2022 Free-Response Questions © 2022 College Board. College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered trademarks of College Board. Visit College Board on the web: collegeboard.org. AP Central is the official online home for the AP Program: apcentral.collegeboard.org.AP® Calculus AB
CALCULUS AB SECTION II, Part A Time—30 minutes 2 Questions A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. GO ON TO THE NEXT PAGE. AP® Calculus AB 2022 Free-Response Questions© 2022 College Board. Visit College Board on the web: collegeboard.org. 2
__________________________________________________________ 1.From 5A.M. to 10A.M., the rate at which vehicles arrive at a certain toll plaza is given byAt()=450sin(0.62t), where tis the number of hours after 5A.M. and At()is measured in vehicles per hour.Traffic is flowing smoothly at 5A.M. with no vehicles waiting in line.(a)Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at thetoll plaza from 6A.M. (t=1) to 10A.M. (t=5).(b)Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from6A.M. (t=1) to 10A.M. (t=5).(c)Is the rate at which vehicles arrive at the toll plaza at 6A.M. (t=1) increasing or decreasing? Give a reasonfor your answer.(d)A line forms whenever At()≥400. The number of vehicles in line at time t, for a££t4, is given by()=(()−N tAx400)dx∫at, where ais the time when a line first begins to form. To the nearest wholenumber, find the greatest number of vehicles in line at the toll plaza in the time interval a£t4£. Justify your answer. Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part. GO ON TO THE NEXT PAGE. AP® Calculus AB 2022 Free-Response Questions© 2022 College Board. Visit College Board on the web: collegeboard.org. 3
__________________________________________________________ 2.Let fand gbe the functions defined by f()x=ln(x+3)and 43g()x=x+2x. The graphs of fand g, shown in the figure above, intersect at x=−2and x=B, where B>0. (a)Find the area of the region enclosed by the graphs of fand g.(b)For −2££x B, let hx()be the vertical distance between the graphs of fand g. Is hincreasing ordecreasing at x=−0.5 ? Give a reason for your answer.(c)The region enclosed by the graphs of fand gis the base of a solid. Cross sections of the solid takenperpendicular to the x-axis are squares. Find the volume of the solid.(d)A vertical line in the xy-plane travels from left to right along the base of the solid described in part (c). Thevertical line is moving at a constant rate of 7units per second. Find the rate of change of the area of thecross section above the vertical line with respect to time when the vertical line is at position x=−0.5.Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part. GO ON TO THE NEXT PAGE. AP® Calculus AB 2022 Free-Response Questions© 2022 College Board. Visit College Board on the web: collegeboard.org. 4
END OF PART A AP® Calculus AB 2022 Free-Response Questions© 2022 College Board. Visit College Board on the web: collegeboard.org. 5
CALCULUS AB SECTION II, Part B Time—1 hour 4 Questions NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS. AP® Calculus AB 2022 Free-Response Questions© 2022 College Board. Visit College Board on the web: collegeboard.org. 6
