lOMoARcPSD|41419381

AP Calc BC 2019 - Please take for reference

Material science and engineering (Shanghai University)

Scan to open on Studocu

Studocu is not sponsored or endorsed by any college or university

Downloaded by Jamilah Modahi (jnamqq@gmail.com)
 lOMoARcPSD|41419381

20

Please note: Some of the questions in this former practice exam may
no longer perfectly align with the AP exam. Even though these
questions do not fully represent the 2020 exam, teachers indicate
that imperfectly aligned questions still provide instructional value.
Teachers can consult the Question Bank to determine the degree to
which these questions align to the 2020 Exam.

7 L PP E F E L
F LF L LE 7 L P L S R LG G E
ROO R G R PS S LR 7 F S PL G R
GR OR G P L O GP FRSL R L L G L
FO RRP L R O 7R P L L F L R L P F
R OG FROO F OO P L O L GPL L LR G S PL
F ORF LR

L LE L P L L F
L L F FL F
T L F P L GGL LR O GL LE LR L L
LRO LR R ROO R G FRS L SROLFL GP O L
PL LR R F LF P FF R R F RRO OO
PR O R FF RR R OL LF F 7 F
RPP L G 2 OL FR SR

k 20 R R VW R R R W : R R R
AP Central is the official online home for the AP Program: apcentral.collegeboard.org

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

URP W H PL L WU WLR

OF O
p

U FWLFH P

F F
F F
L HOH H PL F F F
SUR L H E W H ROOH H R U RU T F
P SUHS U WLR H F HU UH LWLR O L WULE WLR L L
SHUPLWWH WR R OR W H P WHUL O LRO WLR R W H ROOH H R U
P H FRSLH WR H LW W HLU FRS UL W SROLFLH P UH OW
W H W L FO URRP HWWL R O L W H WHUPL WLR R U FWLFH P
R P L W L W H HF ULW R W L H P FFH RU R U F RRO HOO W H
WH F HU R O FROOHFW OO P WHUL O UHPR O R FFH WR RW HU R OL H
WHU W HLU PL L WU WLR HHS HU LFH F W H H F HU
W HP L HF UH ORF WLR RPP LW 2 OL H FRUH HSRUW

k P
V P V 9V
AP Central is the o cial online home for the AP Program: apcentral.collegeboard.org.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

DP ,Q W FW RQ

W G QW Q K W IR WK 0 W S KR F FW RQ

FW RQ , 0 W S KR F W RQ

FW RQ ,, SRQ W RQ

0 WS KR F Q .

R DP RN QP QW DQG DW RQD

SRQ FR Q G Q

FR Q :R N K W

W RQ F SWR DQG IR PDQF DWD

1RW K S E FDW RQ KR WK SD Q PE WKDW DSS D G Q

WK − P ERRN DQG Q WK DFW D DP
K S E FDW RQ D QRW SD QDW G WR E Q WK SD

k K R RD G R RD G G DQF G DF P QW R DP DQG WK DFR Q R R D

W G W DG PD N RI WK R RD G RWK S RG FW DQG F PD E W DG PD N RI WK
S FW R Q P RQ WR FRS KW G R RD G PDW D PD E T W G RQ Q DW
FR ERD G R T W IR P

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

®
AP Calculus BC Exam
SECTION I: Multiple Choice 2019
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.

Instructions
At a Glance
Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For
Total Time Part A, fill in only the circles for numbers 1 through 30 on the answer sheet. For Part B,
1 hour and 45 minutes fill in only the circles for numbers 76 through 90 on the answer sheet. Because Part A and
Number of Questions Part B offer only four answer options for each question, do not mark the (E) answer circle
45 for any question. The survey questions are numbers 91 through 94.
Percent of Total Score
50% Indicate all of your answers to the multiple-choice questions on the answer sheet. No
Writing Instrument credit will be given for anything written in this exam booklet, but you may use the booklet
Pencil required 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
Part A each question. If you change an answer, be sure that the previous mark is erased
Number of Questions completely. Here is a sample question and answer.
30
Time
1 hour
Electronic Device
None allowed

Part B
Number of Questions
15
Time
45 minutes Use your time effectively, working as quickly as you can without losing accuracy. Do not
Electronic Device spend too much time on any one question. Go on to other questions and come back to
Graphing calculator the ones you have not answered if you have time. It is not expected that everyone will
required 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 4PBP4-S

68

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

CALCULUS BC
SECTION I, Part A
Time—1 hour
Number of questions—30

NO CALCULATOR IS ALLOWED FOR 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 this exam booklet. 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).

