AP Calculus BC Unit 7 Progress Check FRQ Part B focuses on evaluating logistic differential equations related to liquid fertilizer injection in hydroponics systems. It includes problems on finding derivatives, determining rates of change, and applying Euler’s method for approximations. This resource is essential for AP Calculus students preparing for their exams, providing practice with real-world applications of calculus concepts. The guide emphasizes the importance of showing work and justifying answers, aligning with the AP exam's scoring criteria.

Key Points

  • Includes logistic differential equations modeling liquid fertilizer injection in hydroponics.
  • Covers finding derivatives and rates of change for functions related to fertilizer application.
  • Applies Euler's method to approximate total fertilizer injected over time.
  • Emphasizes the importance of justifying answers and showing work for AP exam scoring.
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AP® Calculus BC
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AP® Calculus BC
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AP Calculus BC Scoring Guide
Unit 7 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 5
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.
Liquid fertilizer is injected into a hydroponics growing system via a pumping system. The total
amount of liquid fertilizer injected into the growing system by time is modeled by the function
that satisfies the logistic differential equation , where is measured in
months and is measured in liters. At time , 3 liters of liquid fertilizer are injected into the
growing system. (Note: Hydroponics is the process of growing plants in sand, gravel, or liquid,
with added nutrients but without soil.)
(a)
(i) Find .
(ii) Find .
Please respond on separate paper, following directions from your teacher.
(b) Find the value of at the time when is increasing most rapidly. Give a reason for your
answer, and indicate units of measure.
AP Calculus BC Scoring Guide
Unit 7 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 5
Please respond on separate paper, following directions from your teacher.
(c) Find in terms of .
Please respond on separate paper, following directions from your teacher.
(d)
(i) Use Euler’s method, starting at with two steps of equal size, to approximate the total
amount of liquid fertilizer injected into the growing system by time month. Show the
computations that lead to your answer.
(ii) Is the approximation an overestimate or an underestimate for the total amount of liquid
fertilizer injected into the growing system by time month? Give a reason for your answer.
Please respond on separate paper, following directions from your teacher.
Part A
No supporting work is required to earn either point.
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.
Solution:
AP Calculus BC Scoring Guide
Unit 7 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 5
(i)
(ii)
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.
answer with units
Solution:
satisfies a logistic differential equation with carrying capacity 6. grows most rapidly when
liters per month
Part C
Algebraic simplification is not required for the second point.
Select a point value to view scoring criteria, solutions, and/or examples to score the response.
0
1 2
/ 5
End of Document
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FAQs

What is the logistic differential equation used in this unit?
The logistic differential equation models the growth of a quantity, in this case, liquid fertilizer injected into a hydroponics system. It typically takes the form dF/dt = kF(M - F), where F represents the amount of fertilizer, k is a constant, and M is the carrying capacity of the system. Understanding this equation is crucial for analyzing how the fertilizer quantity changes over time and reaches its maximum capacity.
How does Euler's method apply to this calculus unit?
Euler's method is a numerical technique used to approximate solutions to differential equations. In this unit, students apply Euler's method to estimate the total amount of liquid fertilizer injected into the system over a specified time period. By taking equal steps and calculating the function's value at each step, students can observe how the approximation evolves and assess its accuracy compared to the actual solution.
What is the significance of finding when the function is increasing most rapidly?
Determining when the function is increasing most rapidly is essential for understanding the dynamics of the fertilizer injection process. This point corresponds to the maximum rate of change, which can be found by analyzing the derivative of the function. Identifying this point helps students grasp the concept of growth rates in logistic models and their implications for real-world applications in agriculture and resource management.
What are the key components of the scoring guide for this unit?
The scoring guide for AP Calculus BC Unit 7 Progress Check emphasizes correctness and completeness in students' responses. It requires clear labeling of functions, graphs, and tables, as well as justifications for the methods used. Students are encouraged to show all work, as answers without supporting calculations may not receive credit. This approach aligns with the expectations of the AP exam, where demonstrating understanding is as important as arriving at the correct answer.
What types of problems are included in the progress check?
The progress check includes a variety of problems related to logistic growth, derivatives, and numerical methods. Students are tasked with finding specific values of functions, determining rates of change, and applying Euler's method to approximate solutions. These problems are designed to reinforce key calculus concepts and prepare students for the types of questions they will encounter on the AP exam.

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