AP Calculus BC Released Exam 2003 Multiple-Choice Questions (College Board)

Name______________________Period ____

AP Calculus BC

2003 Calculus BC Multiple Choice Exam Part A

1) If sin(3 )y x= then dydx

=

(A) 3cos(3 )x−

(B) cos(3 )x−

(C) 1

cos(3 )3

x−

(D) cos(3 )x

(E) 3cos(3 )x

2) 20

cos 2lim2

x

x

e x xx x→

− −−

is

(A) 1

2−

(B) 0

(C) 1

2

(D) 1

(E) nonexistent

3) 5(3 1)x dx+ =∫

(A) 6(3 1)

18

xC

+ +

(B) 6(3 1)

6

xC

+ +

(C) 6(3 1)

2x

C+ +

(D)

263

( )2

2

xx

C+

+

(E) 2

53( )

2

xx C+ +

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4) For 0 13t≤ ≤ an object travels along an elliptical path given by the

parametric equations 3cosx t= and 4siny t= . At the point where 13t = ,

the object leaves the path and travels along the line tangent to the path at

that point. What is the slope of the line on which the object travels?

(A) 4

3−

(B) 3

4−

(C) 4 tan13

3−

(D) 4

3tan13−

(E) 3

4 tan13−

5) Let ( )y f x= be the solution to the differential equation dy x ydx

= + with

the initial condition (1) 2.f = What is the approximation for (2)f if

Euler’s method is used, starting at 1x = with a step size of 0.5?

(A) 3

(B) 5

(C) 6

(D) 10

(E) 12

6) What are all values of p for which 21

1p dx

x

∞∫ converges?

(A) 1p < −

(B) 0p >

(C) 1

2p >

(D) 1p >

(E) There are no values of p for which this integral converges.


7) The position of a particle moving in the xy-plane is given by the

parametric equations 3 23x t t= − and 3 22 3 12 .y t t t= − − For what values of

t is the particle at rest?

(A) 1only−

(B) 0 only

(C) 2 only

(D) 1 2and only−

(E) 1, 0, 2and−

8) ( )2 3cosx x dx =∫

(A) ( )31sin

3x C− +

(B) ( )31sin

3x C+

(C) ( )3

3sin3

xx C− +

(D) ( )3

3sin3

xx C+

(E) 3 4

sin3 4

x xC

+

9) If 3( ) ln( 4 ),xf x x e−= + + then (0)f ′ is

(A) 2

5−

(B) 1

5

(C) 1

4

(D) 2

5

(E) nonexistent


10) What is the value of 1

1

23n

nn

∞ +

=∑ ?

(A) 1

(B) 2

(C) 4

(D) 6

(E) The series diverges.

11) The Maclaurin series for 1

1 x− is

0.n

nx

∞

=∑ Which of the following is a

power series expansion for 2

21

x

x− ?

(A) 2 4 6 81 x x x x+ + + + + ⋅⋅⋅

(B) 2 3 4 5x x x x+ + + + ⋅⋅ ⋅

(C) 2 3 4 52 3 4x x x x+ + + + ⋅⋅⋅

(D) 2 4 6 8x x x x+ + + + ⋅⋅ ⋅

(E) 2 4 6 8x x x x− + − + ⋅⋅⋅

12) The rate of change of the volume, ,V of water in a tank with respect to

time, ,t is directly proportional to the square root of the volume. Which

of the following is a differential equation that describes this relationship?

(A) ( )V t k t=

(B) ( )V t k V=

(C) dV

k tdt

=

(D) dV k

dt V=

(E) dV

k Vdt

=


13) The graph of a function f is shown above. At which value of x is f

continuous, but not differentiable?

(A) a (B) b (C) c (D) d (E) e

−3 3

−3

3

x

y

14) Shown above is a slope field for which of the following differential

equations?

(A) dy x

dx y=

(B) 2

2

dy x

dx y=

(C) 3dy x

dx y=

(D) 2dy x

dx y=

(E) 3

2

dy x

dx y=

x

y

a b c d eO d

3

3

3−

3−

x

y


15) The length of a curve from 1x = to 4x = is given by 4 4

11 9 .x dx+∫ If

the curve contains the point (1,6), which of the following could be an

equation for this curve?

(A) 23 3y x= +

(B) 35y x= +

(C) 36y x= +

(D) 36y x= −

(E) 516 9

5 5y x x= + +

16) If the line tangent to the graph of the function f at the point (1,7)

passes through the point ( 2, 2),− − then (1)f ′ is

(A) 5−

(B) 1

(C) 3

(D) 7

(E) undefined

17) A curve C is defined by the parametric equations 2 4 1x t t= − + and 3.y t= Which of the following is an equation of the line tangent to the

graph of C at the point ( 3,8)− ?

