AP Calculus AB Rates of Change in Context and Straight-Line Motion Worksheet (PDF)

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

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This rates motion worksheet 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.

These 21 problems practice interpreting derivatives in context, and straight-line motion with position, velocity, acceleration and speed (AP Calculus AB, topics 4.1, 4.2, 4.3). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.

Later problems use sign charts, velocity graphs, tables of values, total distance and real situations such as a draining tank and bacteria growth. Work in pencil, show every step, and check the answer key only after a full attempt.

AP Calculus AB · Common Core State Standards: AP CALC AB 4.1, AP CALC AB 4.2, AP CALC AB 4.3

Rates motion worksheet: exercises 1 to 10

Exercise 1 – Interpreting a cost derivative

The cost in dollars of printing \(x\) flyers is \(C(x)\), and \(C^{\prime}(500)=0.12\). Interpret this value.

Exercise 2 – Interpreting a temperature derivative

A greenhouse temperature in °F is \(T(t)\) after \(t\) hours, and \(T^{\prime}(3)=-4.5\). Interpret this value.

Exercise 3 – Velocity and direction

A bicycle’s position in meters along a path is \(s(t)=t^{2}-6t+5\) for \(t\ge 0\) seconds. Find \(v(t)\), when it is at rest, and when it moves left or right.

Exercise 4 – Velocity and acceleration

A drone’s height in feet is \(s(t)=t^{3}-6t^{2}+5t\). Find \(v(2)\) and \(a(2)\) with units.

Exercise 5 – Speeding up or slowing down

A particle moves with \(s(t)=2t^{3}-9t^{2}+12t\). Decide whether it is speeding up or slowing down at \(t=0.5\), \(t=1.5\) and \(t=3\).

Exercise 6 – Reading a velocity graph

Velocity of a particle in feet per second.

The graph shows the velocity \(v(t)\) of a particle. (a) When is the particle at rest? (b) On which interval is it moving left? (c) When is the speed greatest on \([0,8]\)? (d) Find the acceleration on \(0 < t < 2\).

Exercise 7 – Sign chart for direction

Sign chart for velocity.

A particle has velocity \(v(t)=(t-2)(t-5)\) for \(t\ge 0\). Describe its direction of motion.

Exercise 8 – Average versus instantaneous velocity

A runner’s position is \(s(t)=t^{2}+2t\) meters. Find the average velocity on \([1,3]\) and the instantaneous velocity at \(t=1\) and \(t=3\).

Exercise 9 – Estimating velocity from a table

Position of a go-kart.

Use the table to estimate the velocity at \(t=2\) with the average velocity over \([1,3]\), and give the units.

Exercise 10 – Area of a circle

A circular oil spill has area \(A(r)=\pi r^{2}\) square feet when the radius is \(r\) feet. Find \(A^{\prime}(5)\) and interpret it.

Exercises 11 and up

Exercise 11 – Volume of a cube

The volume of a cube with side \(s\) inches is \(V(s)=s^{3}\). Find the rate of change of volume with respect to \(s\) at \(s=4\).

Exercise 12 – Population growth

Population of a town.

A town’s population is \(P(t)=500+20t-0.5t^{2}\) people after \(t\) years. Find \(P^{\prime}(10)\) and \(P^{\prime}(20)\) and interpret them.

Exercise 13 – Marginal revenue

The revenue in dollars from selling \(x\) skateboards is \(R(x)=120x-0.4x^{2}\). Find \(R^{\prime}(50)\) and interpret it.

Exercise 14 – Oscillating spring

A spring’s position in centimeters is \(s(t)=3\sin t\). Find the velocity and acceleration at \(t=\frac{\pi}{3}\).

Exercise 15 – Bacteria growth

A culture has \(N(t)=200e^{0.3t}\) bacteria after \(t\) hours. Find \(N^{\prime}(5)\) and interpret it.

Exercise 16 – Total distance

A hiker’s position along a trail in feet is \(s(t)=2t^{3}-15t^{2}+24t\) after \(t\) minutes. Find the total distance traveled from \(t=0\) to \(t=5\) minutes.

Exercise 17 – Draining tank

Gallons left in the tank.

Water drains from a tank so that the volume in gallons is \(V(t)=500\left(1-\frac{t}{20}\right)^{2}\) after \(t\) minutes. Find \(V^{\prime}(10)\) and interpret it.

Exercise 18 – Pauses and speeding up

A particle has \(s(t)=t^{4}-8t^{3}+18t^{2}\) for \(t\ge 0\). Find when it is at rest, whether it ever changes direction, and when its speed is increasing.

Exercise 19 – Displacement versus distance

A cart has position \(s(t)=t^{3}-3t^{2}\) meters. Find its displacement and its total distance traveled from \(t=0\) to \(t=3\) seconds.

Exercise 20 – Model rocket

Height of a model rocket in feet.

Noah launches a model rocket from a platform. Its height in feet is \(s(t)=-16t^{2}+96t+80\) after \(t\) seconds. Find the velocity function, the maximum height, and the velocity when it hits the ground.

Exercise 21 – Estimating a rate from a table

Cumulative sales of a new product.

The table shows cumulative sales \(S(t)\) in thousands of dollars. Estimate \(S^{\prime}(2)\) using the interval \([1,3]\) and interpret the result with units.

Key facts behind this rates motion worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

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

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