Practices
for AP Calculus
MPAC 4: Connecting
multiple representations
MPAC 2: Connecting
concepts
Return to
Table of Contents
5
Sample Questions
AP Calculus AB/BC Exam
6. Let f be the piecewise-linear function defined above. Which of the following statements are
true?
I.
II.
III.
(A) None
(B) II only
(C) I and II only
(D) I, II, and III
Learning Objectives
Essential Knowledge
Mathematical Practices
for AP Calculus
LO 2.1A: Identify the derivative
of a function as the limit of a
difference quotient.
EK 2.1A2: The instantaneous rate of change of a
function at a point can be expressed by
MPAC 2: Connecting
concepts
MPAC 5: Building
notational fluency
or
provided that the limit exists. These are common
forms of the definition of the derivative and are
denoted
LO 1.1A(b): Interpret limits
expressed symbolically.
© 2016 The College Board
EK 1.1A2: The concept of a limit can be extended
to include one-sided limits, limits at infinity, and
infinite limits.
Return to
Table of Contents
6
Sample Questions
7. If
AP Calculus AB/BC Exam
for
then
(A)
(B)
(C)
(D)
Mathematical Practices
for AP Calculus
Learning Objectives
Essential Knowledge
LO 3.3A: Analyze functions
defined by an integral.
EK 3.3A2: If f is a continuous function on the
interval
then
where x
is between a and b.
LO 2.1C: Calculate derivatives.
© 2016 The College Board
EK 2.1C4: The chain rule provides a way to
differentiate composite functions.
MPAC 1: Reasoning with
definitions and theorems
MPAC 3: Implementing
algebraic/computational
processes
Return to
Table of Contents
7
Sample Questions
AP Calculus AB/BC Exam
8. Which of the following limits is equal to
(A)
(B)
(C)
(D)
Learning Objective
Essential Knowledge
LO 3.2A(a): Interpret the
definite integral as the
limit of a Riemann sum.
EK 3.2A3: The information in a definite integral
can be translated into the limit of a related
Riemann sum, and the limit of a Riemann
sum can be written as a definite integral.
© 2016 The College Board
Mathematical Practices
for AP Calculus
MPAC 1: Reasoning with
definitions and theorems
MPAC 5: Building
notational fluency
Return to
Table of Contents
8
Sample Questions
AP Calculus AB/BC Exam
9. The function f is continuous for
The graph of f shown above consists of five line
segments. What is the average value of f on the interval
(A)
(B)
(C)
(D)
Learning Objectives
Essential Knowledge
LO 3.4B: Apply definite
EK 3.4B1: The average value of a function f over
integrals to problems involving
an interval
is
the average value of a function.
LO 3.2C: Calculate a definite
integral using areas and
properties of definite integrals.
© 2016 The College Board
EK 3.2C1: In some cases, a definite integral can
be evaluated by using geometry and the
connection between the definite integral and area.
Mathematical Practices
for AP Calculus
MPAC 1: Reasoning with
definitions and theorems
MPAC 4: Connecting
multiple representations
Return to
Table of Contents
9
Sample Questions
10. Let
AP Calculus AB/BC Exam
be a solution to the differential equation
where k is a constant. Values
of f for selected values of t are given in the table above. Which of the following is an expression
for
(A)
(B)
(C)
(D)
Learning Objective
Essential Knowledge
Mathematical Practices
for AP Calculus
LO 3.5B: Interpret, create and
solve differential equations
from problems in context.
EK 3.5B1: The model for exponential growth
and decay that arises from the statement “The
rate of change of a quantity is proportional
MPAC 3: Implementing
algebraic/computational
processes
to the size of the quantity” is
MPAC 4: Connecting
multiple representations
© 2016 The College Board
Return to
Table of Contents
10
Sample Questions
AP Calculus AB/BC Exam
Multiple Choice: Section I, Part B
A graphing calculator is required for some questions on this part of the exam.
11. The graph of
the derivative of the function f, is shown above. Which of the following could
be the graph of f ?
(A)
(B)
(C)
(D)
© 2016 The College Board
Return to
Table of Contents
11
Sample Questions
AP Calculus AB/BC Exam
Learning Objectives
Essential Knowledge
LO 2.2A: Use derivatives to
analyze properties of
a function.
