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AP Calculus AB 2014 Free Response Questions - Complete Paper (pdf)

AP Calculus AB 2014 Free Response Question 1

**A graphing calculator is required for these problems.**

- Grass clippings are placed in a bin, where they decompose. For 0 ≤ t ≤ 30, the amount of glass clippings remaining in the bin is modeled by A(t) = 6.687(0.931)
^{t}, where A(t) is measured on pounds and t is measured in days.

(a) Find the average rate of change A(t) over the interval 0 ≤ t ≤ 30. Indicate units of measure.

(b) Find the value of A'(15). Using correct units, interpret the meaning of the value in the context of the problem.

(c) Find the time t for which the amount of grass clipping in the bin is equal to the average amount of grass clipping in the bin over the interval 0 ≤ t ≤ 30.

(d) For t > 30, L(t), the linear approximation to A at t = 30, is a better model for the amount of grass clippings remaining in the bin. Use L(t) to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.

AP Calculus AB 2014 Free Response Question 2

- Let R be the region enclosed by the graph of f(x) = x
^{4}- 2.3x^{3}+ 4 and the horizontal line y = 4, as shown in the figure above.

(a) Find the volume of the solid generated when R is rotated about the horizontal line y = -2.

(b) Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid.

(c) The vertical line x = k divides R into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value k.

AP Calculus AB 2014 Free Response Question 3

AP Calculus AB 2014 Free Response Question 4

- Train A runs back and forth on an east-west section of railroad track. Train A’s velocity, measured in meters per minute, is given by a differentiable function V
_{A}(t), where time t is measured in minutes. Selected values for V_{A}(t) are given in the table above.

(a) Find the average acceleration of train A over the interval 2 ≤ t ≤ 8.

(b) Do the data in the table support the conclusion that train A’s velocity is -100 meters per minute at some time t with 5 < t < 8 ? Give a reason for your answer.

(c) At time t = 2, train A’s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train A, in meters from the Origin Station, at time t = 12. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time t = 12.

(d) A second train, train B, travels north from the Origin Station. At time t the velocity of train B is given by V_{B}(t) = -5t^{2}+ 60t + 25 and at time t = 2 the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train A and train B is changing at time t = 2.

AP Calculus AB 2014 Free Response Question 5

AP Calculus AB 2014 Free Response Question 6

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