AP Calculus BC 2023 Free Response Questions and Scoring Guidelines

AP Calculus BC 2023 Free Response Questions and Scoring Guidelines

AP Calculus BC 2023 Free Response Questions provide a comprehensive overview of the exam format and scoring criteria. This resource includes detailed solutions and scoring guidelines for each question, helping students understand the expectations for their responses. Ideal for AP Calculus students preparing for the exam, it covers various topics such as derivatives, integrals, and the application of calculus concepts. The document serves as a valuable study aid for mastering the material and improving exam performance.

Key Points

  • Includes detailed solutions for AP Calculus BC free response questions.
  • Covers scoring guidelines for each question to aid student understanding.
  • Focuses on key calculus concepts such as derivatives and integrals.
  • Provides insights into the expectations for AP exam responses.
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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.
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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
• 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 Free Response Questions and Scoring Guidelines

What topics are covered in the AP Calculus BC 2023 exam?
The AP Calculus BC 2023 exam covers a range of topics including limits, derivatives, integrals, and series. Specific areas of focus include the Fundamental Theorem of Calculus, techniques of integration, and applications of derivatives. Students are expected to demonstrate their understanding of these concepts through free response questions that require both analytical and graphical reasoning.
How are the free response questions scored in AP Calculus BC?
Free response questions in AP Calculus BC are scored based on a rubric that assigns points for correct answers, appropriate methods, and clear communication of mathematical reasoning. Each question typically has a maximum score, and partial credit may be awarded for correct steps even if the final answer is incorrect. The scoring guidelines help students understand how their responses will be evaluated.
What is the significance of the scoring guidelines for AP Calculus BC?
The scoring guidelines for AP Calculus BC provide essential insights into how students' responses will be assessed. They outline the criteria for earning points, which helps students focus on the most important aspects of their answers. Understanding these guidelines can significantly improve a student's ability to perform well on the exam by aligning their preparation with the expectations of the examiners.
What types of problems are included in the AP Calculus BC free response section?
The AP Calculus BC free response section includes a variety of problem types, such as those requiring the application of calculus concepts to real-world scenarios, theoretical questions that explore the properties of functions, and problems that involve multiple steps to solve. These questions challenge students to apply their knowledge creatively and demonstrate a deep understanding of calculus principles.
How can students use the scoring guidelines to improve their exam performance?
Students can use the scoring guidelines to identify key areas where they need improvement and to practice writing clear, concise mathematical explanations. By reviewing previous free response questions and their scoring criteria, students can learn how to structure their answers effectively and ensure they meet the expectations of the examiners. This targeted practice can lead to better performance on the actual exam.

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