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2012 AP Calculus BC 模拟
试卷
云南省高中会考试卷哪里下载南京英语小升初试卷下载电路下试卷下载上海试卷下载口算试卷下载
北京新东方 罗勇
luoyong2@xdf.cn
2012-3-1
说明
关于失联党员情况说明岗位说明总经理岗位说明书会计岗位说明书行政主管岗位说明书
：请严格按照实际考试时间进行模拟，考试时间共 195分钟。
Multiple-Choice section 总计 45
题
快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题
/105分钟
A部分：无计算器 28题，55分钟
B部分：有计算器 17题，50分钟
每个选择题答对得 1分，不答得
0分，答错不扣分，卷面得分乘
以系数 1.2为最后得分。如果所
有题都对，则得 54分，占总分
的 50%。
Free-response section 总计 6题/90分钟
A部分：有计算器 2题，30分钟
B部分：无计算器 4题，60分钟
每题 9分，共 54分，占总分的
50%。
(2012年 5月 9日校正版)
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SECTION Ⅰ
Multiple- Choice Questions
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Calculus BC
SECTION Ⅰ, PART A
Time—55minutes
Number of questions—28
A calculator may not be used 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 place the letter of your choice in the corresponding box on the student answer
sheet. 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
notation 1f or with the prefix “arc” (e.g., 1sin arcsinx x ).
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1. Given 2 ( 2 )3 4 4xy e xy , then
dy
dx
(A)
( 2 )4
6
xe y
y x
(B)
( 2 )8
6
xe
y x
(C)
( 2 )8
6
xe
y x
(D)
( 2 )8
6
xe y
y x
(E)
( 2 )8
6
xe y
y x
________________________________________________________________________
2.
2012
2012
cos( )d x
dx
(A) cos x (B) cos x (C) sin x (D) sin x (E) 2sec x
________________________________________________________________________
3.
0
cot 5
lim( 1)
csc3x
x
x
(A)
5
3
(B)
2
3
(C)
8
5
(D) 1 (E) 0
________________________________________________________________________
4. What are all the values of x for which the series
1
n
n
x
n
converges?
(A) 1 1x
(B) 1 1x
(C) 1 1x
(D) 1 1x
(E) All real x
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5.
9
1
5
dx
x
(A) 90 (B) 89 (C) 30 (D) 20 (E) 10
________________________________________________________________________
6. The function is defined by the formula 2
0
( )
x
tg x e dt . What is the slope of the tangent
line at 1x ?
(A) 2e (B) 22e (C)
2
2
e
(D)
2 1
2
e
(E) 2 1e
________________________________________________________________________
7. The solution to the differential equation 10
dy
xy
dx
with the initial condition (0) 2y is
(A) 2ln(5 2)x
(B)
25 2xe
(C)
25 1xe
(D) 22 ln(5 )x
(E)
252 xe
________________________________________________________________________
8. Determine if the function ( ) 6f x x x satisfies the hypothesis of the MVT on the
interval [0,6] , and if does, find all numbers c satisfying the conclusion of that theorem.
(A) 2, 3c
(B) 4, 5c
(C) 5c
(D) 3c
(E) 4c
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9. If
2 3
( ) ... ...
2! 3! !
nx x x
f x x
n
and ( ) ( )F x f x dx and (0) 1F , then ( )F x
(A) xe
(B) 1xe
(C) xe x
(D) xe x
(E) 1xe x
________________________________________________________________________
10. In the right-triangle shown above, the angle is increasing at a constant rate of 2
radians per hour. At what rate is the side length of x increasing when 4x feet?
(A) 8 ft/hour
(B) 4 ft/hour
(C) 10 ft/hour
(D) 6 ft/hour
(E) 2 ft/hour
________________________________________________________________________
11. Which of the following integrals gives the length of the graph of tany x between
x a and x b , where 0
2
a b
?
(A) 2 2tan
b
a
x xdx
(B) tan
b
a
x xdx
(C) 21 sec
b
a
xdx
(D) 21 tan
b
a
xdx
(E) 41 sec
b
a
xdx
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12. The base of a solid S is the region enclosed by the graph of lny x , the line x e ,
and the x-axis. If the cross section of S perpendicular to the x-axis are squares, then the
volume of S is
(A)
1
2
(B)
2
3
(C) 1
(D) 2
(E) 3
1
( 1)
3
e
________________________________________________________________________
13. A circle, centered at (0, 3) , has a radius of 3. What is the polar representation of the
circle?
(A) 6sin cosr
(B) 3sin 3cosr
(C) sin 3r
(D) 6sinr
(E) 6cosr
________________________________________________________________________
14.
1
20
1
4
dx
x
(A) 0
(B) 4
(C)
6
(D)
3
(E) Divergence
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15. Suppose a continuous function f and its derivative 'f have values that are given in
the above table. What is the approximation for (3)f if Euler’s method is used with a step
of 0.5 , starting at 2x ?
(A) 4.5 (B) 5.3 (C) 5.4 (D) 5.5 (E) 5.8
________________________________________________________________________
16.
