AP
®
Calculus AB/BC 2023 Scoring Commentary
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Question 1
Note: Student samples are quoted verbatim and may contain spelling and grammatical errors.
Overview
In this problem students were given a table of times
in seconds and values of a function
which models the
rate of flow of gallons of gasoline pumped into a gas tank.
In part (a) students were asked to interpret the meaning of
using correct units. Then students were
asked to use a right Riemann sum with three subintervals to approximate the value of this integral. A correct
response will indicate that the integral represents the accumulated gallons of gasoline pumped into the tank during
the time interval from
to time
seconds. The approximation is found using the following expression:
(
)
( )
( ) ( ) ( ) ( )
90 60 90 120 90 120 135 120 135 .
ff f
⋅⋅ ⋅− +− +−
In part (b) students were asked to justify whether there must be a value of
with
such that
Students are expected to note that because the function
is known to be differentiable on the interval
it must be continuous on the subinterval
Therefore, because the average rate of change of
on the interval
is equal to
such a value of
is guaranteed by the Mean Value Theorem.
In part (c) the function
( )
( ) (
)
2
cos
500 120
tt
gt
=
was introduced as a second function that modeled the rate of
flow of the gasoline. Students were asked to use the model
to find the average rate of flow of the gasoline over
the time interval
A correct response will show the setup
and then use a
calculator to find the value
gallon per second.
In part (d) students were asked to find the value of
and interpret the meaning of this value in the context of
the problem. A correct response will use a calculator to find
and report that at time
seconds the rate at which gasoline is flowing into the tank is decreasing at a rate of
gallon per second per
second.
Sample: 1A
Score: 9
The response earned 9 points: 3 points in part (a), 2 points in part (b), 2 points in part (c), and 2 points in part (d).
In part (a) the response earned the first point with the statement “the amount of gas pumped, in gallons, from
to
seconds.” The response earned the second point for the correct form of the Riemann sum. The response
earned the third point for the correct answer.
In part (b) the response earned the first point for “
” The response earned the second point
because it earned the first point, states that “
is always differentiable, and therefore it must be continuous on
&
” and states the correct conclusion.
In part (c) the response earned the first point with the inclusion of the average value formula. The response earned
the second point with the correct answer.