-3- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

∫1 (4x )
2 3
1. − x dx =

27
(A) (B) 27 (C) 36 (D) 57
2

2. Let f be the function defined by f (x ) = x 3 − 3x 2 − 9x + 11. At which of the following values of x does f
attain a local minimum?

(A) 3 (B) 1 (C) −1 (D) −3

-4- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

( )
d
3. 2(sin x )2 =
dx
 1  2 sin x 2 sin x cos x
(A) 4 cos   (B) 4 sin x cos x (C) (D)
 2 x  x x

( )
4. The position of a particle is given by the parametric equations x(t ) = ln t 2 + 1 and y(t ) = e3−t . What is the
velocity vector at time t = 1 ?

1 2 1
(A) 1, e2 (B) 1, −e2 (C) ,e (D) , −e2
2 2

-5- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

∞
en
5. ∑ n is
n=1 p

p e e
(A) (B) (C) (D) divergent
p−e p−e
p ln ()
p
e

x f (x ) f ¢(x ) f ≤(x ) g(x ) g ¢(x ) g ≤(x )

2 4 −3 3 −2 5 1

6. The table above gives values of the twice-differentiable functions f and g and their derivatives at x = 2. If h is
f ¢(x )
the function defined by h(x ) = , what is the value of h¢(2) ?
g(x )
9 3 3 21
(A) (B) (C) − (D) −
4 5 2 4

-6- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

7. Which of the following is the Maclaurin series for x cos x 2 ? ( )

x5 x9 x13
(A) x − + − +L
2! 4! 6!
x3 x5 x7
(B) x − + − +L
2! 4! 6!
x 7 x11 x15
(C) x 3 − + − +L
3! 5! 7!
x5 x7 x9
(D) x 3 − + − +L
3! 5! 7!

10 − 6x 2
8. lim is
x→∞ 5 + 3e x
(A) −2 (B) 0 (C) 2 (D) nonexistent

-7- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

9. The function f is not differentiable at x = 5. Which of the following statements must be true?

(A) f is not continuous at x = 5.

(B) lim f (x ) does not exist.
x→5

f (x ) − f (5)
(C) lim does not exist.
x→5 x−5
5
(D) ∫0 f (x) dx does not exist.

10. The second derivative of a function f is given by f ≤(x ) = x(x − 3)5 (x − 10)2. At which of the following values
of x does the graph of f have a point of inflection?
(A) 3 only
(B) 0 and 3 only
(C) 3 and 10 only
(D) 0, 3, and 10

-8- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

p
⌠ ex − 1
11.  x dx =
⌡0 e − x

(A) e p − p − 1 (B) ln(e p − p) − 1 (C) p − ln p (D) ln(e p − p)

x 0 4 8 12 16
f (x ) 8 0 2 10 1

12. The table above gives selected values for the differentiable function f. In which of the following intervals must
there be a number c such that f ¢(c) = 2 ?

(A) (0, 4) (B) (4, 8) (C) (8, 12) (D) (12, 16)

-9- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

dy
13. Let y = f (x ) be the solution to the differential equation= x + 2y with initial condition f (0) = 2. What is
dx
the approximation for f (−0.4) obtained by using Euler’s method with two steps of equal length starting

at x = 0 ?

