Rates of Change Practice
Practice understanding rates of change. A rate of change compares how much one quantity changes compared with another quantity. This idea connects slope, speed, acceleration, graphs, and the beginning of calculus.
Skill 1: Average Rate of Change
Average rate of change compares the change in output to the change in input. In real life, this often appears as speed, production rate, cost per item, or change over time.
Worked Example
Problem: A car travels 120 miles in 2 hours. Find the average speed.
Step 1: Average rate means change in distance divided by change in time.
Average speed = distance / time
Step 2: Substitute the values.
Average speed = 120 miles / 2 hours
Step 3: Divide.
Average speed = 60 miles per hour
Answer: The average speed is 60 mph.
Try These
- A runner travels 10 km in 50 minutes. Find the average speed in km/min.
- A machine produces 500 parts in 5 hours. Find the production rate.
- A car travels 180 miles in 3 hours. Find the average speed.
- A student reads 80 pages in 4 hours. Find the reading rate.
- A tank fills with 60 liters in 12 minutes. Find the filling rate.
Answer Key
- 0.2 km/min
- 100 parts/hour
- 60 mph
- 20 pages/hour
- 5 liters/minute
Mastery Check
You are ready to move on when you can divide change in output by change in input and include the correct units.
Skill 2: Rate of Change from a Table
A table can show how one quantity changes compared with another. To find the rate of change, choose two points from the table and divide the change in output by the change in input.
Worked Example
Problem: Use the table to find the rate of change.
| Time (s) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Distance (m) | 0 | 4 | 8 | 12 | 16 |
Step 1: Choose two points from the table.
(0, 0) and (4, 12)
Step 2: Use change in output divided by change in input.
Rate = (12 - 0) / (4 - 0)
Step 3: Simplify.
Rate = 12 / 4 = 3
Answer: The rate of change is 3 m/s.
Try These
| Time (s) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Distance (m) | 0 | 4 | 8 | 12 | 16 |
- Find the rate of change from 0 s to 1 s.
- Find the rate of change from 1 s to 2 s.
- Find the rate of change from 2 s to 3 s.
- Find the rate of change from 3 s to 4 s.
- Is the rate of change constant?
Answer Key
- 4 m/s
- 4 m/s
- 4 m/s
- 4 m/s
- Yes, the rate of change is constant.
Mastery Check
You are ready to move on when you can use two table values to calculate change in output divided by change in input.
Skill 3: Rate of Change from a Graph
On a graph, the rate of change is the slope. Slope tells how much the output changes when the input changes. For a distance-time graph, slope represents speed.
Worked Example
Problem: A distance-time graph passes through the points (0, 0) and (5, 20). Find the rate of change.
Step 1: Identify the two points.
(0, 0) and (5, 20)
Step 2: Use the slope formula.
Rate of change = change in distance / change in time
Step 3: Substitute the values.
Rate = (20 - 0) / (5 - 0)
Step 4: Simplify.
Rate = 20 / 5 = 4
Answer: The rate of change is 4 units per second.
Try These
- A graph passes through (0, 0) and (2, 10). Find the rate of change.
- A graph passes through (1, 3) and (4, 12). Find the rate of change.
- A graph passes through (0, 5) and (5, 5). Find the rate of change.
- A graph passes through (2, 10) and (6, 2). Find the rate of change.
- On a distance-time graph, what does slope represent?
Answer Key
- 5
- 3
- 0
- -2
- Speed
Mastery Check
You are ready to move on when you can choose two points from a graph and calculate slope as change in output divided by change in input.
Skill 4: Instantaneous Rate of Change
Average rate of change describes what happens over an interval. Instantaneous rate of change describes what is happening at one exact moment. In calculus, instantaneous rate of change becomes the derivative.
Worked Example
Problem: Is a speedometer showing average speed or instantaneous speed?
Step 1: A speedometer shows how fast the car is moving right now.
Step 2: “Right now” means at one instant, not over a long interval.
Answer: A speedometer shows instantaneous speed.
Try These
- Does average speed describe motion over an interval or at one exact moment?
- Does instantaneous speed describe motion over an interval or at one exact moment?
- What does a speedometer show?
- In calculus, instantaneous rate of change becomes what idea?
- Why is instantaneous rate important for motion and safety?
Answer Key
- Over an interval
- At one exact moment
- Instantaneous speed
- The derivative
- It shows what is happening right now, which helps describe speed, braking, acceleration, and safe design.
Mastery Check
You are ready to move on when you can explain the difference between average rate of change and instantaneous rate of change.
Skill 5: Acceleration as a Rate of Change
Acceleration measures how quickly velocity changes over time. If speed increases, acceleration is positive. If speed decreases, acceleration is negative.
Worked Example
Problem: A car increases its speed from 10 m/s to 30 m/s in 5 seconds. Find the average acceleration.
Step 1: Use the acceleration formula.
Acceleration = change in velocity / change in time
Step 2: Find the change in velocity.
30 - 10 = 20 m/s
Step 3: Divide by time.
Acceleration = 20 / 5 = 4 m/s²
Answer: The average acceleration is 4 m/s².
Try These
- A bike speeds up from 2 m/s to 10 m/s in 4 seconds. Find the acceleration.
- A car slows from 25 m/s to 5 m/s in 10 seconds. Find the acceleration.
- A train speeds up from 0 m/s to 20 m/s in 5 seconds. Find the acceleration.
- A runner slows from 8 m/s to 4 m/s in 2 seconds. Find the acceleration.
- If velocity does not change, what is the acceleration?
Answer Key
- 2 m/s²
- -2 m/s²
- 4 m/s²
- -2 m/s²
- 0 m/s²
Mastery Check
You are ready to move on when you can calculate acceleration as change in velocity divided by change in time and explain what positive, negative, and zero acceleration mean.