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

Definition of Differentiation (College Board AP® Calculus AB): Exam Questions

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Definition of Differentiation

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1
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t

(hours)

0

3

7

10

12

r to the power of apostrophe open parentheses t close parentheses

(centimeters per hour)

-3.2

-2.5

-2.1

-1.6

-1.4

The radius of a puddle as it dries up is given by a twice-differentiable function r, where r open parentheses t close parentheses is measured in centimeters and t is measured in hours. The table above gives selected values of r to the power of apostrophe open parentheses t close parentheses, the rate of change of the radius, over the time interval 0 less or equal than t less or equal than 12.

Approximate r to the power of apostrophe apostrophe end exponent open parentheses 5 close parentheses using the average rate of change of r apostrophe over the interval 3 less or equal than t less or equal than 7. Show the computations that lead to your answer, and indicate units of measure.

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2
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The line y equals 7 minus 3 over 2 open parentheses x minus 5 close parentheses is a tangent to the differentiable function h at the point where x equals 2.

Find h to the power of apostrophe open parentheses 2 close parentheses.

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3
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t

(minutes)

0

12

20

24

40

v open parentheses t close parentheses

(meters per minute)

0

240

280

250

180

A dog runs along a straight path. For 0 less or equal than t less or equal than 40, the dog's velocity is given by a differentiable function v.

Selected values of v open parentheses t close parentheses, where t is measured in minutes and v open parentheses t close parentheses is measured in meters per minute, are given in the table above.

Use the data in the table to estimate the value of v to the power of apostrophe open parentheses 22 close parentheses.

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4
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Graph of function x(t) with the following points labeled: (5, 2), (7, 5), (11, 16), (15, 32) and (18, 41).

The graph above shows the displacement x of a particle along a straight line, where x open parentheses t close parentheses is measured in meters and t is measured in minutes. Selected times and the associated displacements of the particle are displayed on the graph on the interval 0 less or equal than t less or equal than 42.

Approximate the rate of change of the particle's displacement, at t equals 13, over the interval 11 less or equal than t less or equal than 15. Show the computations that lead to your answer, and indicate units of measure.

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5
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Graph of function f. The function starts at the top left of the graph and comes down in a straight line to the point (-2, 2). It then changes to a curve. It continues to go down through the x-axis and lower coming close to the line x=1. A separate section of graph comes down from above just to the right of the line x=1 and curves towards the line y=2.

The graph of the function f is shown above. At which values of x is f not differentiable?

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1a
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Let f be the function defined by f open parentheses x close parentheses equals e to the power of x space cos open parentheses x over 2 close parentheses.

Find the average rate of change of f on the interval 0 less or equal than x less or equal than pi.

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1b
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What is the slope of the line tangent to the graph of f open parentheses x close parentheses equals e to the power of x space cos open parentheses x over 2 close parentheses at x equals 3 pi ?

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2
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A curve has the equation y equals a x cubed plus b x squared plus c x, where a comma space b and c are constants. The point open parentheses p comma space q close parentheses, where p and q are constants, lies on this curve.

Find the equation of the tangent to this graph at the point open parentheses p comma space q close parentheses in terms of x and y, and the constants a comma space b comma space c comma space p and q.

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3
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A new eco-friendly plastic decomposes over time. The mass of the material is modeled by f open parentheses t close parentheses equals 4.98 e to the power of negative 0.027 t end exponent for 0 less or equal than t less or equal than 90, where f open parentheses t close parentheses is measured in pounds and t is measured in days.

Find the average rate of decomposition of the material over the interval 0 less or equal than t less or equal than 90. Indicate units of measure.

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4
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A function f is defined by f left parenthesis x right parenthesis equals sin open parentheses negative x over 2 close parentheses, and f apostrophe open parentheses pi over 2 close parentheses equals negative fraction numerator square root of 2 over denominator 4 end fraction.

Find the equation of the line that is tangent to the graph of g at x equals pi over 2.

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5
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The function f is defined by

f left parenthesis x right parenthesis equals open curly brackets table row cell x squared minus 3 end cell cell for space x less than 1 end cell row cell 4 x minus 5 end cell cell for space x greater or equal than 1 end cell end table close

Is f differentiable at x equals 1? Justify your answer.

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1
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x

negative 5

negative 3

0

2

6

g open parentheses x close parentheses

negative 103

negative 11

7

9

205

The table above gives some values of g open parentheses x close parentheses for a function g.

Find the equation of a line that is tangent to the graph of g at x equals 4.

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2
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The function f is defined by

f left parenthesis x right parenthesis equals open curly brackets table row cell 2 x plus 1 end cell cell for space x less than 3 end cell row cell 3 x minus 2 end cell cell for space x greater or equal than 3 end cell end table close

Is f differentiable for all values of x?

Use the definition of the derivative of the function, f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction, at x equals a to justify your answer.

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3
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Using the definition of the derivative of the function,

f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction, at x equals a,

find the equation of the line that is normal to the graph f open parentheses x close parentheses equals 4 x squared at x equals negative 1

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4
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The velocity of a particle is defined by the function

v open parentheses t close parentheses equals p t squared plus q t plus 4,

where v open parentheses t close parentheses is measured in meters per second and t is measured in seconds.

The velocity of the particle at t equals 3 is 7 meters per second. The average rate of change of the velocity of the particle over the interval 3 less or equal than t less or equal than 6 is negative 11 meters per second per second. What are the values of p and q?

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5
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A function f shown on a graph with two points, p and q, labelled. The function starts at the bottom left of the graph and goes up towards the top right. There are two turning points before p and q are highlighted.

A function f is shown on the graph above. Two points, p and q, are indicated on the graph. Point p has coordinates open parentheses 5 comma space 8 close parentheses and point q is positioned where x equals 7.

A tangent to the graph at the midpoint between p and q has equation y equals 3 x minus 7. What is f open parentheses 7 close parentheses?

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