Practice Test 2: AP

Calculus AB
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practice test 2
SECTION I, PART A

55 Minutes • 28 Questions

A CALCULATOR MAY NOT BE USED FOR THIS PART OF THE EXAMINATION.

Directions: Solve each of the following problems, using the available

space for scratchwork. After examining the form of the choices, decide
which is the best of the choices given and fill in the corresponding oval on
the answer sheet. No credit will be given for anything written in the test
book. Do not spend too much time on any one problem.

In this test: 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. What is the instantaneous rate 2.

of change for f(x) 5 100
cars per minute

x3 1 3x2 1 3x 1 1
at x 5 2?
x11
50
(A) 227
(B) 26
(C) 6 6 12 18 24
(D) 9 time of day
(E) 27 The rate at which cars cross a
bridge in cars per minute is
given by the preceding graph. A
good approximation for the total
number of cars that crossed the
bridge by 12:00 noon is
(A) 50.
(B) 825.
(C) 1,200.
(D) 45,000.
(E) 49,500.

537
 538 PART III: Four Practice Tests
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5⎛ 3x ⎞
3.
∫1 ⎜⎝ x 3 ⎟⎠ dx = 6. What is the slope of the curve de-
fined by 3x2 1 2xy 1 6y2 2 3x 2 8y 5
18 0 at the point (1,1)?
(A) 2
5
5
72 (A) 2
(B) 2 6
25
1
124 (B) 2
(C) 2
125
126 (C) 0
(D)
125
1
12 (D)
(E) 2
5
(E) It is undefined.
4.
x 0 1 2

∫( )(x
f (x) 3 4 9
)
1
2
7. x + 3x − 8 dx =
0

The function f is continuous on the (A) 2 4

closed interval [0,2] and has values
as defined by the table above. Which −404
of the following statements must be (B)
105
true?
−8
(A) f must be increasing on [0,2]. (C)
5
(B) f must be concave up on (0,2).

(C) f ′
3
2 SD SD
. f′
1
2
. (D)
−28
105
(D) The average rate of increase of
f over [0,2] is 3. (E) 1
(E) f has no points of inflection on
[0,2].
8. The radius of a sphere is increasing
at a rate of 2 inches per minute. At
π3 ⎛ tan xe sec x ⎞
5. ∫0
⎜⎝ cos x ⎟⎠ dx = what rate (in cubic inches per
minute) is the volume increasing
when the surface area of the sphere
(A) e2
is 9p square inches?
(B) e2 2 1 (A) 2
(C) =e (B) 2p
(C) 9p
(D) =e2 e (D) 18
2
(E) e 2 e (E) 18p

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 Practice Test 2: AP Calculus AB 539
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practice test
9. 12. Let f be defined as

y=x+6

y = x3 What is the average value of f over

[24,4]?
2
(A)
3
8
(B)
3
The area of the shaded region in the 10
(C)
preceding diagram is equivalent to 3
16
∫ ( x + 6 − x ) dx .
8
(A) 3 (D)
0 3
80
(E)
∫ (x )
8
(B) 3
− x − 6 dx . 3
0

13.
∫( )
2
3
(C) x + 6 − x dx .
0

f
∫( )
2
(D) x 3 − x − 6 dx .
0

∫ ( x + 6 + x ) dx .
2
3
(E)
0

f is a twice differentiable function

10. What is the average rate of change of
with a horizontal tangent line at x 5
f(x) 5 x3 2 3x2 1 x 2 1 over [21,4]?
1, as shown in the diagram above.
13 Which of these statements must be
(A)
5 true?
(B) 3 (A) f ′ (1) < f (1) < f ′′ (1)
(C) 5 (B) f (1) < f ′′ (1) < f ′ (1)
(D) 10
(C) f (1) < f ′ (1) < f ′′ (1)
(E) 25
(D) f ′′ (1) < f (1) < f ′ (1)
11. If the graph of the second derivative (E) f ′′ (1) < f ′ (1) < f (1)
of some function, f, is a line of slope
6, then f could be which type of func-
tion?
(A) constant
(B) linear
(C) quadratic
(D) cubic
(E) quartic

