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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 2: Differentiation Definition & Fundamental PropertiesFundamental Properties of Differentiation

Fundamental Properties of Differentiation (College Board AP® Calculus AB)

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Fundamental Properties of Differentiation

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Fundamental Properties of Differentiation

1
Sme Calculator3 marks

Function h is a differentiable function with h open parentheses 5 close parentheses equals 2. The line y equals 5 plus 4 over 5 open parentheses x minus 7 close parentheses is a tangent to the graph of h at x equals 5.

Let a be the function given by a open parentheses x close parentheses equals 4 x squared space h open parentheses x close parentheses.

Write an expression for a to the power of apostrophe open parentheses x close parentheses and find a to the power of apostrophe open parentheses 5 close parentheses.

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2
Sme Calculator2 marks

x

0

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

3

-4

8

g open parentheses x close parentheses

-6

2

-7

-1

g to the power of apostrophe open parentheses x close parentheses

4

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 k be a differentiable function such that k equals open parentheses f open parentheses x close parentheses close parentheses squared cross times g open parentheses x close parentheses.

Find the value of k apostrophe open parentheses 5 close parentheses.

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3
Sme Calculator3 marks
Line graph, showing a peak at (4,5) and a trough at (8,-4), extending from x=0 to x=13 and y=-6 to y=6 within grid lines. The line ends at (12, 0).
Graph of f

Let function f be a continuous function defined on the closed interval 0 less or equal than x less or equal than 12. The graph of f consisting of three line segments is shown above. Let G be the function defined by G open parentheses x close parentheses equals integral subscript 0 superscript x f open parentheses t close parentheses space italic d t.

Let P be the function defined by P open parentheses x close parentheses equals fraction numerator G open parentheses x close parentheses over denominator f open parentheses x close parentheses end fraction. Find P to the power of apostrophe open parentheses 7 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 10 space sin space x space cos space x over denominator x squared minus x plus 3 end fraction.

Find the value of f to the power of apostrophe open parentheses 0 close parentheses.

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5
Sme Calculator2 marks

The function f is defined as f open parentheses x close parentheses equals 4 to the power of x cross times e to the power of 3 x end exponent.

Find the slope of the tangent line to the graph of y equals f open parentheses x close parentheses at the point where x equals 0.

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