AP Unit 7 Progress Check: FRQ Part A Answers

AP Unit 7 Progress Check: FRQ Part A Answers

AP Calculus AB Unit 7 Progress Check focuses on Free Response Questions (FRQ) for students preparing for the AP exam. It includes detailed solutions and explanations for various calculus problems, particularly those involving differential equations and tangent line approximations. Designed for AP Calculus students, this resource helps reinforce understanding of key concepts and problem-solving techniques. The content aligns with the latest AP curriculum, making it a valuable tool for exam preparation.

Key Points

  • Includes solutions for AP Calculus AB Unit 7 FRQs focusing on differential equations.
  • Provides step-by-step explanations for tangent line approximations and initial conditions.
  • Covers the flow of solutions in a saltwater vat scenario, enhancing real-world application of calculus concepts.
  • Designed for AP Calculus students aiming to improve their exam performance with practice problems.
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AP Calculus AB 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 4
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.
A large vat is initially filled with a saltwater solution. A solution with a higher concentration of
salt flows into the vat, and solution flows out of the vat at the same rate. The number of pounds
of salt in the vat at time minutes is modeled by the function that satisfies the differential
equation . At time minutes, the vat contains 50 pounds of salt.
ⅆ
ⅆ
(a) Write an equation for the line tangent to the graph of at . Use the tangent line to
approximate the number of pounds of salt in the vat at time minutes.
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 flow of solution into the vat is stopped, and the solution is drained. The depth of
solution in the vat is modeled by the function that satisfies the differential equation
AP Calculus AB 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 4
, where is measured in meters, is the number of minutes since draining
ⅆ
ⅆ
began, and is a constant. If the depth of the solution is 16 meters at time minutes and 4
meters at time minutes, 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 minutes, the vat contains approximately pounds of salt.
Part B
Select a point value to view scoring criteria, solutions, and/or examples and to score the response.
AP Calculus AB 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 4
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
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End of Document
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FAQs of AP Unit 7 Progress Check: FRQ Part A Answers

What types of problems are included in the AP Unit 7 Progress Check?
The AP Unit 7 Progress Check includes various Free Response Questions (FRQs) that focus on differential equations, tangent line approximations, and real-world applications of calculus. Students will encounter problems involving the modeling of saltwater solutions and the behavior of functions over time. Each question is designed to test understanding of calculus concepts and the ability to apply them in different contexts. The solutions provided offer detailed explanations to help students grasp the underlying principles.
How does the document assist students in preparing for the AP Calculus exam?
This resource assists students by providing comprehensive solutions to AP Calculus AB Unit 7 FRQs, which are crucial for exam preparation. It breaks down complex problems into manageable steps, allowing students to follow along and understand the reasoning behind each solution. By practicing with these problems, students can enhance their problem-solving skills and gain confidence in their ability to tackle similar questions on the exam. The alignment with the AP curriculum ensures that students are studying relevant material.
What is the significance of tangent line approximations in calculus?
Tangent line approximations are significant in calculus as they provide a way to estimate the value of a function at a point using the slope of the function at that point. This concept is fundamental in understanding derivatives and their applications in real-world scenarios. By using tangent lines, students can approximate values and analyze the behavior of functions near specific points. This technique is particularly useful in optimization problems and in understanding the instantaneous rate of change.
What concepts are reinforced through the saltwater vat problem?
The saltwater vat problem reinforces concepts related to differential equations and the modeling of dynamic systems. It illustrates how changes in concentration and volume can be represented mathematically, allowing students to apply calculus to real-life situations. This problem also emphasizes the importance of initial conditions and how they affect the behavior of solutions over time. By working through this scenario, students gain a deeper understanding of how calculus can be used to solve practical problems.

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