AP Calculus BC 2023 FRQ Scoring Guidelines and Solutions

AP Calculus BC 2023 FRQ Scoring Guidelines and Solutions

AP Calculus BC 2023 FRQ Scoring Guidelines provides detailed scoring criteria for free-response questions on the AP Calculus BC exam. It includes model solutions, scoring notes, and interpretations for each question, helping students understand how to effectively approach the exam. This resource is essential for AP Calculus BC students preparing for the May exam, offering insights into problem-solving strategies and scoring rubrics. The guidelines cover various topics, including Riemann sums, the Mean Value Theorem, and average rates of change, ensuring comprehensive preparation.

Key Points

  • Includes scoring criteria for AP Calculus BC free-response questions.
  • Provides model solutions and detailed scoring notes for each question.
  • Covers essential calculus concepts like Riemann sums and the Mean Value Theorem.
  • Helps students prepare for the AP Calculus BC exam with effective strategies.
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AP
®
Calculus BC
Scoring Guidelines
2023
© 2023 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 ofÏcial online home for the AP Program: apcentral.collegeboard.org.
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AP Calculus BC 2023 FRQ Scoring Guidelines and Solutions
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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
f t
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
( )
f t
is measured in gallons per second and
t
is measured in seconds since
pumping began. Selected values of
( )
f t
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
f f f≈ − + − + −
= + + =
∫
Form of Riemann
sum
1 point
Answer 1 point
Scoring notes:
• To earn the first point the time the response must reference gasoline added/pumpedgallons of and
interval
60t =
to
135.t =
• To earn the second point at least factors in the Riemann sum must be correct. five of the six
• 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.
5/11/26, 2:03 AM
AP Calculus BC 2023 FRQ Scoring Guidelines and Solutions
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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 8.25f f 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,f f f+ + =
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 ?f c
′
=
Justify your answer.
f
is differentiable.
f⇒
is continuous on
[ ]
60, 120 .
( ) ( )
120 60
0.1 0.1
0
120 60 60
f f−
−
= =
−
By the Mean Value Theorem, there must exist a
,c
for
60 120,c< <
such that
( )
0.f c
′
=
( ) ( )
120 60 0f f− =
1 point
Answer with
justification
1 point
Scoring notes:
• To earn the first point a response must present either
( ) ( )
120 60 0,f f− =
0.1 0.1 0− =
(perhaps
as the numerator of a quotient), or
( ) ( )
60 120 .f f=
• 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
t t
g t
=
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
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Faqs of AP Calculus BC 2023 FRQ Scoring Guidelines and Solutions
What scoring criteria are used for AP Calculus BC free-response questions?
The scoring criteria for AP Calculus BC free-response questions include specific points awarded for correct answers, appropriate use of calculus concepts, and clear communication of mathematical reasoning. Each question is broken down into parts, with points allocated for correct interpretations, calculations, and justifications. The guidelines emphasize the importance of showing work and using proper notation, which is crucial for maximizing scores on the exam.
How are Riemann sums applied in the AP Calculus BC exam?
Riemann sums are used in the AP Calculus BC exam to approximate the area under curves and to interpret definite integrals. Students are required to set up Riemann sums using given data points and calculate approximate values for integrals. Understanding the application of Riemann sums is essential for solving problems related to rates of change and total accumulation, which are common themes in the exam.
What is the Mean Value Theorem and its significance in calculus?
The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative equals the average rate of change over that interval. This theorem is significant because it provides a formal framework for understanding the relationship between a function's behavior and its derivatives, which is a key concept in calculus. It is often used in problems involving motion, optimization, and curve sketching.
What topics are covered in the AP Calculus BC scoring guidelines?
The AP Calculus BC scoring guidelines cover a range of topics including limits, derivatives, integrals, and series. Specific concepts such as the Fundamental Theorem of Calculus, differential equations, and polynomial approximations are also included. Each topic is addressed through various free-response questions, allowing students to demonstrate their understanding and application of calculus principles in real-world scenarios.