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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 3: Differentiation of Composite, Implicit & Inverse FunctionsDifferentiation of Composite & Inverse Functions

Differentiation of Composite & Inverse Functions (College Board AP® Calculus AB)

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Differentiation of Composite & Inverse Functions

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Differentiation of Composite & Inverse Functions

1
Sme Calculator2 marks

x

2

3

5

8

f open parentheses x close parentheses

-5

0

7

1

f to the power of apostrophe open parentheses x close parentheses

-2

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-4

8

g open parentheses x close parentheses

-6

2

-7

-1

g to the power of apostrophe open parentheses x close parentheses

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5

6

-8

The functions f and g are differentiable. The table shown gives the values of the functions and their first derivatives at selected values of x.

Let h be a differentiable function such that h open parentheses x close parentheses equals f open parentheses g open parentheses x close parentheses close parentheses.

Find the value of h apostrophe open parentheses 3 close parentheses.

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2a
Sme Calculator2 marks

The function f is defined by f open parentheses x close parentheses equals square root of 64 minus x squared end root for negative 8 less or equal than x less or equal than 8.

Find f to the power of apostrophe open parentheses x close parentheses.

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2b
Sme Calculator2 marks

Write an equation for the line tangent to the graph of f at x equals 2 square root of 7.

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31 mark

x

2

3

5

8

f open parentheses x close parentheses

6

0

7

5

f apostrophe open parentheses x close parentheses

2

3

-4

9

The function f is a one-to-one, differentiable function. The table shown gives the values of the function and its first derivatives at selected values of x.

Let g be a differentiable function such that g open parentheses x close parentheses equals f to the power of negative 1 end exponent open parentheses x close parentheses.

Find the value of g to the power of apostrophe open parentheses 5 close parentheses.

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4
Sme Calculator3 marks

The function f is defined as f open parentheses x close parentheses equals fraction numerator x over denominator 2 pi end fraction cos open parentheses x squared over pi close parentheses.

f to the power of apostrophe open parentheses pi over 2 close parentheses can be written as fraction numerator square root of a open parentheses b minus pi close parentheses over denominator c pi end fraction where a comma space b and c are integers. Find the values of a comma space b and c.

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5
Sme Calculator1 mark

The function f is defined as f open parentheses x close parentheses equals arcsin space open parentheses 2 x close parentheses for negative pi over 2 less or equal than x less or equal than pi over 2.

Find the exact value of f to the power of apostrophe open parentheses 1 fourth close parentheses.

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