2023
AP
®
Calculus BC
Sample Student Responses
and Scoring Commentary
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Inside:
Free-Response Question 1
Scoring Guidelines
Student Samples
Scoring Commentary
AP® Calculus AB/BC 2023 Scoring Guidelines
© 2023 College Board
Part A (AB or BC): Graphing calculator required
Question 1 9 points
General Scoring Notes
The model solution is presented using standard mathematical notation.
Answers (numeric or algebraic) need not be simplified. Answers given as a decimal approximation should be
correct to three places after the decimal point. Within each individual free-response question, at most one
point is not earned for inappropriate rounding.
( )
0 60 90 120 135 150
(seconds)
0 0.1 0.15 0.1 0.05 0
(gallons per second)
t
ft
A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a
differentiable function
,f
where
( )
ft
is measured in gallons per second and
t
is measured in seconds since
pumping began. Selected values of
(
)
ft
are given in the table.
Model Solution Scoring
(a)
Using correct units, interpret the meaning of
( )
135
60
f t dt
∫
in the context of the problem. Use a right
Riemann sum with the three subintervals
[ ]
60, 90 ,
[ ]
90, 120 ,
and
[ ]
120, 135
to approximate the
value of
( )
135
60
.f t dt
∫
( )
135
60
f t dt
∫
represents the total number of gallons of gasoline
pumped into the gas tank from time
60t =
seconds to time
135t =
seconds.
Interpretation with
units
1 point
( )
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )
135
60
90 90 60 120 120 90 135 135 120
0.15 30 0.1 30 0.05 15 8.25
f t dt
ff f≈ −+ −+ −
= ++ =
∫
Form of Riemann
sum
1 point
Answer 1 point
Scoring notes:
• To earn the first point the response must reference gallons of gasoline added/pumped and the time
interval
60t =
to
135.t
=
• To earn the second point at least five of the six factors in the Riemann sum must be correct.
• If there is any error in the Riemann sum, the response does not earn the third point.
• A response of
( )( ) ( )( ) ( )( )
0.15 30 0.1 30 0.05 15++
earns both the second and third points, unless
there is a subsequent error in simplification, in which case the response would earn only the
second point.
AP® Calculus AB/BC 2023 Scoring Guidelines
© 2023 College Board
• A response that presents a correct value with accompanying work that shows the three products in
the Riemann sum but does not show all six of the factors and/or the sum process, does not earn the
second point but does earn the third point. For example, responses of either
4.5 3.0 0.75
++
or
( )( ) (
) ( )
0.15 30 , 0.1 30 , 0.05 15 8.25→
earn the third point but not the second.
• A response of
( )( ) ( )( ) ( )( )
90 90 60 120 120 90 135 135 120 8.25ff f
−+ −+ − =
earns both the
second and the third points.
• A response that presents an answer of only
8.25
does not earn either the second or third point.
• A response that provides a completely correct left Riemann sum with accompanying work,
( )(
) ( )( ) ( )( )
60 30 90 30 120 15 9,fff++ =
or
( )( ) ( )( ) ( )( )
0.1 30 0.15 30 0.1 15++
earns 1 of the
last 2 points. A response with any errors or missing factors in a left Riemann sum earns neither of
the last 2 points.
Total for part (a) 3 points
(b)
Must there exist a value of
,
c
for
60 120,
c<<
such that
( )
0?
fc
′
=
Justify your answer.
f
is differentiable.
f
⇒
is continuous on
[ ]
60, 120 .
( ) (
)
120 60
0.1 0.1
0
120 60 60
ff−
−
= =
−
By the Mean Value Theorem, there must exist a
,c
for
60 120,c
<<
such that
( )
0.fc
′
=
( ) ( )
120 60 0ff−=
1 point
Answer with
justification
1 point
Scoring notes:
• To earn the first point a response must present either
( ) ( )
120 60 0,ff−=
0.1 0.1 0−=
(perhaps
as the numerator of a quotient), or
( ) ( )
60 120 .ff=
• To earn the second point a response must:
o have earned the first point,
o state that
f
is continuous because
f
is differentiable (or equivalent), and
o answer “yes” in some way.
• A response may reference either the Mean Value Theorem or Rolle’s Theorem.
• A response that references the Intermediate Value Theorem cannot earn the second point.
Total for part (b) 2 points
(c)
The rate of flow of gasoline, in gallons per second, can also be modeled by
( )
( ) ( )
2
cos
500 120
tt
gt
=
for
0 150.t≤≤
Using this model, find the average rate of flow of
gasoline over the time interval
0 150.t≤≤
Show the setup for your calculations.
( )
150
0
1
150 0
g t dt
−
∫
Average value
formula
1 point
0.0959967=
Answer 1 point
AP® Calculus AB/BC 2023 Scoring Guidelines
© 2023 College Board
The average rate of flow of gasoline, in gallons per second, is
0.096
(or
0.095
).
Scoring notes:
• The exact value of
( )
150
0
1
150
g t dt
∫
is
( )
12 25
sin .
125 16
• A response may present the average value formula in single or multiple steps. For example, the
following response earns both points:
( )
150
0
14.399504g t dt =
∫
so the average rate is
0.0959967.
• A response that presents the average value formula in multiple steps but provides incorrect or
incomplete communication (e.g.,
( )
150
0
14.399504
0.0959967
150
g t dt = =
∫
) earns 1 out of
2 points.
