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): Exam Questions

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Differentiation of Composite & Inverse Functions

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1
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3 marks

Let f be a differentiable function where f open parentheses negative 3 close parentheses equals 5 and f apostrophe open parentheses negative 3 close parentheses equals negative 2.

The function g is defined by g open parentheses x close parentheses equals ln open parentheses space f open parentheses x close parentheses close parentheses.

Find g to the power of apostrophe open parentheses negative 3 close parentheses. Show the computations that lead to your answer.

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2
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3 marks

Let f be a differentiable function where f open parentheses 5 close parentheses equals negative 1 half and f to the power of apostrophe open parentheses 5 close parentheses equals 3.

The function k is defined by k open parentheses x close parentheses equals open parentheses space f open parentheses x close parentheses close parentheses to the power of 4.

Find the equation of the line tangent to the graph of k at x equals 5.

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3
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3 marks

If f to the power of negative 1 end exponent is the inverse function of f; f open parentheses 2 close parentheses equals 3 and f to the power of space apostrophe end exponent open parentheses 2 close parentheses equals negative 1, write an equation for the line tangent to the graph of y equals f to the power of negative 1 end exponent open parentheses x close parentheses at x equals 3.

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4
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1 mark

The rate at which the temperature of a cup of coffee decreases is proportional to the difference between its current temperature and the surrounding room temperature of 68 degree straight F. At time t equals 0 the coffee's temperature s 176 degree straight F. If T open parentheses t close parentheses is the coffee's temperature, in degrees Fahrenheit, at time t minutes after it was first measured, then

fraction numerator d T over denominator d t end fraction equals negative 1 over 10 open parentheses T minus 68 close parentheses.

Find fraction numerator d squared T over denominator d t squared end fraction in terms of T.

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5
1 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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6
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2 marks

An ice sculpture melts in such a way that it can be modelled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function r, where r(t) is measured in centimeters and t is measured in days. The table below gives selected values of r'(t), the rate of change of the radius, over the time interval 0 \leq t \leq 12.

t (days)

0

3

7

10

12

r'(t) (cm per day)

-6.1

-5.0

-4.4

-3.8

-3.5

Approximate r''(8.5) using the average rate of change of r' over the interval 7 \leq t \leq 10. Show the computations that lead to your answer, and indicate units of measure.

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1
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2 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

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 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
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2 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
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2 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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3a
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2 marks

Let f be the function defined by f open parentheses x close parentheses equals e to the power of cos space x end exponent minus sin space 2 x.

Find the slope of the line tangent to the graph of f at x equals 2 pi.

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3b
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2 marks

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

Graph of a piecewise linear function on a coordinate grid. The x and y axes are labelled with x=1 and y=1. A straight line goes through the origin and joins the point (-4, -1) to (4, 1).

Let h be the function defined by h open parentheses x close parentheses equals g open parentheses f open parentheses x close parentheses close parentheses. Find h to the power of apostrophe open parentheses 2 pi close parentheses.

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4
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2 marks

Let f be the function defined by f open parentheses x close parentheses equals sin open parentheses square root of x squared plus 1 end root close parentheses.

Find the equation of the line tangent to the graph of f at x equals 0.

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5
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4 marks

The function f is defined by f left parenthesis x right parenthesis equals open parentheses 2 x minus 5 close parentheses cubed.

Show that the inverse of f left parenthesis x right parenthesis exists and find the derivative of the inverse of f open parentheses x close parentheses at the point where x equals negative 27.

Show all necessary steps involved in solving this problem.

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6
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2 marks

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

x

0

2

4

7

f(x)

10

7

4

5

f'(x)

\frac{3}{2}

-8

3

6

g(x)

1

2

-3

0

g'(x)

5

4

2

8

Let h be the function defined by h(x) = f(g(x)). Find h'(7). Show the work that leads to your answer.

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1
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3 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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2
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1 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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3
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2 marks

Let f be the function defined by f open parentheses x close parentheses equals k square root of 3 x plus 1 end root minus ln open parentheses 2 x close parentheses for x greater than 0 comma where k is a positive constant.

Find f to the power of space apostrophe end exponent open parentheses x close parentheses and f to the power of space apostrophe apostrophe end exponent open parentheses x close parentheses.

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4a
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2 marks

Let f be the function defined by f open parentheses x close parentheses equals sin squared open parentheses e to the power of x squared end exponent close parentheses.

Differentiate f open parentheses x close parentheses with respect to x comma showing all working clearly.

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4b
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3 marks

Let g open parentheses x close parentheses equals ln space f open parentheses x close parentheses comma where f open parentheses x close parentheses equals sin squared open parentheses e to the power of x squared end exponent close parentheses.

Show that g to the power of apostrophe open parentheses x close parentheses table row blank equals blank end table table row blank blank 4 end table table row blank blank x end table e to the power of x squared end exponent table row blank blank cot end table table row blank blank cell open parentheses e to the power of x squared end exponent close parentheses end cell end table.

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5a
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3 marks

A roller coaster travels along a track, and the height of the coaster above the ground at time t seconds is modeled by the function h open parentheses t close parentheses equals tan open parentheses ln open parentheses t squared plus 1 close parentheses close parentheses comma where t represents the time in seconds since the coaster enters a section of the track just before it hits a water feature. The model is valid for negative 1 less or equal than t less or equal than 1 commarepresenting a brief window of time just before and after the coaster splashes into the water.

Determine the value of h to the power of apostrophe open parentheses t close parentheses comma and explain what it tells us about the roller coaster's velocity at the moment it hits the water.

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5b
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3 marks

Determine the value of h to the power of space apostrophe apostrophe end exponent open parentheses 0 close parentheses and use this to describe the change in the behavior of the roller coaster's height at the moment it hits the water.

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6a
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2 marks

Let f be the function defined by f(x) = \text{cos} \; (2x) + e^{\text{sin} \; x}. Let g be a differentiable function. The table below gives values of g and its derivative g' at selected values of x.

x

g open parentheses x close parentheses

g apostrophe open parentheses x close parentheses

-5

10

-3

-4

5

-1

-3

2

4

-2

3

1

-1

1

-2

0

0

-3

Let h be the function whose graph, consisting of five line segments, is shown below.

Piecewise linear graph of function h on x–y axes, rising, flat, falling through origin, then dipping below x-axis before sharply rising again on the right.

Find the slope of the line tangent to the graph of f at x = \pi.

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6b
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2 marks

Let k be the function defined by k(x) = h(f(x)). Find k'(\pi).

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6c
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3 marks

Let m be the function defined by m(x) = g(-2x)\,h(x). Find m'(2).

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7
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3 marks

The functions f and g have continuous second derivatives. The table below gives values of the functions and their derivatives at selected values of x.

x

f \left( x \right)

f ' \left( x \right)

g \left( x \right)

g ' \left( x \right)

1

-6

3

2

8

2

2

-2

-3

0

3

8

7

6

2

6

4

5

3

-1

Let k \left( x \right) = f \left( g \left( x \right) \right). Write an equation for the line tangent to the graph of k at x = 3.

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