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AP? Calculus AB 2013 Free-Response Questions

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2013 AP? CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A

Time— 30 minutes Number of problems— 2 A graphing calculator is required for these problems.

1. On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by G t

90

45cos

t2 , where t is measured in hours and 0 18

t

8. At the beginning of the

t 8, the

workday t 0 , the plant has 500 tons of unprocessed gravel. During the hours of operation, 0 plant processes gravel at a constant rate of 100 tons per hour. (a) Find G 5 . Using correct units, interpret your answer in the context of the problem.

(b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday. (c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time t work that leads to your answer.

5 hours? Show the

(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer.

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2013 AP? CALCULUS AB FREE-RESPONSE QUESTIONS

2. A particle moves along a straight line. For 0

t

5, the velocity of the particle is given by

vt

2

t2

3t

6 5

t 3 , and the position of the particle is given by s t . It is known that s 0

10.

(a) Find all values of t in the interval 2

t

4 for which the speed of the particle is 2.

(b) Write an expression involving an integral that gives the position s t . Use this expression to find the position of the particle at time t 5. (c) Find all times t in the interval 0

t

5 at which the particle changes direction. Justify your answer.

4 ? Give a reason for your answer.

(d) Is the speed of the particle increasing or decreasing at time t

END OF PART A OF SECTION II

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2013 AP? CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part B

Time— 60 minutes Number of problems—4 No calculator is allowed for these problems. t (minutes) C t (ounces)

0 0

1 5.3

2 8.8

3

4

5

6

11.2 12.8 13.8 14.5

3. Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at time t, 0 t 6, is given by a differentiable function C, where t is measured in minutes. Selected values of C t , measured in ounces, are given in the table above. (a) Use the data in the table to approximate C 3.5 . Show the computations that lead to your answer, and indicate units of measure. (b) Is there a time t, 2

t

4, at which C t

2 ? Justify your answer.

(c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate 1 6 1 6 the value of C t dt. Using correct units, explain the meaning of C t dt in the context of the 6 0 6 0 problem.

16 16e (d) The amount of coffee in the cup, in ounces, is modeled by B t rate at which the amount of coffee in the cup is changing when t 5.

0.4 t

. Using this model, find the

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2013 AP? CALCULUS AB FREE-RESPONSE QUESTIONS

4. The figure above shows the graph of f , the derivative of a twice-differentiable function f, on the closed interval 0 x 8. The graph of f has horizontal tangent lines at x 1, x 3, and x 5. The areas of the regions between the graph of f and the x-axis are labeled in the figure. The function f is defined for all real numbers and satisfies f 8 4. (a) Find all values of x on the open interval 0 your answer. (c) On what open intervals contained in 0 Explain your reasoning. (d) The function g is defined by g x of g at x

x

8 for which the function f has a local minimum. Justify x 8. Justify your answer.

(b) Determine the absolute minimum value of f on the closed interval 0

x f x

3

8 is the graph of f both concave down and increasing? . If f 3

5 , find the slope of the line tangent to the graph 2

3.

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2013 AP? CALCULUS AB FREE-RESPONSE QUESTIONS

5. Let f x

2 x2

6x

4 and g x

4 cos

1 p x . Let R be the region bounded by the graphs of f and g, as 4

shown in the figure above. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y 4. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Write, but do not evaluate, an integral expression that gives the volume of the solid.

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2013 AP? CALCULUS AB FREE-RESPONSE QUESTIONS

6. Consider the differential equation

dy e y 3x2 dx differential equation that passes through 1, 0 .

6 x . Let y

f x be the particular solution to the

(a) Write an equation for the line tangent to the graph of f at the point 1, 0 . Use the tangent line to approximate f 1.2 . (b) Find y

f x , the particular solution to the differential equation that passes through 1, 0 .

STOP END OF EXAM

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