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 540 PART III: Four Practice Tests
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14. Let f be a continuous function on [2 17.
4, 12]. If f(2 4) 5 2 2 and f(12) 5 6, x 1 2 3 4
then the mean value theorem guar- f 3 0 1 2
antees that 2 3 4 1
′ −1 −1 0 2
(A) f(4) 5 2
g x −2 2 1 −1
1
(B) f′(4) 5
2
1 Let f, g, and their derivatives be de-
(C) f′(c) 5 for at least one c fined by the table above. If h(x) 5
2
f(g(x)), then for what value, c, is h(c)
between 2 4 and 12
5 h′(c)?
(D) f(c) 5 0 for at least one c (A) 1
between 2 4 and 12 (B) 2
(C) 3
(E) f(4) 5 0
(D) 4
(E) None of the above
⎛ d ⎞ 2 x et dt =
( )
2

⎝ dx ⎠ ∫3
15.
18. Let f be a differentiable function over
2
[0,10] such that f(0) 5 0 and f(10) 5
(A) e2x 3. If there are exactly two solutions
2 to f(x) 5 4 over (0,10), then which of
(B) 4xe2x
these statements must be true?
2
(C) e2x 2 e3 (A) f ′(c) 5 0 for some c on (0,10).
(D) 4xe2x 2 e3
2
(B) f has a local maximum at x 5
5.
(E) ex (C) f ′′(c) 5 0 for some c on (0,10).
(D) 0 is the absolute minimum of f.
16. Let f(x) 5 ex. If the rate of change of f (E) f is strictly monotonic.
at x 5 c is e3 times its rate of change
at x 5 2, then c 5 19. The normal line to the curve
(A) 1 y = 8 − x 2 at the point (2,2) has
(B) 2 slope
(C) 3 (A) 22
(D) 4 1
(E) 5 (B) 2
2
1
(C)
2
(D) 1

(E) 2

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 Practice Test 2: AP Calculus AB 541
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practice test
20. What are all the values for k such 24. Below is the slope field graph of
k dy
∫−2 x dx = 0 ?
3
that some differential equation = f′~x!.
dx
(A) 0 (Note: Each dot on the axes marks
(B) 2 one unit.)
(C) 22 and 2 Which of the following equations is
the easiest possible differential
(D) 22, 0, and 2
equation for the characteristics
(E) 0 and 2 shown in the graph?
21. If the rate of change of y is directly
proportional to y, then it’s possible
that
(A) y = 3 te 2 3

(B) y 5 5e1.5t

(C) y = 23 t 2

(D) y = ln ( t)
3
2

(E) y = t 3 2 (A) x2(y 1 1)

(B) xy + x 2 y 2 1
22. The graph of y 5 3x3 2 2x2 1 6x 2 2
is decreasing for which interval(s)? (C) xy 1 y

(A) ⎛ −∞ , ⎞
2 x21
⎝ (D)
9⎠ y11

(B) ⎛ , ∞ ⎞
2 (E) xy 1 3xy 2 1
⎝9 ⎠

(C) ⎡ 0 , ⎤
2
⎢⎣ 9 ⎥⎦

(D) ( 2 `, `)

(E) None of the above

23. Determine the value for c on [2,5]

that satisfies the mean value theo-
x2 − 3
rem for f(x) 5 .
x −1
(A) 21
(B) 2
(C) 3
(D) 4
(E) 5

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 542 PART III: Four Practice Tests
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25. 27. The water level in a cylindrical bar-
f rel is falling at a rate of one inch per
minute. If the radius of the barrel is
g ten inches, what is the rate that wa-
ter is leaving the barrel (in cubic
inches per minute) when the volume
is 500p cubic inches?

a b
(A) 1
(B) p
(C) 100p
The area of the shaded region in the (D) 200p
preceding diagram is (E) 500p

(A) ∫ ( f ( x ) − g ( x )) dx
a
b
28. If f(x) 5 arctan(x2), then f ′ ( 3) =
1
∫ ( f ( x ) − g ( x )) dx
(B)
a (A)
5
b

1
(B)
∫ ( g ( x ) + f ( x )) dx
b
(C) 4
a

(C)
3
∫ ( g ( x ) − f ( x )) dx
a
(D) 4
b

(D)
3
∫ ( g ( x ) + f ( x )) dx
a
(E) 5
b

(E)
2 3
26. The function f is continuous on the 5
closed interval [0,2]. It is given that
f(0)5 2 1 and f(2) 5 2. If f′(x) . 0 for
all x on [0,2] and f′′(x) , 0 for all x on
(0,2), then f(1) could be
(A) 0
1
(B)
2
(C) 1

(D) 2
5
(E)
2

END OF SECTION I, PART A. IF YOU HAVE ANY TIME LEFT, GO OVER

STOP YOUR WORK IN THIS PART ONLY. DO NOT WORK IN ANY OTHER PART
OF THE TEST.

