2012 AP Calculus BC 模拟试卷

Transcription

1 0 AP Calculus BC 模拟试卷 北京新东方罗勇 0-3- 说明 : 请严格按照实际考试时间进行模拟, 考试时间共 95 分钟 Multiple-Choice section A 部分 : 无计算器 B 部分 : 有计算器 Free-response section A 部分 : 有计算器 B 部分 : 无计算器 总计 45 题 /05 分钟 8 题,55 分钟 7 题,50 分钟总计 6 题 /90 分钟 题,30 分钟 4 题,60 分钟 每个选择题答对得 分, 不答得 0 分, 答错不扣分, 卷面得分乘以系数. 为最后得分 如果所有题都对, 则得 54 分, 占总分的 50% 每题 9 分, 共 54 分, 占总分的 50% (0 年 5 月 9 日校正版 ) - -

2 SECTION Ⅰ Multiple- Choice Questions - -

3 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Calculus BC SECTION Ⅰ, PART A Time 55minutes Number of questions 8 A calculator may not be used on this part of the eam. Directions: Solve each of the following problems, using the available space for scratch work. After eamining the form of the choices, decide which is the best of the choices given and place the letter of your choice in the corresponding bo on the student answer sheet. Do not spend too much time on any one problem. In this eam: () Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f ( ) is a real number. () The inverse of a trigonometric function f may be indicated using the inverse notation f or with the prefi arc (e.g., sin arcsin )

4 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ( ). Given 3y 4e y 4, then dy d ( ) 4e y (A) 6y (B) (C) (D) (E) 8e 6y ( ) ( ) 8e 6y ( ) 8e y 6y ( ) 8e y 6y 0 d cos( ). 0 d (A) cos (B) cos (C) sin (D) sin (E) sec cot 5 3. lim( ) 0 csc3 (A) 5 (B) (C) 8 (D) (E) n 4. What are all the values of for which the series converges? n n (A) (B) (C) (D) (E) All real - 4 -

5 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA d (A) 90 (B) 89 (C) 30 (D) 0 (E) 0 t 6. The function is defined by the formula g( ) e dt. What is the slope of the tangent 0 line at? e e (A) e (B) e (C) (D) (E) e dy 7. The solution to the differential equation 0y with the initial condition (0) d y is (A) ln(5 ) 5 (B) e 5 (C) e (D) ln(5 ) 5 (E) e 8. Determine if the function f ( ) 6 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) c, 3 (B) c 4, 5 (C) c 5 (D) c 3 (E) c 4-5 -

6 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 3 n 9. If f ( ) and F( ) f ( ) d! 3! n! and F(0), then F( ) (A) e (B) e (C) e (D) e (E) e 0. In the right-triangle shown above, the angle is increasing at a constant rate of radians per hour. At what rate is the side length of increasing when 4 feet? (A) 8 ft/hour (B) 4 ft/hour (C) 0 ft/hour (D) 6 ft/hour (E) ft/hour. Which of the following integrals gives the length of the graph of y tan between a and b, where 0 a b? b (A) tan d (B) (C) (D) (E) a b a b a b a b a tan d sec tan sec 4 d d d - 6 -

7 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. The base of a solid S is the region enclosed by the graph of y ln, the line e, and the -ais. If the cross section of S perpendicular to the -ais are squares, then the volume of S is (A) (B) 3 (C) (D) 3 (E) ( ) 3 e 3. A circle, centered at (0, 3), has a radius of 3. What is the polar representation of the circle? (A) r 6sin cos (B) r 3sin 3cos (C) r sin 3 (D) r 6sin (E) r 6cos 4. d 0 4 (A) 0 (B) 4 (C) 6 (D) 3 (E) Divergence - 7 -

8 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 5. Suppose a continuous function f and its derivative f ' have values that are given in the above table. What is the approimation for f (3) if Euler s method is used with a step of 0.5, starting at? (A) 4.5 (B) 5.3 (C) 5.4 (D) 5.5 (E) d 7 (A) 3ln( 3 ) 4ln( 4 ) C (B) ln(( 4)( 3)) C (C) 3ln( 4 ) 4ln( 3 ) C (D) ln( ( 4)( 3) ) C (E) 4ln( 4 ) 3ln( 3 ) C 7. The position of a particle in the y-plane is given by t and y ln(t 3) for all t 0. The acceleration vector of the particle is 4 (A), (t 3) (B), (t 3) 4 (C), (t 3) (D), t t 3 4 (E) t, (t 3) - 8 -