(A) 3x = −

(B) 2x =

(C) 8y =

(D) 27

( 3) 810

y x= − + +

(E) 12( 3) 8y x= + +


1 2 3 4 5 6 7

−3

−2

−1

1

2

3

4

5

6

7

x

y

Graph of f

18) The graph of the function f shown in the figure above has horizontal

tangents at 3x = and 6.x = If 2

0( ) ( ) ,

xg x f t dt= ∫ what is the value of (3)g ′ ?

(A) 0

(B) 1−

(C) 2−

(D) 3−

(E) 6−


19) A curve has slope 2 3x + at each point ( , )x y on the curve. Which of

the following is an equation for this curve if it passes through the point

(1,2) ?

(A) 5 3y x= −

(B) 2 1y x= +

(C) 2 3y x x= +

(D) 2 3 2y x x= + −

(E) 22 3 3y x x= + −

20) A function f has Maclaurin series given by 4 5 6 3

.2! 3! 4! ( 1)!

nx x x x

n

+

+ + + ⋅⋅⋅ + + ⋅⋅⋅+

Which of the following is an expression for ( )f x ?

(A) ( ) 23 sin 3x x x− +

(B) ( )2cos 1x− +

(C) ( )2 2cosx x x− +

(D) 2 3 2xx e x x− −

(E) 2 2 1xe x− −

21) The number of moose in a national park is modeled by the function

M that satisfies the logistic differential equation 0.6 (1 ),200

dM MM

dt= − where

t is the time in years and (0) 50.M = What is lim ( )t

M t→∞

?

(A) 50

(B) 200

(C) 500

(D) 1000

(E) 2000


22) What are all values of p for which the infinite series 1 1p

n

nn

∞

= +∑

converges?

(A) 0p >

(B) 1p ≥

(C) 1p >

(D) 2p ≥

(E) 2p >

23) ( )sin 6x x dx =∫

(A) cos(6 ) sin(6 )x x x C− + +

(B) 1

cos(6 ) sin(6 )6 36

xx x C− + +

(C) 1

cos(6 ) sin(6 )6 6

xx x C− + +

(D) 1

cos(6 ) sin(6 )6 36

xx x C+ +

(E) 6 cos(6 ) sin(6 )x x x C− +

24) Which of the following series diverge?

0

31

1

sin2.

1.

.1

n

n

n

n

nn

I

IIn

eIIIe

π∞

=∞

=∞

=

+

∑

∑

∑

(A) III only

(B) I and II only

(C) I and III only

(D) II and III only

(E) I, II, and III


2 5 10 14

( ) 12 28 34 30

x

f x

25) The function f is continuous on the closed interval [2,14] and has

values as shown in the table above. Using the subintervals [2,5], [5,10],

and [10,14], what is the approximation of 14

2( )f x dx∫ found by using a

right Riemann sum?

(A) 296

(B) 312

(C) 343

(D) 374

(E) 390

26) 2

( 2)( 1)x dx

x x=

+ +∫

(A) ln | 2 | ln | 1|x x C+ + + +

(B) ln | 2 | ln | 1| 3x x x C+ + + − +

(C) 4ln | 2 | 2ln | 1|x x C− + + + +

(D) 4ln | 2 | 2ln | 1|x x C+ − + +

(E) 22 12ln | |

3 2x x x C+ + +


27) 3

2

0ln( 1)

xd t dtdx

+ =∫

(A) 3

6

2

1

x

x +

(B) 2

6

3

1

x

x +

(C) 6ln( 1)x +

(D) 3 62 ln( 1)x x +

(E) 2 63 ln( 1)x x +

28) What is the coefficient of 2x in the Taylor series for 21

(1 )x+ about

0x = ?

(A) 1

6

(B) 1

3

(C) 1

(D) 3

(E) 6


Name ______________________ Period _____

2003 Calculus BC Multiple Choice Exam Part B

76) The graph of the function f is shown above. Which of the following

statements must be false?

(A) ( )f a exists

(B) ( )f x is defined for 0 x a< <

(C) f is not continuous at x a=

(D) lim ( )x a

f x→

exists

(E) lim '( )x a

f x→

exists

77) Let 2 3 4 5( ) 3 5 7 3P x x x x x= − + + be the fifth-degree Taylor polynomial

for the function f about 0.x = What is the value of (0)f ′′′ ?

(A) 30−

(B) 15−

(C) 5−

(D) 5

6−

(E) 1

6−

x

y

aO

Graph of f


78) The radius of a circle is increasing at a constant rate of 0.2 meters per

second. What is the rate of increase in the area of the circle at the instant

when the circumference of the circle is 20π meters?

