[PDF] Sample AP Calculus AB and BC Exam Questions

15317_62019_AP_Calculus_AB_and_BC_Practice_Exam_And_Answers.pdf

Sample AP Calculus

AB and BC Exam

Questions

Section I: Multiple-Choice

ǥǦ

1. 1c-os 2 (2x)lim x0 (2x) 2 = (A) 1 4 (C) 1 2 (D)

2for1x<-xfx()=xx

2 --3for12

43xx->for2

2.f be the function de?ned above. At what values of x, if any, is f not

di?erentiable? x = -1 only x = 2 only x = -1 and x = -2 f is di?erent iable for all values of x.

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2283/12/19 1:44 PM

̝

Course and Exam Description

© 2019 College Board

̝

̞|̞228

x(x)fʹ(x)g(x)gʹ(x)  --  -8

3. f and g and their

derivatives at selected values of x. If h is the function de?ned by h(x) = f(x)g(x) + 2g(x), then hʹ (1) = -6 -16 dyx 3 - 2xy + 3y 2 = 7, then = dx (A) 34xy
2 + 2x (B) 32xy
2 26xy
3x 2 26xy-
(D) 3x 2 26-y
5. r and height h of the cylinder? V of a cylinder with radius r and height h is V = πr 2 h.) -20πr (B)-2πrh (C)4πrh - 5πr 2

πrh + 5πr

2 - -

̝̞|̞AP Calculus AB and BC̝

00762-114-CED-Calculus-AB/BC_Exam Information.indd 229

© 2019 College Board

44
n k n 1 k n 1 1 6.

6xdx?2

(A) (B) (C) (D)

Graph of g12

O345678

7.g on the interval

[0, 8]. Let h be the func tion de?ned by xh(x) = gt()dt. 3On what intervals is h increasing? xdx=19-x 2 (A)--19xC 2 +9 (B)

1--ln19

xC 2 +18 (C)

1arcsin(3xC)+3

(D) xarcsin(3xC)+3

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2303/12/19 1:44 PM

̝

Course and Exam Description

© 2019 College Boar

d ̝

̞|̞230

y x 2 -edy y xdxl)- 3 -1 -2 -33 2 1 xy

21-1-2-3

9. dyy-2=dx2 (B) dyy 2 -4=dx4 (C) dyx-2=dx2 (D) dyx 2 -4=dx4

10.R be the region bounded by the graph of x = e

y , x = 10, and the horizontal lines y = 1 and y = 2. Which of the following gives the area of R? 2edy1 (B) e lnxde (C)

2(10)1

(D) 10(n1 e

̝̞|̞̝Course and Exam Description

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2313/12/19 1:44 PM

© 2019 College Board

+

ǩǪ

Graph of fx

4 3 2 1 -1 -2-1-2-3-41234 -3 -4O y

11.f is shown in the ?gure above. ?e value

of lim(fx) x1 is -2 -1 v(t) = 1.3tln (0.2 t + 0.4) for time t ≥

0. t = 1.2?

-0.580 -0.548 -0.093

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2323/12/19 1:44 PM

̝

Course and Exam Description

© 2019 College Boar

d ̝

̞|̞232

x- fʹ ( x ) 9

13.f be a twice-di?erentiable function. Values of fʹ,f, at

selected values of x are given in the table above. Which of the following statements must be true? f is increasing for -1 ≤ x ≤ 5. f is con cave down for -1 < x < 5. c, where x  f ( x ) 7 fʹ ( x ) 

3 -1 < c < 5, such that fc()=-.2

(D)c, where3-1 < c < 5, such that fc()=-.2

14.f be t

he function with derivative de?ned by fx()=+2(28xx-+)sin(3). f have on the interval 0 < x < 9 ? r(t) = 4e -0.35t ounces per minute, where t is measured in minutes. How many ounces of honey are poured through the funnel from time t = 0 to time t = 3?

ǥǦ

16.f and its derivative fʹ

x.

5fxdx=14,2 what is the value of 5xf()xdx?2

(A) 63
2 (D) ()

̝̞|̞AP Calculus AB and BC̝

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2333/12/19 1:44 PM

© 2019 College Board

1! xxx

17.F that satis?es the

logistic di?eren tial equation dFF=-0.

04F1,dt5000

w here t is the time in mont hs and F(0) = 2000. lim(Ft)? t (A) x(t) = t 2 + 3 and y(t) = sin (t 2 ). dy 2 dx 2 in terms of t ? -sin (t 2 ) -2tsin (t 2 ) t 2 ) - 2t 2 sin ( t 2 ) t 2 ) - 4t 2 sin ( t 2 ) k 5(1-) k k 3 =1 +1 (B) k 5(1-) k=1 k+1 (C) k 5k (1-) k=1 k+1 (D) 2 k 5k (1-) k=1 k+1

20. f be the fun

ction de?ned by f(x) = e 2x . fʹ , f ? xx 23
x n ++x++++2!3n! (B) 2xx 23
2 2x n

22++x++++2!3!n!

