AP® Calculus AB 2008 Free-Response Questions Form B 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. © 2008 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. Permission to use copyrighted College Board materials may be requested online at: www.collegeboard.com/inquiry/cbpermit.html. Visit the College Board on the Web: www.collegeboard.com. AP Central is the official online home for the AP Program: apcentral.collegeboard.com. www.mymathscloud.com
2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2008 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- CALCULUS AB 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 region in the first quadrant bounded by the graphs of yx and .3xy(a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the vertical line 1. x(c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are squares. Find the volume of this solid. WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2008 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- 2. For time 0t hours, let 210120 1trte represent the speed, in kilometers per hour, at which a car travels along a straight road. The number of liters of gasoline used by the car to travel x kilometers is modeled by 20.05 1. xgxxe(a) How many kilometers does the car travel during the first 2 hours? (b) Find the rate of change with respect to time of the number of liters of gasoline used by the car when 2t hours. Indicate units of measure. (c) How many liters of gasoline have been used by the car when it reaches a speed of 80 kilometers per hour? WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2008 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). -4- Distance from the river’s edge (feet) 0 8 14 22 24 Depth of the water (feet)0 7 8 2 0 3. A scientist measures the depth of the Doe River at Picnic Point. The river is 24 feet wide at this location. The measurements are taken in a straight line perpendicular to the edge of the river. The data are shown in the table above. The velocity of the water at Picnic Point, in feet per minute, is modeled by 162 sin10 vtt for 0120
t minutes. (a) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the area of the cross section of the river at Picnic Point, in square feet. Show the computations that lead to your answer. (b) The volumetric flow at a location along the river is the product of the cross-sectional area and the velocity of the water at that location. Use your approximation from part (a) to estimate the average value of the volumetric flow at Picnic Point, in cubic feet per minute, from 0t to 120t minutes. (c) The scientist proposes the function f, given by 8sin,24pxfx as a model for the depth of the water, in feet, at Picnic Point x feet from the river’s edge. Find the area of the cross section of the river at Picnic Point based on this model. (d) Recall that the volumetric flow is the product of the cross-sectional area and the velocity of the water at a location. To prevent flooding, water must be diverted if the average value of the volumetric flow at Picnic Point exceeds 2100 cubic feet per minute for a 20-minute period. Using your answer from part (c), find the average value of the volumetric flow during the time interval 4060
t minutes. Does this value indicate that the water must be diverted? WRITE ALL WORK IN THE EXAM BOOKLET. END OF PART A OF SECTION II www.mymathscloud.com
2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2008 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- CALCULUS AB SECTION II, Part B Time — 45 minutes Number of problems — 3 No calculator is allowed for these problems. 4. The functions f and g are given by 3204 Ôxfxt dt and sin.gxfx(a) Find fx and .gx(b) Write an equation for the line tangent to the graph of ygx at .px(c) Write, but do not evaluate, an integral expression that represents the maximum value of g on the interval 0.p
x Justify your answer. WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2008 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. -6- 5. Let g be a continuous function with 25.g The graph of the piecewise-linear function ,g the derivative of g, is shown above for 37.x
(a) Find the x-coordinate of all points of inflection of the graph of ygx for 37.x Justify your answer. (b) Find the absolute maximum value of g on the interval 37.x
Justify your answer. (c) Find the average rate of change of gx on the interval 37.x
(d) Find the average rate of change of gx on the interval 37.x
Does the Mean Value Theorem applied on the interval 37x
guarantee a value of c, for 37,c such that gc is equal to this average rate of change? Why or why not? WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
