Post on 14-Apr-2015
description
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+ +
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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.
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
=
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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= + +
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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−
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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+ + +
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
−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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
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
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com