Acing AP Calculus AB and BC

Practice Test 341

CALCULUS AB
SECTION I, Part A

Time ─ 60 minutes
Number of questions ─ 30

No calculator is allowed for problems on this part of the exam.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining
the choices given, decide which is the best answer choice and fill in the corresponding circle 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) 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 ).

1. An equation of the line normal to the graph of y = sec x at the point π , 2) is
(
4

(A) y − 2= 2(x − π )
4

(B) y − 2 =− 1 (x − π )
24

(C) y − 1 =− 1 (x − π )
2 24

(D) y − 1 = 1 (x − π )
22 4

342 Calculus AB

2. The first derivative f ′ of a function f is given by f ′(x)= 1 − e−x . On which of the following intervals
2

is f increasing?

(A) (−∞, − ln 2] (B) [− ln 2, ln 2] (C) [ln 2, ∞) (D) [e, ∞)

y bc x

4
3
2
1

Oa

3. The graph of the function f is shown in the figure above. Which of the following statements
about f is not true?

(A) lim f (x) = 3

x→a

(B) f (a) = 4
(C) lim f (x) = 1

x→b

(D) f (b) = 2

4. If =f (θ ) tan2 (2 −θ ) , then f ′(0) =

(A) 2 tan 2
(B) −2 tan 2
(C) −2 tan 2 sec 2
(D) −2 tan 2 sec2 2

Practice Test 2 343

5. If f (=x) (x2 +1)x , then f ′(1) =

(A) 2 + ln 4 (B) 2 + ln 2 (C) 1+ ln 4 (D) 1+ ln 2

6. If x = −2 is the vertical asymptote and y = 1 is the horizontal asymptote for the graph of the function f ,
which of the following is the equation of the curve?

(A) f (x) = x2 − 2
x2 + 4

(B) f (x) = x2 − 2x
x2 − 4

(C) f (x) = x2 − 4x + 4
−x2 + 4

(D) f (x) = x2 + 2x
x2 − 4

7. If g(x) = 2x f ( x ) , then g′(x) =

(A) 2  f ′( x ) + f ( x )
(B) 2 x f ′( x ) + f ( x )
(C) x f ′( x ) + 2 f ( x )
(D) 2 x f ′( x ) + 4 f ( x )

344 Calculus AB

8. The slope of the tangent to the curve y3 + x2 y2 − ye2x =6 at (0, 2) is

(A) − 9 (B) − 5 (C) 4 (D) 15
4 11 11 22

y

2

x

−3 O 3

−2
Graph of f ′

9. The graph f ′ is shown above. Which of the following statements is not true about f ?

(A) f is decreasing for −3 ≤ x ≤ 1 .
(B) f is increasing for −4 ≤ x ≤ −1 or 2 ≤ x ≤ 4 .
(C) f has a local minimum at x = 2 .
(D) f has a local maximum at x = −1 .

10. Which of the following is the antiderivative of f (=x) 1 + sec2 x ?
x

(A) −2 + tan x + C
x

(B) 2 x + tan x + C
(C) −2 x + sec x tan x + C
(D) 2 x + sec x tan x + C

Practice Test 2 345

11. If f ′′(x) = 10x−1 3 , which of the following could be true?

I. f (x) =−9x5 3 + 5
II. =f ′(x) 15x2 3 −12
III. f ′′′(x) =− 10 x−4 3 + 7

3

(A) None (B) I only (C) II only (D) I and II only

y

−3 B x
A O
C4

12. The shaded regions A , B , and C in the figure above are bounded by the graph of y = f (x) and

the x-axis. If the area of region A is 4, region B is 3, and region C is 2, what is the value of

∫4 [ f (x) + 2] dx ?
−3

(A) 8 (B) 9 (C) 11 (D) 13

13. If F ′(x) = f (x) for all real numbers x , and if k is a constant, then 2 f (kx) dx =

∫1

[F (2) − F (1)]

(A)
k

[F(2k) − F(k)]

(B)
k

(C) F (2k) − F (k)

(D) k [F (2) − F (1)]

346 Calculus AB

14. If lim 3 − kx − 3 = 1 what is the value of k ?
x→0 x 3

(A) −3 (B) −2 (C) − 3 (D) − 2

y

y = f ′(x)

Oa x

b

15. The graph of f ′ , the derivative of function f , is shown above. If f is a twice differentiable function
which of the following statements must be true?

I. f (a) > f (b)
II. The graph of f has a point of inflection at x = b .
III. The graph of f concaves down on the interval a < x < b .

