洋桃课堂AP微积分2006真题

AP® Calculus BC 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns. © 2006 The College Board. All rights reserved. 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AP Central is the official online home for the AP Program: apcentral.collegeboard.com. 2006 AP® CALCULUS BC FREE-RESPONSE QUESTIONS CALCULUS BC SECTION II, Part A Time—45 minutes Number of problems—3 A graphing calculator is required for some problems or parts of problems. 1. Let R be the shaded region bounded by the graph of y = ln x and the line y = x - 2, as shown above. (a)​ Find the area of R. (b)​ Find the volume of the solid generated when R is rotated about the horizontal line y = -3. (c)​ Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis. WRITE ALL WORK IN THE PINK EXAM BOOKLET. © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 2 2006 AP® CALCULUS BC FREE-RESPONSE QUESTIONS 2.​ At an intersection in Thomasville, Oregon, cars turn left at the rate L (t ) = 60 t sin2 ( 3t ) cars per hour over the time interval 0 £ t £ 18 hours. The graph of y = L ( t ) is shown above. (a)​ To the nearest whole number, find the total number of cars turning left at the intersection over the time interval 0 £ t £ 18 hours. (b)​ Traffic engineers will consider turn restrictions when L (t ) ≥ 150 cars per hour. Find all values of t for which L (t ) ≥ 150 and compute the average value of L over this time interval. Indicate units of measure. (c)​ Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion. WRITE ALL WORK IN THE PINK EXAM BOOKLET. © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 3 2006 AP® CALCULUS BC FREE-RESPONSE QUESTIONS 3. An object moving along a curve in the xy-plane is at position ( x ( t ) , y ( t )) at time t, where dx = sin -1 1 - 2e-t ) and dy = 4t 1 + t3 dt ( dt for t ≥ 0. At time t = 2, the object is at the point ( 6, -3 ). (Note: sin -1 x = arcsin x ) (a)​ Find the acceleration vector and the speed of the object at time t = 2. (b)​ The curve has a vertical tangent line at one point. At what time t is the object at this point? (c)​ Let m (t ) denote the slope of the line tangent to the curve at the point ( x ( t ) , y ( t )). Write an expression for m t in terms of t and use it to evaluate lim m t . ( ) t Æ• ( ) (d) The graph of the curve has a horizontal asymptote y = c. Write, but do not evaluate, an expression involving an improper integral that represents this value c. WRITE ALL WORK IN THE PINK EXAM BOOKLET. END OF PART A OF SECTION II © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). 4 2006 AP® CALCULUS BC FREE-RESPONSE QUESTIONS CALCULUS BC SECTION II, Part B Time—45 minutes Number of problems—3 No calculator is allowed for these problems. t 0 10 20 30 40 50 60 70 80 (seconds) v t ( ) 5 14 22 29 35 40 44 47 49 (feet per second) 4.​ Rocket A has positive velocity v (t ) after being launched upward from an initial height of 0 feet at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 £ t £ 80 seconds, as shown in the table above. (a)​ Find the average acceleration of rocket A over the time interval 0 £ t £ 80 seconds. Indicate units of measure. (b)​ Using correct units, explain the meaning of Ú1070 v (t ) dt in terms of the rocket’s flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate Ú1070 v ( t ) dt. (c) Rocket B is launched upward with an acceleration of a (t ) = 3 feet per second per second. At time t + 1 t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time t = 80 seconds? Explain your answer. WRITE ALL WORK IN THE PINK EXAM BOOKLET. © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 5 2006 AP® CALCULUS BC FREE-RESPONSE QUESTIONS 5.​ Consider the differential equation dydx = 5 x2 - y 6- 2 for y π 2. Let y = f ( x ) be the particular solution to this differential equation with the initial condition f ( -1) = -4. (a) Evaluate dy and d 2 y at ( -1, -4 ). dx dx2 (b)​ Is it possible for the x-axis to be tangent to the graph of f at some point? Explain why or why not. (c)​ Find the second-degree Taylor polynomial for f about x = -1. (d)​ Use Euler’s method, starting at x = -1 with two steps of equal size, to approximate f ( 0 ). Show the work that leads to your answer. 6. The function f is defined by the power series f ( x ) = - x + 2 x 2 - 3 x3 + + ( -1)n nxn + n + 1 2 3 4 for all real numbers x for which the series converges. The function g is defined by the power series g ( x ) = 1 - x + x 2 - x 3 + + ( -1) n xn + 2! 4! 6! ( 2 n )! for all real numbers x for which the series converges. (a)​ Find the interval of convergence of the power series for f. Justify your answer. (b)​ The graph of y = f ( x ) - g ( x ) passes through the point ( 0, -1) . Find y¢( 0 ) and y¢¢( 0 ). Determine whether y has a relative minimum, a relative maximum, or neither at x = 0. Give a reason for your answer. WRITE ALL WORK IN THE PINK EXAM BOOKLET. END OF EXAM © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). 6