212 7
x
dx
x x
(A) 3ln(| 3 |) 4 ln(| 4 |)x x C
(B)
1
ln(( 4)( 3))
2
x x C
(C) 3ln(| 4 |) 4 ln(| 3 |)x x C
(D)
1
ln(| ( 4)( 3) |)
2
x x C
(E) 4 ln(| 4 |) 3ln(| 3 |)x x C
________________________________________________________________________
17. The position of a particle in the xy-plane is given by 2 1x t and ln(2 3)y t for all
0t . The acceleration vector of the particle is
(A)
2
4
2,
(2 3)t
(B)
2
2
2,
(2 3)t
(C)
2
4
2,
(2 3)t
(D)
2
2 ,
2 3
t
t
(E)
2
4
2 ,
(2 3)
t
t
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18. Suppose a function f is approximated with a fourth-degree Taylor polynomial about
1x . If the maximum value of the fifth derivative between 1x and 3x is 0.015, that
is, (5)| ( ) | 0.015f x , then the maximum error incurred using this approximation to
compute (3)f is
(A) 24.000 10
(B) 34.000 10
(C) 12.667 10
(D) 22.667 10
(E) 32.667 10
________________________________________________________________________
19. If sin( )y x x , then
2
2
d y
dx
(A) sin 2cosx x x
(B) sin cosx x x
(C) sin cosx x x
(D) 3sin cosx x x
(E) cos sinx x x
________________________________________________________________________
20. The graph of the piecewise function f is shown in the figure above. Which of the
following statements is true?
(A) f has a limit at 1x .
(B) f has an absolute minimum value at 2x .
(C) f is not continuous at 3x .
(D) f is differentiable at 3x .
(E) f is continuous on the interval [ 2,4] .
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21. The average value of 2( ) 3 1f x x x on [ 1,2] equals
(A) 3
(B) 2
(C) 1.5
(D) 1
(E) None of above
________________________________________________________________________
22. If 2( ) xf x xe , which of the following statements is correct?
(A) Absolute maximum at
1
( , )
2 2
e
(B) Absolute maximum at
1 1
( , )
2 2e
(C) Absolute minimum at
1
( , )
2 2
e
(D) Absolute minimum at
1 1
( , )
2 2e
(E) None of above
________________________________________________________________________
23. If ( ) 2 | 3 |f x x for all x , then the value of the derivative ( )f x at 3x is
(A) 1
(B) 0
(C) 1
(D) 2
(E) Nonexistent
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24. Which of the following series is absolutely convergent?
Ⅰ.
1
( 1) ( 1)n
n
n
n n
Ⅱ.
0
( 1)n n
n
e
Ⅲ.
2
1
( 1)
2
n
n
n
n
(A) None
(B) Ⅱ only
(C) Ⅲ only
(D) Ⅰ and Ⅱ only
(E) Ⅱ and Ⅲ only
________________________________________________________________________
25. The above graph of a function f consists of a semicircle and two line segments. Let
1
( ) ( )
x
g x f t dt , which of the following is true?
(A) (1) 1g
(B)
1
(2)
2
g
(C) ( 1)g
(D) The slope of the tangent line of function f at 1x is 1 .
(E) The function g has one relative minimum value on the interval [-1, 4].
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26. Which of the following is the coefficient of 4x in the Maclaurin polynomial
generated by
2
1
(1 )x
?
(A) 5
(B) 5
(C) 1
(D) 1
(E) 0
________________________________________________________________________
27. The figure above shows the graph of the polynomial function f . Which of the
following statements is false?
(A) ''( ) ( )f a f a
(B) ''( ) '( )f b f b
(C) '( ) ( )f c f c
(D) '( ) ( )f d f d
(E) ''( ) '( )f e f e
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28. Suppose g is the inverse function of a differentiable function f and
1
( )
( )
G x
g x
. If
(3) 7f and
1
'(3)
9
f , then '(7)G ?
(A) 5
(B) 4
(C) 6
(D) 1
(E) 4
________________________________________________________________________
END OF PART A OF SECTION Ⅰ
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WOK ON PART A ONLY.
DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.
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Calculus BC
SECTION Ⅰ, PART B
Time—50 minutes
Number of questions—17
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 place the letter of your choice in the corresponding box on the student answer
sheet. Do not spend too much time on any one problem.
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 values.
(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
notation 1f or with the prefix “arc” (e.g., 1sin arcsinx x ).
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76.
2
1
21
lim
1
x
t
x
e dt
x
(A) 0
(B) 1
(C)
2
e
(D)
2
e
(E) e
________________________________________________________________________
77. The function f is differentiable and ( ) tanf x x . If ( ) 0.01df x at 1x , then dx ?
(A) 31.000 10
(B) 32.919 10
(C) 35.403 10
(D) 37.081 10
(E) 32.916 10
________________________________________________________________________
78. If 35 40 ( )
x
c
x f t dt , what’s the value of c?
(A) 40 (B) 0 (C) 15 (D) 2 (E) 5
________________________________________________________________________
79. What are the coordinates of the inflection point on the graph of ( 1)arctany x x ?
(A) ( 1, 0)
(B) (0, 0)
(C) (0,1)
(D) (1, )
4
(E) (1, )
2
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80.
