AP Calc BC 2024 Frq #1

1. The temperature of coffee in a cup at time t minutes is modeled by a decreasing differentiable function C,
where C t( ) is measured in degrees Celsius. For 0 £ £t 12, selected values of C t( ) are given in the table
shown.
(a) Approximate C¢( )5 using the average rate of change of C over the interval 3 £ £t 7. Show the work that
leads 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 the
value of C t d
0
12 ∫ ( ) t. Interpret the meaning of C t d 1
12 0
12 ∫ ( ) t in the context of the problem.
(c) For 12 £ £t 20, the rate of change of the temperature of the coffee is modeled by C t
e
t
24.55 0.01t
¢( ) = − ,
where C t ¢( ) is measured in degrees Celsius per minute. Find the temperature of the coffee at time t = 20.
Show the setup for your calculations.
(d) For the model defined in part (c), it can be shown that C t
e t
t
0.2455 100 0.01t
2 ( ) ( − ) ¢¢ = . For 12 is less than t is less than 20,
determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate.
Give a reason for your answer.