Step by step
The question
Question: find the average rate of change of f(x) = x³ − 2x from x = 0 to x = 2.
1 Find the two points
We put 0 in place of x.
f(0) = 0.
We put 2 in place of x.
f(2) = 4.
So the two points are (0, 0) and (2, 4).
2 Change over change
Average rate of change means: how much y changes, divided by how much x changes.
On top: f(2) minus f(0).
At the bottom: 2 − 0.
End minus start, on top and at the bottom.
Careful!
The change in y goes on top, not the change in x.
We put in the numbers: 4 minus 0, over 2 − 0.
4 minus 0 is 4.
2 − 0 is 2.
4 divided by 2 is 2.
So the average rate of change is 2.
On average, f(x) goes up 2 for each 1 that x goes up.
3 Check
Let's check on a graph.
This is the graph of f(x).
Our two points.
The line through them.
Its slope is the average rate of change.
Run: 2 across.
Rise: 4 up.
Start at f(0) = 0.
Add the rate, 2, 2 times.
We land on 4, which is f(2).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.