Derivatives in Context: Motion and Related Rates: AP Calculus AB Math Test Answer Key – PDF Download

AP Calculus Contextual Test Answer Key: questions 1–5

Question 1: Particle motion (3 points)

1. \(v(t)=3t^{2}-18t+24\), \(a(t)=6t-18\)
2. \(v=3(t-2)(t-4)=0\): t = 2 and t = 4.
3. \(v(5)=75-90+24=9>0\) and \(a(5)=12>0\). Same sign: speed is increasing.

Scoring: (a) 1 point for both functions. (b) 1 point for both times . (c) 1 point for evaluating both signs and the correct conclusion.

Question 2: Interpreting a derivative (2 points)

1. At 10 minutes the soup is cooling at about 1.8 °F per minute.
2. \(H(12)\approx 140+(-1.8)(2)=136.4\). About 136.4 °F.

Scoring: (a) 1 point for °F per minute and the meaning of the sign. (b) 1 point for the linearization value.

Question 3: Local linearization (3 points)

1. \(f^{\prime}(x)=\frac{1}{3}x^{-2/3}\), \(f^{\prime}(27)=\frac{1}{27}\): \(L(x)=3+\frac{1}{27}(x-27)\).
2. \(L(28)=3+\frac{1}{27}\approx 3.037\). \(\sqrt[3]{28}\approx 3.037\)
3. \(f^{\prime\prime}(x)=-\frac{2}{9}x^{-5/3}< 0\) so \(f\) is concave down and the tangent line lies above the graph: overestimate.

Scoring: 1 point per part. In (c) the reason must refer to concavity or the sign of the second derivative.

Question 4: Melting ice cube (2 points)

\(\frac{dV}{dt}=3s^{2}\frac{ds}{dt}=3(25)(-0.2)=-15\).

\(-15\text{ in}^{3}/\text{min}\)

Scoring: 1 point for differentiating with the chain rule factor. 1 point for the value with the negative sign and units.

Question 5: Plane and radar station (3 points)

1. \(s^{2}=x^{2}+9\)
2. \(2s\frac{ds}{dt}=2x\frac{dx}{dt}\). At \(x=4\), \(s=\sqrt{25}=5\). \(\frac{ds}{dt}=\frac{4}{5}(400)=320\). 320 mph

Scoring: (a) 1 point. (b) 1 point for the differentiated equation, 1 point for s = 5 and the final value.

AP Calculus Contextual Test Answer Key: questions 6–8

Question 6: Limits with L’Hôpital (3 points)

1. Form \(\frac{0}{0}\): \(\frac{4\sin(4x)}{2x}\), still \(\frac{0}{0}\); then \(\frac{16\cos(4x)}{2}\to 8\). 8
2. Form \(\frac{\infty}{\infty}\): \(\frac{1/x}{1/(2\sqrt{x})}=\frac{2}{\sqrt{x}}\to 0\). 0

Scoring: (a) 1 point for naming the form and applying the rule once, 1 point for the value. (b) 1 point.

Question 7: Marginal revenue (2 points)

1. \(R^{\prime}(x)=50-0.2x\), so \(R^{\prime}(100)=30\).
2. Selling the 101st scarf adds about $30 in revenue.

Scoring: 1 point for each part; the interpretation must include dollars per scarf.

Question 8: Speeding up and slowing down (2 points)

\(v=(t-2)(t-4)\) and \(a(t)=2t-6\). Signs: \((0,2)\): \(v>0,a<0\); \((2,3)\): \(v<0,a<0\); \((3,4)\): \(v<0,a>0\); \((4,\infty)\): \(v>0,a>0\).

Speed increases on (2, 3) and (4, ∞).

Scoring: 1 point for the sign analysis of v and a, 1 point for both intervals.

Scoring summary

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