AP Calculus AB Unit 5 Progress Check FRQ Part B

AP Calculus AB Unit 5 Progress Check FRQ Part B

AP Calculus AB Unit 5 Progress Check FRQ Part B focuses on critical points, relative extrema, and points of inflection in calculus functions. It includes detailed questions that require students to analyze a continuous function's behavior, identify critical points, and justify their classifications. This resource is essential for AP Calculus students preparing for the exam, providing practice with free-response questions aligned with the curriculum. The document emphasizes the importance of showing work and justifying answers, which is crucial for scoring well on the AP exam.

Key Points

  • Analyzes critical points and classifies them as relative maxima or minima.
  • Identifies points of inflection and explains their significance in function behavior.
  • Includes practice problems that align with AP Calculus AB exam standards.
  • Encourages detailed justifications for answers to enhance understanding of calculus concepts.
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AP Calculus AB Scoring Guide
Unit 5 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 9
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 graph of the continuous function is shown above for . The function is twice
AP Calculus AB Scoring Guide
Unit 5 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 9
differentiable, except at .
Let be the function with and derivative given by .
(a) Find the -coordinate of each critical point of . Classify each critical point as the location
of a relative minimum, a relative maximum, or neither. Justify your answers.
Please respond on separate paper, following directions from your teacher.
(b) Find all values of at which the graph of has a point of inflection. Give reasons for your
answers.
Please respond on separate paper, following directions from your teacher.
(c) Fill in the missing entries in the table below to describe the behavior of and on the
interval . Indicate Positive or Negative. Give reasons for your answers.
Please respond on separate paper, following directions from your teacher.
(d) Let be the function defined by . Is increasing or decreasing at
? Give a reason for your answer.
Please respond on separate paper, following directions from your teacher.
Part A
AP Calculus AB Scoring Guide
Unit 5 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 9
Note: Sign charts are a useful tool to investigate and summarize the behavior of a function. By itself a
sign chart for or is not a sufficient response for a justification.
The first point requires reference to
A maximum of 1 out of 3 points is earned for only one correct critical point with correct identification and
justification, and no incorrect critical points are included.
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.
critical points
relative maximum at with justification
relative minimum at with justification
Solution:
has a relative maximum at because changes from positive to negative there.
has a relative minimum at because changes from negative to positive there.
Part B
Note: Sign charts are a useful tool to investigate and summarize the behavior of a function. By itself a
sign chart for or is not a sufficient response for a justification.
A maximum of 1 out of 2 points is earned if only one point of inflection with reason and no incorrect points
of inflection.
Select a point value to view scoring criteria, solutions, and/or examples and to score the response.
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End of Document
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FAQs of AP Calculus AB Unit 5 Progress Check FRQ Part B

How do you find critical points in calculus?
To find critical points in calculus, you first need to take the derivative of the function and set it equal to zero. This helps identify where the function's slope is zero, indicating potential maxima or minima. Additionally, you should check where the derivative does not exist, as these points can also be critical. Once identified, you can use the first or second derivative test to classify each critical point as a relative maximum, minimum, or neither.
What is the significance of points of inflection?
Points of inflection are significant because they indicate where a function changes its concavity, which can affect the overall shape of the graph. At these points, the second derivative of the function is either zero or undefined. Understanding points of inflection helps in sketching the graph of a function and analyzing its behavior, particularly in optimization problems and curve sketching.
What is the first derivative test?
The first derivative test is a method used to determine whether a critical point is a relative maximum or minimum. By evaluating the sign of the derivative before and after the critical point, you can conclude if the function is increasing or decreasing. If the derivative changes from positive to negative at a critical point, it indicates a relative maximum. Conversely, if it changes from negative to positive, it indicates a relative minimum.
How do you justify answers in AP Calculus FRQs?
Justifying answers in AP Calculus free-response questions involves providing mathematical reasoning and supporting work for each conclusion. This includes showing calculations, explaining the application of relevant theorems, and discussing the behavior of functions based on their derivatives. Clear and logical reasoning is essential, as it demonstrates a deep understanding of calculus concepts, which is crucial for scoring well on the exam.
What topics are covered in AP Calculus AB Unit 5?
AP Calculus AB Unit 5 covers topics related to derivatives and their applications, including critical points, relative extrema, and concavity. Students learn how to analyze functions to find maxima and minima, as well as points of inflection. This unit emphasizes the importance of understanding the behavior of functions through their derivatives, which is foundational for solving complex calculus problems.

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