The average rate of change over an interval is the slope of the line joining the two endpoints: (f(b) − f(a))/(b − a). A curve has no single rate of change. Ask over which interval, or the question is incomplete.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The calculation
Change in output over change in input. For f(x) = x³ on [1, 3]: (27 − 1)/(3 − 1) = 26/2 = 13. Rise over run, with the two points read from the function.
It depends on the interval
x³ on [0, 1]: (1 − 0)/1 = 1. On [1, 2]: (8 − 1)/1 = 7. On [2, 3]: 19. The same function, and the rate grows as the interval moves right, because the curve steepens.
What the sign says
Positive means the function ended higher than it started. It does not mean it rose the whole way. x² on [−1, 2]: (4 − 1)/3 = 1, positive, though the curve first falls to 0.
With units
A car at 20 km at 1 pm and 80 km at 3 pm: (80 − 20)/(3 − 1) = 30 km per hour. The units of the rate are output units over input units, always.
When it is constant
For a linear function the average rate of change is the same on every interval: its slope. A curve is exactly a function whose average rate changes from interval to interval.
Shrinking the interval
x² on [2, 3]: 5. On [2, 2.1]: (4.41 − 4)/0.1 = 4.1. On [2, 2.01]: 4.01. The rates close in on 4 as the interval shrinks. That limit is where calculus begins.
Step 2: Try It Yourself
Tap and try it out.
For f(x) = x³ − 3x, find the average rate of change on [0, 2], on [−1, 1], and on [1, 1.1]
- 1f(0) = 0, f(2) = 8 − 6 = 2endpoints of [0, 2]
Step 0 of 4
- Point(-2, 0)
- Second point(3, 5)
- Slope between them1
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: In Real Life
A share price over a month
A share went from 40 to 52 dollars in 30 days: 0.40 a day on average. That is the slope between two points on the curve, whatever happened in between.
Step 4: Watch an Example
One step at a time.
Watch Priya Compare Two Intervals
For f(x) = x², Priya finds the average rate of change on [0, 2] and on [−2, 0].
- Step 1
On [0, 2]: f(2) − f(0) = 4 − 0, over 2 − 0. That gives 2.
Step 5: Your Turn
Practice makes it stick.
The Interval
Problem 1 of 2
For f(x) = x², what is the average rate of change on [1, 4]?
Reading It Off the Graph
Problem 2 of 2
The secant shown joins x = −2 and x = 3 on y = x² − 4. What is its slope?
- Point(-2, 0)
- Second point(3, 5)
- Slope between them1
a
b
c
First point
Second point
Rates Across Intervals
1 of 8
f(x) = x². Average rate of change on [2, 5]?
2 of 8
f(x) = 3x + 1. Average rate of change on [0, 10]?
3 of 8
f(x) = x². Average rate of change on [−3, 3]?
4 of 8
f(x) = x³. Average rate of change on [0, 2]?
5 of 8
Select every statement that must be true when the average rate of change on [a, b] is 0.
6 of 8
Put these steps in the order you would carry them out.
- 1Subtract to get the change in output.
- 2Subtract to get the change in input.
- 3Divide output change by input change.
- 4Work out f(a) and f(b).
7 of 8
f(x) = −x². Average rate of change on [1, 3]?
8 of 8
Does a positive average rate of change guarantee the function increased throughout the interval?
Step 6: Quick Check
Show what you know.
Question 1 of 1
f(x) = x². Average rate of change on [3, 6]?
What You Learned
- The average rate of change is the slope of the secant between the endpoints.
- It is (f(b) − f(a))/(b − a), and it depends on the interval chosen.
- It compares endpoints only, and its units are output over input.
- As the interval shrinks, the rates close in on the instantaneous rate.