新东方罗勇-2012-AP微积分BC模考试卷

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2012 AP Calculus BC 模拟试卷

北京新东方罗勇 [email protected]

2012-3-1

说明：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/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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GO ON TO THE NEXT PAGE - 3 -

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 ).



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 5lim( 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



5. 9

1

5dxx

(A) 90 (B) 89 (C) 30 (D) 20 (E) 10 ________________________________________________________________________

6. The function is defined by the formula 2

0( )

xtg 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 10dy

xydx

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



9. If 2 3

( ) ... ...2! 3! !

nx x xf 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 02

a b

?

(A) 2 2tanb

ax xdx

(B) tanb

ax xdx

(C) 21 secb

axdx

(D) 21 tanb

axdx

(E) 41 secb

axdx



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) 31( 1)

3e

________________________________________________________________________ 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

4dx

x

(A) 0 (B) 4

(C) 6

(D) 3

(E) Divergence



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

xdx

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

42,

(2 3)t

(B) 2

22,

(2 3)t

(C) 2

42,

(2 3)t

(D) 2

2 ,2 3

tt

(E) 2

42 ,

(2 3)t

t



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] .



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



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

nn

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].



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



28. Suppose g is the inverse function of a differentiable function f and 1

( )( )

G xg 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


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 ).



76.

2

121

lim1

xt

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

cx 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



80.

8 8

0

1 18 8

2 2limh

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

2t

?

(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) 3dy

y xdx

(B) 3

dy xy

dx

(C) 3

dy xy

dx

(D) 3

dy yx

dx

(E) 3

dy yx

dx



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

03 cos d

(B) 2

03 cos d

(C) 22

0

3cos

2d

(D) 2

03 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 12 1( ) ... ( 1) ...

3! 5! (2 1)!

nnx x x

f x xn

(B) 4 2 2

2 12 (2 )( ) 1 2 ... ( 1) ...

3 (2 2)!

nnx x

f x xn

(C) 4 2 2

2 22 (2 )( ) 2 ... ( 1) ...

3 (2 2)!

nnx x

f x xn

(D) 4 2 2

2 2 (2 )( ) ... ( 1) ...

3 2(2 2)!

nnx x

f x xn

(E) 4 2 2

2 1 (2 )( ) 1 ... ( 1) ...

3 2(2 2)!

nnx x

f x xn



85. A population of animals is modeled by a function P that satisfies the logistic

differential equation 0.01 (100 )dP

P Pdt

, 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 te

(B) 100

( )1 t

P te

(C) 0.01

1( )

1 2 tP t

e

(D) 0.01

100( )

1 2 tP t

e

(E) 0.01

100( )

1 tP 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



88. The rate at which customers arrive at a counter to be served by the function F

defined by ( ) 12 6cost

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



91. Which of the following is the solution to the differential equation 2secdy

y xdx

,

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

4x

. 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.


SECTION Ⅱ

Free- Response Questions


Calculus BC

SECTION Ⅱ, PART A

Time—30 minutes


A graphing calculator is required for some problems or parts of problems.

________________________________________________________________________

1. An object moving along a curve in the xy-plane has position ( ), ( )x t y t at time t with

2 sin( )tdy

edt . The derivative of

dx

dt is not explicitly given. At 3t , the object is at the

point (4, 5) .

(a) Find the y-coordinate of the position at time 1t .

(b) At time 3t , the value of dy

dx is 1.8 . Find the value of

dx

dt when 3t .

(c) Find the speed of the object at 3t .


2. Let S be the region in the first quadrant bounded by the graphs of 2xy e and

22y x and the y-axis.

(a) Find the area of the region S . (b) Find the volume of the solid generated when the region S is rotated about the x-axis. (c) The region S is the base of a solid for which each cross section perpendicular to the x-axis is a semi-circle with diameter in the xy-plane. Find the volume of this solid.


Calculus BC

SECTION Ⅱ, PART B

Time—60 minutes


No calculator is allowed for these problems. ________________________________________________________________________

3. The graphs of the polar curves 1 2 2sinr and 2 1r for 0 2 are shown

above:

(a) Write an integral expression for the area inside 1r and outside 2r .

(b) Write expressions for dx

d and

dy

d in terms of for the 1r curve.

(c) Write an equation in terms of x and y for the line tangent to the graph of the polar

curve 1r at the point where 0 .


4. The graph of the function f , consisting of three segments, is shown above. Let

1( ) ( )

x

g x f t dt .

(a) Find (1)g and '(1)g .

(b) Find the instantaneous rate of change of g , which respect to x , at 2x .

(c) Find the absolute minimum value of g on the closed interval [ 2, 4] . Justify your

answer. (d) The second derivative of g is not defined at 1x and 2x . Which of these values

are x-coordinates of points of inflection of the graph of g ? Justify your answer.


5. Consider the differential equation 2 4dy

y xdx

.

(a) The slope field for the differential equation is shown above. Sketch the solution curve that passes through the point (0, 1) and sketch the solution curve that passes through the point (0, -1). (b) There is a value of b for which 2y x b is a solution to the differential equation.

Find this value of b . Justify your answer. (c) Let g be the function that satisfies the given differential equation with the initial

condition (0) 0g . It appears from the slope field that g has a local maximum at the

point (0, 0). Using the differential equation, prove analytical that this is so. (d) Write the second-degree Taylor polynomial for g about 0a .

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6. Let f be the function given by 22( ) xf x e

(a) Find the first four nonzero terms and the general term of the power series for ( )f x

about 0x . (b) Find the interval of convergence of the power series for ( )f x about 0x . Show the

analysis that leads to your conclusion. (c) Let g be the function given by the sum of the first four nonzero term of the power

series for ( )f x about 0x . Show that 6| ( ) ( ) | 1.8 10f x g x for 0.2 0.2x .

STOP

END OF EXAM

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新东方罗勇-2012-AP微积分BC模考试卷