(A) 0.76 (B) 1.20 (C) 1.29 (D) 3.96

9 1 
14. What is the slope of the line tangent to the curve x + y = 2 at the point  ,  ?
 4 4 
1 4
(A) −3 (B) − (C) 1 (D)
3 3

-10- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

∞
⌠ 6
15.  dx is

⌡1 (x + 3) /
3 2

3
(A) (B) 3 (C) 6 (D) divergent
4

dy
16. If = 2 − y , and if y = 1 when x = 1, then y =
dx

(A) 2 − e x−1 (B) 2 − e1−x (C) 2 − e−x (D) 2 + e−x

-11- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

17. Which of the following series converges?

∞
1 − n 
(A) ∑ (−1)n  n 

n=1
∞
 n + 1
(B) ∑ (−1)n  
2n 
n=1
∞  2 
n n 
(C) ∑ (−1)  

n=1  3 n
∞
2 n 
(D) ∑ (−1)n  
n 
n=1

-12- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

18. Let R be the region in the first and second quadrants between the graphs of the polar curves f (q ) = 3 + 3 cos q
and g(q ) = 4 + 2 cos q , as shaded in the figure above. Which of the following integral expressions gives the
area of R ?

6
(A) ∫− 2 ( g(q ) − f (q)) dq
p
(B) ∫0 ( g(q ) − f (q)) dq
1 p
∫0 ( g(q ) − f (q))
2
(C) dq
2

∫0 (( g(q)) )
1 p 2
(D) − ( f (q ))2 dq
2

-13- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

∞
(− 1)n n
19. Which of the following statements about the series ∑ is true?
n=1 n2 + 1
∞
1
(A) The series can be shown to diverge by comparison with ∑ n.
n=1
∞
1
(B) The series can be shown to diverge by limit comparison with ∑ n.
n=1
∞
1
(C) The series can be shown to converge by comparison with ∑ 2
.
n=1 n
(D) The series can be shown to converge by the alternating series test.

-14- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

4 4
20. If ∫1 f (x ) dx = 8 and ∫1 g(x ) dx = −2, which of the following cannot be determined from the information
given?

1
(A) ∫4 g(x ) dx
4
(B) ∫1 3f (x) dx
4
(C) ∫1 3 f (x) g(x) dx
4
(D) ∫1 (3f (x) + g(x)) dx

-15- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

∞ (x + 4)n
21. Which of the following is the interval of convergence for the series ∑ ?
n=1 n ◊ 5n+1
(A) [−9, 1) (B) [−5, 5) (C) [1, 9) (D) (−∞, ∞)

p
22. What is the slope of the line tangent to the polar curve r = 3q at the point where q = ?
2
p 2
(A) − (B) − (C) 0 (D) 3
2 p

-16- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

4
23. The definite integral ∫0 x dx is approximated by a left Riemann sum, a right Riemann sum, and a trapezoidal
sum, each with 4 subintervals of equal width. If L is the value of the left Riemann sum, R is the value of the
right Riemann sum, and T is the value of the trapezoidal sum, which of the following inequalities is true?

4
(A) L < ∫0 x dx < T < R

4
(B) L < T < ∫0 x dx < R

4
(C) R < ∫0 x dx < T < L

4
(D) R < T < ∫0 x dx < L

-17- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

12
24. The graph of the piecewise linear function f is shown above. What is the value of ∫0 f ¢(x ) dx ?

(A) −8 (B) −6 (C) 0 (D) 22

-18- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

∫0 x ◊ f ¢(x) dx ?
2 2
25. The function f has a continuous derivative. If f (0) = 1, f (2) = 5, and ∫0 f (x ) dx = 7, what is

(A) 3 (B) 6 (C) 10 (D) 17

26. Which of the following series are conditionally convergent?

∞
(− 1)n
I. ∑
n=1 n!
∞
(− 1)n
II. ∑ n
n=1
∞
(− 1)n
III. ∑
n=1 n + 2

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

-19- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

27. Let R be the region in the first quadrant bounded by the graph of y = x − 1 , the x-axis, and the vertical line
x = 10. Which of the following integrals gives the volume of the solid generated by revolving R about the
y-axis?

