The Average Rate of Change Over an Interval

A curve has no single gradient

A straight line has the same gradient everywhere. A curve does not. The curve $y=x^2/4$y = x² / 4 is nearly flat near x = 0 and gets steeper and steeper to the right, so no single number says how steep it is.

What can be measured is how fast y changes on average between two points. Choose two points on the curve, (2, 1) and (6, 9), and join them with a straight line. A straight line joining two points of a curve is called a chord.

Rise over run of the chord

Find the gradient of the chord exactly as for any line. From (2, 1) to (6, 9), the rise is 9 − 1 = 8 and the run is 6 − 2 = 4. The gradient is 8 ÷ 4 = 2.

This is the average rate of change of y from x = 2 to x = 6: across that interval, y goes up by 2 for each 1 across, on average. In general, the average rate of change from one point to another is (change in $y)/($y) / (change in x), or $\left(y_2-y_1\right)/\left(x_2-x_1\right)$(y₂ − y₁) / (x₂ − x₁) for the points $\left(x_1,y_1\right)$(x₁, y₁) and $\left(x_2,y_2\right)$(x₂, y₂).

The chord from (2, 1) to (6, 9) on the curve $y=x^2/4$y = x² / 4 rises 8 over a run of 4, so its gradient is 2.

What the average hides

Between x = 2 and x = 6, the curve does not rise at a steady 2 per step. Work out y at x = 2, 3, 4, 5 and 6: it is 1, 2.25, 4, 6.25 and 9. From x = 2 to x = 3 the curve rises only 1.25, and from x = 5 to x = 6 it rises 2.75.

So the curve is less steep than the chord at first, and steeper than the chord at the end. The four rises add up to 8, the rise of the chord, and 8 shared over the 4 steps is 2. The chord replaces the changing steepness by its average.

The rise of $y=x^2/4$y = x² / 4 over each step of 1 from x = 2 to x = 6. The rises grow, and they add up to 8.

A different interval, a different average

Change the interval and the answer changes. From x = 0 to x = 4, the curve goes from (0, 0) to (4, 4): a rise of 4 over a run of 4, so the average rate of change is 1.

This interval is further left, where the curve is less steep, so its average is smaller. An average rate of change always belongs to an interval, so always say which interval it is for.

The chord from (0, 0) to (4, 4) on the same curve has gradient 4 ÷ 4 = 1.

Rates with units

When the two quantities have units, so does the rate. A pot of soup is at 20°C at 1 pm and 80°C at 4 pm. Over those 3 hours its average rate of change of temperature is $\left(80-20\right)/\left(4-1\right)=60/3=20^{\circ }C$(80 − 20) / (4 − 1) = 60 / 3 = 20°C per hour.

On a graph of distance against time, the average rate of change is the change in distance divided by the change in time, which is the average speed.

The usual mistakes

Giving the rise alone. From (2, 1) to (6, 9) the rise is 8, but the average rate of change is 8 shared over the run of 4, which is 2.

Dividing the y-value at the end by the x-value at the end. 9 ÷ 6 = 1.5 is the gradient of the line from the origin to (6, 9), not of the chord from (2, 1).

Using how steep the curve is at one end. At x = 6 the curve is steeper than the chord; the average uses both ends of the interval.

Subtracting in different orders. Take both differences from the same end: (9 − 1) ÷ (6 − 2), not (9 − 1) ÷ (2 − 6).