(A) I only (B) II only (C) III only (D) II and III only

16. Let f be a differentiable function with f π ) = −2 and f ′(π ) = 3 . If g is the function defined
(
44

by =g(x) cos2 x ⋅ f (x) , then g′(π ) =
4

(A) − 3 2 (B) − 3 (C) 7 32
2 2 2 (D)

2

Practice Test 2 347

17. The expression 1 [ln(1.1) + ln(1.2) + ln(1.3) + . . . + ln(2)] is a Riemann sum approximation for

10

∫(A) 1 1

ln x dx
10 0

∫1

(B) ln x dx

0

∫1 2

(C) ln x dx
10 1

∫2

(D) ln x dx

1

y

3

2

1
x

O 1 23 4
Graph of f

18. The graph of f is shown above for 0 ≤ x ≤ 4 . Let L , R , and T be the left Riemann sum,
right Riemann sum, and the trapezoidal sum approximation respectively, of f (x) on [0, 4]
with 4 subintervals of equal length. Which of the following statements is true?

∫(A) L < 4 f (x) dx < T < R
0

∫(B) L < 4 f (x) dx < R < T
0

∫(C) R < 4 f (x) dx < L < T
0

∫(D) T < L < 4 f (x) dx < R
0

348 Calculus AB

y y = f (x)

−2 A O B

x

4

19. The graph of y = f (x) is shown in the figure above. The shaded region A has area a and the shaded

region B has area b . If g=(x) f (x) + 3 what is the average value of g on the interval [−2, 4] ?

(A) a + b + 3 (B) −a + b + 3 (C) −a + b + 3 (D) a + b + 3
6 6 6 6

20. Let f be the function given by f (x) = ecos x . Which of the following statements are true?

∫I. d π 3 f (x) dx = 0
dx 0

∫II. π 3 d f (x) d=x e − e
0 dx

∫III. d x f (t) dt = ecos x
dx 0

(A) I only (B) I and II only (C) I and III only (D) I , II, and III

∫21. e2 ( x −1 ) dx = (B) e2 − 2 (C) e3 − e (D) e2 − 3
1x
(A) e2 −1

Practice Test 2 349

∫9

22. Using the substitution u = x , sin( x ) dx is equivalent to

0

∫3

(A) 2 u sin(u) du

0

∫(B) 2 3 sin u du
0u

∫1 3

(C) u sin(u) du
20

∫(D) 1 3 sin u
du
20 u

23. If dy = 1+ y2 and y(−1) =0 , then y =
dx

(A) 1− tan x (B) 1+ sec x (C) tan(x +1) (D) 1 − etan(x+1)

x −1 1 4 6 9
f (x) 12 9 5 8 10

24. A function f is continuous on the closed interval [−1,9] and has values that are given in the table above.

Using subintervals [−1,1] , [1, 4] , [4, 6] , and [6,9] , what is the trapezoidal approximation of ∫9 f (x) dx ?
−1

(A) 76 (B) 82 (C) 92 (D) 98

350 Calculus AB

25. lim (ex + x)1 x =

x→∞

(A) −e (B) −1 (C) 1 (D) e

∫π4 1 dx =

26.
0 etan x cos2 x

(A) −e (B) 1− e (C) 1 (D) 1− 1
e e

y

y = f (x)

Oa bc d e x

x

27. The graph of the function f is shown in the figure above. If h(x) = f (t) dt , which of the

∫a

following is true?

(A) h(x) has a minimum at x = b and has a maximum at x = d .
(B) h(x) has a minimum at x = a and has a maximum at x = e .
(C) h(x) has a minimum at x = e and has a maximum at x = c .
(D) h(x) has a minimum at x = c and has a maximum at x = e .

Practice Test 2 351

28. Let R be the region in the first quadrant bounded by the graph of y = a x ( a > 0 ), x = 1 , and x = 4 .
If the area of the region R is 7, what is the value of a ?

(A) 4 (B) 3 (C) 7 (D) 5
3 2 3 2

∫29. If F (x) = 0 dt , then F ′(x) =
tan x 1+ t2

(A) sin x (B) cos x (C) −1 (D) cos2 x

∫30. If the substitution x = 5sinθ is made for 1 dx , where 0 < θ < π , then
x2 25 − x2 2

∫ 1 dx =

x2 25 − x2

(A) 1 ∫ cscθ cotθ dθ
5

(B) − 1 ∫ csc2θ dθ
5

(C) 1 ∫ csc2θ dθ
25

(D) 1 ∫ sec2θ dθ
25

352 Calculus AB

CALCULUS AB
SECTION I, Part B

Time ─ 45 minutes
Number of questions ─ 15

A graphing calculator is required for some problems on this part of the exam.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining
the choices given, decide which is the best answer choice and fill in the corresponding circle 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. If this
occurs, select the number that best approximates the exact numerical value from the choices given.

(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 ).

31. What is the area of the region enclosed by the graphs of f (=x) x +1 and g(x) = 1 from x = 0
to x = 2 ? 1+ x

(A) 1.362 (B) 1.475 (C)1.699 (D) 1.833

32. The maximum acceleration attained on the interval 0 ≤ x ≤ 5 by the particle whose velocity is given
by v(t) = t3 − 4t2 + 7t + 3 is

(A) 28 (B) 32 (C) 37 (D) 42

Practice Test 2 353

33. A particle moves along the x-axis with a(t) = 1.2 units /sec2 . When t = 1, the particle is at the
point (0,1) . Which of the following could be the graph of the distance s(t) of the particle as a
function of time t .

