Average rate of change
The average rate of change (AvRC) of a function measures how the output of the function changes, on average, over a specific interval of the input. It is a way to describe the overall change of the function without considering the exact behaviour at every point within the interval.
For instance, consider a car travelling along a straight road. If the car covers \(100\) kilometers in \(2\) hours, its average speed (rate of change of distance with respect to time) is \(50\text{ km/h}\). During that time, its speed any point it time may vary – at some points the car my be travelling at \(100 \text{ km/h}\), and at other times it may be travelling more slowly, or even be stationary. The average rate of change of distance, however, remains constant at \(50\text{ km/h}\) over the 2-hour period, as it represents the overall change in distance divided by the total time taken.
For a function \(f\left(x\right)\), the average rate of change between two points \(x = a\) and \(x = b\) is given by
\[AvRC = \frac{f\left(b\right) - f\left(a\right)}{b - a}\]
This formula calculates the slope of the secant line connecting the points \(\left(a,f\left(a\right)\right)\) and \(\left(b,f\left(b\right)\right)\).
NoteIn Australia, the terms slope and gradient are often used interchangeably when referring to the rate of change of a function or steepness of a line. In other regions, only slope is used in this way, and gradient is used only for multivariable calculus. |
Worked Example
If \(f\left(x\right) = x^2 + 3\), then the average rate of change between \(x = 1\) and \(x = 3\) is
\[\frac{f\left(3\right) - f\left(1\right)}{3 - 1} = \frac{\left(3^{2} +3 \right) - \left(1^{2} + 3\right)}{3 - 1} = \frac{12 - 4}{2} =4\]
This is graphed below.
Alternative notation
\[AvRC = \frac{\Delta{y}}{\Delta{x}}\ \textsf{(rise over run)}\]
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If \(f\left(x\right) = 5\), the graph will form a horizontal line. The slope of a horizontal line is zero.
If the line were vertical, then slope would be undefined.
To solve this, first find the \(f\left(x\right)\) when \(x = 2\) and \(x = 5\)
\[\begin{align}&f(2) = 2^{2} - 5 = -1 \\ &f(5) = 5^{2} - 5 = 20\end{align}\]
The average rate of change is given by
\[\frac{f(5) - f(2)}{5 - 2} = \frac{20 - (-1)}{5 - 2} = \frac{21}{3} = 7\]
Therefore, the rate of change is 7, not 6.
First, find the values for \(f\left(0\right)\) and \(f\left(\frac{\pi}{2}\right)\):
\[\begin{array}{II}f\left(0\right) = \sin\left(0\right) = 0 \\ f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1\end{array}\]
The average rate of change of \(f\left(x\right) = \sin\left(x\right)\) between \(x = 0\) and \(x = \frac{\pi}{2}\) is \(\frac{2}{\pi}\).
To find the average rate of change between two points, we use the formula
\[\frac{f\left(b\right) - f\left(a\right)}{b - a}\]
\(a = 2\) is found by the \(x\)-coordinate of the point \(A\), and \(f\left(a\right) = \sqrt{3}\) is found by the \(y\)-coordinate at point \(A\). Similarly for point \(B\), \(b = 6\) and f\(\left(b\right) = \sqrt{7}.\)
\[\frac{\sqrt{7} - \sqrt{3}}{6 - 2} = \frac{\sqrt{7} - \sqrt{3}}{4}\]