(A) 20.04 /secmπ

(B) 20.4 /secmπ

(C) 24 /secmπ

(D) 220 /secmπ

(E) 2100 / secmπ

( ) ( ) ( ) ( )

1 6 5 3 2

1 3 3 1 2

3 1 2 2 3

x f x f x g x g x′ ′

− −

− −

−

79) The table above gives values of , , ,f f g′ and g′ at selected values of

.x If ( ) ( ( )),h x f g x= then (1)h′ =

(A) 5

(B) 6

(C) 9

(D) 10

(E) 12

80) Insects destroyed a crop at the rate of 0.1

31002

t

tee

−

−− tons per day, where

time t is measured in days. To the nearest ton, how many tons did the

insects destroy during the time interval 7 14t≤ ≤ ?

(A) 125

(B) 100

(C) 88

(D) 50

(E) 12


−1 1 2 3

−1

1

2

3

4

81) The graph of the function f is shown in the figure above. The value

of ( )1

lim sin ( )x

f x→

is

(A) 0.909

(B) 0.841

(C) 0.141

(D) 0.416−

(E) nonexistent

82) The rate of change of the altitude of a hot-air balloon is given by 3 2( ) 4 6r t t t= − + for 0 8.t≤ ≤ Which of the following expressions gives

the change in altitude of the balloon during the time the altitude is

decreasing?

(A) 3.514

1.572( )r t dt∫

(B) 8

0( )r t dt∫

(C) 2.667

0( )r t dt∫

(D) 3.514

1.572( )r t dt′∫

(E) 2.667

0( )r t dt′∫

Graph of f

x

y

O


0 1 2 3 4

( ) 2 3 4 3 2

x

f x

83) The function f is continuous and differentiable on the closed interval

[0,4]. The table above gives selected values of f on this interval. Which

of the following statements must be true?

(A) The minimum value of f on [0,4] is 2.

(B) The maximum value of f on [0,4] is 4.

(C) ( ) 0f x > for 0 4x< <

(D) ( ) 0f x′ < for 2 4x< <

(E) There exists c , with 0 4,c< < for which ( ) 0.f c′ =

84) A particle moves in the xy-plane so that its position at any time t is

given by 2( )x t t= and sin(4 ).( ) ty t = What is the speed of the particle when

3t = ?

(A) 2.909

(B) 3.062

(C) 6.884

(D) 9.016

(E) 47.393


85) If a trapezoidal sum overapproximates 4

0,( )f x dx∫ and a right

Riemann sum underapproximates 4

0,( )f x dx∫ which of the following

could be the graph of ( )y f x= ?

(A) 1 2 3 4

1

2

3

4

x

y

(B) 1 2 3 4

1

2

3

4

x

y

(C) 1 2 3 4

1

2

3

4

x

y

(D) 1 2 3 4

1

2

3

4

x

y

(E) 1 2 3 4

1

2

3

4

x

y


86) Let f be the function with derivative defined by ( )3( ) sinf x x′ = on

the interval 1.8 1.8.x− < < How many points of inflection does the graph

of f have on this interval?

(A) Two

(B) Three

(C) Four

(D) Five

(E) Six

87) A particle moves along the x-axis so that at any time 0,t ≥ its velocity

is given by ( )22 .( ) cos tv t −= The position of the particle is 3 at time 0.t =

What is the position of the particle when its velocity is first equal to 0?

(A) 0.411

(B) 1.310

(C) 2.816

(D) 3.091

(E) 3.411


88) On the closed interval [2,4], which of the following could be the

graph of a function f with the property that 4

2

1 ( ) 14 2

f t dt =− ∫ ?

(A) 1 2 3 4

1

2

3

4

x

y

(B) 1 2 3 4

1

2

3

4

x

y

(C) 1 2 3 4

1

2

3

4

x

y

(D) 1 2 3 4

1

2

3

4

x

y

(E) 1 2 3 4

1

2

3

4

x

y


89) The region bounded by the graph of 22y x x= − and the x-axis is the

base of a solid. For this solid, each cross-section perpendicular to the

x-axis is an equilateral triangle. What is the volume of this solid?

(A) 1.333

(B) 1.067

(C) 0.577

(D) 0.462

(E) 0.267

90) The graph of ',f the derivative of the function ,f is shown above. If

(0) 0,f = which of the following must be true?

I. (0) (1)

II. (2) (1)

III. (1) (3)

f f

f f

f f

>>>

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) II and III only

y

x1 2 3 4


91) The height ,h in meters, of an object at time t is given by 3 2 2.( ) 24 24 16h t t t t= + − What is the height of the object at the instant when

it reaches its maximum upward velocity?

(A) 2.545 meters

(B) 10.263 meters

(C) 34.125 meters

(D) 54.889 meters

(E) 89.005 meters

92) Let f be the function defined by .( ) lnf x x x= + What is the value of

c for which the instantaneous rate of change of f at x c= is the same as

the average rate of change of f over [1,4] ?

(A) 0.456

(B) 1.244

(C) 2.164

(D) 2.342

(E) 2.452


AP Calculus BC Released Exam 2003 Multiple-Choice Questions (College Board)

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