EK 2.2A3: Key features of the graphs of
are related to one another.
LO 2.2B: Recognize the
connection between
differentiability and continuity.
EK 2.2B2: If a function is differentiable at a
point, then it is continuous at that point.
© 2016 The College Board
Mathematical Practices
for AP Calculus
and
MPAC 4: Connecting
multiple representations
MPAC 2: Connecting
concepts
Return to
Table of Contents
12
Sample Questions
AP Calculus AB/BC Exam
12. The derivative of a function f is given by
intervals is f decreasing?
(A)
and
(B)
and
(C)
and
(D)
and
for
Learning Objective
Essential Knowledge
LO 2.2A: Use derivatives
to analyze properties of
a function.
EK 2.2A1: First and second derivatives of a
function can provide information about the
function and its graph including intervals of
increase or decrease, local (relative) and global
(absolute) extrema, intervals of upward or
downward concavity, and points of inflection.
© 2016 The College Board
On what
Mathematical Practices
for AP Calculus
MPAC 4: Connecting
multiple representations
MPAC 2: Connecting
concepts
Return to
Table of Contents
13
Sample Questions
AP Calculus AB/BC Exam
13. The temperature of a room, in degrees Fahrenheit, is modeled by , a differentiable function
of the number of minutes after the thermostat is adjusted. Of the following, which is the best
interpretation of
(A) The temperature of the room is 2 degrees Fahrenheit, 5 minutes after the thermostat is
adjusted.
(B) The temperature of the room increases by 2 degrees Fahrenheit during the first 5 minutes
after the thermostat is adjusted.
(C) The temperature of the room is increasing at a constant rate of
minute.
degree Fahrenheit per
(D) The temperature of the room is increasing at a rate of 2 degrees Fahrenheit per minute,
5 minutes after the thermostat is adjusted.
Mathematical Practices
for AP Calculus
Learning Objectives
Essential Knowledge
LO 2.3A: Interpret the
meaning of a derivative
within a problem.
EK 2.3A1: The unit for
divided by the unit for x.
LO 2.3D: Solve problems
involving rates of change
in applied contexts.
EK 2.3D1: The derivative can be used to express
information about rates of change in
applied contexts.
© 2016 The College Board
is the unit for f
MPAC 2: Connecting
concepts
MPAC 5: Building
notational fluency
Return to
Table of Contents
14
Sample Questions
AP Calculus AB/BC Exam
14. A function f is continuous on the closed interval
with
and
Which
of the following additional conditions guarantees that there is a number c in the open interval
such that
(A) No additional conditions are necessary.
(B) f has a relative extremum on the open interval
(C) f is differentiable on the open interval
(D)
exists.
Learning Objective
Essential Knowledge
LO 2.4A: Apply the Mean
Value Theorem to describe
the behavior of a function
over an interval.
EK 2.4A1: If a function f is continuous over
the interval
and differentiable over
the interval
the Mean Value Theorem
guarantees a point within that open interval
where the instantaneous rate of change equals
the average rate of change over the interval.
© 2016 The College Board
Mathematical Practices
for AP Calculus
MPAC 1: Reasoning with
definitions and theorems
MPAC 5: Building
notational fluency
Return to
Table of Contents
15
Sample Questions
AP Calculus AB/BC Exam
15. A rain barrel collects water off the roof of a house during three hours of heavy rainfall. The
height of the water in the barrel increases at the rate of
feet per hour, where t is
the time in hours since the rain began. At time
hour, the height of the water is
foot.
What is the height of the water in the barrel at time
hours?
(A)
(B)
(C)
(D)
Learning Objectives
Essential Knowledge
LO 3.4E: Use the definite
integral to solve problems
in various contexts.
EK 3.4E1: The definite integral can be used
to express information about accumulation
and net change in many applied contexts.
LO 3.3B(b): Evaluate
definite integrals.
EK 3.3B2: If is continuous on the
interval
and is an antiderivative
Mathematical Practices
for AP Calculus
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/computational
processes
of , then
© 2016 The College Board
Return to
Table of Contents
16
Sample Questions
AP Calculus AB/BC Exam
16. A race car is traveling on a straight track at a velocity of 80 meters per second when the brakes
seconds. From time
to the moment the race car stops, the
are applied at time
acceleration of the race car is given by
meters per second per second. During
this time period, how far does the race car travel?
