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More videos, activities and worksheets that are suitable for Calculus

Questions and Worked Solutions for AP Calculus AB 2011.

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 (e^{1/4}) + 1. The acceleration of the particle is given by

a(t) = 1/2 e^{1/4} cos (e^{1/4}) 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) = 8x^{3} 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 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. 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].

More videos, activities and worksheets that are suitable for Calculus

Questions and Worked Solutions for AP Calculus AB 2011.

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

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 (e

a(t) = 1/2 e

(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

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) = 8x

(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 d

(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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