AP Calculus BC Unit 7 Progress Check FRQ Part A

AP Calculus BC Unit 7 Progress Check FRQ Part A

Unit 7 of the AP Calculus BC course focuses on differential equations and their applications. This section includes free-response questions that require students to demonstrate their understanding of tangent lines, initial conditions, and the effects of antiviral medication on viral populations. Designed for AP Calculus BC students preparing for the exam, it emphasizes problem-solving and analytical skills necessary for success. The content is aligned with the 2024 AP exam format, providing practice with real exam scenarios.

Key Points

  • Includes free-response questions on differential equations and tangent lines.
  • Covers initial conditions and their verification in calculus problems.
  • Explores the impact of antiviral medication on viral populations.
  • Designed for AP Calculus BC students preparing for the May exam.
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AP Calculus BC Scoring Guide
Unit 7 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.
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 population of a virus in a host can be modeled by the function that satisfies the
differential equation , where is measured in millions of virus cells and is
measured in days for . At time days, there are 10 million cells of the virus in
the host.
(a) Write an equation for the line tangent to the graph of at . Use the tangent line to
approximate the number of virus cells in the host, in millions, at time days.
Please respond on separate paper, following directions from your teacher.
(b) Show that satisfies the differential equation
ⅆ
ⅆ
with initial condition .
Please respond on separate paper, following directions from your teacher.
(c) The host receives an antiviral medication. The amount of medication in the host is
modeled by the function that satisfies the differential equation , where is
AP Calculus BC Scoring Guide
Unit 7 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
measured in milligrams, is measured in days since the host received the medication, and is
a positive constant. If the amount of medication in the host is 30 milligrams at time days
and 15 milligrams at time days, what is in terms of ?
Please respond on separate paper, following directions from your teacher.
Part A
For the second point, it is incorrect to state rather than
Select a point value to view scoring criteria, solutions, and/or examples to score the response.
0
1 2
The student response accurately includes both of the criteria below.
tangent line equation
approximation
Solution:
An equation for the line tangent to the graph of at is
At time days, there are approximately million virus cells in the host.
Part B
Select a point value to view scoring criteria, solutions, and/or examples and to score the response.
AP Calculus BC Scoring Guide
Unit 7 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
0
1 2 3
The student response accurately includes all three of the criteria below.
verification of initial condition
verification that
Solution:
Part C
Zero out of 4 points earned if no separation of variables.
At most 2 out of 4 points earned [1-1-0-0] if no constant of integration.
Both antiderivatives must be correct to earn the second point.
The fourth point requires an expression for The domain of is included with the solution; this
is not a requirement to earn the fourth point.
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.
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End of Document
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FAQs of AP Calculus BC Unit 7 Progress Check FRQ Part A

What types of problems are included in the Unit 7 FRQ?
The Unit 7 FRQ includes problems related to differential equations, specifically focusing on modeling populations and the effects of medication. Students are required to write equations for tangent lines, verify initial conditions, and solve for variables in context. These problems test both conceptual understanding and practical application of calculus principles.
How does the antiviral medication affect the viral population?
The section discusses how the amount of antiviral medication in a host can influence the population of a virus. It models the medication's effect through a differential equation, allowing students to analyze how changes in medication levels impact viral cell counts over time. Understanding this relationship is crucial for applying calculus to real-world biological scenarios.
What is the significance of initial conditions in calculus?
Initial conditions are vital in solving differential equations as they provide specific values that help determine the unique solution of a problem. In the context of the AP Calculus BC Unit 7 FRQ, students must verify initial conditions to ensure their solutions are accurate and applicable to the given scenarios. This concept reinforces the importance of boundary values in mathematical modeling.
What concepts are emphasized in the AP Calculus BC Unit 7?
Unit 7 emphasizes key concepts such as differential equations, the behavior of functions, and their applications in real-world contexts. Students learn to analyze and interpret mathematical models, particularly in biological systems, which enhances their problem-solving skills. The unit prepares students for the complexities of the AP exam by integrating theoretical knowledge with practical applications.

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