10
(A) p ∫1 (x − 1) dx

10
(B) p ∫1 (100 − (x − 1)) dx

3
⌠
( ))
2
(C) p  10 − y 2 + 1
⌡0
( dy

3
⌠  2
(D) p 
⌡0  ( )  dy
100 − y 2 + 1

-20- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

d 2y
28. If
dx
dt
= 5 and
dy
dt
()
= sin t 2 , then 2 is
dx

()
2 t cos t 2 2t cos t 2 ()
()
(A) 2 t cos t 2
(B)
5
(C)
25
(D) undefined

-21- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

1 1 1 1 1 
29. Which of the following expressions is equal to lim  + + +L+ n  ?
n →∞ n  1 2 3 2 + 
 2 + 2 + 2 + n 
n n n
2
⌠ 1
(A)  dx
⌡1 x
1
⌠ 1
(B)  dx
⌡0 2 + x
2
⌠ 1
(C)  dx
⌡0 2 + x
3
⌠ 1
(D)  dx
⌡2 2 + x

-22- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

1 3
30. A function f has a Maclaurin series given by 2 + 3x + x 2 + x + L, and the Maclaurin series converges to
3
f (x ) for all real numbers x. If g is the function defined by g(x ) = e f (x), what is the coefficient of x 2 in the

Maclaurin series for g ?

1 2 5 2 11 2
(A) e (B) e2 (C) e (D) e
2 2 2

END OF PART A
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.

-23-

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B

CALCULUS BC
SECTION I, Part B
Time—45 minutes
Number of questions—15

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 this exam booklet. Do not spend too much time on any one
problem.

BE SURE YOU FILL IN THE CIRCLES ON THE ANSWER SHEET THAT CORRESPOND TO
QUESTIONS NUMBERED 76–90.

YOU MAY NOT RETURN TO QUESTIONS NUMBERED 1–30.

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).

-26- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B

76. The function f is continuous on the closed interval [0, 5]. The graph of f ¢, the derivative of f, is shown above.
On which of the following intervals is f increasing?

(A) [0, 1] and [2, 4]

(B) [0, 1] and [3, 5]
(C) [0, 1] and [4, 5] only
(D) [0, 2] and [4, 5]

dy 2
77. If = 6e−0.08 (t − 5) , by how much does y change as t changes from t = 1 to t = 6 ?
dt
(A) 3.870 (B) 8.341 (C) 18.017 (D) 22.583

-27- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B
78. The function f has the property that lim− f (x ) = +∞ , lim+ f (x ) = −∞ , lim f (x ) = 2, and lim f (x ) = 2 .
x→1 x→1 x→−∞ x→+∞
Of the following, which could be the graph of f ?

(A) (B)

(C) (D)

-28- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B
79. Tara’s heart rate during a workout is modeled by the differentiable function h, where h(t ) is measured in beats
per minute and t is measured in minutes from the start of the workout. Which of the following expressions gives
Tara’s average heart rate from t = 30 to t = 60 ?

60
(A) ∫30 h(t ) dt
1 60
(B)
30 ∫30 h(t) dt
1 60
(C)
30 ∫30 h¢(t) dt
h¢(30) + h¢(60)
(D)
2

-29- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B
80. The function f has derivatives of all orders for all real numbers with f (2) = −1, f ¢(2) = 4 , f ≤(2) = 6,
and f ¢¢¢(2) = 12 . Using the third-degree Taylor polynomial for f about x = 2, what is the approximation
of f (2.1) ?

(A) −0.570 (B) −0.568 (C) −0.566 (D) −0.528

81. Let f be a function with derivative given by f ¢(x ) = x 3 + 1 . What is the length of the graph of y = f (x) from
x = 0 to x = 1.5 ?

(A) 4.266 (B) 2.497 (C) 2.278 (D) 1.976

-30- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B
82. Let h be a continuous function of x. Which of the following could be a slope field for a differential equation of
dy
the form = h(x ) ?
dx
(A) (B)

(C) (D)

-31- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B

 3
 k + x for x < 3

f (x ) = 
 16

 2 for x ≥ 3

k − x


83. Let f be the function defined above, where k is a positive constant. For what value of k, if any, is f continuous?

(A) 2.081 (B) 2.646 (C) 8.550 (D) There is no such value of k.

84. Let f be a differentiable function. The figure above shows the graph of the line tangent to the graph of f
at x = 0 . Of the following, which must be true?

