AP Calculus AB Unit 4 Progress Check FRQ Part B focuses on free-response questions designed to assess students' understanding of calculus concepts. It includes problems related to rates of change, concavity, and particle motion, providing a comprehensive review for AP exam preparation. This resource is essential for students aiming to excel in the AP Calculus AB exam, covering critical topics and problem-solving techniques. It is suitable for high school students enrolled in AP Calculus courses.

Key Points

  • Includes free-response questions on rates of change and concavity.
  • Covers particle motion concepts relevant to AP Calculus AB.
  • Provides detailed explanations for calculus problem-solving techniques.
  • Designed for students preparing for the AP Calculus AB exam.
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AP® Calculus AB
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AP® Calculus AB
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AP Calculus AB Scoring Guide
Unit 4 Progress Check: FRQ Part B
Copyright © 2017. The College Board. These materials are part of a College Board program. Use or distribution of these materials online or
in print beyond your school’s participation in the program is prohibited.
Page 1 of 8
1.
NO CALCULATOR IS ALLOWED FOR THIS QUESTION.
Show all of your work, even though the question may not explicitly remind you to do so. Clearly
label any functions, graphs, tables, or other objects that you use. Justifications require that you
give mathematical reasons, and that you verify the needed conditions under which relevant
theorems, properties, definitions, or tests are applied. Your work will be scored on the
correctness and completeness of your methods as well as your answers. Answers without
supporting work will usually not receive credit.
Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your
answer is given as a decimal approximation, it should be correct to three places after the
decimal point.
Unless otherwise specified, the domain of a function is assumed to be the set of all real
numbers for which is a real number.
The depth of a river at a certain point is modeled by the function defined above, where
is measured in feet and time is measured in hours.
(a) Find . Using correct units, explain the meaning of in the context of the
problem.
Please respond on separate paper, following directions from your teacher.
(b) The graph of is concave down for . Use the line tangent to the graph of
at to show that .
Please respond on separate paper, following directions from your teacher.
(c) Find .
AP Calculus AB Scoring Guide
Unit 4 Progress Check: FRQ Part B
Copyright © 2017. The College Board. These materials are part of a College Board program. Use or distribution of these materials online or
in print beyond your school’s participation in the program is prohibited.
Page 2 of 8
Please respond on separate paper, following directions from your teacher.
Part A
The first 2 points may be earned with the presence of a correct numerical answer.
The second point may be earned with a maximum of one sign error in
The third point requires a time reference, rate, and units. The response is eligible for the third point if the
second point is earned AND provided the behavior described is consistent with the sign of the
approximation, the numerical value, and the time reference.
Select a point value to view scoring criteria, solutions, and/or examples and to score the response.
0
1 2 3
The student response accurately includes all three of the criteria below.
interpretation with units
Solution:
For
At time hours, the depth of the river is changing at a rate of foot per hour.
Part B
The second point may be earned with a maximum of one error in
For the second and third points, trigonometric function values do not need to be evaluated.
AP Calculus AB Scoring Guide
Unit 4 Progress Check: FRQ Part B
Copyright © 2017. The College Board. These materials are part of a College Board program. Use or distribution of these materials online or
in print beyond your school’s participation in the program is prohibited.
Page 3 of 8
The third point may be earned if the response is consistent with previous results.
Select a point value to view scoring criteria, solutions, and/or examples and to score the response.
0
1 2 3 4
The student response accurately includes all four of the criteria below.
approximation
explanation
Solution:
For
Because the graph of is concave down for approximation of is an overestimate of
the actual value of .
Therefore,
Part C
The first point requires showing that an indeterminate form 0 is present in order to apply L’Hospital’s
Rule. Any error in mathematical communication (e.g., writing “ " ” or not using limit notation) impacts
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FAQs

What types of problems are included in the AP Calculus AB Unit 4 Progress Check?
The AP Calculus AB Unit 4 Progress Check includes various free-response questions that test students on topics such as rates of change, concavity, and particle motion. These problems require students to apply calculus concepts and demonstrate their understanding through detailed solutions. Each question is designed to mimic the style and rigor of the actual AP exam, helping students prepare effectively.
How does the progress check assess understanding of concavity?
The progress check assesses understanding of concavity by presenting problems that require students to analyze the second derivative of functions. Students must determine where functions are concave up or down and use this information to make inferences about the behavior of the graph. This understanding is crucial for interpreting the shape of graphs and solving real-world problems involving acceleration and deceleration.
What is the significance of particle motion questions in the AP Calculus AB exam?
Particle motion questions are significant in the AP Calculus AB exam as they integrate concepts of derivatives and integrals to describe the position, velocity, and acceleration of moving objects. These questions help students apply calculus to real-world scenarios, enhancing their problem-solving skills. Understanding particle motion is essential for grasping the broader applications of calculus in physics and engineering.
How can students use this progress check to prepare for the AP exam?
Students can use the AP Calculus AB Unit 4 Progress Check to identify areas of strength and weakness in their understanding of calculus concepts. By practicing with free-response questions, they can develop their problem-solving skills and become familiar with the exam format. Additionally, reviewing the solutions and explanations provided will reinforce their learning and help them master the material.
What topics are emphasized in the AP Calculus AB Unit 4 Progress Check?
The AP Calculus AB Unit 4 Progress Check emphasizes key topics such as rates of change, the concept of concavity, and the analysis of particle motion. These topics are fundamental to understanding calculus and are frequently tested on the AP exam. Mastery of these concepts is crucial for students aiming to achieve a high score on the exam.

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