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A short class with Sarah and her students. The other ways are under More.
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For f(x) = x² from x = 1 to x = 3, what are f(1) and f(3)?
f(1) is 1 and f(3) is 9.
So the change in y is 8 and the change in x is 2. What is the average rate of change?
8 divided by 2, which is 4.
Does the function go up by exactly 4 for every step of 1?
No. From 1 to 2 it goes up 3, and from 2 to 3 it goes up 5. The average of those is 4.
Plot the points (1, 1) and (3, 9) on the curve y = x² and join them. What is the gradient of that line?
Rise over run is 8 over 2, so 4. It's the same number as the AROC.
That line is called a secant. Now try the interval from 3 to 5.
f(5) is 25 and f(3) is 9, so 16 over 2, which is 8. That secant is steeper.
And from −2 to 0?
0 minus 4 is −4, over 2, so −2. It's negative because the curve is going down there.
A ball's height is h(t) = −5t² + 20t + 2. Find the AROC from t = 1 to t = 3.
h(1) is 17 and h(3) is 17, so the AROC is 0 metres per second.
So the ball didn't move?
It did. It went up and then came back down to 17 m, so on average its height didn't change.
Exactly. An average can hide what happens in between. What are the units of an AROC?
Output units per input unit, like metres per second or people per year.
(f(b) − f(a))/(b − a), the change in output divided by the change in input.
the AROC is the gradient of the straight line joining the two points on the graph.
a curve's AROC changes with the interval, while a straight line's AROC always equals its gradient.
a positive AROC means increasing on average, a negative one decreasing, and zero means equal start and end values.
an AROC is measured in output units per input unit, such as km/h or people per year.
as the interval shrinks, the AROC approaches the gradient of the tangent at a point.
For f(x) = x² from x = 1 to x = 3: (9 − 1)/(3 − 1) = 8/2 = 4, the gradient of the secant through (1, 1) and (3, 9).
For f(x) = x² from x = −2 to x = 0: (0 − 4)/(0 − (−2)) = −4/2 = −2, so the function decreases on average over this interval.
A car is 50 km from home after 1 hour and 180 km after 3 hours, so its average speed is (180 − 50)/(3 − 1) = 65 km/h.
For h(t) = −5t² + 20t + 2 from t = 1 to t = 3: h(1) = h(3) = 17, so the average velocity is 0 m/s even though the ball moved.
End y minus start y, over end x minus start x; change in y over change in x is the average step.
Two hikers stand on a hill at x = 1 and x = 3 and stretch a rope between them. The rope's slope is 4, even though the path beneath it climbs 3 and then 5.
"Out over In" - the change in the output goes over the change in the input.
E.S.D. - Evaluate both ends, Subtract, Divide.
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What is the AROC of f over the interval from a to b?
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Find the AROC of f(x) = x² from x = 1 to x = 3.
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What does the AROC represent on a graph?
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What is the AROC of f(x) = 3x − 2 over any interval?
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What does an AROC of zero tell you?
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(f(b) − f(a))/(b − a), the change in output divided by the change in input over an interval.
The set of x-values between two endpoints a and b, written [a, b].
A straight line through two points on a curve; its gradient equals the AROC between them.
Rise divided by run; the steepness of a straight line.
The rate of change at a single point, the gradient of the tangent line there.
A line that touches a curve at a point and has the same steepness as the curve there.
The AROC of position with respect to time, measured in units such as m/s.
This lesson works through examples of the average rate of change (AROC), starting with f(x) = x² from x = 1 to x = 3, where AROC = (9 − 1)/(3 − 1) = 4. It shows how to calculate, interpret and picture an AROC in graphs, tables and real situations.
The average rate of change of a function over an interval [a, b] is (f(b) − f(a))/(b − a): the change in output divided by the change in input. For f(x) = x² from x = 1 to x = 3 it is 8/2 = 4, so on average y rises 4 units for each unit increase in x, and 4 is the gradient of the secant line joining (1, 1) and (3, 9). This lesson repeats the calculation over other intervals to show that the AROC of a curve changes, compares it with a straight line whose AROC never changes, and applies it to tables of data, a ball's height and population growth. It ends by shrinking the interval to see the AROC approach the rate at a single point.
The definition of the average rate of change was introduced in the previous lesson. Here we practise using it.
The formula. For a function f on the interval from x = a to x = b:
AROC = (f(b) − f(a)) / (b − a) = change in y ÷ change in x
Worked example: f(x) = x² from x = 1 to x = 3.
Interpreting the answer. Over the interval from 1 to 3, the function increases on average by 4 units for every 1 unit increase in x. It does not rise by exactly 4 for each step: from 1 to 2 it rises by 3, and from 2 to 3 it rises by 5. The average of those two steps is 4.
Picturing it. The AROC is the gradient of the secant line through the points (1, 1) and (3, 9) on the curve. That line has equation y = 4x − 3, which passes through both points.
The AROC of a curve depends on the interval. For the same f(x) = x²:
A straight line has a constant AROC. For f(x) = 3x − 2, the AROC over any interval equals the gradient 3. For example, from 1 to 4: (10 − 1)/(4 − 1) = 9/3 = 3.
Rates from data. A car's distance from home is 50 km after 1 hour and 180 km after 3 hours. Its average speed over that time is (180 − 50)/(3 − 1) = 65 km/h. A town's population grows from 12 000 in 2015 to 15 600 in 2021, an average rate of 3600 ÷ 6 = 600 people per year. The units of an AROC are always output units per input unit.
A zero AROC does not mean nothing happened. A ball's height is h(t) = −5t² + 20t + 2 metres after t seconds. Since h(1) = 17 and h(3) = 17, the AROC from t = 1 to t = 3 is 0 m/s. The ball went up and came back down to the same height, so its average velocity over that time is zero.
From average to instantaneous. For f(x) = x², take shorter and shorter intervals starting at x = 2: from 2 to 3 the AROC is 5, from 2 to 2.5 it is 4.5, from 2 to 2.1 it is 4.1, and from 2 to 2.01 it is 4.01. The values approach 4, the instantaneous rate of change at x = 2, which is the gradient of the tangent line there. This idea leads to the derivative.