AP Calculus AB Rates of Change in Context and Straight-Line Motion Test with Point Values (PDF)

AP Calculus AB math test: Rates of Change in Context and Straight-Line Motion – Maths PDF

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This rates motion test for AP Calculus AB covers rates of change in context and straight-line motion (Common Core AP CALC AB 4.1, AP CALC AB 4.2, AP CALC AB 4.3), with a free PDF to print.

This test checks interpreting derivatives in context, and straight-line motion with position, velocity, acceleration and speed (AP Calculus AB, 4.1, 4.2, 4.3). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.

Time and points

About this test

Grade: AP Calculus AB
Chapter: Rates of Change in Context and Straight-Line Motion
Standards: AP CALC AB 4.1, AP CALC AB 4.2, AP CALC AB 4.3
Time: 45 minutes
Total: 20 points

Rates motion test: questions

Question 1 – Interpreting derivatives [2 pts]

(a) The height of a plant in cm is \(H(t)\) after \(t\) days, and \(H^{\prime}(8)=1.5\). Interpret. (1 point) (b) The price of a share in dollars is \(p(t)\) after \(t\) months and \(p^{\prime}(6)=-2.3\). Interpret. (1 point)

Question 2 – Position, velocity, acceleration [3 pts]

A skater’s position in meters is \(s(t)=t^{3}-12t+4\) for \(t\ge 0\) seconds. (a) Find \(v(t)\) and \(a(t)\). (1 point) (b) When is the skater at rest? (1 point) (c) Find \(a\) at that time. (1 point)

Question 3 – Direction from a sign chart [3 pts]

Sign chart for the velocity of a particle.

The sign chart shows the velocity \(v(t)=(t-1)(t-4)\) of a particle, \(t\ge 0\). (a) When does the particle change direction? (1 point) (b) On which intervals does it move left? (1 point) (c) If \(a(t)=2t-5\), is it speeding up or slowing down at \(t=2\)? (1 point)

Question 4 – Geometric rates [2 pts]

(a) Find the rate of change of the area of a circle with respect to its radius when \(r=8\) cm. (1 point) (b) The volume of a sphere is \(V(r)=\frac{4}{3}\pi r^{3}\). Find \(V^{\prime}(3)\). (1 point)

Question 5 – Velocity from a table [3 pts]

Position of a sled.

Use the table. (a) Find the average velocity on \([0,6]\). (1 point) (b) Estimate the velocity at \(t=4\) using the interval \([2,6]\). (1 point) (c) Is the sled speeding up? Explain using average velocities on \([0,2]\), \([2,4]\) and \([4,6]\). (1 point)

Question 6 – Reading a velocity graph [3 pts]

Velocity in feet per second.

The graph shows \(v(t)\). (a) When is the particle at rest on \((0,6)\)? (1 point) (b) On which interval is it slowing down? (1 point) (c) Find the acceleration on \(4 < t < 6\). (1 point)

Question 7 – Exponential rate [2 pts]

A pond has \(N(t)=50e^{0.4t}\) algae cells (in thousands) after \(t\) days. Find \(N^{\prime}(3)\) and interpret it.

Question 8 – Total distance [2 pts]

A robot’s position in feet is \(s(t)=t^{3}-9t^{2}+24t\). Find the total distance traveled from \(t=0\) to \(t=5\) seconds.

Key facts behind this rates motion test

Key facts

A derivative has units of output per input, such as dollars per pencil or feet per second.
Velocity is s′(t), acceleration is v′(t), and speed is the absolute value of velocity.
Speed increases when velocity and acceleration have the same sign.
Total distance adds the distances between turning points, where velocity changes sign.

How to use this rates motion test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

Download the rates motion test as a PDF with the button below and keep it with your notes.

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