# AP Calculus AB 2011 Questions and Answers

### Questions and Worked Solutions for AP Calculus AB 2011

AP Calculus AB 2011 Free Response Question 1
Determining whether speed is increasing. Difference between speed and acceleration.

1. For 0 ≤ t ≤ 6, a particle is moving along the x-axis. The particle’s position, x(t), is not explicitly given.
The velocity of the particle is given by v(t) = 2 sin (e1/4) + 1. The acceleration of the particle is given by
a(t) = 1/2 e1/4 cos (e1/4) and x(0) = 2.
(a) Is the speed of the particle increasing or decreasing at time t = 5.5? Give a reason for your answer.
(b) Find the average velocity of the particle for the time period 0 ≤ t ≤ 6.
(c) Find the total distance traveled by the particle from time t = 0 to t = 6.
(d) For 0 ≤ t ≤ 6, the particle changes direction exactly once. Find the position of the particle at that time.

AP Calculus AB 2011 Free Response Question 2
Approximating rate of change and total area under a curve. Trapezoidal sums to approximate integrals.
2. 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?

AP Calculus AB 2011 Free Response Question 3
Equation of a tangent line and area between curves.
3. Let R be the region in the first quadrant enclosed by the graphs of f(x) = 8x3 and g(x) = sin (πx), as shown in the figure above.
(a) Write an equation for the line tangent to the graph of f at x = 1/2.
(b) Find the area of R.
(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y = 1.

AP Calculus AB 2011 Free Response Question 4
Taking derivatives and integrals of strangely defined functions. Absolute maximum over an interval. Critical points and differentiability. Finding the points of inflection for a strangely defined function. Mean Value Theorem and differentiability.
4. 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.

AP Calculus AB 2011 Free Response Question 5
5. 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 d2W/dt2 in terms of W. Use d2W/dt2 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.

AP Calculus AB 2011 Free Response Question 6
6. Let f be a function defined by f(x) =.
(a) Show that f is continuous at x = 0.
(b) For x ≠ 0, express f'(x) as a piecewise-defined function. Find the value of x for which f'(x) = -3.
(c) Find the average value of f on the interval [-1, 1].

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