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 essential concepts in calculus, including critical points, points of inflection, and the behavior of functions. This assessment is designed for students preparing for the AP Calculus AB exam, providing practice with free-response questions. It covers topics such as differentiability, relative maxima and minima, and the interpretation of derivatives in real-world contexts. Students will engage with problems that require justification of their answers and application of calculus theorems.

Key Points

  • Analyzes critical points and classifies them as relative maxima or minima.
  • Identifies points of inflection based on the second derivative test.
  • Includes a detailed examination of function behavior on specified intervals.
  • Explores the implications of derivative values on function monotonicity.
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  • MR ALi
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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
What are critical points and how are they found?
Critical points occur where the derivative of a function is zero or undefined. In the context of AP Calculus, students learn to find these points by setting the first derivative equal to zero and solving for the variable. These points are crucial for determining the local maxima and minima of the function, which are essential for graphing and understanding function behavior.
How do you determine points of inflection?
Points of inflection are found where the second derivative of a function changes sign. This indicates a change in the concavity of the graph. In the AP Calculus curriculum, students are taught to calculate the second derivative and analyze its sign to identify these points, which helps in sketching accurate graphs and understanding the function's behavior.
What is the significance of relative maxima and minima?
Relative maxima and minima are important for understanding the local behavior of functions. A relative maximum is a point where the function value is higher than all nearby points, while a relative minimum is lower. Identifying these points helps in optimization problems and is a key skill tested in AP Calculus exams.
What role does the first derivative play in function analysis?
The first derivative of a function provides information about its slope and rate of change. By analyzing the first derivative, students can determine where the function is increasing or decreasing. This analysis is crucial for identifying critical points and understanding the overall behavior of the function, which is a fundamental concept in calculus.
How is the behavior of functions described on specific intervals?
Describing the behavior of functions on specific intervals involves evaluating the first derivative to determine whether the function is increasing or decreasing. Students learn to create sign charts to visualize this behavior, which aids in understanding the overall shape of the graph and identifying important features such as peaks and troughs.