8 8
0
1 1
8 8
2 2
lim
h
h
h
?
(A) 0
(B) 1
(C)
1
2
(D)
1
2
(E) Cannot be determined from the information given
________________________________________________________________________
81. A particle moves along the x-axis so that any time 0t , its acceleration is given by
( ) cos(2 )a t t . If the position of the particle is ( ) 4x and the velocity of the particle is
( ) 0v at time t , what is the particle’s position at time
3
2
t
?
(A) 4.5 (B) 4.25 (C) 4 (D) 3.75 (E) 5.25
________________________________________________________________________
82. Which of the following differential equations would produce the slope field shown
above?
(A) 3
dy
y x
dx
(B)
3
dy x
y
dx
(C)
3
dy x
y
dx
(D)
3
dy y
x
dx
(E)
3
dy y
x
dx
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83. Which of the following is equal to the area of the region inside the polar curve
2cosr and outside the polar curve cosr ?
(A) 22
0
3 cos d
(B) 2
0
3 cos d
(C) 22
0
3
cos
2
d
(D) 2
0
3 cos d
(E)
0
3 cos d
________________________________________________________________________
84. Which of the following is an expression of Maclaurin series for 2( ) sinf x x ?
(A)
2 2 23 5 2 1
2 1( ) ... ( 1) ...
3! 5! (2 1)!
n
nx x xf x x
n
(B)
4 2 2
2 12 (2 )( ) 1 2 ... ( 1) ...
3 (2 2)!
n
nx xf x x
n
(C)
4 2 2
2 22 (2 )( ) 2 ... ( 1) ...
3 (2 2)!
n
nx xf x x
n
(D)
4 2 2
2 2 (2 )( ) ... ( 1) ...
3 2(2 2)!
n
nx xf x x
n
(E)
4 2 2
2 1 (2 )( ) 1 ... ( 1) ...
3 2(2 2)!
n
nx xf x x
n
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85. A population of animals is modeled by a function P that satisfies the logistic
differential equation 0.01 (100 )
dP
P P
dt
, where t is measured in years and P is
measured in millions. Which of the following equations is the solution of P as a function
of t if (0) 50P ?
(A)
1
( )
1 t
P t
e
(B)
100
( )
1 t
P t
e
(C)
0.01
1
( )
1 2 t
P t
e
(D)
0.01
100
( )
1 2 t
P t
e
(E)
0.01
100
( )
1 t
P t
e
________________________________________________________________________
86. What is the value of
3 3 3 3
... ...
10 100 1000 10n
?
(A) 0
(B)
1
3
(C) 1
(D) 3
(E) The series diverges.
________________________________________________________________________
87. If 2 cos ( ) 2 sinx xdx h x x xdx , then ( )h x ?
(A) 2sin 2 cosx x x C
(B) 2 sinx x C
(C) 22 cos sinx x x x C
(D) 4cos 2 sinx x x C
(E) 2(2 )cos 4sinx x x C
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88. The rate at which customers arrive at a counter to be served by the function F
defined by ( ) 12 6cos
t
F t
for 0 60t , where ( )F t is measured in customers
per minute and t is measured in minutes. To the nearest whole number, how many
customers arrive at the counter over the 60-minute period?
(A) 720 (B) 725 (C) 732 (D) 744 (E) 756
________________________________________________________________________
89. Let R be the region enclosed by the graphs of xy e , sin( 3 )y x and the y-axis.
Which of the following gives the approximate area of the region R ?
(A) 1.139 (B) 1.334 (C) 1.869 (D) 2.114 (E) 2.340
________________________________________________________________________
t (sec) 0 3 6 9 12 15
( )a t (ft/sec2) 4 8 6 9 10 10
90. The data for the acceleration ( )a t of a car from 0 to 15 seconds are given in the above
table. If the velocity at 0t is 5 ft/sec, which of the following gives the approximate
velocity at 15t using the Trapezoidal Rule?
(A) 47 ft/sec (B) 52 ft/sec (C) 120 ft/sec (D) 125 ft/sec (E) 141 ft/sec
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91. Which of the following is the solution to the differential equation 2sec
dy
y x
dx
,
where (0) 1y ?
(A) tan xy e
(B) tan xy e
(C) tan xy e
(D) tan 1xy e
(E) tan 1xy e
________________________________________________________________________
92. Let S be the region enclosed by the graphs of siny x and cosy x for
5
4
x
. What is the volume of the solid generated when S is revolved about the line
2y ?
(A)
5
24 (sin cos )x x dx
(B)
5
24 (sin cos 2)x x dx
(C)
5
2 24 [(sin 2) (cos 2) ]x x dx
(D)
5
2 24 [(cos 2) (sin 2) ]x x dx
(E)
5
2 24 [(sin 2) (cos 2) ]x x dx
________________________________________________________________________
END OF SECTION Ⅰ
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY
CHECK YOUR WOK ON PART B ONLY.
DO NOT GO ON TO SCTION Ⅱ UNTIL YOU ARE TOLD TO DO SO.
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SECTION Ⅱ
Free- Response Questions
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- 22 -
Calculus BC
SECTION Ⅱ, PART A
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