Practice a College Board-style free response question on parametric motion — velocity, speed, slope, and total distance. Write your response, then reveal the model answer to see exactly what earns each point.
Why it scores: Correct componentwise velocity ⟨2t, 3t² − 3⟩, evaluated at t = 2, and speed as the magnitude √97.
B
Find the slope dy/dx of the path at t = 2.
✓ Model answer
dy/dx = (dy/dt)/(dx/dt) = (3t² − 3)/(2t). At t = 2: (3(4) − 3)/(2·2) = (12 − 3)/4 = 9/4.
Why it scores: Uses dy/dx = (dy/dt)/(dx/dt), substitutes t = 2, and simplifies to 9/4. Taking only dy/dt, or inverting the ratio, loses the point.
C
Write, but do not evaluate, an integral expression for the total distance traveled by the particle from t = 0 to t = 3.
✓ Model answer
Total distance is the integral of speed: ∫ from 0 to 3 of √((2t)² + (3t² − 3)²) dt = ∫₀³ √(4t² + (3t² − 3)²) dt.
Why it scores: Writes distance as ∫|v| dt with the correct speed integrand √((dx/dt)² + (dy/dt)²) and limits 0 to 3. Omitting the square root, or using displacement instead, loses credit.
How to score points on AP Calculus FRQs
Velocity is componentwise: v = ⟨x′(t), y′(t)⟩; speed is its magnitude.
Slope divides the rates: dy/dx = (dy/dt)/(dx/dt).
Total distance integrates speed: ∫√((dx/dt)² + (dy/dt)²) dt.
Keep the square root in arc-length and distance integrals.
Answer the verb. "Find" wants the value; "write, do not evaluate" wants only the integral.