(A) f ¢(0) = − f (0)

(B) f ¢(0) < f (0)
(C) f ¢(0) = f (0)
(D) f ¢(0) > f (0)

-32- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B
85. A referee moves along a straight path on the side of an athletic field. The velocity of the referee is given by
v(t ) = 4(t − 6) cos(2t + 5), where t is measured in minutes and v(t) is measured in meters per minute. What is
the total distance traveled by the referee, in meters, from time t = 2 to time t = 6 ?

(A) 3.933 (B) 14.578 (C) 21.667 (D) 29.156

-33- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B

86. The graph of the function f shown above consists of three line segments. Let h be the function defined by
x
h(x ) = ∫0 f (t) dt . At what value of x does h attain its absolute maximum on the interval [−4, 3] ?
(A) −4 (B) −2 (C) 0 (D) 3

-34- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B
87. The position of a particle moving in the xy-plane is given by the parametric equations x(t ) = e− t and
y(t ) = sin(4t) for time t ≥ 0. What is the speed of the particle at time t = 1.2 ?

(A) 1.162 (B) 1.041 (C) 0.462 (D) 0.221

1 4 2 3 1 2 1
88. Let f be the function defined by f (x ) = x − x + x − x . For how many values of x in the open
4 3 2 2
interval (0, 1.565) is the instantaneous rate of change of f equal to the average rate of change of f on the closed

interval [0, 1.565] ?

(A) Zero (B) One (C) Three (D) Four

-35- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B
89. The population P of rabbits on a small island grows at a rate that is jointly proportional to the size of the rabbit
population and the difference between the rabbit population and the carrying capacity of the population. If the
carrying capacity of the population is 2400 rabbits, which of the following differential equations best models
the growth rate of the rabbit population with respect to time t, where k is a constant?

dP
(A) = 2400 − kP
dt
dP
(B) = k(2400 − P)
dt
dP 1
(C) = k (2400 − P)
dt P
dP
(D) = kP(2400 − P)
dt

90. A region is bounded by two concentric circles, as shown by the shaded region in the figure above. The radius of
the outer circle, R, is increasing at a constant rate of 2 inches per second. The radius of the inner circle, r, is
decreasing at a constant rate of 1 inch per second. What is the rate of change, in square inches per second, of
the area of the region at the instant when R is 4 inches and r is 3 inches?

(A) 3p (B) 6p (C) 10p (D) 22p

-36- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

B B B B B B B B B

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.

_______________________________________________________

MAKE SURE YOU HAVE DONE THE FOLLOWING.

• PLACED YOUR AP NUMBER LABEL ON YOUR ANSWER SHEET

• WRITTEN AND GRIDDED YOUR AP NUMBER CORRECTLY ON YOUR
ANSWER SHEET
• TAKEN THE AP EXAM LABEL FROM THE FRONT OF THIS BOOKLET
AND PLACED IT ON YOUR ANSWER SHEET

-37-

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

®
AP Calculus BC Exam
SECTION II: Free Response 2019
DO NOT OPEN THIS BOOKLET OR BREAK THE SEALS ON PART B UNTIL YOU ARE TOLD TO DO SO.

At a Glance
Total Time
1 hour and 30 minutes
Number of Questions
6
Percent of Total Score
50%
Writing Instrument
Either pencil or pen with
black or dark blue ink
Weight
The questions are
weighted equally, but
the parts of a question
are not necessarily
given equal weight.