(A) (B)
s s

1 1 t
O1 t

(C) O1
s
(D)
s

1 1 t
O1
t
O1

34. Let f be the function given=by f (x) cos ln(x2 +1) . On the closed interval [0,8] , how many

values of c satisfies the conclusion of Mean Value Theorem?

(A) 1 (B) 2 (C) 3 (D) 4

354 Calculus AB

35. Let f be a function given by f (x) = x2 −1 . Which of the following statements are true about f ?
x −1

I. lim f (x) = 2

x→1+

II. lim f (x) = −2

x→1−

III. lim f (x) = lim f (x)
x→3 x→ −5

(A) I only (B) I and II only (C) II and III only (D) I , II, and III

36. The base of a solid is the region enclosed by the graph of y = sin(x2 ) and the x-axis for 0 ≤ x ≤ π .
If cross sections of the solid perpendicular to the x-axis are squares, what is the volume of the solid?

(A) 0.670 (B) 0.783 (C) 0.835 (D) 1.032

x f (x) g(x) f ′(x) g′(x)

1 3 5 −1 −3
2 3 −2 −2 1
3 2 4 −1 7

37. The table above gives values of f , f ′ , g , and g′ at selected values of x . If h(x) = g  f (x2 ) ,
what is the value of h′(1) ?

(A) −14 (B) −8 (C) −3 (D) 6

Practice Test 2 355

38. The volume V of a sphere is decreasing at a rate of 12 in3/sec . What is the rate of decrease of the
radius of the sphere, in inches per second, at the instant when the surface area S becomes 36π

square inches? (V = 4 π r3 , and S = 4π r2 )
3

(A) 1 (B) 1 (C) 1 (D) 1
6π 4π 3π 2π

39. Water is leaking from a tank at the rate of te(−0.1t) gallons per hour. If there are 100 gallons of water
in the tank at time t = 0 , how many gallons of water are in the tank at time t = 10 ?

(A) 58.379 (B) 60.455 (C) 68.702 (D) 73.576

y = cos x y

(x, y)

x

O

40. The figure above shows a rectangle that has its base on the x-axis and its other two vertices on
the curve y = cos x .What is the largest possible area of such a rectangle?

(A) 1.074 (B) 1.122 (C) 1.384 (D) 1.678

356 Calculus AB

41. A particle travels along a straight line with a constant acceleration of 2 ft/sec2 . If the velocity
of the particle is 5 ft/sec at time t = 3 seconds, how far does the particle travel during the time
interval when the velocity increases from 5 ft/sec to 15 ft/sec ?

(A) 28 ft (B) 36 ft (C) 42 ft (D) 50 ft

42. Let R be the region enclosed by the graph of y = 4 and the line y= 5 − x . The volume of the
x

solid obtained by revolving R about the y-axis is given by

∫(A) π 4 (5 − y − 4 )2dy

1y

∫(B) π 4  4 )2 − (5 − y)2  dy
1 ( y 

∫(C) π 4  − y)2 − ( 4)2  dy
1 (5 y 
 

∫(D) 2π 4 (5 − x − 4)2  dx
1 x 

43. Oil is being pumped from an oil well at a rate proportional to the amount of oil left in the well; that

is dy = ky , where y is the amount of oil left in the well at any time t measured in years. There
dt

were 2,000,000 gallons of oil in the well at time t = 0 , and 1,200,000 gallons remaining at time t = 5 .
To the nearest thousand, how many gallons of oil will be left in the well at time t = 10 ?

(A) 570,000 (B) 720,000 (C) 840,000 (D) 920,000

Practice Test 2 357

y (1,1)

y = x2

x

O

=y 2x −1

44. The figure above shows a shaded region bounded by the x-axis and the graphs of y = x2 and =y 2x −1.
If the shaded region is rotated about the x-axis, what is the volume of the solid generated?

π π π π
(A) (B) (C) (D)

30 24 12 8

y
6

4

2 x
24 6
−6 −4 −2
−2

−4
−6

45. The slope field for a certain differential equation is shown above. Which of the following could be
a specific solution to that differential equation?

(A) y = 2e−x (B) y= x + ex (C) y= x + e−x (D) y= x − ex

358 Calculus AB

CALCULUS AB
SECTION II, Part A

Time ─ 30 minutes
Number of problems ─ 2
A graphing calculator is required for these problems.

1. Let f be the function given by f (x=) x3 + 2 , and let g be the function given by g(x) = mx ,
where m is the nonzero constant such that the graph of g is tangent to the graph of f .

(a) Find the x-coordinate of the point of tangency and the value of m .
(b) Let R be the region enclosed by the graphs of f and g . Find the area of R .
(c) Find the volume of the solid generated when R is rotated about the line y = 3 .

Practice Test 2 359

2. The rate at which people enter a movie theater on a given day is modeled by the function S
defined by S(t=) 80 −12 cos( t ) . The rate at which people leave the same movie theater is
5
modeled by the function R defined by=R(t) 12et 10 + 20 . Both S(t) and R(t) are measured
in people per hour and these functions are valid for 10 ≤ t ≤ 22 . At time t = 10 , there are no
people in the movie theater.

