AP Calculus BC Unit 8 Progress Check FRQ Part A focuses on evaluating student understanding of calculus concepts through free-response questions. It includes problems related to velocity, acceleration, and distance, requiring the application of derivatives and integrals. This resource is essential for AP Calculus students preparing for the exam, offering practice in critical thinking and problem-solving skills. The questions cover topics such as average velocity, total distance, and the interpretation of acceleration, providing a comprehensive review for students.

Key Points

  • Includes free-response questions on velocity and acceleration concepts.
  • Covers average velocity calculations over specified time intervals.
  • Requires interpretation of acceleration and its significance in motion.
  • Focuses on total distance calculations during a defined time period.
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AP® Calculus BC
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AP Calculus BC Scoring Guide
Unit 8 Progress Check: FRQ Part A
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.
A GRAPHING CALCULATOR IS REQUIRED FOR THIS QUESTION.
You are permitted to use your calculator to solve an equation, find the derivative of a function
at a point, or calculate the value of a definite integral. However, you must clearly indicate the
setup of your question, namely the equation, function, or integral you are using. If you use
other built-in features or programs, you must show the mathematical steps necessary to
produce your results. Your work must be expressed in standard mathematical notation rather
than calculator syntax.
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.
Gwen runs back and forth along a straight track. During the time interval seconds,
Gwen’s velocity, in feet per second, is modeled by the function given by .
(a) What is the first time, , that Gwen changes direction? Find Gwen’s average velocity over
the time interval .
Please respond on separate paper, following directions from your teacher.
(b) At time , how far is Gwen from her position at time ?
AP Calculus BC Scoring Guide
Unit 8 Progress Check: FRQ Part A
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
Please respond on separate paper, following directions from your teacher.
(c) Find the total distance that Gwen runs during the 45 seconds from time to time
.
Please respond on separate paper, following directions from your teacher.
(d) Let represent Gwen’s acceleration during the time interval seconds. Find
the value of . Using correct units, explain the meaning of in the context
ⅆ
ⅆ
of the problem.
Please respond on separate paper, following directions from your teacher.
Part A
For the first point, supporting work stating that changes sign is required.
For the second point, supporting work that includes the average value expression with an appropriate
definite integral is required. To be eligible for the second point, the presented value of may be incorrect
but must be positive.
The third point requires the first two points to have been earned.
Select a point value to view scoring criteria, solutions, and/or examples to score the response.
0
1 2 3
The student response accurately includes all three of the criteria below.
average value integral
AP Calculus BC Scoring Guide
Unit 8 Progress Check: FRQ Part A
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
answer
Solution:
Gwen changes direction when changes sign. The first time this happens is at
Gwen’s average velocity over the time interval is 7.264 feet per second.
Part B
Select a point value to view scoring criteria, solutions, and/or examples and to score the response.
0
1 2
The student response accurately includes both of the criteria below.
definite integral
answer
Solution:
At time Gwen is 116.538 (or 116.537) feet from her position at time
Part C
Select a point value to view scoring criteria, solutions, and/or examples and to score the response.
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FAQs

What types of problems are included in the AP Calculus BC Unit 8 Progress Check?
The AP Calculus BC Unit 8 Progress Check includes free-response questions that assess students' understanding of key calculus concepts. Problems focus on velocity, acceleration, and distance, requiring students to apply their knowledge of derivatives and integrals. Each question challenges students to demonstrate their problem-solving skills and critical thinking in a calculus context.
How does the progress check help students prepare for the AP exam?
This progress check is designed to help students prepare for the AP Calculus BC exam by providing practice with free-response questions similar to those on the actual test. By working through these problems, students can enhance their understanding of calculus concepts and improve their ability to articulate their reasoning. The check also allows students to identify areas where they may need further study or practice.
What is the significance of calculating average velocity in the context of this progress check?
Calculating average velocity is significant as it helps students understand the relationship between distance and time in motion problems. It requires the application of definite integrals and the fundamental theorem of calculus, reinforcing key concepts in calculus. Understanding average velocity also lays the groundwork for more complex topics such as instantaneous velocity and acceleration.
What concepts are tested in the acceleration-related questions?
The acceleration-related questions in the progress check test students' understanding of how acceleration relates to changes in velocity over time. Students are required to interpret acceleration values and explain their significance in the context of motion. This reinforces the concept that acceleration is the rate of change of velocity, which is a fundamental idea in calculus and physics.
How can students use this progress check to improve their calculus skills?
Students can use the AP Calculus BC Unit 8 Progress Check to improve their calculus skills by practicing problem-solving techniques and applying theoretical concepts to real-world scenarios. By reviewing their answers and understanding the reasoning behind each solution, students can identify their strengths and weaknesses. This targeted practice is crucial for mastering calculus concepts and performing well on the AP exam.

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