4 Q - AP® Calculus AB 2008 Free-Response Questions Form B...

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Unformatted text preview: 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. 2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 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 y (a) Find the area of R. x and y x . 3 (b) Find the volume of the solid generated when R is rotated about the vertical line x 1. (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. © 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- 2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 10 t 2 2. For time t 0 hours, let r t 120 1 e 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 0.05 x 1 e x 2 . by g x (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 t 2 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. © 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- 2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) Distance from the river’s edge (feet) Depth of the water (feet) 0 0 8 7 14 8 22 2 24 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 v t 16 2sin t 10 for 0 t 120 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 t 0 to t 120 minutes. (c) The scientist proposes the function f, given by f x px , as a model for the depth of the water, 24 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. 8 sin (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 40 t 60 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 © 2008 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). -4- 2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 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 f x (a) Find f x and g x . 3x 0 4 t 2 dt and g x f sin x . (b) Write an equation for the line tangent to the graph of y g x at x p. (c) Write, but do not evaluate, an integral expression that represents the maximum value of g on the interval 0 x p . Justify your answer. WRITE ALL WORK IN THE EXAM BOOKLET. © 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- 2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 5. Let g be a continuous function with g 2 of g, is shown above for 3 x 7. 5. The graph of the piecewise-linear function g , the derivative g x for 3 x 7. Justify your (a) Find the x-coordinate of all points of inflection of the graph of y answer. (b) Find the absolute maximum value of g on the interval 3 (c) Find the average rate of change of g x on the interval 3 x x 7. Justify your answer. 7. (d) Find the average rate of change of g x on the interval 3 x 7. Does the Mean Value Theorem applied on the interval 3 x 7 guarantee a value of c, for 3 c 7, such that g c is equal to this average rate of change? Why or why not? WRITE ALL WORK IN THE EXAM BOOKLET. © 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- 2008 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 6. Consider the closed curve in the xy-plane given by x2 (a) Show that 2x y4 4y 5. dy dx x 2y 3 1 1 . 2, 1 . (b) Write an equation for the line tangent to the curve at the point (c) Find the coordinates of the two points on the curve where the line tangent to the curve is vertical. (d) Is it possible for this curve to have a horizontal tangent at points where it intersects the x-axis? Explain your reasoning. WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM © 2008 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). -7- ...
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