(a) To the nearest whole number, how many people have entered the movie theater by 8:00 PM ( t = 20 )?

(b) To the nearest whole number, how many people are in the movie theater at time t = 20 ?

∫(c) L=et P(t) t [S(t) − R(t)] dt for 10 ≤ t ≤ 22 . Find the value of P′(20) and explain the meaning
10
of P′(20) .

(d) At what time t , for 10 ≤ t ≤ 22 , is the number of people in the movie theater a maximum?

360 Calculus AB

CALCULUS AB
SECTION II, Part B

Time ─ 60 minutes
Number of problems ─ 4
No calculator is allowed for these problems.

t 0 10 20 30 40 50 60 70 80 90
(sec) 21 23 25 15 6 0 −12 −10 −8 −12

v(t)
(ft/sec)

3. A car is traveling on a straight road. The car’s velocity v , measured in feet per second, is continuous
and differentiable. The table above shows selected values of the velocity function during the time
interval 0 ≤ t ≤ 90 seconds.

(a) Find the average acceleration of the car over the time interval 0 ≤ t ≤ 90 .

(b) Using correct units explain the meaning of ∫ 70 v(t) dt . Use a trapezoidal approximation with
40

three subintervals of equal length to approximate ∫ 70 v(t) dt .
40

(c) For 0 < t < 90 , must there be a time t when v(t) = 10 ? Justify your answer.

(d) For 0 < t < 90 , must there be a time t when a(t) = 0 ? Justify your answer.

4. Consider the curve given by x2 + xy + y2 =12 .

(a) Find dy .
dx

(b) Find the two points where the curve crosses the x-axis, and write an equation for the tangent
line at each of these two points.

(c) Find the x-coordinate of each point on the curve where the tangent line is horizontal.

Practice Test 2 361

y

4

2

−2 O 2 4 6 x

8

Graph of f

5. The graph of a differentiable function f on the closed interval [−2,8] is shown in the figure above.

The graph of f has a horizontal tangent line at x = 0 and x = 6 . Let h(x) =−3 + ∫0x

f (t) dt

for −2 ≤ x ≤ 8 .

(a) Find h(0) , h′(0) , and h′′(0) .

(b) On what interval is h decreasing? Justify your answer.
(c) On what intervals does the graph of h concave up? Justify your answer.

∫(d) Find a trapezoidal approximation of 8 f (t) dt using five subintervals of length ∆t =2 .
−2

362 Calculus AB

6. Consider the differential equation dy = −4x3 y2 .
dx 3

(a) On the axis provided, sketch a slope field for the given differential equation at the nine
points indicated.

y

2

1 x
−1 O 1

(b) Let y = f (x) be the particular solution to the differential equation with the initial condition
y(−1) =3 . Write an equation for the line tangent to the graph of f at (−1, 3) and use it to
22
approximate f (−1.1) .

(c) Find the particular solution y = f (x) to the differential equation with the initial condition
y(−1) =3 .
2

AP Calculus BC
Practice Test 1

1ABC D Answer Sheet 31 A B C D
2A BC D 32 A B C D
3A B C D 16 A B C D 33 A B C D
4A B C D 17 A B C D 34 A B C D
5A B C D 18 A B C D 35 A B C D
6A B C D 19 A B C D 36 A B C D
7A B C D 20 A B C D 37 A B C D
8A B C D 21 A B C D 38 A B C D
9A B C D 22 A B C D 39 A B C D
10 A B C D 23 A B C D 40 A B C D
11 A B C D 24 A B C D 41 A B C D
12 A B C D 25 A B C D 42 A B C D
13 A B C D 26 A B C D 43 A B C D
14 A B C D 27 A B C D 44 A B C D
15 A B C D 28 A B C D 45 A B C D
29 A B C D
30 A B C D

Practice Test 1 365

CALCULUS BC
SECTION I, Part A

Time ─ 60 minutes
Number of questions ─ 30

No calculator is allowed for problems on this part of the exam.

.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining
the choices given, decide which is the best answer choice and fill in the corresponding circle 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) 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 ).