(A)
(B)
(C)
(D)
Learning Objectives
Essential Knowledge
LO 3.4C: Apply definite
integrals to problems
involving motion.
EK 3.4C1: For a particle in rectilinear motion
over an interval of time, the definite integral of
velocity represents the particle’s displacement
over the interval of time, and the definite
integral of speed represents the particle’s total
distance traveled over the interval of time.
LO 3.1A: Recognize
antiderivatives of
basic functions.
EK 3.1A2: Differentiation rules provide the
foundation for finding antiderivatives.
© 2016 The College Board
Mathematical Practices
for AP Calculus
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/computational
processes
Return to
Table of Contents
17
Sample Questions
AP Calculus AB/BC Exam
Answers and Rubrics (AB)
Answers to Multiple-Choice Questions
1.
B
2.
B
3.
C
4.
A
5.
A
6.
B
7.
D
8.
D
9.
B
10.
A
11.
A
12.
A
13.
D
14.
C
15.
D
16.
B
© 2016 The College Board
Return to
Table of Contents
22
AP Calculus AB and
AP Calculus BC
®
®
Course and Exam Description
Effective Fall 2016
New York, NY
Sample Exam Questions
AP Calculus AB Sample Exam Questions
Multiple Choice: Section I, Part A
A calculator may not be used on questions on this part of the exam.
is
AP CALCULUS AB SAMPLE EXAM QUESTIONS
1. The graphs of the functions f and g are shown above. The value of
(A) 1
(B) 2
(C) 3
(D) nonexistent
Learning Objective
Essential Knowledge
LO 1.1C: Determine
limits of functions.
EK 1.1C1: Limits of sums, differences,
products, quotients, and composite
functions can be found using the basic
theorems of limits and algebraic rules.
AP Calculus AB/BC Course and Exam Description
Mathematical
Practice for
AP Calculus
MPAC 4: Connecting
multiple representations
MPAC 2: Connecting
concepts
Return to
Table of Contents
© 2015 The College Board
47
Sample Exam Questions
2.
(A) 6
(B) 2
(C) 1
(D) 0
Learning Objective
Essential Knowledge
LO 1.1C: Determine
limits of functions.
EK 1.1C3: Limits of the indeterminate forms
and
may be evaluated using L’Hospital’s Rule.
Mathematical
Practice for
AP Calculus
MPAC 3: Implementing
algebraic/computational
processes
AP CALCULUS AB SAMPLE EXAM QUESTIONS
MPAC 5: Building
notational fluency
48
AP Calculus AB/BC Course and Exam Description
Return to
Table of Contents
© 2015 The College Board
Sample Exam Questions
3. If
then
(A)
(B)
(C)
(D)
Learning Objective
Essential Knowledge
LO 2.1C: Calculate
derivatives.
EK 2.1C4: The chain rule provides a way
to differentiate composite functions.
Mathematical
Practice for
AP Calculus
MPAC 3: Implementing
algebraic/computational
processes
AP Calculus AB/BC Course and Exam Description
Return to
Table of Contents
© 2015 The College Board
AP CALCULUS AB SAMPLE EXAM QUESTIONS
MPAC 5: Building
notational fluency
49
Sample Exam Questions
4. Three graphs labeled I, II, and III are shown above. One is the graph of f, one is the graph of
and one is the graph of
Which of the following correctly identifies each of the three
graphs?
AP CALCULUS AB SAMPLE EXAM QUESTIONS
f
50
(A)
I
II
III
(B)
II
I
III
(C)
II
III
I
(D)
III
I
II
Learning Objective
Essential Knowledge
Mathematical
Practice for
AP Calculus
LO 2.2A: Use
derivatives to analyze
properties of a function.
EK 2.2A3: Key features of the graphs of f,
and
are related to one another.
MPAC 2: Connecting
concepts
AP Calculus AB/BC Course and Exam Description
MPAC 4: Connecting
multiple representations
Return to
Table of Contents
© 2015 The College Board
Sample Exam Questions
5. The local linear approximation to the function g at
is
What is the value of
(A) 4
(B) 5
(C) 6
(D) 7
Learning Objective
Essential Knowledge
LO 2.3B: Solve
problems involving the
slope of a tangent line.