(C) (2xx) 23
(2)(2x) n

12++x++++2!3!n!

(D) 2(2) 23

2(2)2(2)

n 22++(2x)++++2!3!n!

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2343/12/19 1:44 PM

̝

Course and Exam Description

© 2019 College Boar

d ̝

̞|̞234

ǥǦ

y x O

21. r = 2 + 4sin θ.

2.174 2.739 37.699
?e function f has deriva tives of all orders for all real numbers. It is known that (4 12fx )

3() and fx

(5) ()for0x2. Let 52
Px 4 () f about x = 0. ?e Taylor series for f about x = 0 converges at x = 2. Of the following, which is the smallest value of k for which the Lagrange error bound guarantees that fP(2)( 4 2)k? 2 5 3 5!2 (B) 2 5 12 5!5 (C) 2 4 3 4!2 (D) 2 4 12 4!5 ̝ ̞|̞AP Calculus AB and BC̝Course and Exam Description

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2353/12/19 1:44 PM

© 2019 College Boar

d

Section II: Free-Response

ǥǦ

t (hours) 6 R(t) (vehicles per hour)  1.

R for 0 ≤ t ≤ 12, where R(t)

t is the number of hours since 7:00 a.m. (t = 0). Values of R(t) t are given in the table above. Rʹ (5). Rʹ (5) 12 ∫

Rt()dt.0

H de?ned by H(t) = -t

3 - 3t 2 + 288t + 1300 for

0 ≤ t ≤ 17,H(t) t is the number

of hours since 7:00 a.m. (t = 0). According to this model, what is the average number of vehicles crossing the bridge per hour on the weekend day for 0 ≤ t ≤ 12? < t < 17, L(t),

H given

in part (c) at t = 12, is a better model for the rate at which vehicles cross the bridge on the weekend day. Use

L(t)t, for 12 < t < 17,

00762-114-CED-Calculus-AB/BC_Exam Information.indd 236

̝

Course and Exam Description

© 2019 College Boar

d ̝

̞|̞236

ǥǦ

1O234xy

Graph of f´

2. fʹ,

f, on the closed interval [0, 4]. ?e areas of the regions bounded by the graph of fʹ x-axis on the intervals [0, 1], [1, 2], [2, 3], and [3, 4] are 2, 6, 10, and 14, respectively. ?e graph of fʹ has horizontal tangents at x = 0.6, x = 1.6, x = 2.5, and x = 3.5. It is known that f(2) = 5. On what open interva ls contained in (0, 4) is the graph of f both decreasing an d concave down? Give a reason for your answer. Find the absolut e minimum value of f on the interval [0, 4]. Justify your answer. Evaluate 4fx()fx()dx.0(d) ?e function g is de?n ed by g(x) = x 3 f(x). Find gʹ (2).

ǥǦ

3. For 0 ≤ t ≤ 5, t

is (x(t), y(t))t = 1, the par ticle is at position (2, -7). dxtdy=sin and =e cost .dtt+3dt (a) -7). Find the y-coordina te of the position of the particle at time t = 4. t = 1 to time t = 4. Find the time at which t he speed of the particle is 2.5. Find the acceleration vector of the par ticle at this time. ̝ ̞|̞AP Calculus AB and BC̝Course and Exam Description

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2373/12/19 1:44 PM

© 2019 College Boar

d

ǥǦ

4. ?e Maclaurin series for the function f is given by

(1-) kk+1 xxx 23
f (x) = x k k 2 =-+ =1 49
-on its interval of convergence. f. Show the work that leads to your answer. ?e Maclaurin series fo r f evaluated at

1x=4is an alternating series whose

terms decrease in absolute value to 0. ?e approximation for

1f4 using

the ?rst two nonzero terms of this series is

15.64

11f by less than .4500

(c) Let h be the function de?ned by xhx()=ft()dt.0 h.

00762-114-CED-Calculus-AB/BC_Exam Information.indd 2383/12/19 1:44 PM

̝

Course and Exam Description

© 2019 College Boar

d ̝

̞|̞238

Answer Key and Question

Alignment to Course Framework

Multiple-

Choice

Question

1

Answer

Unit 1

Free-Response

QuestionUnit

1

CHA-2.D, CHA-3.A,

CHA-3.C, CHA-3.F,

CHA-4.B, LIM-5.A

CHA-3.G, FUN-8.B

̝

̞|̞

00762-114-CED-Calculus-AB/BC_Exam Information.indd 239

AP Calculus AB and BC

̝

Course and Exam Description

© 2019 College Boar

d