• A response of
150
0
0.0959967()dtgt =
∫
does not earn either point.
• Degree mode: A response that presents answers obtained by using a calculator in degree mode
does not earn the first point it would have otherwise earned. The response is generally eligible for
all subsequent points (unless no answer is possible in degree mode or the question is made simpler
by using degree mode). In degree mode,
( )
150
0
1
0.149981
150
g t dt =
∫
or
0.002618.
Total for part (c) 2 points
(d)
Using the model
g
defined in part (c), find the value of
( )
140 .g
′
Interpret the meaning of your
answer in the context of the problem.
( )
140 0.004908
g
′
≈−
( )
140 0.005g
′
= −
(or
0.004
−
)
( )
140g
′
1 point
The rate at which gasoline is flowing into the tank is decreasing at
a rate of
0.005
(or
0.004
) gallon per second per second at time
140
t =
seconds.
Interpretation 1 point
Scoring notes:
• The exact value of
( )
140g
′
is
( ) ( )
1 49 49 49
cos sin .
500 36 9000 36
−
• The value of
(
)
140g
′
may appear only in the interpretation.
• To be eligible for the second point a response must present some numerical value for
( )
140 .g
′
• To earn the second point the interpretation must include “the rate of flow of gasoline is changing
at a rate of [the declared value of
( )
140g
′
]” and “at
140t =
” (or equivalent).
• An interpretation of “decreasing at a rate of
–0.005
” or “increasing at a rate of
0.005
” does not
earn the second point.
• Degree mode: In degree mode,
(
)
140 0.001997g
′
=
or
0.00187.
Total for part (d) 2 points
Total for question 1 9 points
1 of 2
Sample 1A
2 of 2
Sample 1A
1 of 2
Sample 1B
2 of 2
Sample 1B
1 of 2
Sample 1C
2 of 2
Sample 1C
AP
®
Calculus AB/BC 2023 Scoring Commentary
© 2023 College Board.
Visit College Board on the web: collegeboard.org.
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
t
in seconds and values of a function
( )
,ft
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
(
)
135
60
f t dt
∫
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
60
t =
to time
135t =
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
,c
with
60 120,c<<
such that
( )
0.
fc
′
=
Students are expected to note that because the function
f
is known to be differentiable on the interval
( )
0, 150 ,
it must be continuous on the subinterval
[ ]
60, 120 .
Therefore, because the average rate of change of
f
on the interval
[ ]
60, 120
is equal to
0,
such a value of
c
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
g
to find the average rate of flow of the gasoline over
the time interval
0 150.t
≤≤
A correct response will show the setup
( )
150
0
1
150 0
g t dt
⋅
−
∫
and then use a
calculator to find the value
0.096
gallon per second.
In part (d) students were asked to find the value of
( )
140g
′
and interpret the meaning of this value in the context of
the problem. A correct response will use a calculator to find
( )
140 0.005g
′
= −
and report that at time
140t =
seconds the rate at which gasoline is flowing into the tank is decreasing at a rate of
0.005
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
60t =
to
135t =
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 “
( ) (
)
60 .1 120 .ff= =
” The response earned the second point
because it earned the first point, states that “
( )
ft
is always differentiable, and therefore it must be continuous on
[ ]
,ab
60a =
&
120,b =
” 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.
AP
®
Calculus AB/BC 2023 Scoring Commentary
© 2023 College Board.
Visit College Board on the web: collegeboard.org.
Question 1 (continued)
In part (d) the response earned the first point for the correct value of
( )
140 .g
′
The response earned the second point
with the statement “at
140t =
seconds, the rate of flow of gasoline is changing at a rate of
.005.
−
”
Sample: 1B
Score: 5
The response earned 5 points: 2 points in part (a), no points in part (b), 2 points in part (c), and 1 point in part (d).
In part (a) the response earned the first point with the statement “the time frame (from
60
to
135
seconds) in
which a certain amount of gallons of gasoline are pumped into a gas tank.” The response earned the second point
for the correct form of the Riemann sum with five of the six factors correct. The third point was not earned
because the response contains an error in the Riemann sum.
In part (b) the response did not earn the first point because the expression
( )
( )
120 60 0ff−=
is not included.
Because the first point was not earned, the response is not eligible for the second point. In addition, the response
did not earn the second point because the response does not state that
f
is continuous because
f
is
differentiable.
In part (c) the response earned the first point because the response includes the average value formula. The
response earned the second point with the correct answer.
In part (d) the response earned the first point with the presence of the correct value of
( )
140 .g
′
The response did not
earn the second point because the response does not interpret the declared value of
( )
140g
′
correctly (it needs to
discuss a rate of a rate). The words acceleration and velocity should be used to refer to an object in motion.
Sample: 1C
Score: 2
The response earned 2 points: 2 points in part (a), no points in part (b), no points in part (c), and no points in
part (d).
In part (a) the response did not earn the first point. 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 did not earn the first point because the response does not include
( ) ( )
120 60 0.ff
−=
Because the first point was not earned, the response is not eligible for the second point. In addition, the response
did not earn the second point because the response does not state that
f
is continuous because
f
is
differentiable.
In part (c) the response did not earn the first point because the response does not include the average value
formula. The response did not earn the second point because the response does not include the correct answer.
In part (d) the response did not earn the first point because the response does not include the value of
( )
140 .g
′
The
response did not earn the second point because the response does not include the correct interpretation.