9 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 8. Suppose a function f is approimated with a fourth-degree Taylor polynomial about. If the maimum value of the fifth derivative between and 3 is 0.05, that (5) is, f ( ) 0.05, then the maimum error incurred using this approimation to compute f (3) is (A) (B) (C) (D) (E) d y 9. If y sin( ), then d (A) sin cos (B) sin cos (C) sin cos (D) 3sin cos (E) cos sin 0. 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. (B) f has an absolute minimum value at. (C) f is not continuous at 3. (D) f is differentiable at 3. (E) f is continuous on the interval [,4]

10 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. The average value of (A) 3 (B) (C).5 (D) (E) None of above f ( ) 3 on [,] equals. If f ( ) e, which of the following statements is correct? e (A) Absolute maimum at (, ) (B) Absolute maimum at (, ) e e (C) Absolute minimum at (, ) (D) Absolute minimum at (, ) e (E) None of above 3. If f ( ) 3 for all, then the value of the derivative f ( ) at 3 is (A) (B) 0 (C) (D) (E) Noneistent - 0 -

11 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 4. Which of the following series is absolutely convergent? n ( ) ( n ) Ⅰ. n n Ⅱ. Ⅲ. n n0 n ( ) n e n n n ( ) (A) None (B) Ⅱ only (C) Ⅲ only (D) Ⅰ and Ⅱ only (E) Ⅱ and Ⅲ only n 5. The above graph of a function f consists of a semicircle and two line segments. Let g( ) f ( t) dt, which of the following is true? (A) g() (B) g() (C) g( ) (D) The slope of the tangent line of function f at is. (E) The function g has one relative minimum value on the interval [-, 4]. - -

12 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 6. Which of the following is the coefficient of generated by? ( ) (A) 5 (B) 5 (C) (D) (E) 0 4 in the Maclaurin polynomial 7. 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) - -

13 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 8. Suppose g is the inverse function of a differentiable function f and f (3) 7 and (A) 5 (B) 4 (C) 6 (D) (E) 4 f '(3), then G '(7)? 9 G( ). If g( ) 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

14 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB Calculus BC SECTION Ⅰ, PART B Time 50 minutes Number of questions 7 A graphing calculator is required for some questions on this part of the eam. Directions: Solve each of the following problems, using the available space for scratch work. After eamining the form of the choices, decide which is the best of the choices given and place the letter of your choice in the corresponding bo on the student answer sheet. Do not spend too much time on any one problem. In this eam: () The eact 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 approimates the eact numerical values. () Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f ( ) is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse notation f or with the prefi arc (e.g., sin arcsin )

15 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 76. lim (A) 0 (B) e (C) e (D) t e dt (E) e 77. The function f is differentiable and f ( ) tan. If df ( ) 0.0at, then d? (A) (B) (C) (D) (E) If 5 40 f ( t) dt, what s the value of c? c (A) 40 (B) 0 (C) 5 (D) (E) What are the coordinates of the inflection point on the graph of y ( )arctan? (A) (, 0) (B) (0, 0) (C) (0,) (D) (, ) 4 (E) (, ) - 5 -

16 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB h 80. lim? h 0 h (A) 0 (B) (C) (D) (E) Cannot be determined from the information given 8. A particle moves along the -ais so that any time t 0, its acceleration is given by a( t) cos( t). If the position of the particle is ( ) 4 and the velocity of the particle is 3 v( ) 0 at time t, what is the particle s position at time t? (A) 4.5 (B) 4.5 (C) 4 (D) 3.75 (E) Which of the following differential equations would produce the slope field shown above? dy (A) y 3 d dy (B) y d 3 dy (C) y d 3 dy y (D) d 3 dy y (E) d 3-6 -

17 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 83. Which of the following is equal to the area of the region inside the polar curve r cos and outside the polar curve r cos? (A) 3 cos d 0 (B) 3 cos d (C) cos d (D) 3 cos d (E) 0 3 cosd Which of the following is an epression of Maclaurin series for (A) (B) (C) (D) (E) 3 5 n n f ( )... ( )... 3! 5! (n )! 4 n n ( ) f ( )... ( )... 3 (n )! 4 n n ( ) f ( )... ( )... 3 (n )! 4 n n ( ) f ( )... ( )... 3 (n )! 4 n n ( ) f ( )... ( )... 3 (n )! f ( ) sin? - 7 -

