Unit 3 Progress Check: FRQ Part B - Mr Guillen's Mathematics

AP Calculus AB

Unit 3 Progress Check: FRQ Part B

Scoring Guide

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.

Consider the curve given by the equation

.

(a) Show that

.

Please respond on separate paper, following directions from your teacher.

(b) Write an equation for the line tangent to the curve at the point

.

Please respond on separate paper, following directions from your teacher.

(c) Evaluate at the point

.

Please respond on separate paper, following directions from your teacher.

(d) The point

is on the curve. Find the value of

.

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 8

AP Calculus AB

Unit 3 Progress Check: FRQ Part B

Please respond on separate paper, following directions from your teacher.

Scoring Guide

Part A

The first point is earned for

or equivalent.

The second point cannot be earned without the first point. The second point is earned for a response that arrives at the given expression rather than an algebraically equivalent expression.

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. implicit differentiation verification

Solution:

Part B

The second point may be earned if response contains an incorrect value for slope based on one computational error.

Select a point value to view scoring criteria, solutions, and/or examples and to score the response.

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 8

AP Calculus AB

Unit 3 Progress Check: FRQ Part B

Scoring Guide

0

1

2

The student response accurately includes both of the criteria below.

slope equation for tangent line

Solution:

An equation for the tangent line is

Part C

The first point is earned with

or equivalent.

The second point is earned with an algebraic or numerical substitution for The response does not require an explicit expression for The third point does not require a simplified answer.

Select a point value to view scoring criteria, solutions, and/or examples to score the response.

0

1

2

3

The student responses accurately includes all three of the criteria below. implicit differentiation (2 points for full credit) answer

Solution:

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 8

AP Calculus AB

Unit 3 Progress Check: FRQ Part B

Scoring Guide

Part D

The second point may be earned if response contains an incorrect value based on one computational error. Substitution of function values is required.

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.

at answer

Solution:

Because

is on the curve,

is on the inverse.

2. NO CALCULATOR IS ALLOWED FOR THIS QUESTION.

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 4 of 8

AP Calculus AB

Scoring Guide

Unit 3 Progress Check: FRQ Part B

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 table above gives values of the differentiable function and its derivative for selected values of .

(a) Let be the function defined by

. Find

.

Please respond on separate paper, following directions from your teacher.

Let be the function defined by

. Find

.

Please respond on separate paper, following directions from your teacher.

Let be the function defined by

. Find

.

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 5 of 8

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