1. If lim f (x) = 2 and lim g(x) = −1 , then lim x − f (x) =
x→1 x→1 x→1 [ g(x)] 2 +1

(A) − 1 (B) − 1 (C) 1 (D) nonexistent
5 2 2

2. If f (x) = 2tan x , then f ′(π ) =
4

(A) ln 4 (B) ln 8 (C) ln16 (D) ln 32

366 Calculus BC

3. If f (x) =  ax − 3, if x ≤ 2
 + a, is continuous on (−∞, ∞) , what is the value of a ?
 x 2
if x > 2

(A) 7 (B) 5 (C) 3 (D) 1

4. If four equal subdivisions are used for the internal [1, 2] , what is the trapezoidal approximation

∫of 2 e1 x dx ?
1

(A) 1 (e + e1 1.25 + e1 1.5 + e1 1.75 + e1 2 )
4

(B) 1 (e + 2e1 1.25 + 2e1 1.5 + 2e1 1.75 + e1 2 )
4

(C) 1 (e + 2e1 1.25 + 2e1 1.5 + 2e1 1.75 + e1 2 )
8

(D) 1 (e + e1 1.25 + e1 1.5 + e1 1.75 + e1 2 )
8

5. d 1 sec3 x − sec x + 3 
dx  3 

(A) tan4 x
(B) sec2 x + tan2 x
(C) sec2 x − sec x tan x
(D) tan3 x sec x

Practice Test 1 367

6. Which of the following sequences converge?

I.  4n  II.  2n − 9  III. n sin( 1 )
     n 
 3n + 2   en 

(A) I only (B) I and II only (C) II and III only (D) I, II, and III

y

x

O (c, 0)
y = f (x)
R

(a,b)
y = g(x)

7. The curves y = f (x) and y = g(x) shown in the figure above intersect at point (a,b) . The volume
of the solid obtained by revolving R about the x-axis is given by

(A) ∫π c 2 ∫dx − π c 2 dx
0 0
[g(x)] [ f (x)]

(B) ∫π a 2 ∫dx − π c 2 dx
0 a
[ f (x)] [g(x)]

(C) ∫π c 2 dx
0
[ f (x) − g(x)]

(D) ∫π a 2 ∫dx + π c 2 dx
0 a
[g(x)] [ f (x)]

368 Calculus BC
(D) − f (x)
8. If f ′(0) = −1 and f (0) = 1 then lim f (h) −1 =
h→0 h

(A) −1 (B) 0 (C) 1

∫9. The length of a curve from x = a to x = b is given by b 1+ sin2 (2x) dx . Which of the following
a
could be the equation for this curve?

(A) y = sin(2x) (B) y = cos(2x) (C) y = − 1 cos(2x) (D) y = 1 sin(2x)
22

10. If n is a positive integer, then lim 3  1+ 3 1+ 6 ++ 1+ 3n  can be expressed as
e n 
n→∞ n  +e n en



∫(A) 3 ex dx ∫(B) 3 e1+x dx ∫(C) 4 ex/3 dx ∫(D) 4 ex dx
0 0 1 1

∞

∑11. e−n+1 ⋅ 2n =
n=1

(A) e (B) 2e (C) e + 2 (D) e + 2
2e −1 e−2 e−2 2e

Practice Test 1 369

12. ∫ 1 dx =
x2 + 2x + 2

(A) arc tan(x +1) + C
(B) arccot(x +1) + C
(C) − 1 (x2 + 2x + 2)−2 + C

2
(D) ln(x2 + 2x + 2) + C

y

r =1

x

O

r = 1 − sinθ

13. Which of the following gives the area of the region inside the polar curve r = 1− sinθ and outside the
polar curve r = 1 , as shown in the figure above?

∫(A) 1 2π (1− sinθ )2 −1 dθ
π
2

∫1 2π (1− sinθ )2 −1 dθ
0
(B)
2

∫(C) 1 π (1− sinθ )2 dθ

2 π2

∫(D) π (1− sinθ )2 −1 dθ
0

370 Calculus BC

14. A population is modeled by a function P that satisfies the logistic differential equation

=dP P  2 − P  , where the initial population P(0) = 360 and t is the time in years.
dt 3  60 

What is lim P(t) ?

t→∞

(A) 30 (B) 60 (C) 120 (D) 240

15. If x = et and y= (t +1)2 , then d 2 y at t = 1 is
dx2

(A) −2 (B) −2 (C) 2 (D) 2
e e2 e e2

16. A particle moves on the curve y = ln( x ) so that the x-component has velocity x′(t=) et +1

for t > 0 . At time t = 0 , the particle is at the point (2, 1 ln 2) . At time t = 1, the particle is at
2

the point

( )(A) e2,1

( )(B) e4, 2

(C)  (e + 1), 1 ln(e + 1) 
 2 

(D)  (e + 2), 1 ln(e + 2) 
 2 

Practice Test 1 371

t 0 123 45

v(t) 2 0 −2 − 1 0 1
2 2

17. The table above gives selected values of the velocity, v(t) , of a particle moving along the x- axis.
At time t = 0 , the particle is at the point (1, 0) . Which of the following could be the graph of the
position x(t) , of the particle for 0 ≤ t ≤ 5 ?

(A) x(t) (B) x(t)
3 3
2
1 2
x
O 12345 1
−1 x

O 12345
−1

(C) x(t) (D) x(t)
3 3

2 2

1 1
x x

O 12345 O 12345
−1 −1

18. An object moves along a curve in the xy-plane so that its position at any time t ≥ 0 is given
by (t2 +1, tet / 2 ) . What is the speed of the object at time t = 2 ?

(A) 3.946 (B) 4.822 (C) 6.749 (D) 8.615

372 Calculus BC

∫19. What are all values of p for which ∞ 1 dx converges?

