Related Rates Test Answer Key with Scoring – AP Calculus AB

Question 1 Direction change (2 points)

v(t) = 3t² − 12 = 3(t − 2)(t + 2). For t ≥ 0 the velocity changes from negative to positive at t = 2. Answer (B).

Scoring: 2 points for the correct choice; 1 point for finding v(t) = 0.

Question 2 L’Hôpital multiple choice (2 points)

The form is 0/0. First pass: e^x − 12x, still 0/0. Second pass: e^x2 → 1/2. Answer (B).

Scoring: 2 points for the correct choice; 1 point for a correct first application.

Question 3 Rectangle related rates (3 points)

(a) A = ℓw, so dA/dt = ℓ′w + ℓw′ = 3(6) + 10(−2) = −2 in²/s.

(b) The rate is negative, so the area is decreasing.

(c) D² = ℓ² + w² gives D = √136 = 2√34 and dD/dt = (ℓℓ′ + ww′)/D = (30 − 12)/(2√34) = 9/√34 ≈ 1.54 in/s.

Scoring: 1 point per part.

Question 4 Linearization of a square root (3 points)

(a) f(49) = 7 and f′(x) = 1/(2√x), so f′(49) = 1/14. L(x) = 7 + (x − 49)/14.

(b) √50 ≈ L(50) = 7 + 1/14 ≈ 7.0714.

(c) f″ < 0, so f is concave down and the tangent line lies above the graph. The approximation is greater than √50.

Scoring: 1 point per part.

Question 5 Velocity graph analysis (3 points)

(a) v = 0 at t = 4.

(b) On (4, 6) the graph goes from 0 to −3 in 2 seconds, so a(5) = −3/2 m/s².

(c) v(5) < 0 and a(5) < 0, the same sign, so the particle is speeding up.

Scoring: 1 point per part.

Question 6 Marginal analysis (3 points)

(a) C′(x) = 0.02x + 5.

(b) C′(200) = 9. When 200 backpacks are made, cost is rising at $9 per backpack.

(c) C(201) − C(200) = 0.01(40,401 − 40,000) + 5 = $9.01, very close to the estimate of $9.

Scoring: 1 point per part.

Question 7 L’Hôpital with a logarithm (2 points)

The form is ∞/∞. Differentiate: 1/x1/(2√x) = 2√x, which tends to 0. The limit is 0.

Scoring: 1 point for naming the form, 1 point for the derivative quotient and the limit.

Question 8 Sliding ladder test (2 points)

x² + y² = 100. When y = 6, x = 8. Differentiate: x dxdt + y dydt = 0, so 8 dxdt + 6(−1.2) = 0 and dxdt = 7.2/8 = 0.9 ft/s.

Scoring: 1 point for the correct equation and derivative, 1 point for the value 0.9.