Part A Instructions
Number of Questions The questions for Section II are printed in this booklet. Do not break the seals on Part B
2 until you are told to do so. Write your solution to each part of each question in the space
Time provided. Write clearly and legibly. Cross out any errors you make; erased or crossed-out
30 minutes
work will not be scored.
Electronic Device
Graphing calculator Manage your time carefully. During Part A, work only on the questions in Part A. You
required are permitted to use your calculator to solve an equation, find the derivative of a function
Percent of Section II Score at a point, or calculate the value of a definite integral. However, you must clearly indicate
33.33%
the setup of your question, namely the equation, function, or integral you are using. If you
Part B use other built-in features or programs, you must show the mathematical steps necessary
to produce your results. During Part B, you may continue to work on the questions in
Number of Questions
4 Part A without the use of a calculator.
Time As you begin each part, you may wish to look over the questions before starting to work
1 hour on them. It is not expected that everyone will be able to complete all parts of all questions.
Electronic Device
None allowed • Show all of your work, even though a question may not explicitly remind you to do so.
Percent of Section II Score Clearly label any functions, graphs, tables, or other objects that you use. Justifications
66.67% require that you give mathematical reasons, and that you verify the needed conditions
under which relevant theorems, properties, definitions, or tests are applied. Your work
will be scored on the correctness and completeness of your methods as well as your
answers. Answers without supporting work will usually not receive credit.
• Your work must be expressed in standard mathematical notation rather than calculator
5
syntax. For example, x2 dx may not be written as fnInt(X2, X, 1, 5).
1

• Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If you
use decimal approximations in calculations, your work will be scored 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.

Form I
Form Code 4PBP4-S

68
Downloaded by Jamilah Modahi (jnamqq@gmail.com)
 lOMoARcPSD|41419381

CALCULUS BC
SECTION II, Part A
Time—30 minutes
Number of questions—2

A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS.

-3- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

1 1 1 1 1 1 1 1 1 1

1. The rate at which cars enter a parking lot is modeled by E (t ) = 30 + 5(t − 2)(t − 5)e− 0.2t . The rate at
which cars leave the parking lot is modeled by the differentiable function L. Selected values of L(t ) are given
in the table above. Both E (t ) and L(t ) are measured in cars per hour, and time t is measured in hours after
5 A.M. (t = 0). Both functions are defined for 0 £ t £ 12.

(a) What is the rate of change of E (t ) at time t = 7 ? Indicate units of measure.

Do not write beyond this border.

Do not write beyond this border.

(b) How many cars enter the parking lot from time t = 0 to time t = 12 ? Give your answer to the nearest
whole number.

-4- Continue question 1 on page 5.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

1 1 1 1 1 1 1 1 1 1

(c) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate
12 12
∫2 L(t ) dt . Using correct units, explain the meaning of ∫2 L(t ) dt in the context of this problem.

Do not write beyond this border.

Do not write beyond this border.

(d) For 0 £ t < 6 , 5 dollars are collected from each car entering the parking lot. For 6 £ t £ 12, 8 dollars are
collected from each car entering the parking lot. How many dollars are collected from the cars entering the
parking lot from time t = 0 to time t = 12 ? Give your answer to the nearest whole dollar.

-5- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

2 2 2 2 2 2 2 2 2 2

2. A laser is a device that produces a beam of light. A design, shown above, is etched onto a flat piece of metal

using a moving laser. The position of the laser at time t seconds is represented by (x(t), y(t )) in the xy-plane.

Do not write beyond this border.

Do not write beyond this border.

Both x and y are measured in centimeters, and t is measured in seconds. The laser starts at position (0, 0) at
dx
time t = 0, and the design takes 3.1 seconds to complete. For 0 £ t £ 3.1,
dt
( )
= 3 cos t 2 and
dy
= 4 cos(2.5t).
dt

(a) Find the speed of the laser at time t = 3 seconds.

(b) Find the total distance traveled by the laser from time t = 1 to time t = 3 seconds.

-6- Continue question 2 on page 7.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

2 2 2 2 2 2 2 2 2 2

(c) The laser is farthest to the right at time t = 1.253 seconds. Find the x-coordinate of the laser’s rightmost
position.

(d) What is the difference between the y-coordinates of the laser’s highest position and lowest position

Do not write beyond this border.

for 0 £ t £ 3.1 ? Justify your answer.
Do not write beyond this border.

-7- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

END OF PART A
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.

-8-

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

CALCULUS BC
SECTION II, Part B
Time—1 hour
Number of questions—4

NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS.

DO NOT BREAK THE SEALS UNTIL YOU ARE TOLD TO DO SO.

