ap微积分bc官方练习题及详解

1、Advanced Placement ProgramPractice Exam© 2008 The CollegeACollegeAdvanced Placement Program, AP, AP Central,SAT, and the acorn logo are registered trademarks of the Colleg. PSAT/NMSQT is a registered trade-mark of the Collegand National Merit Scholarship Corporation. All other products and serv

2、ices maybe trademarks of their respective owners. Visit the Collegon the Web:.The questions contained in this AP® Calculus BC Practice Exam are written to the content specifications of AP Exams for this subject. Taking this practice exam should provides with an idea of their general areas of st

3、rengths and weaknesses in preparing for the actual AP Exam. Because this AP Calculus BC Practice Exam has never been administered as an operational AP Exam, statistical data are not available for calculating potential raw scores or conversions into AP grades.This AP Calculus BC Practice Exam is prov

4、ided by the Collegfor AP Exam preparation. Teachers are permitted to download the materials and make copies to use with theirs in a classroom setting only. To maintain the security of this exam, teachers should collect all materials after their administration and keep them in a secure location. Teac

5、hers may not redistribute the files electronically for any reason.AP® Calculus BCSection IMultiple-Choice Questions-1-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A ACALCULUS BCSECTION I, Part ATime55 minutes Number of questions28A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXA

6、M.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 thecorresponding box on theanswer sheet. Do not spend too much time on any one

7、 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 whichf (x) is a real number.f -1(2)The inverse of a trigonometric function f may be indicated using the inverse function notationor with theprefix “arc” (e.g., sin-1 x =

8、 arcsin x ).GO ON TO THE NEXT PAGE.-2-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A3x - 2 ,If f (x) =then f ¢(x) =1.2x + 3-13(A)(2x + 3)2 3(2x + 3)2 5(2x + 3)2 13(2x + 3)2 12x + 5 (2x + 3)2(B)(C)(D)(E)v(t) = sin (2t ).A particle moves along the x-axis so that at any time t 0, it

9、s velocity is given by2.p2If the position of the particle at time t =is x = 4,what is the particles position at time t = 0 ?- 1(A)(B) 2(C) 3(D) 5(E) 82GO ON TO THE NEXT PAGE.-3-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A AÂ()n?- 23.What is the value of3n =0- 235(A) -2(B)(C)(D) 3

10、(E) The series diverges.54.For values of h verytan(x + h) - tan xto 0, which of the following functions best approximates?f (x) =h(A)sin xsin x xtan x xsec x(B)(C)(D)sec2 x(E)GO ON TO THE NEXT PAGE.-4-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A Ax4y =fromx = 1 tox = 5 is given by5.Th

11、e length of the curve5Ú1 + 4x3 dx(A)15Ú1 + x4 dx(B)15Ú1 + 4x6 dx(C)15Ú1 + 16 x6 dx(D)15Ú1 + x8 dx(E)1ÛÙexdx =6. 1 + exln Ê 1 + 1 + C(A)ÁË ex¯ln(1 + ex ) + Cx - ln (1 + ex ) + C(B)(C)ex+ x + C(D)tan-1(ex ) + C(E)GO ON TO THE NEXT PAGE.-5-A A A A

12、A A A A A A A A A A A A A A A A A A A A A A A A A A Ady dxLet y = f (x) be the solution to the differential equation= x - y - 1 with the initial condition f (1) = -2.7.f (1.4)if Eulers method is used, starting at x = 1 with two steps of equal size?What is the approximation for(A) -2(B) -1.24(C) -1.2

13、(D) -0.64(E) 0.2d interval 0, 68.The function f is continuous on theand has the values given in the table above.6Ú0The trapezoidal approximation forf (x) dx found with 3 subintervals of equal length is 52. Whatis the value of k ?(A)2(B)6(C)7(D)10(E)14GO ON TO THE NEXT PAGE.-6-x0246f (x)4k812A A

14、 A A A A A A A A A A A A A A A A A A A A A A A A A A A A A9. The function f is twice differentiable, and the graph of f has no points of inflection. If f (6) = 3, f ¢(6) = - 1 ,2and f ¢¢(6) = -2,which of the following could be the value of f (7) ?(A) 2(B)2.5(C) 2.9(D)3(E) 4 x66!x2n(2n

15、)! 4!10. A function f has Maclaurin series given by 1 +expression for f (x) ?+ " + ". Which of the following is an2!(A)cos xexex- sin x+ sin x(B)(C)(x + e- x )12(D)eex 2(E)GO ON TO THE NEXT PAGE.-7-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A Ax = 4 and11. The sides and diag

16、onal of the rectangle above are strictly increasing with time. At the instant whendxdzdyk dz .y = 3,=and=What is the value of k at that instant?dtdtdtdt1413(A)(B)(C) 3(D) 4(E) It cannot be determined from the information given.2xf (e ) = 5,f ¢(x) =f (e) =12. Ifandthen2 - 2 (C) 5 +(A) 2(B) ln 25