EK 2.3B2: The tangent line is the graph
of a locally linear approximation of the
function near the point of tangency.
Mathematical
Practice for
AP Calculus
MPAC 2: Connecting
concepts
MPAC 1: Reasoning
with definitions
and theorems
AP CALCULUS AB SAMPLE EXAM QUESTIONS
AP Calculus AB/BC Course and Exam Description
Return to
Table of Contents
© 2015 The College Board
51
Sample Exam Questions
6. For time
the velocity of a particle moving along the x-axis is given by
At what values of t is the acceleration of the particle equal to 0?
(A) 2 only
(B) 4 only
(C) 2 and 4
(D) 2 and 5
Essential Knowledge
LO 2.3C: Solve
problems involving
related rates,
optimization, rectilinear
motion, (BC) and
planar motion.
EK 2.3C1: The derivative can be used to
solve rectilinear motion problems involving
position, speed, velocity, and acceleration.
LO 2.1C: Calculate
derivatives.
EK 2.1C3: Sums, differences, products,
and quotients of functions can be
differentiated using derivative rules.
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/computational
processes
AP CALCULUS AB SAMPLE EXAM QUESTIONS
Learning Objective
Mathematical
Practice for
AP Calculus
52
AP Calculus AB/BC Course and Exam Description
Return to
Table of Contents
© 2015 The College Board
Sample Exam Questions
7. The cost, in dollars, to shred the confidential documents of a company is modeled by C, a
differentiable function of the weight of documents in pounds. Of the following, which is the
best interpretation of Cʹ(500) = 80?
(A) The cost to shred 500 pounds of documents is $80.
(B) The average cost to shred documents is
dollar per pound.
(C) Increasing the weight of documents by 500 pounds will increase the cost to shred the
documents by approximately $80.
(D) The cost to shred documents is increasing at a rate of $80 per pound when the weight of
the documents is 500 pounds.
Learning Objective
Essential Knowledge
LO 2.3D: Solve
problems involving
rates of change in
applied contexts.
EK 2.3D1: The derivative can be used
to express information about rates
of change in applied contexts.
Mathematical
Practice for
AP Calculus
MPAC 2: Connecting
concepts
MPAC 5: Building
notational fluency
AP CALCULUS AB SAMPLE EXAM QUESTIONS
AP Calculus AB/BC Course and Exam Description
Return to
Table of Contents
© 2015 The College Board
53
Sample Exam Questions
8. Which of the following integral expressions is equal to
(A)
(B)
(C)
(D)
Mathematical
Practice for
AP Calculus
Learning Objective
Essential Knowledge
LO 3.2A(b): Express the
limit of a Riemann sum
in integral notation.
EK 3.2A2: The definite integral of a continuous
denoted by
function f over the interval
is the limit of Riemann sums as the
widths of the subintervals approach 0. That is,
AP CALCULUS AB SAMPLE EXAM QUESTIONS
where
MPAC 1: Reasoning
with definitions
and theorems
MPAC 5: Building
notational fluency
is a
value in the ith subinterval,
is the width
of the ith subinterval, n is the number of
subintervals, and
is the width of the
largest subinterval. Another form of the definition
is
and
54
where
is a value in the ith subinterval.
AP Calculus AB/BC Course and Exam Description
Return to
Table of Contents
© 2015 The College Board
Sample Exam Questions
9.
If f is the function defined above, then
is
(A)
(B)
(C)
(D) undefined
Essential Knowledge
LO 3.2C: Calculate a
definite integral using
areas and properties
of definite integrals.
EK 3.2C3: The definition of the definite
integral may be extended to functions with
removable or jump discontinuities.
AP Calculus AB/BC Course and Exam Description
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/computational
processes
Return to
Table of Contents
© 2015 The College Board
AP CALCULUS AB SAMPLE EXAM QUESTIONS
Learning Objective
Mathematical
Practice for
AP Calculus
55
Sample Exam Questions
10.
(A)
(B)
(C)
(D)
Essential Knowledge
LO 3.3B(a): Calculate
antiderivatives.
