2024AP® Calculus AB Free-Response Questions © 2024 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.
CALCULUS ABSECTION II, Part ATime—30 minutes2 QuestionsA GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS.GO ON TO THE NEXT PAGE.AP® Calculus AB 2024 Free-Response Questions © 2024 College Board. Visit College Board on the web: collegeboard.org. 2
t(minutes)03712Ct(degrees Celsius)()1008569551.The temperature of coffee in a cup at time tminutes is modeled by a decreasing differentiable function C,where Ct()is measured in degrees Celsius. For t012££, selected values of Ct()are given in the tableshown.(a)Approximate C5¢()using the average rate of change of Cover the interval t37££. Show the work thatleads to your answer and include units of measure.(b)Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate thevalue of Ctd012∫()t. Interpret the meaning ofCtd112012∫()tin the context of the problem.(c)For t1220££, the rate of change of the temperature of the coffee is modeled byCtet24.55t0.01¢=()−,where Ct¢()is measured in degrees Celsius per minute. Find the temperature of the coffee at time t20=.Show the setup for your calculations.(d)For the model defined in part (c), it can be shown thatCtett0.2455100t0.012()(−)¢¢=. For t1220<<,determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate.Give a reason for your answer.__________________________________________________________Write your responses to this question only on the designated pages in the separate Free Responsebooklet. Write your solution to each part in the space provided for that part.GO ON TO THE NEXT PAGE.AP® Calculus AB 2024 Free-Response Questions © 2024 College Board. Visit College Board on the web: collegeboard.org. 3
2.A particle moves along the x-axis so that its velocity at time t0≥is given by vtttln450.22=+()()−−t.(a)There is one time, t tR=, in the interval t02<<when the particle is at rest (not moving). Find tR. Fortt0R<<, is the particle moving to the right or to the left? Give a reason for your answer.(b)Find the acceleration of the particle at time t1.5=. Show the setup for your calculations. Is the speed of theparticle increasing or decreasing at time t1.5=? Explain your reasoning.(c)The position of the particle at time tis xt(), and its position at time t1=is x13()−=. Find the position ofthe particle at time t4=. Show the setup for your calculations.(d)Find the total distance traveled by the particle over the interval t14££. Show the setup for yourcalculations.__________________________________________________________Write your responses to this question only on the designated pages in the separate Free Responsebooklet. Write your solution to each part in the space provided for that part.GO ON TO THE NEXT PAGE.AP® Calculus AB 2024 Free-Response Questions © 2024 College Board. Visit College Board on the web: collegeboard.org. 4
END OF PART AAP® Calculus AB 2024 Free-Response Questions © 2024 College Board. Visit College Board on the web: collegeboard.org. 5
CALCULUS ABSECTION II, Part BTime—1 hour4 QuestionsNO CALCULATOR IS ALLOWED FOR THESE QUESTIONS.AP® Calculus AB 2024 Free-Response Questions © 2024 College Board. Visit College Board on the web: collegeboard.org. 6
