AP Calculus BC FRQ

1. Differential Equations and Velocity

Question:

The velocity of a particle moving along the x-axis is given by v(t) = 3t^2 - 2t + 1, where t is time in seconds and v(t) is the velocity in meters per second.
The particle is initially at position x(0) = 4 meters.

(a) Find the position function x(t) for the particle.

(b) What is the position of the particle at t = 3 seconds?

Response Guidelines:

• Part (a):
  To find the position function, integrate the velocity function v(t) to obtain x(t).
  x(t) = ∫v(t) dt = ∫(3t^2 - 2t + 1) dt.
  After integrating, use the initial condition x(0) = 4 to solve for the constant of integration.
• Part (b):
  Substitute t = 3 into the position function x(t) to find the position of the particle at t = 3.

2. Improper Integrals

Question:

Evaluate the following improper integral:
∫ from 1 to ∞ of (1/x^2) dx.

Response Guidelines:

• Recognize that this is an improper integral because the upper limit is infinite.
• To evaluate the integral, first rewrite it as the limit:
  lim from b to ∞ of ∫ from 1 to b of (1/x^2) dx.
• Perform the integration and take the limit as b → ∞.

3. Parametric Equations and Area

Question:

The parametric equations for a curve are given by x(t) = t^2 - 1 and y(t) = 2t + 3, for 0 ≤ t ≤ 2.

(a) Find the area enclosed by the curve and the x-axis.

(b) What is the length of the curve from t = 0 to t = 2?

Response Guidelines:

• Part (a):
  To find the area, use the formula for the area under a curve with parametric equations:
  A = ∫ from a to b of y(t) * dx/dt dt,
  where dx/dt is the derivative of x(t) with respect to t. Compute the integral from t = 0 to t = 2.
• Part (b):
  The formula for the length of a curve with parametric equations is:
  L = ∫ from a to b of √( (dx/dt)^2 + (dy/dt)^2 ) dt.
  Compute this integral from t = 0 to t = 2.

Grading Rubric for AP Calculus BC FRQs

1. Differential Equations and Velocity

• 0-1 points: The response demonstrates little understanding of differential equations or incorrect integration of the velocity function.
• 2 points: The response correctly integrates the velocity function, but may have minor mistakes in applying the initial condition or calculating the position at t = 3.
• 3 points: The response correctly solves for the position function, applies the initial condition, and finds the correct position at t = 3, with clear steps shown.
• 4 points: The response provides a complete and correct solution, with all steps clearly explained and correctly computed.

2. Improper Integrals

• 0-1 points: The response demonstrates little understanding of improper integrals or incorrectly sets up the limit.
• 2 points: The response correctly identifies the improper integral and sets up the limit but may contain minor errors in evaluation or integration.
• 3 points: The response correctly evaluates the improper integral using limits and provides the correct result.
• 4 points: The response includes a clear and accurate evaluation of the improper integral, with all necessary steps and correct limits applied.

3. Parametric Equations and Area

• 0-1 points: The response demonstrates little understanding of parametric equations or contains major errors in integration or applying the formulas.
• 2 points: The response sets up the integrals for both the area and the length of the curve but contains errors in the calculations or application of the formulas.
• 3 points: The response correctly sets up and computes the integrals for both the area and the curve length but may contain minor mistakes.
• 4 points: The response provides clear, correct solutions for both the area and the curve length, with all steps properly shown and correctly calculated.

Sample Grading Breakdown (for one FRQ)

• Differential Equations and Velocity: 4 points
• Improper Integrals: 4 points
• Parametric Equations and Area: 4 points
• Total: 12 points (for all three FRQs combined)