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

AP Calculus Contextual Applications Test: questions 1–5

Question 1: Particle motion (3 points)

A particle moves along a line with position \(s(t)=t^{3}-9t^{2}+24t\) feet, for \(0\le t\le 6\) seconds.

1. Find \(v(t)\) and \(a(t)\). (1 point)
2. Find the times when the particle is at rest. (1 point)
3. Is the speed increasing or decreasing at \(t=5\)? (1 point)

Question 2: Interpreting a derivative (2 points)

The temperature of a bowl of soup is \(H(t)\) °F, \(t\) minutes after serving. \(H(10)=140\) and \(H^{\prime}(10)=-1.8\).

1. Interpret \(H^{\prime}(10)\) with units. (1 point)
2. Use the tangent line at \(t=10\) to estimate \(H(12)\). (1 point)

Question 3: Local linearization (3 points)

Let \(f(x)=\sqrt[3]{x}\). The graph and its tangent line at \(x=27\) are shown.

1. Write the tangent line equation at \(x=27\). (1 point)
2. Approximate \(\sqrt[3]{28}\). (1 point)
3. Is the approximation an overestimate or an underestimate? Justify. (1 point)

Question 4: Melting ice cube (2 points)

A cube of ice melts so that its edge length decreases at 0.2 inch per minute. Find the rate of change of its volume when the edge length is 5 inches. The volume of a cube is \(V=s^{3}\).

Question 5: Plane and radar station (3 points)

An airplane flies horizontally at a constant altitude of 3 miles, directly away from a radar station, at 400 mph. Let \(x\) be the horizontal distance from the point on the ground below the plane to the station, and \(s\) the straight-line distance.

1. Write an equation relating \(s\) and \(x\). (1 point)
2. Find \(\frac{ds}{dt}\) when \(x=4\) miles. (2 points)

AP Calculus Contextual Applications Test: questions 6–8

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

1. Evaluate \(\lim_{x\to 0}\frac{1-\cos(4x)}{x^{2}}\). (2 points)
2. Evaluate \(\lim_{x\to\infty}\frac{\ln x}{\sqrt{x}}\). (1 point)

Question 7: Marginal revenue (2 points)

The revenue in dollars from selling \(x\) scarves is \(R(x)=50x-0.1x^{2}\).

1. Find \(R^{\prime}(100)\). (1 point)
2. Interpret the value in context. (1 point)

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

A particle moves on a line with velocity \(v(t)=t^{2}-6t+8\) ft/s for \(t\ge 0\). Find all intervals on which the speed is increasing.

How to use this ap calculus contextual applications test

Take this ap calculus contextual applications test without notes, then grade it with the test answer key out of 20 points.

Printable PDF · Test

AP Calculus AB Derivatives in Context: Motion and Related Rates test

• 📄 US Letter
• 💾 203 KB
• 🖨️ Black and white friendly

Download the PDF

See every chapter on the AP Calculus AB math page.