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 Practice Test 2: AP Calculus AB 543
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practice test
SECTION I, PART B

50 Minutes • 17 Questions

A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS IN THIS PART OF

THE EXAMINATION.

Directions: Solve each of the following problems, using the available space
for scratchwork. After examining the form of the choices, decide which is the
best of the choices given and fill in the corresponding oval on the answer
sheet. No credit will be given for anything written in the test book. Do not
spend too much time on any one problem.
In this test: (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.

29. A particle starts at the origin and moves along the x-axis with decreasing positive
velocity. Which of these could be the graph of the distance, s(t), of the particle
from the origin at time t?
(A) (B)

s s

t t

(C) (D) (E)

s s s

t t t

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 544 PART III: Four Practice Tests
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30. Let f be the function given by f(x) 5 3 33.
ln 2x, and let g be the function given x 3 6 9 12
by g(x) 5 x3 1 2x. At what value of x f (x) 3 2 4 5
do the graphs of f and g have parallel
tangent lines?
Let f be a continuous function with
(A) 2 0.782 values as represented in the table
12
(B) 20.301 above. Approximate ∫3 f ( x )dx us-
(C) 0.521
ing a right-hand Riemann sum with
(D) 0.782 three subintervals of equal length.
(E) 1.000
(A) 14
31. Let f be some function such that the (B) 27
rate of increase of the derivative of f (C) 33
is 2 for all x. If f ′(2) 5 4 and f(1) 5 2,
(D) 42
find f(3).
(E) 48
(A) 3
(B) 6 34.
(C) 7
(D) 9 f′
(E) 10
x−a
32. lim 3 = a b
x→a x −a
3

1
(A)
a2 The graph of f′, the derivative of f, is
1 shown above. Which of the following
(B) describes all relative extrema of f on
3a 2 (a,b)?
1 (A) One relative maximum and one
(C)
4a2 relative minimum
(B) Two relative maximums and
(D) 0 one relative minimum
(C) One relative maximum and no
(E) It is nonexistent. relative minimum
(D) No relative maximum and two
relative minimums
(E) One relative maximum and two
relative minimums

(et )dt . What value

2x
35. Let f ( x) = ∫
0
on [0,4] satisfies the mean value
theorem for f?
(A) 2.960
(B) 2.971
(C) 3.307
(D) 3.653
(E) 4.000

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 Practice Test 2: AP Calculus AB 545
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practice test
36. The position for a particle moving on 39. Population y grows according to the
the x-axis is given by dy
1 equation
dt = ky, where k is a con-
s(t ) = − t + 2 t + . At what time, t,
3 2

2 stant and t is measured in years. If

on [0,3] is the particle’s instanta-
the population triples every five
neous velocity equal to its average
velocity over [0,3]? years, then k 5
(A) 0.535
(A) 0.110.
(B) 1.387
(B) 0.139.
(C) 1.821
(C) 0.220.
(D) 1.869
(D) 0.300.
(E) 2.333
(E) 1.099.
37. Let f be defined as
40. The circumference of a circle is in-
creasing at a rate of 25π inches per
minute. When the circumference is
10p inches, how fast is the area of
the circle increasing in square inches
per minute?
1
(A)
5
p
(B)
5
(C) 2
and g be defined as
x (D) 2p
g( x ) = ∫ f ( x )dx . Which of the
−10
following statements about f and g is (E) 25p
false?
(A) g′(23) 5 0
(B) g has a local minimum at
x 5 23.
(C) g(210) 5 0
(D) f′(1) does not exist.
(E) g has a local maximum at
x 5 1.

