AP Calculus BC 2011 Free Response Questions - Complete Paper (pdf)

AP Calculus BC 2011 Free Response Questions - Scoring Guide (pdf)

- At time t, a particle moving in the xy-plane is at position (x(t), y(t)) where x(t) and y(t) are not explicitly given.
- As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for 0 ≤ t ≤ 10, where time t is measured in minutes and temperature H(t) is measured in degrees Celsius. Values of H(t) at selected values of time t are shown in the table above.

(a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time t = 3.5. Show the computations that lead to your answer.

(b), (c)

(d) At time t = 0, biscuits with temperature 100°C were removed from an oven. The temperature of the biscuits at time t is modeled by a differentiable function B for which it is known that B'(t) = -13.84e^{-0.173t}. Using the given models, at time t = 10, how much cooler are the biscuits than the tea?

- Let f(x) = e
^{2x}Let R be the region in the first quadrant bounded by the graph of f, the coordinate axes, and the vertical line x = k , where k > 0. The region R is shown in the figure above. - The continuous function f is defined on the interval -4 ≤ x ≤ 3. The graph of f consists of two quarter circles and one line segment, as shown in the figure above. Let g(x) =

(a) Find g(-3). Find g'(x) and evaluate g'(-3).

(b) Determine the x-coordinate of the point at which g has an absolute maximum on the interval -4 ≤ x ≤ 3. Justify your answer.

(c) Find all values of x on the interval -4 ≤ x ≤ 3 for which the graph of g has a point of inflection. Give a reason for your answer.

(d) Find the average rate of change of f on the interval -4 ≤ x ≤ 3. There is no point c, -4 < c < 3, c for which f'(c) is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem. - At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential equation dW/dt = 1/25(w - 300) for the next 20 years. W is measured in tons, and t is measured in years from the start of 2010.

(a) Use the line tangent to the graph of W at t = 0 to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time t = 1/4).

(b) Find d^{2}W/dt^{2}in terms of W. Use d^{2}W/dt^{2}to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time t = 1/4.

(c) Find the particular solution W = W(t) to the differential equation dW/dt = 1/25(W - 300) with initial condition W(0) = 1400. - Let f(x) = sin(x
^{2}) + cos x.

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