1 px

(A) p < 1 (B) p > 1 (C) p < 0 (D) p < −1

20. ∫ ln x dx =
x2

(A) x ln x − 1 + C
x3

(B) x ln x + 1 + C
x

(C) −x ln x − 1 + C
x

(D) − ln x − 1 + C
xx

21. Which of the following series converge?

∑∞ n − 2 ∑II.∞ 3n + 4n ∞
n=1 5n
I. ∑III. ne−n
n=1 n(n + 7) n=1

(A) I only (B) II only (C) I and II only (D) II and III only

Practice Test 1 373

22. d (arcsin ex 2 ) =
dx

(A) ex 2 (B) − ex 2 (C) ex 2 (D) − ex 2
1− ex 2 1+ ex 2 1− ex 2 1− ex

∑∞ (4x −1)n converges?

23. What are all values of x for which the series
n=1 n4n

(A) − 3 < x < 5 (B) − 3 ≤ x < 5 (C) − 3 < x ≤ 5 (D) − 3 ≤ x ≤ 5
44 44 44 44

24. Let f be a function that is differentiable on the open interval (−1,8) . If f (−1) =7 , f (5) = 7 ,

and f (8) = −2 . Which of the following must be true?

I. There exists a number k in the interval (−1,5) , such that f (k) = 3 .
II. f has at least one zero.
III. The graph of f has at least one horizontal tangent.

(A) None
(B) II only
(C) I and II only
(D) II and III only

374 Calculus BC

25. If dy = y2 sec2 x and y(0) = 1 , then y =
dx 2

(A) 1 (B) 1 (C) 1 (D) 1
2 cos x 1+ cos x 2 − sin x 2 − tan x

y II
III
O
I x

26. Three graphs labeled I, II, and III are shown above. They are the graphs of f , f ′ , and f ′′ . Which of
the following correctly identifies each of the three graphs?

f f ′ f ′′
(A) I II III
(B) II I III
(C) III I II
(D) I III II

∫27. ∞ xe−x2 dx = (B) 0 (C) 1 (D) ∞
0 2
(A) − 1
2

Practice Test 1 375

x f (x) f ′(x) f ′′(x)

1 −2 −3 4

21 2 −1

28. The table above gives values of f , f ′ , and f ′′ at selected values of x . If f ′′ is continuous everywhere,

∫then 2 f ′′(t) dt =
1

(A) 5 (B) 3 (C) −3 (D) −5

29. The position of a particle moving along a line is given by s(t) =t3 −15t2 +14 for t ≥ 0 . For what
values of t is the speed of the particle increasing?

(A) t > 10 only
(B) 5 < t < 10 only
(C) 3 < t < 5 and t > 10
(D) 0 < t < 5 and t > 10

30. A series expansion of 1− cos x is
x

(A) 1 − x + x2 − + (−1)n−1 xn−1 +
3! 5! 7! (2n +1)!

(B) 1 − x + x2 − + (−1)n−1 xn−1 +
2! 4! 6! (2n)!

(C) 1− x − x2 − + (−1)n−1 xn−1 +
2! 4! (n −1)!

(D) x − x2 + x3 − + (−1)n−1 xn +
2! 4! 6! (2n)!

376 Calculus BC

CALCULUS BC
SECTION I, Part B

Time ─ 45 minutes
Number of questions ─ 15

A graphing calculator is required for some problems on this part of the exam.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining
the choices given, decide which is the best answer choice and fill in the corresponding circle 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. If this
occurs, select the number that best approximates the exact numerical value from the choices given.

(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 ).

31. The rate of consumption of a certain commodity, in thousand units per month, is given
by C(x) = 12e0.112x , where x represents the number of months. What is the average rate

of consumption of the commodity, in thousand units, over the first 6 month period?

(A) 1.085 (B) 1.384 (C) 1.506 (D) 1.916

32. A particle moving in the xy-plane has velocity vector given by v(t) = 1 , −t for time
1−t2 1−t2

t ≥ 0 . What is the magnitude of the displacement of the particle between time t = 0 to t = 0.8 ?

(A) 0.877 (B) 1.058 (C) 1.099 (D) 1.206

Practice Test 1 377

33. Let f be the function given by f (x) = x2 + 3x −1. If the tangent line to the graph of f at x = 1 is
used to find an approximate value of f , which of the following is the greatest value of x for which
the error resulting from this tangent line approximation is less than 0.3?

(A) 1.3 (B) 1.4 (C) 1.5 (D) 1.6

Light Ball at time t = 0
10 ft

40 ft ⇓

Shadow
⇐

34. A light shines from the top of a pole 40 feet high. A ball is dropped from the same height from a point
10 feet away from the light, as shown in the figure above. If the position of the ball at time t is given
by y(t=) 40 −16 t2 , how fast is the shadow moving one second after the ball is released?

