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: Picture It
Start with something you already know.
A hike up and down a trail
- You start a hike at 200 meters above sea level.
- The trail climbs, then drops into a valley, then climbs again.
- Three hours later you are at 500 meters.
- You climbed 300 meters in 3 hours: 100 meters an hour, on average.
- That average never mentions the valley in the middle.
The average rate of change is the change in height over the change in time, between two moments you choose.
It joins the start and the end with one straight line and ignores every bump between them.
Choose a different stretch of the hike and you get a different rate, sometimes a negative one, where the trail went down.
The same hike with numbers: your height at each hour.
| Hour | Height (meters) |
|---|---|
| 0 | 200 |
| 1 | 350 |
| 2 | 260 |
| 3 | 500 |
From hour 0 to hour 3: (500 − 200) ÷ 3 = 100 meters per hour. From hour 1 to hour 2: (260 − 350) ÷ 1 = −90 meters per hour. Same hike, different stretch, different rate.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
The points (1, 1) and (3, 9) both lie on the curve y = x². A straight line is drawn through these two points. What is the slope of that line?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Ravi Find Two Rates on One Curve
A fresh function: f(x) = x² − 4x. Ravi finds its average rate of change on [1, 5], then on [0, 2], and checks the first one against a straight line.
- Step 1
Ravi starts with the two ends of [1, 5], because they are the only inputs the average uses. f(1) = 1 − 4 = −3, and f(5) = 25 − 20 = 5. So the curve passes through (1, −3) and (5, 5).
Watch Maya Start It, Then Finish It Yourself
A ball is thrown upward. Its height after t seconds is h(t) = 20t − 5t² meters. Maya finds the average rate of change on [0, 2] and on [1, 3]; the later steps are yours.
- Step 1
Maya starts with the ends of [0, 2]. At t = 0 the ball is at h(0) = 0 meters. At t = 2 it is at h(2) = 40 − 20 = 20 meters.
Step 6: Your Turn
Practice makes it stick.
The Coffee Cart
Problem 1 of 15
A coffee cart keeps a running total of the day’s sales. By 9 am it has taken $120, and by noon it has taken $390. What was its average rate of sales between 9 am and noon, in dollars per hour?
dollars per hour
The Draining Battery
Problem 2 of 15
A phone’s battery shows 85% at 2 pm and 40% at 5 pm. What was the average rate of change of the battery level over that time, in percentage points per hour?
percentage points per hour
Rates Across Intervals
Find the two ends first. Then change in output over change in input.
Problem 3 of 15
f(x) = x². What is the average rate of change on [1, 4]?
Problem 4 of 15
f(x) = 3x + 1. What is the average rate of change on [0, 10]?
Problem 5 of 15
f(x) = x³. What is the average rate of change on [0, 2]?
Problem 6 of 15
f(x) = −x². What is the average rate of change on [1, 3]?
Problem 7 of 15
The graph shows y = x² − 4, with a secant joining the points at x = −1 and x = 3. What is the slope of the secant?
Whose Step?
One line on each page is a slip. Find it, then put it right.
Problem 8 of 15
Noor found the average rate of change of f(x) = x² + 1 on [2, 4]. One line is a slip. Which?
Noor’s working
- 1f(x) = x² + 1 on [2, 4]
- 2f(2) = 5 and f(4) = 17
- 3change in output = 17 − 5 = 12
- 4change in input = 4 − 0 = 4
- Answeraverage rate = 12 ÷ 4 = 3
Problem 9 of 15
What is the average rate of change of f(x) = x² + 1 on [2, 4], correctly?
Problem 10 of 15
Leo found the average rate of change of f(x) = x³ on [−2, 1]. One line is a slip. Which?
Leo’s working
- 1f(x) = x³ on [−2, 1]
- 2f(−2) = −8 and f(1) = 1
- 3change in output = 1 − (−8) = 9
- 4change in input = −2 − 1 = −3
- Answeraverage rate = 9 ÷ (−3) = −3
Problem 11 of 15
What is the average rate of change of f(x) = x³ on [−2, 1], correctly?
Pick the Move
Decide what to do, or what the number means, before doing any arithmetic.
Problem 12 of 15
f(x) = 5x − 2. Ravi wants its average rate of change on [3, 97]. What is the quickest move?
Problem 13 of 15
A pool holds W(t) gallons of water t hours after a hose is turned on. W(2) = 300 and W(5) = 900. What does the average rate of change of W on [2, 5] tell you?
Same Rate?
Judge these by thinking about the function first, then check any you are unsure of.
Problem 14 of 15
Select every function whose average rate of change on [0, 2] is the same as its average rate of change on [5, 7].
Problem 15 of 15
f(x) = x² − 2x. Select every interval on which f has the same average rate of change as it has on [0, 2].
Step 7: Quick Check
Show what you know.
Question 1 of 1
f(x) = x². Average rate of change on [3, 6]?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
f(x) = x². For which number b is the average rate of change of f on [1, b] equal to 6?
Stretch 2 of 2
A function has an average rate of change of 5 on [0, 2] and an average rate of change of −1 on [2, 6]. What is its average rate of change on [0, 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.
For the grown-up
The misconception this lesson targets is reading an average rate of change as a description of the whole journey: a positive rate taken to mean the function rose the whole way, or a rate of 0 taken to mean nothing happened. The average uses only the two ends of the interval, which is why the ball in the second example can climb and fall and still score 0. Two smaller slips travel with it: measuring the change in input from 0 instead of from the interval’s left end, and starting the two subtractions from different ends, which flips the sign. Asking "over which interval?" and "did it end higher or lower?" settles all three.