-13-

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

3 3 3 3 3 3 3 3 3 3
NO CALCULATOR ALLOWED




 9 − x2 for −3 £ x £ 0

f (x ) =  px
−x + 3 cos



 2( ) for 0 < x £ 4

3. Let f be the function defined above.

(a) Find the average rate of change of f on the interval − 3 £ x £ 4.

Do not write beyond this border.

Do not write beyond this border.

(b) Write an equation for the line tangent to the graph of f at x = 3.

-14- Continue question 3 on page 15.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

3 3 3 3 3 3 3 3 3 3
NO CALCULATOR ALLOWED

(c) Find the average value of f on the interval − 3 £ x £ 4.

Do not write beyond this border.

Do not write beyond this border.

(d) Must there be a value of x at which f (x ) attains an absolute maximum on the closed interval
− 3 £ x £ 4 ? Justify your answer.

-15- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

4 4 4 4 4 4 4 4 4 4
NO CALCULATOR ALLOWED

Do not write beyond this border.

Do not write beyond this border.

4. The continuous function f is defined for −4 £ x £ 4. The graph of f, shown above, consists of two line
1 1 5
segments and portions of three parabolas. The graph has horizontal tangents at x = − , x = , and x = .
2 2 2
It is known that f (x ) = −x 2 + 5x − 4 for 1 £ x £ 4 . The areas of regions A and B bounded by the graph
x
of f and the x-axis are 3 and 5, respectively. Let g be the function defined by g(x ) = ∫− 4 f (t) dt .
(a) Find g(0) and g(4).

-16- Continue question 4 on page 17.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

4 4 4 4 4 4 4 4 4 4
NO CALCULATOR ALLOWED

(b) Find the absolute minimum value of g on the closed interval [−4, 4]. Justify your answer.

Do not write beyond this border.

Do not write beyond this border.

(c) Find all intervals on which the graph of g is concave down. Give a reason for your answer.

-17- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

5 5 5 5 5 5 5 5 5 5
NO CALCULATOR ALLOWED

5. The graph of the curve C, given by 4x 2 + 3y 2 + 6y = 9, is shown in the figure above.

Do not write beyond this border.

dy −4x
Do not write beyond this border.

(a) Show that = .

dx 3(y + 1)

d 2y
(b) Using the information from part (a), find in terms of x and y.
dx 2

-18- Continue question 5 on page 19.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

5 5 5 5 5 5 5 5 5 5
NO CALCULATOR ALLOWED

3 dr
(c) In polar coordinates, the curve C is given by r = for 0 £ q £ 2p . Find .
2 + sin q dq
As q increases, on what intervals is the distance between the origin and the point (r, q) increasing?

Give a reason for your answer.

Do not write beyond this border.

Do not write beyond this border.

(d) Let S be the region inside curve C, as defined in part (c), but outside the curve r = 2 . Write, but do not
evaluate, an integral expression for the area of S.

-19- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

6 6 6 6 6 6 6 6 6 6
NO CALCULATOR ALLOWED

∞ (−1)n+1(x − 3)n
6. Consider the series ∑ , where p is a constant and p > 0.
n=1 ◊
5n n p

(a) For p = 3 and x = 8, does the series converge absolutely, converge conditionally, or diverge? Explain
your reasoning.

Do not write beyond this border.

Do not write beyond this border.

(b) For p = 1 and x = 8, does the series converge absolutely, converge conditionally, or diverge? Explain
your reasoning.

-20- Continue question 6 on page 21.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

6 6 6 6 6 6 6 6 6 6
NO CALCULATOR ALLOWED

(c) When x = −2, for what values of p does the series converge? Explain your reasoning.

Do not write beyond this border.

Do not write beyond this border.

(d) When p = 1 and x = 3.1 , the series converges to a value S. Use the first two terms of the series to

approximate S. Use the alternating series error bound to show that this approximation differs from S by
1
less than .
300,000

-21- GO ON TO THE NEXT PAGE.

Downloaded by Jamilah Modahi (jnamqq@gmail.com)

 lOMoARcPSD|41419381

I
P ,

4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q 4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q
4 Q

Downloaded by Jamilah Modahi (jnamqq@gmail.com)