EK 3.3B5: Techniques for finding antiderivatives
include algebraic manipulation such as long
division and completing the square, substitution
of variables, (BC) integration by parts, and
nonrepeating linear partial fractions.
MPAC 3: Implementing
algebraic/computational
processes
MPAC 5: Building
notational fluency
AP CALCULUS AB SAMPLE EXAM QUESTIONS
Learning Objective
Mathematical
Practice for
AP Calculus
56
AP Calculus AB/BC Course and Exam Description
Return to
Table of Contents
© 2015 The College Board
Sample Exam Questions
11. At time t, a population of bacteria grows at the rate of
grams per day, where t is
measured in days. By how many grams has the population grown from time
days to
days?
(A)
(B)
(C)
(D)
Learning Objective
Essential Knowledge
LO 3.4A: Interpret
the meaning of a
definite integral
within a problem.
EK 3.4A2: The definite integral of the rate of
change of a quantity over an interval gives the
net change of that quantity over that interval.
Mathematical
Practice for
AP Calculus
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/computational
processes
AP CALCULUS AB SAMPLE EXAM QUESTIONS
AP Calculus AB/BC Course and Exam Description
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Sample Exam Questions
12. The right triangle shown in the figure above represents the boundary of a town that is bordered
by a highway. The population density of the town at a distance of x miles from the highway is
where
is measured in thousands of people per square mile.
modeled by
According to the model, which of the following expressions gives the total population, in
thousands, of the town?
AP CALCULUS AB SAMPLE EXAM QUESTIONS
(A)
(B)
(C)
(D)
58
Learning Objective
Essential Knowledge
Mathematical
Practice for
AP Calculus
LO 3.4A: Interpret
the meaning of a
definite integral
within a problem.
EK 3.4A3: The limit of an approximating Riemann
sum can be interpreted as a definite integral.
MPAC 2: Connecting
concepts
AP Calculus AB/BC Course and Exam Description
MPAC 5: Building
notational fluency
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Sample Exam Questions
13. Which of the following is the solution to the differential equation
p
initial condition yÊ ˆ = -1 ?
Ë 4¯
with the
(A)
(B)
(C)
(D)
Learning Objective
Essential Knowledge
LO 3.5A: Analyze
differential equations
to obtain general and
specific solutions.
EK 3.5A2: Some differential equations can
be solved by separation of variables.
Mathematical
Practice for
AP Calculus
MPAC 3: Implementing
algebraic/computational
processes
MPAC 2: Connecting
concepts
AP CALCULUS AB SAMPLE EXAM QUESTIONS
AP Calculus AB/BC Course and Exam Description
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Sample Exam Questions
14. The graph of the function f is shown in the figure above. For how many values of x in the open
interval
is f discontinuous?
(A) one
AP CALCULUS AB SAMPLE EXAM QUESTIONS
(B) two
(C) three
(D) four
60
Learning Objective
Essential Knowledge
LO 1.2A: Analyze
functions for intervals
of continuity or points
of discontinuity.
EK 1.2A3: Types of discontinuities include
removable discontinuities, jump discontinuities,
and discontinuities due to vertical asymptotes.
AP Calculus AB/BC Course and Exam Description
Mathematical
Practice for
AP Calculus
MPAC 2: Connecting
concepts
MPAC 4: Connecting
multiple representations
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Sample Exam Questions
15.
x
0
1
5
2
2
The table above gives selected values of a differentiable and decreasing function f and its
derivative. If g is the inverse function of f, what is the value of
(A)
(B)
(C)
(D) 5
Essential Knowledge
LO 2.1C: Calculate
derivatives.
EK 2.1C6: The chain rule can be used to find
the derivative of an inverse function, provided
the derivative of that function exists.
MPAC 3: Implementing
algebraic/computational
processes
AP CALCULUS AB SAMPLE EXAM QUESTIONS
Learning Objective
Mathematical
Practice for
AP Calculus
MPAC 4: Connecting
multiple representations
AP Calculus AB/BC Course and Exam Description
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Sample Exam Questions
Multiple Choice: Section I, Part B
A graphing calculator is required for some questions on this part of the exam.
