2013 AP CALCULUS AB FREE RESPONSE QUESTIONS
2013 AP CALCULUS AB FREE RESPONSE QUESTIONS (Scroll to View Content)
2013 AP CALCULUS AB SCORING GUIDELINES (Scroll to View Content)
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(t^2/18)
where t is measured in hours and 0 t 8. At the beginning of the
workday t 0, the plant has 500 tons of unprocessed gravel. During the hours of operation, 0 t 8, the
plant processes gravel at a constant rate of 100 tons per hour.
(a) Find G5. 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 5 hours? Show the
work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this
workday? Justify your answer.2. A particle moves along a straight line. For 0 t 5, the velocity of the particle is given by
v(t)=-2+(t^2+3t)^(6/5)-t^3 , and the position of the particle is given by st. It is known that s0 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 st. 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.
(d) Is the speed of the particle increasing or decreasing at time t 4 ? Give a reason for your answer.
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
Ct, measured in ounces, are given in the table above.
(a) Use the data in the table to approximate C3.5. Show the computations that lead to your answer, and
indicate units of measure.
(b) Is there a time t, 2 t 4, at which Ct 2 ? Justify your answer.
(c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate
the value of 6
C t dt in the context of the
problem.
(d) The amount of coffee in the cup, in ounces, is modeled by Bt 16 16e0.4t . Using this model, find the
rate at which the amount of coffee in the cup is changing when t 5.
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 x 8 for which the function f has a local minimum. Justify
your answer.
(b) Determine the absolute minimum value of f on the closed interval 0 x 8. Justify your answer.
(c) On what open intervals contained in 0 x 8 is the graph of f both concave down and increasing?
Explain your reasoning.
(d) The function g is defined by 3 g x f x . If 3 5 ,
2
f find the slope of the line tangent to the graph
of g at x 3.
5. Let f(x)=2x^2-6x+4 and g(x)=4cos(1/4 pi x)
px Let R be the region bounded by the graphs of f and g, as
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.
6. Consider the differential equation dy/dx=e^y(3x^2-6x)
e x x
dx
Let y f x be the particular solution to the
differential equation that passes through 1, 0.
(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.
is modeled by G(t)=90+45cos(t^2/18)
where t is measured in hours and 0 t 8. At the beginning of the
workday t 0, the plant has 500 tons of unprocessed gravel. During the hours of operation, 0 t 8, the
plant processes gravel at a constant rate of 100 tons per hour.
(a) Find G5. 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 5 hours? Show the
work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this
workday? Justify your answer.2. A particle moves along a straight line. For 0 t 5, the velocity of the particle is given by
v(t)=-2+(t^2+3t)^(6/5)-t^3 , and the position of the particle is given by st. It is known that s0 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 st. 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.
(d) Is the speed of the particle increasing or decreasing at time t 4 ? Give a reason for your answer.
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
Ct, measured in ounces, are given in the table above.
(a) Use the data in the table to approximate C3.5. Show the computations that lead to your answer, and
indicate units of measure.
(b) Is there a time t, 2 t 4, at which Ct 2 ? Justify your answer.
(c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate
the value of 6
C t dt in the context of the
problem.
(d) The amount of coffee in the cup, in ounces, is modeled by Bt 16 16e0.4t . Using this model, find the
rate at which the amount of coffee in the cup is changing when t 5.
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 x 8 for which the function f has a local minimum. Justify
your answer.
(b) Determine the absolute minimum value of f on the closed interval 0 x 8. Justify your answer.
(c) On what open intervals contained in 0 x 8 is the graph of f both concave down and increasing?
Explain your reasoning.
(d) The function g is defined by 3 g x f x . If 3 5 ,
2
f find the slope of the line tangent to the graph
of g at x 3.
5. Let f(x)=2x^2-6x+4 and g(x)=4cos(1/4 pi x)
px Let R be the region bounded by the graphs of f and g, as
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.
6. Consider the differential equation dy/dx=e^y(3x^2-6x)
e x x
dx
Let y f x be the particular solution to the
differential equation that passes through 1, 0.
(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.