17、(D)6(E) 25ee2GO ON TO THE NEXT PAGE.-8-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A13. For time t > 0, the position of a particle moving in the xy-plane is given by the parametric equations 1.4t + t2 and y =x =What is the acceleration vector of the particle at time t = 1 ?3t + 1(

18、) 1 (A)2,32() 9 (B)2,32(5,) 3 16 1 1614(C)()6, -(D)(6, -(E)ÛÙ 8 dx=14. x2- 4( ) x-1+ C(A)4 tan2x2- 4+ C(B)8 lnx - 2x + 2 x + 2x - 2x + 2+ C(C)2 ln+ C(D)2 ln+ 2 lnx - 2+ C(E)2 lnGO ON TO THE NEXT PAGE.-9-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A15. The slope field for a

19、certain differential equation is shown above. Which of the following could be a solution to the differential equation with the initial condition y(0) = 1 ?y = cos x y = 1 - x2 y = exy = 1 - x2(A)(B)(C)(D) 11 + x2y =(E)GO ON TO THE NEXT PAGE.-10-A A A A A A A A A A A A A A A A A A A A A A A A A A A A

20、 A A Af ¢(x) =f (x) ?x - 2 , which of the following could be the graph ofy =16. If(A)(B)(C)(D)(E)GO ON TO THE NEXT PAGE.-11-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A(x - 3)2nnÂ17. The radius of convergence for the power seriesis equal to 1. What is the interval of conve

21、rgence?n=1(A) -4 £ x < -2 (B) -1 < x < 1(C) -1 £ x < 1(D) 2 < x < 4(E) 2 £ x < 4f (x) = arccos(x2 ),f ¢(x) =18. Ifthen 11 - x4(A) -2x 1 - x4(B) 2x1 - x4(C)-4 x3(D)1 - x44x3(E)1 - x4GO ON TO THE NEXT PAGE.-12-A A A A A A A A A A A A A A A A A A A A A A A A A A

22、 A A A A Ax22(2, 0) ?19. What is the slope of the line tangent to the curve y + 2 =- 2 sin y at the point1223(A) -2(B) 0(C)(D)(E)220. Which of the following series converge?1nÂI.n=13nn!ÂII.n=1( ) neÂIII.pn=1(A) None(B) II only(C) III only(D) I and II only(E)II and III onlyGO ON TO THE

23、 NEXT PAGE.-13-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A3 - 6 has a relative minimum at x =21. The function f given by f- 1(A) -8(B) -3 2(C) -1(D)(E) 08x = 0 and22. The function f has a continuous derivative. The table above gives values of f and its derivative for44Ú0what i

24、s the value of Ú0 x f ¢(x) dx ?x = 4. Iff (x) dx = 8,(A) -20(B) -13(C) -12(D) -7(E)36GO ON TO THE NEXT PAGE.-14-xf (x)f ¢(x)0254311A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A Ap23. What is the slope of the line tangent to the polar curve r = 2q at the point q =?2- p- 2

25、p2(A)(B)(C) 0(D)(E) 22p24. The radius of a circle is increasing. At a certain instant, the rate of increase in the area of the circle is numerically equal to twice the rate of increase in its circumference. What is the radius of the circle at that instant?12(A)(B)1(C)2(D)2(E)4GO ON TO THE NEXT PAGE.

26、-15-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A AÛ1 125. The table above shows several Riemann sum approximations todx using right-hands of n0 xsubintervals of equal length of the interval 0, 1.Which of the following statements best describes the limitof the Riemann sums as n ap

27、proaches infi?(A) The limit of the Riemann sums is a finite number less than 10.(B) The limit of the Riemann sums is a finite number greater than 10.( )Ê 1 1(C) The limit of the Riemann sums does not exist becausedoes not approach 0.ÁË x ¯ nn(D) The limit of the Riemann sums does

28、 not exist because it is a sum of infinitely many positive numbers.Û1 1(E)The limit of the Riemann sums does not exist becausedxdoes not exist.0 xGO ON TO THE NEXT PAGE.-16-nÂÁË x ¯ (n )n Ê 1 1k =1k1005.192005.883006.284006.575006.79A A A A A A A A A A A A A A A A A A A

29、 A A A A A A A A A A A A()2n + 1Â26. The coefficients of the power seriesa (x - 2)nsatisfy a = 5 and a =for all n 1. Thean -10- 1nn3nn =0radius of convergence of the series is2332(A) 0(B)(C)(D) 2(E) infinite2 xÚf (x) =f ¢(2) =t2- t dt, then27. If f is the function given by4 7 (A)0(B)(

30、C)2(D)12(E)212212GO ON TO THE NEXT PAGE.-17-A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A28. The function f is given by f (x) = sin Ê x + 1. Which of the following statements are true?˯x2I. The graph of f has a horizontal asymptote at y = 0.II. The graph of f has a ho