(A) −16 ft /sec (B) −32 ft /sec (C) −40 ft /sec (D) −50 ft /sec

35. The base of the solid is an elliptical region enclosed by the graph of x2 + y2 =1 . If cross sections of
4

the solid perpendicular to the y-axis are isosceles right triangles with the hypotenuse in the base, what
is the volume of the solid?

(A) 16 (B) 20 (C) 8 (D) 26
3 3 3

378 Calculus BC

36. If f (x) = 3x2 and g is an antiderivative of f such that g(4) = 6 , then g(1) =
x4 + 5

(A) 1.655 (B) 2.704 (C) 3.862 (D) 4.704

37. A curve is defined by the polar equation r = 1+ 3sinθ . When θ = 5π , which of the following
6

statements is true of the polar curve?

(A) The curve is closest to the origin.
(B) The curve is getting farther from the origin.
(C) The curve is getting closer to the origin.
(D) The curve has a horizontal tangent.

38. The graph of the function represented by the Maclaurin series x − x3 + x5 − x7 + + (−1)n x2n+1 +
2! 4! 6! (2n)!

intersects the graph of ln x at x =

(A) 0.735 (B) 0.916 (C) 1.347 (D) 1.466

Practice Test 1 379

39. The equation of a polar curve is given by r= 3 + sin 5θ . What is the angle θ that corresponds to
the point on the curve with x-coordinate 2?

(A) 0.516 (B) 0.628 (C) 0.705 (D) 0.844

40. What is the length of the curve y = x ln x from x = 1 to x = 2 ?

(A) 1.548 (B) 1.713 (C) 1.952 (D) 2.043

∫41. Let h be the function given by=h(x) x 4(x − 2) cos( x ) dx . Which of the following statements

02
about h must be true?

I. h is increasing on (0, 2) .
II. h′(3) > 0 .

III. h(3) < 0 .

(A) I only (B) II only (C) III only (D) II and III only

380 Calculus BC

42. The number of bacteria in a colony increases at a rate proportional to the number present. If the
colony starts with one bacterium and doubles every half-hour, how many bacteria will the colony
contain at the end of 12 hours?

(A) 4096 (B) 65,536 (C) 1.049 ×106 (D) 1.678×107

43. The velocity of a particle moving along the y-axis is given by v(t) = t3 − 5t2 + 2t + 8 for 0 ≤ t ≤ 10 .
Which of the following expressions gives the change in position of the particle during the time the
particle is moving downward?

∫(A) 4 (t3 − 5t2 + 2t + 8) dt
2

∫(B) 4 (t3 − 5t2 + 2t + 8) dt
0

∫(C) 4 (3t2 −10t + 2) dt
2

∫(D) 3.23 (t3 − 5t2 + 2t + 8) dt
0

Practice Test 1 381

44. Let y = f (x) be the solution to the differential equation dy = 1− xy with the initial condition
dx 2

f (0) = 1. What is the approximation for f (1) if Euler’s method is used, starting at x = 0 with
a step size of 0.5?

(A) 1.5 (B) 1.65 (C) 1.762 (D) 1.813

45. Let P(x) =x − 2x2 + 2x3 − 4 x4 be the fourth-degree Taylor polynomial for the function f
3

about x = 0 . What is the value of f (4) (0) ?

(A) −32 (B) −16 (C) − 32 (D) − 16
3 3

382 Calculus BC

CALCULUS BC
SECTION II, Part A

Time ─ 30 minutes
Number of problems ─ 2
A graphing calculator is required for these problems.

y y=3x

y= 2 − x2 x
4

S
R

O

1. Let R and S be the region in the first quadrant as shown in the figure. The region R is bounded
by the x- axis and the graph of y= 2 − x2 and y = 3 x . The region S is bounded by the y-axis
4
and the graph of y= 2 − x2 and y = 3 x .
4

(a) Find the area of R .

(b) Find the area of S .

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the
x-axis is a square. Find the volume of this solid.

Practice Test 1 383

2. A particle moves along the x-axis with its velocity given by v(t) = t ln(t2 ) for t ≥ 0 .
At time t = 0 , the position of the particle is x(0) = −1 .

(a) Find the acceleration of the particle at time t = 0.5 .
(b) Is the speed of the particle increasing or decreasing at time t = 0.5 ?

Give a reason for your answer.
(c) Find the time, t ≥ 0 , at which the particle is farthest to the left. What is the distance

between the particle and the origin when it is farthest to the left?
(d) Find the position of the particle at time t = 0.5 . Is the particle moving toward the origin

or away from the origin at time t = 0.5 ? Justify your answer.

384 Calculus BC

CALCULUS BC
SECTION II, Part B

Time ─ 60 minutes
Number of problems ─ 4
No calculator is allowed for these problems.

y r= 4
1 + sinθ

2

R x
O2
−4 −2 4

−2

3. The figure above shows the graph of the polar curve r = 4 . Let R be the shaded region
1+ sinθ

bounded by the curve and the x-axis.