38. Let f(x) 5 x2 1 3. Using the trapezoi-

dal rule, with n 5 5, approximate
3
∫0 f ( x ) dx .
(A) 11.34
(B) 17.82
(C) 18.00
(D) 18.18
(E) 22.68

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 546 PART III: Four Practice Tests
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41. 43. The first derivative of a function, f, is
−x
given by f ′ ( x ) = e 2 − sin x . How
x
many critical values does f have on
the open interval (0,10)?
(A) One
(B) Two
(C) Three
(D) Four
The base of a solid is the region in
the first quadrant bounded by the (E) Five
x-axis and the parabola y 5 2x2 1
6x, as shown in the figure above. If
cross sections perpendicular to the
44. d
dx (∫
2x
3
)
f ′ (t ) dt =
x-axis are equilateral triangles, what
is the volume of the solid? (A) f′(3)
(B) 2f( 2 2x)
(A) 15.588
(C) 22f(2x)
(B) 62.354
(D) 2f′(2x)
(C) 112.237
(E) 22f′(2x)
(D) 129.600
(E) 259.200 45. Let f be defined as

42. Let f be the function given by f(x) 5

x2 1 4x 2 8. The tangent line to the
graph at x 5 2 is used to approxi-
mate values of f. For what value(s) of
x is the tangent line approximation
twice that of f? for a constant, k. For what value of k

I. − will lim f ( x ) = lim+ f(x)?

2 x →1− x →1
II. 1
(A) 2 2
III. 2 (B) 21
(A) I only (C) 0
(B) II only (D) 1
(C) III only (E) None of the above
(D) I and II
(E) I and III

END OF SECTION I, PART B. IF YOU HAVE ANY TIME LEFT, GO OVER

STOP YOUR WORK IN THIS PART ONLY. DO NOT WORK IN ANY OTHER PART
OF THE TEST.

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 Practice Test 2: AP Calculus AB 547
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practice test
SECTION II, PART A

45 Minutes • 3 Questions

A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF

PROBLEMS IN THIS PART OF THE EXAMINATION.

SHOW ALL YOUR WORK. It is important to show your setups for these
problems because partial credit will be awarded. If you use decimal approxi-
mations, they should be accurate to three decimal places.

1. Examine the function, f, defined as 2. A man is observing a horserace. He

x + x for 0 ≤ x ≤ 10. is standing at some point, O, 100 feet
f ( x) = 3 from the track. The line of sight from
(a) Use a Riemann sum with five the observer to some point P located
equal subintervals evaluated at on the track forms a 30° angle with
the midpoint to approximate the the track, as shown in the diagram
area under f from x 5 0 to x 5 below. Horse H is galloping at a con-
10. stant rate of 45 feet per second.
(b) Again using five equal subinter-
vals, use the trapezoidal rule to
approximate the area under f
from x 5 0 to x 5 10.
(c) Using your result from part B,
approximate the average value
of the function, f, from x 5 0 to
x 5 10.
(a) At what rate is the distance
(d) Determine the actual average
from the horse to the observer
value of the function, f, from x
changing 4 seconds after the
5 0 to x 5 10.
horse passes point P?
(b) At what rate is the area of the
triangle formed by P, H, and O
changing 4 seconds after the
horse passes point P?
(c) At the instant the horse gallops
past him, the observer begins
running at a constant rate of 10
feet per second on a line
perpendicular to and toward the
track. At what rate is the
distance between the observer
and the horse changing when
the observer is 50 feet from the
track?

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 548 PART III: Four Practice Tests
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3. Let v(t) be the velocity, in feet per (b) How far does the car travel
second, of a race car at time t sec- before coming to a stop?
onds, t ≥ 0. At time t 5 0, while trav- (c) Write an equation for the
eling at 197.28 feet per second, the tangent line to the velocity
driver applies the brakes such that curve at t 5 9 seconds.
the car’s velocity satisfies the differ-
(d) Find the car’s average velocity
dt = − 25 t − 7 .
ential equation dv 11
from t 5 0 until it stops.
(a) Find an expression for v in
terms of t where t is measured
in seconds.

END OF SECTION II, PART A. IF YOU HAVE ANY TIME LEFT, GO OVER
STOP YOUR WORK IN THIS PART ONLY. DO NOT WORK IN ANY OTHER PART
OF THE TEST.

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