31、rizontal asymptote at y = 1.x = 0.III. The graph of f has a vertical asymptote at(A) I only(B) II only(C) III only(D) I and III only(E)II and III onlyEND OF PART A OF SECTION IIF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY.DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO

32、SO.-18-B B B B B B B B BCALCULUS BCSECTION I, Part BTime50 minutes Number of questions 17A 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

33、, decide which is the best of the choices given and place the letter of your choice in thecorresponding box on theanswer 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 happ

34、ens, select from among the choices the number that best approximates the exact numerical value.(2)Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for whichf (x) is a real number.f -1(3)The inverse of a trigonometric function f may be indicated us

35、ing the inverse function notationor with theprefix “arc” (e.g., sin-1 x = arcsin x ).GO ON TO THE NEXT PAGE.-19-B B B B B B B B B14476. If Ú0 f (x) dx = 2 andf (x) dx = -3, then Ú1 (3 f (x) + 2) dx =Ú0(B) -9(A) -13(C) -7(D)3(E)2177. The figure above shows the graph of the polynomial f

36、unction f. For which value of x is it true thatf ¢¢(x) < f ¢(x) < f (x) ?(A) a(B) b(C) c(D) d(E) eGO ON TO THE NEXT PAGE.-20-B B B B B B B B B78. The graph above shows the polar curve r = 2q + cos qby the curve and the x-axis?for 0 £ q £ p.What is the area of the regio

37、n bounded(A) 3.069(B) 4.935(C) 9.870(D) 17.456(E) 34.912f (3) = 8f ¢(3) = 5,79. If f is a differentiable function such that false?andwhich of the following statements could belim f (x) = 8(A)xÆ3limxÆ3+f (x) =f (x)(B)limxÆ3-lim f (x) - 8= 5(C)x - 3xÆ3lim f (3 + h) - 8= 5(D)hh

38、Æ0lim f ¢(x) = 5(E)xÆ3GO ON TO THE NEXT PAGE.-21-B B B B B B B B Bf, on the interval -3, 6.f ¢,80. The figure above shows the graph ofthe derivative of the functionIf thederivative of the function h is given by h¢(x) = 2 f ¢(x), how many points of inflection does the gr

39、aphhave on the interval -3, 6 ?of h(A) One(B) Two(C) Three(D) Four(E) Five= e- x2 2 ,81. Let R be the region in the first quadrant bounded by the y-axis, the x-axis, the graph of yand the linex = 3. The region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is

40、 a square. What is the volume of the solid?(A)0.886(B)0.906(C)1.078(D)1.245(E)2.784GO ON TO THE NEXT PAGE.-22-B B B B B B B B Bd interval a, b,82. If f is a continuous function on thewhich of the following must be true?(A) There is a number c in the open interval (a, b) such that f (c) = 0.(a, b)f (

41、a) < f (c) < f (b).(B) There is a number c in the open intervalsuch thatd interval a, bf (c) f (x)in a, b.(C) There is a number c in thesuch thatfor all x(D) There is a number c in the open interval (a, b) such that f ¢(c) = 0. f (b) - f (a) .(E) There is a number c in the open interval (

42、a, b) such that f ¢(c) =b - a83. The function f is differentiable and has values as shown in the table above. Both f and f ¢ are strictly increasing on the interval 0 £ x £ 5. Which of the following could be the value of f ¢(3) ?(A)20(B)27.5(C)29(D)30(E)30.5GO ON TO THE NEXT

43、 PAGE.-23-x2.52.83.03.1f (x)31.2539.204548.05B BB B B B B B BdP ,84. The rate of change,equation. Theof the number of people on an ocean beach is med by a logistic differentialdtum number of people allowed on the beach is 1200. At 10 A.M., the number of peopleon the beach is 200 and is increasing at

44、 the rate of 400 people per hour. Which of the following differential equations describes the situation? 1 (1200 - P) + 200dP dtdP dtdP dtdP dtdP dt=(A)4002 (1200 - P)=(B)5 1 P (1200 - P)=(C)500 1 P (1200 - P)=(D)400400P (1200 - P)=(E)85. The third derivative of the function f is continuous on the i

45、nterval (0, 4). Values for f and its firstlim f (x) ?three derivatives at x = 2 are given in the table above. What isxÆ2 (x - 2)252(A) 0(B)(C) 5(D) 7(E) The limit does not exist.GO ON TO THE NEXT PAGE.-24-xf (x)f ¢(x)f ¢¢(x)f ¢¢¢(x)20057B B B B B B BB B86. The Tayl

46、or polynomial of degree 100 for the function f about x = 3 is given by1 (x - 3)2nn!(x - 3)100 50!P (x) = (x - ) -+ " -"1. What is2!3!f (30) (3) ?(B)the value of- 30!- 1 1 30! 1 15!30!15!(A)(C)(D)(E)15!30!f (4)(x) = esin x .87. The function f has derivatives of all orders for all real numbers, andIf the third-degree Tayloron the interval 0, 1,polynomial for f about x = 0 is used to approximate fwhat