18 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 85. A population of animals is modeled by a function P that satisfies the logistic dp differential equation 0.0 P(00 P), where t is measured in years and P is dt measured in millions. Which of the following equations is the solution of P as a function of t if P(0) 50? (A) P( t) t e 00 (B) P( t) t e (C) P( t) 0.0t e 00 (D) P( t) 0.0t e 00 (E) P( t) 0.0t e 86. What is the value of ? n (A) 0 (B) 3 (C) (D) 3 (E) The series diverges. 87. If cos d h( ) sin d h?, then ( ) (A) sin cos C (B) sin C (C) cos sin C (D) 4 cos sin C (E) ( )cos 4sin C - 8 -

19 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 88. The rate at which customers arrive at a counter to be served by the function F t defined by F( t) 6 cos for 0 60 t, 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) 70 (B) 75 (C) 73 (D) 744 (E) Let R be the region enclosed by the graphs of y e, y sin( 3 ) and the y-ais. Which of the following gives the approimate area of the region R? (A).39 (B).334 (C).869 (D).4 (E).340 t (sec) a( t ) (ft/sec ) The data for the acceleration a( t) of a car from 0 to 5 seconds are given in the above table. If the velocity at t 0 is 5 ft/sec, which of the following gives the approimate velocity at t 5 using the Trapezoidal Rule? (A) 47 ft/sec (B) 5 ft/sec (C) 0 ft/sec (D) 5 ft/sec (E) 4 ft/sec - 9 -

20 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 9. Which of the following is the solution to the differential equation where y(0)? tan (A) y e tan (B) y e (C) y e tan tan (D) y e tan (E) y e dy y sec d, 9. Let S be the region enclosed by the graphs of y sin and y cos for 5. What is the volume of the solid generated when S is revolved about the line 4 y? (A) 5 4 (sin cos ) d 5 (B) 4 (sin cos ) d 5 (C) 4 [(sin ) (cos ) ] d 5 (D) 4 [(cos ) (sin ) ] d 5 (E) 4 [(sin ) (cos ) ] d 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

21 SECTION Ⅱ Free- Response Questions - -

22 Calculus BC SECTION Ⅱ, PART A Time 30 minutes Number of questions A graphing calculator is required for some problems or parts of problems.. An object moving along a curve in the y-plane has position ( t), y( t ) at time t with dy d sin( e t ). The derivative of dt dt point (4, 5). (a) Find the y-coordinate of the position at time t. (b) At time t 3, the value of dy d is.8 (c) Find the speed of the object at t 3. is not eplicitly given. At t 3, the object is at the d. Find the value of dt when t

23 . Let S be the region in the first quadrant bounded by the graphs of and the y-ais. y e and y (a) Find the area of the region S. (b) Find the volume of the solid generated when the region S is rotated about the -ais. (c) The region S is the base of a solid for which each cross section perpendicular to the -ais is a semi-circle with diameter in the y-plane. Find the volume of this solid

24 Calculus BC SECTION Ⅱ, PART B Time 60 minutes Number of questions 4 No calculator is allowed for these problems. 3. The graphs of the polar curves r sin and r for 0 are shown above: (a) Write an integral epression for the area inside r and outside r. (b) Write epressions for d dy and d d in terms of for the r curve. (c) Write an equation in terms of and y for the line tangent to the graph of the polar curve r at the point where

25 4. The graph of the function f, consisting of three segments, is shown above. Let g( ) f ( t) dt. (a) Find g () and g '(). (b) Find the instantaneous rate of change of g, which respect to, at. (c) Find the absolute minimum value of g on the closed interval [, 4]. Justify your answer. (d) The second derivative of g is not defined at and. Which of these values are -coordinates of points of inflection of the graph of g? Justify your answer

26 dy 5. Consider the differential equation y 4 d. (a) The slope field for the differential equation is shown above. Sketch the solution curve that passes through the point (0, ) and sketch the solution curve that passes through the point (0, -). (b) There is a value of b for which y 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 g(0) 0. It appears from the slope field that g has a local maimum 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 a

27 6. Let f be the function given by f ( ) e (a) Find the first four nonzero terms and the general term of the power series for f ( ) about 0. (b) Find the interval of convergence of the power series for f ( ) about 0. 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 6 series for f ( ) about 0. Show that f ( ) g( ).8 0 for STOP END OF EXAM (Find answer at (More information at 北京新东方罗勇老师个人网站 是致力于为 AP 和 SAT 考生提供第一手 AP 微积分和 SAT 数学考试资料和备考策略的网络平台, 使在家自学的考生也能和参加培训的考生一样, 分享到宝贵的备考资料 在美国, 每个人都十分方便的能拥有自己的个人网站, 尤其是初高中老师和大学教授 老师的个人网站成为了发布讲义 作业 考试资料的重要平台, 也成为了老师和学生交流的重要阵地 - 7 -