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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): Exam Questions

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Fundamental Properties of Differentiation

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1
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Function h is a differentiable function with h open parentheses 5 close parentheses equals 2. The line y equals 6 plus 4 over 5 open parentheses x minus 10 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

2

4

6

f open parentheses x close parentheses

0

-2

-1

3

f to the power of apostrophe open parentheses x close parentheses

-4

-2

3

7

g open parentheses x close parentheses

4

8

7

0

g to the power of apostrophe open parentheses x close parentheses

5

1

-4

-9

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 equals fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction.

Find the value of h apostrophe open parentheses 2 close parentheses.

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3
Sme Calculator2 marks

The function f is defined as f open parentheses x close parentheses equals tan space 5 x.

Find the value of f to the power of apostrophe open parentheses pi over 15 close parentheses.

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4
Sme Calculator2 marks

The function f is defined by f open parentheses x close parentheses equals 5 open parentheses 2 x minus 4 close parentheses squared minus 18 over x.

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 3.

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5
Sme Calculator2 marks

The function f is defined as g left parenthesis x right parenthesis equals 3 x space ln space open parentheses x close parentheses.

Find the value of g apostrophe open parentheses 1 close parentheses.

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1
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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2
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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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3
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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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4
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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5
Sme Calculator2 marks

x

f open parentheses x close parentheses

f apostrophe open parentheses x close parentheses

g open parentheses x close parentheses

g apostrophe open parentheses x close parentheses

negative 3

0

4

2

negative 5

0

2

negative 1

negative 1

3

2

1

2

4

1

The differentiable functions f and g are defined for all real numbers x. Values of f, f apostrophe, g and g apostrophe for various values of x are given in the table above.

Let h open parentheses x close parentheses equals fraction numerator f open parentheses x close parentheses over denominator 2 g open parentheses x close parentheses end fraction. Find h apostrophe open parentheses negative 3 close parentheses.

Show the computations that lead to your answer.

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1
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Graph of a piecewise function f. A horizontal line for x<-4, followed by straight line segments between (-6, -3) to (0, 3) and (0, 3) to (2, -3), then a curve down to (6,-4).
Graph of f

The function f is defined on the closed interval open square brackets negative 6 comma space 6 close square brackets. The graph of f consists of three line segments and a curve and is shown in the figure above. Let g be the function defined by g open parentheses x close parentheses equals integral subscript negative 4 end subscript superscript x f open parentheses t close parentheses space d t.

The function h is defined by h open parentheses x close parentheses equals fraction numerator 2 x squared over denominator g open parentheses x close parentheses end fraction. Find h apostrophe open parentheses 2 close parentheses.

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2
Sme Calculator3 marks

x

f open parentheses x close parentheses

f apostrophe open parentheses x close parentheses

2

7

3

1

2

2

0

1

0

negative 1

negative 1

2

Graph of the function g, a piecewise function made up of 5 line segments. Line segments go between the following pairs of coordinates: (-5, 0), (-4, -2), (0, -1), (1, 4), (4, -2) and (6, 0).
Graph of g

Let f be a differentiable function. The table above gives values of f and its derivative f apostrophe at selected values of x.

Let g be the function whose graph, consisting of five line segments, is shown in the figure above.

Let h be the function defined by h open parentheses x close parentheses equals fraction numerator open parentheses g open parentheses x close parentheses close parentheses squared over denominator f open parentheses x close parentheses end fraction. Find h apostrophe open parentheses negative 1 close parentheses.

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3
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The functions f and g are differentiable, where f open parentheses x close parentheses equals sin squared space x and g left parenthesis x right parenthesis equals sec space x.

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

Find the slope of the tangent line to the graph of y equals k open parentheses x close parentheses at the point where x equals pi over 4.

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4
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The functions f and g are defined as f open parentheses x close parentheses equals cos squared space x and g open parentheses x close parentheses equals tan space x.

Let the function h be defined by h open parentheses x close parentheses equals fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction. Find the value of h to the power of apostrophe open parentheses pi over 3 close parentheses.

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5
Sme Calculator3 marks

x

0

0 less than x less than 1

1

1 less than x less than 2

2

f open parentheses x close parentheses

9

Positive

4

Positive

7

f apostrophe open parentheses x close parentheses

negative 7

Negative

negative 2

Positive

5

g open parentheses x close parentheses

0

Positive

8

Positive

6

g apostrophe open parentheses x close parentheses

10

Positive

0

Negative

negative 2

The twice-differentiable functions f and g are defined for all real numbers x. Values of f, f apostrophe, g and g apostrophe for various values of x are given in the table above.

The function h is defined by h open parentheses x close parentheses equals ln space open parentheses 5 x close parentheses times f open parentheses x close parentheses plus 4 g open parentheses x close parentheses. Find h apostrophe open parentheses 1 fifth close parentheses. Show the computations that lead to your answer.

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