(a) Find dr at θ = π . What does the value of dr at θ = π say about the curve?
dθ 6 dθ 6

(b) Set up, but do not evaluate, an integral expression that represents the area of the polar region R ,
using the equation r = 4 .
1+ sinθ

(c) Show that r = 4 can be written as the equation y =− 1 x2 + 2 .
1+ sinθ 8

(d) Use the equation y =− 1 x2 + 2 to find the area of the region R .
8

Practice Test 1 385

t 0 3 6 9 12 15 18 21 24
(hours)

P(t) 700 620 760 1040 1200 1120 960 920 680
(gallons / hour)

4. The rate of fuel consumption in a factory, in gallons per hour, recorded during a 24-hour period
is given by a twice differentiable function P of time t . The table of selected values of P(t) ,
for the time interval 0 ≤ t ≤ 24 , is shown above.

(a) Use the data from the table to find an approximation for P′(7.5) . Indicate the units of measure.

(b) The rate of fuel consumption is increasing the fastest at time t = 7.5 minutes. What is the value
of P′′(7.5) ?

(c) Approximate the average value of the rate of fuel consumption on the interval 12 ≤ t ≤ 24 using
a left Riemann sum with the four subintervals indicated by the data in the table above.

(d) For 12 ≤ t ≤ 24 hours, P(t) is strictly a decreasing function of time t . Is the data in the table
consistent with the assertion that P′′(t) < 0 for every x in the interval 12 < t < 24 ?
Explain your answer.

386 Calculus BC

5. Consider the differential equation dy =−x − y .
dx

(a) On the axis provided, sketch a slope field for the given differential equation at the eight
points indicated, and sketch the solution curve that passes through the point (0, −1) .

y

x

−1 O 12
−1

−2

(b) Let f be the function that satisfies the given differential equation with the initial condition
f (0) = −1 . Use Euler’s method, starting at x = 0 with two steps of equal size, to approximate
f (−0.4) . Show the work that leads to your answer.

(c) The solution curve that passes through the point (0, −1) has a local maximum at x = ln 2 .
What is the y- coordinate of this local maximum?

(d) Find d2y in terms of x and y . Determine whether the approximation found in part (b) is
dx2

less than or greater than f (−0.4) . Explain your reasoning.

6. The Maclaurin series for tan−1 x is tan−1 x =x − x3 + x5 − x7 + + (−1)n x2n+1 + .
357 2n +1

The continuous function f is defined by f (x) = tan−1 x for x ≠ 0 and f (0) = 1.
x

(a) Write the first three nonzero terms and the general term for the Maclaurin series of f ′(x) .

(b) Use the result from part (a) to find the sum of the infinite series

−2⋅ 1 + 4 ⋅ 1 + 6 ⋅ 1 3 + + (−1)n (2n) ⋅ 1 + .
3 3 5 33 7 32 2n +1 3n−1 3

∫x

(c) Let g be the function given by g(x) = f (t) dt . Find the first four nonzero terms and

0

the general term for the Maclaurin series representing g(x) .

(d) Show that 1− 1 + 1 approximates g(1) with an error less than 1 .
32 52 40

AP Calculus BC
Practice Test 2

1ABC D Answer Sheet 31 A B C D
2A BC D 32 A B C D
3A B C D 16 A B C D 33 A B C D
4A B C D 17 A B C D 34 A B C D
5A B C D 18 A B C D 35 A B C D
6A B C D 19 A B C D 36 A B C D
7A B C D 20 A B C D 37 A B C D
8A B C D 21 A B C D 38 A B C D
9A B C D 22 A B C D 39 A B C D
10 A B C D 23 A B C D 40 A B C D
11 A B C D 24 A B C D 41 A B C D
12 A B C D 25 A B C D 42 A B C D
13 A B C D 26 A B C D 43 A B C D
14 A B C D 27 A B C D 44 A B C D
15 A B C D 28 A B C D 45 A B C D
29 A B C D
30 A B C D

Practice Test 2 389

CALCULUS BC
SECTION I, Part A

Time ─ 60 minutes
Number of questions ─ 30

No calculator is allowed for problems on this part of the exam.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining
the choices given, decide which is the best answer choice and fill in the corresponding circle 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) 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 ).

1. lim 1− cos x = (B) 0 (C) 1 (D) nonexistent
x→0 1+ x − ex
(A) −1

2. If f (x) = e−x and g(x) = f ( f (x)) , then g′(0) =

(A) 2 (B) − 1 (C) 1 (D) 1
e e e e2

390 Calculus BC

3. If 3sin x cos y = 1 , then dy =
dx

(A) − cot x cot y
(B) cot x cot y
(C) tan x tan y
(D) − tan x tan y

y y = f ′(x)

b x
a O cde

4. The graph of f ′ , the derivative of the function f , is shown in the figure above. For what values
of x does the graph of f concave up ?

(A) b < x < d
(B) a < x < 0 or x > d
(C) b < x < c or x > e
(D) a < x < b or c < x < e

5. I=f h(x) arctan x + arctan(1 ) , then h′(x) =
x

(A) 2 (B) 2x (C) 0 (D) 1
1+ x2 1+ x2