16. The derivative of the function f is given by
At what values of x does f
have a relative minimum on the interval
(A)
and
(B)
(C)
AP CALCULUS AB SAMPLE EXAM QUESTIONS
(D)
62
Learning Objective
Essential Knowledge
LO 2.2A: Use
derivatives to analyze
properties of a function.
EK 2.2A1: First and second derivatives of a
function can provide information about the
function and its graph including intervals of
increase or decrease, local (relative) and global
(absolute) extrema, intervals of upward or
downward concavity, and points of inflection.
AP Calculus AB/BC Course and Exam Description
Mathematical
Practice for
AP Calculus
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/computational
processes
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Sample Exam Questions
17. The second derivative of a function g is given by
on what open intervals is the graph of g concave up?
(A)
(B)
only
only
(C)
(D)
For
only
and
Learning Objective
Essential Knowledge
LO 2.2A: Use
derivatives to analyze
properties of a function.
EK 2.2A1: First and second derivatives of a
function can provide information about the
function and its graph including intervals of
increase or decrease, local (relative) and global
(absolute) extrema, intervals of upward or
downward concavity, and points of inflection.
Mathematical
Practice for
AP Calculus
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/computational
processes
AP CALCULUS AB SAMPLE EXAM QUESTIONS
AP Calculus AB/BC Course and Exam Description
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Sample Exam Questions
18. The temperature, in degrees Fahrenheit
H given by
of water in a pond is modeled by the function
where t is the number of days since January 1
What is the instantaneous rate of change of the temperature of the water at time
days?
(A)
(B)
(C)
(D)
Essential Knowledge
LO 2.3D: Solve
problems involving
rates of change in
applied contexts.
EK 2.3D1: The derivative can be used
to express information about rates
of change in applied contexts.
AP CALCULUS AB SAMPLE EXAM QUESTIONS
Learning Objective
64
AP Calculus AB/BC Course and Exam Description
Mathematical
Practice for
AP Calculus
MPAC 2: Connecting
concepts
MPAC 3: Implementing
algebraic/
computational
processes
Return to
Table of Contents
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Sample Exam Questions
19.
x
0
2
4
8
3
4
9
13
0
1
1
2
The table above gives values of a differentiable function f and its derivative at selected values of
x. If h is the function given by
which of the following statements must be true?
(I) h is increasing on
(II) There exists c, where
such that
(III) There exists c, where
such that
(A) II only
(B) I and III only
(C) II and III only
Learning Objective
Essential Knowledge
LO 2.4A: Apply the
Mean Value Theorem
to describe the
behavior of a function
over an interval.
EK 2.4A1: If a function f is continuous over
and differentiable over
the interval
the interval
the Mean Value Theorem
guarantees a point within that open interval
where the instantaneous rate of change equals
the average rate of change over the interval.
LO 1.2B: Determine
the applicability of
important calculus
theorems using
continuity.
EK 1.2B1: Continuity is an essential condition
for theorems such as the Intermediate
Value Theorem, the Extreme Value
Theorem, and the Mean Value Theorem.
AP Calculus AB/BC Course and Exam Description
AP CALCULUS AB SAMPLE EXAM QUESTIONS
(D) I, II, and III
Mathematical
Practice for
AP Calculus
MPAC 1: Reasoning
with definitions
and theorems
MPAC 4: Connecting
multiple representations
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Sample Exam Questions
20. Let h be the function defined by
If g is an antiderivative of h and
what is the value of
(A)
(B)
(C)
(D)
Learning Objective
Essential Knowledge
LO 3.3B(b): Evaluate
definite integrals.
EK 3.3B2: If f is continuous on the
interval
and is an antiderivative
MPAC 1: Reasoning
with definitions
and theorems
MPAC 2: Connecting
concepts
AP CALCULUS AB SAMPLE EXAM QUESTIONS
of f, then
Mathematical
Practice for
AP Calculus
66
AP Calculus AB/BC Course and Exam Description
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Sample Exam Questions
Answers and Rubrics (AB)
Answers to Multiple-Choice Questions
1
C
2
B
3
B
4
C
5
D
6
C
7
D
8
A
9
B
10
A
11
C
12
D
13
B
14
C
15
A
16
C
17
B
18
B
19
C
20
D
AP Calculus AB/BC Course and Exam Description
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