Chapter 4: Rates of Change

Understand how quantities change over time, from average speed to acceleration and the idea behind derivatives.

What You Will Practice

Rates of change explain how one quantity changes compared to another. This chapter prepares students for derivatives by using motion, tables, graphs, slope, and acceleration before formal calculus notation appears.

Average Rate of Change
Rate from a Table
Slope from a Graph Idea
Instantaneous Rate Concept
Acceleration
Interpreting Motion

Mini Lesson

1. Average Rate of Change

Average rate of change tells how much an output changes per unit change in input.

Average rate = change in output / change in input

Example: A car travels 120 miles in 2 hours. Average speed = 120 / 2 = 60 mph.

2. Rate of Change from a Table

A table can show how a quantity changes. If the change is constant, the relationship is linear.

Rate = (y₂ - y₁) / (x₂ - x₁)

Example: Time: 0, 2, 4 and Distance: 0, 6, 12 gives rate = 12 / 4 = 3 m/s.

3. Slope from a Graph

On a distance-time graph, the slope represents speed.

Steeper slope: faster change.

Horizontal line: no change.

4. Instantaneous Rate of Change

Average rate describes change over an interval. Instantaneous rate describes what is happening right now.

Example: A speedometer shows instantaneous speed, not average speed over the whole trip.

5. Acceleration

Acceleration is the rate of change of velocity.

Acceleration = change in velocity / change in time

Example: Speed changes from 10 m/s to 30 m/s in 5 s. Acceleration = (30 - 10) / 5 = 4 m/s².

Interactive Rates of Change Practice

Choose a topic and practice with instant feedback. Type numbers only unless the answer asks for a concept. Round decimal answers to two decimal places when needed.

Typing tip: For concept questions, type answers like instantaneous, average, no motion, faster, or slower.
Why this matters: Rates of change are the doorway into derivatives. Before students learn derivative notation, they need to understand that calculus is really asking: “How fast is something changing right now?”

Mastery Check

Before moving to Book 2 Chapter 5, students should be able to do the following.

Average Rate

I can calculate average rate of change using output change over input change.

Tables

I can find rate of change from a table.

Graphs

I can explain what slope means on a distance-time graph.

Instantaneous Rate

I understand that instantaneous rate describes what is happening right now.

Acceleration

I can calculate acceleration as change in velocity divided by time.

Go to Chapter 5 Back to Book 2