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

Definition of Differentiation (College Board AP® Calculus AB)

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Definition of Differentiation

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Definition of Differentiation

1
Sme Calculator2 marks

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
Sme Calculator1 mark

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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4a
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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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4b
Sme Calculator2 marks

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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5
Sme Calculator2 marks

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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