AP Calculus AB Sample FRQ

Question:

A ladder 10 feet long is leaning against a wall. The bottom of the ladder is being pulled away from the wall at a rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall?

Response Guidelines:

Draw a right triangle representing the situation. Label the sides with appropriate variables: the length of the ladder (10 feet), the distance from the bottom of the ladder to the wall (x), and the height from the top of the ladder to the ground (y).
Apply the Pythagorean Theorem:
x^2 + y^2 = 10^2.
Differentiate both sides of the equation with respect to time (t). Use the chain rule to express the rates of change of x and y, and solve for the rate at which y is changing when x = 6.

2. Definite Integral and Area Between Curves

Question:

Consider the functions f(x) = x^2 and g(x) = 4 - x^2.

(a) Find the area of the region bounded by the curves y = f(x) and y = g(x) from x = -2 to x = 2.

(b) Find the volume of the solid generated by revolving the region in part (a) around the x-axis.

Response Guidelines:

Part (a):
Set up the integral for the area between the curves. The formula for the area between two curves is:
∫ from a to b of [g(x) - f(x)] dx,
where g(x) is the upper curve and f(x) is the lower curve.
Integrate g(x) - f(x) over the interval [-2, 2].
Part (b):
Use the disk method to find the volume of the solid. The formula for the volume is:
V = π ∫ from a to b of [f(x)]^2 dx.
Apply this formula to find the volume when the region is revolved around the x-axis.

3. Differential Equations and Growth Model

Question:

A population of bacteria grows according to the differential equation dy/dt = ky, where y(t) is the population at time t, and k is a constant.
The initial population at t = 0 is y(0) = 100.

(a) Solve the differential equation for y(t).

(b) If the population doubles in 4 hours, find the value of k.

Response Guidelines:

Part (a):
Separate variables and integrate both sides to solve the differential equation. The solution should be of the form:
y(t) = C * e^(kt),
where C is the constant of integration.
Use the initial condition y(0) = 100 to determine the value of C.
Part (b):
Use the information that the population doubles in 4 hours to find the value of k. This leads to an equation where y(4) = 2 * y(0).
Part (c):
Use the value of k from part (b) to find the population at t = 6 using the formula:
y(t) = 100 * e^(kt).

Grading Rubric for AP Calculus AB FRQs

1. Related Rates

0-1 points: The response demonstrates little understanding of related rates and fails to apply the chain rule correctly.
2 points: The response applies the Pythagorean Theorem but may have minor errors in differentiating or solving for the rate of change.
3 points: The response correctly applies related rates, including differentiation and solving for the desired rate of change.
4 points: The response provides a complete and correct solution, with all steps clearly explained, including a properly labeled diagram and an accurate rate of change.

2. Definite Integral and Area Between Curves

0-1 points: The response demonstrates little understanding of the problem or incorrectly sets up the integrals.
2 points: The response sets up the integrals but may contain errors in the computation or application of the formula for the area or volume.
3 points: The response correctly sets up and computes the integral for the area, with some minor errors in the volume calculation.
4 points: The response includes correct and detailed solutions for both the area and the volume, with accurate integral setups and computations.

3. Differential Equations and Growth Model

0-1 points: The response demonstrates little understanding of differential equations and does not correctly solve or interpret the equation.
2 points: The response attempts to solve the differential equation and may apply the initial condition, but contains errors in integration or solving for the constant.
3 points: The response correctly solves the differential equation, applies the initial condition, and solves for the population after 6 hours, but with minor mistakes.
4 points: The response provides a clear and complete solution for all parts, correctly solving the differential equation, applying the initial condition, and calculating the population for part (c).

Sample Grading Breakdown (for one FRQ)

Related Rates: 4 points
Definite Integral and Area Between Curves: 4 points
Differential Equations and Growth Model: 4 points
Total: 12 points (for all three FRQs combined)