AP Calculus AB Unit 1 Progress Check FRQ Part B

AP Calculus AB Unit 1 Progress Check FRQ Part B

Unit 1 Progress Check for AP Calculus AB focuses on Free Response Questions (FRQ) that assess students' understanding of calculus concepts. This assessment includes questions on limits, continuity, and the application of the Intermediate Value Theorem. Designed for AP Calculus students preparing for the exam, it provides a rigorous evaluation of their problem-solving skills and conceptual knowledge. The document includes detailed scoring guidelines and explanations to help students understand their performance and areas for improvement.

Key Points

  • Includes multiple Free Response Questions assessing limits and continuity concepts.
  • Applies the Intermediate Value Theorem to analyze function behavior.
  • Provides scoring criteria to help students evaluate their answers.
  • Designed for AP Calculus AB students preparing for the May exam.
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AP Calculus AB Scoring Guide
Unit 1 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.
The number of fish in a pond at time years is modeled by the function defined
above, where is a continuous function such that .
(a) Find . Explain the meaning of in the context of the problem.
(b) Is the function continuous at ? Justify your answer.
(c) The function is continuous at . Is there a time , for , at which
? Justify your answer.
AP Calculus AB Scoring Guide
Unit 1 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.
Part A
The second point requires a reference to time increasing and the connection to the number of
fish.
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.
interpretation
Solution:
As time increases, the number of fish in the pond approaches 1600.
Part B
At most 2 out of 3 points if mathematical notation for limits is missing or incorrect.
Select a point value to view scoring criteria, solutions, and/or examples and to score the
response.
AP Calculus AB Scoring Guide
Unit 1 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
0
1 2 3
The student response accurately includes all three of the criteria below.
answer with justification
Solution:
Yes, is continuous at because
Part C
At most 1 out of 2 points if correct justification does not specifically reference
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 of AP Calculus AB Unit 1 Progress Check FRQ Part B
What types of questions are included in the AP Calculus AB Unit 1 Progress Check?
The AP Calculus AB Unit 1 Progress Check includes Free Response Questions (FRQs) that focus on key calculus concepts such as limits and continuity. Students are required to demonstrate their understanding by solving problems that involve applying theorems and properties of functions. Each question is designed to challenge students' analytical and problem-solving skills, ensuring they are well-prepared for the AP exam.
How does the Intermediate Value Theorem apply in this assessment?
The Intermediate Value Theorem is a crucial concept in calculus that states if a function is continuous on a closed interval, then it takes on every value between its endpoints. In this assessment, students are asked to justify the application of this theorem to determine the existence of certain function values within specified intervals. This not only tests their understanding of continuity but also their ability to apply theoretical concepts to practical problems.
What is the significance of continuity in calculus?
Continuity is a fundamental concept in calculus that ensures a function behaves predictably without any breaks, jumps, or holes. In the context of the AP Calculus AB Unit 1 Progress Check, understanding continuity is essential for analyzing limits and applying theorems like the Intermediate Value Theorem. Students must demonstrate their ability to determine whether functions are continuous at given points, which is critical for solving more complex calculus problems.
What scoring criteria are used for the Free Response Questions?
The scoring criteria for the Free Response Questions in the AP Calculus AB Unit 1 Progress Check focus on correctness, completeness, and justification of answers. Each response is evaluated based on the accuracy of the mathematical reasoning provided and whether the student has clearly articulated their thought process. This helps ensure that students not only arrive at the correct answer but also understand the underlying principles of calculus.