Average Rates of Change

Let \(f(x) \colon X \subseteq \mathbb{R} \longrightarrow \mathbb{R}.\) Let \(\exists a, b \in \mathbb{R} \mid a < b, [a,b] \subseteq X.\) Suppose I want to quantify how much \(f\) increases, on average, per unit change in \(x\) over the interval \([a,b]\).

How would we do that? Clearly, we’d want it to be a ratio between change in \(f\) and change in \(x\), so we know that much. But what does it mean for this ratio to be the average rate of change?

It means that, if \(f\) starts at \((a, f(a))\) and increases at a constant rate of \(m\) (\(m\) being its average rate of change on \([a,b]\)) until reaching \(x = b\), the end result should be the same as if we had just traced the graph until reaching \(x = b\). That is, we should end at \((b, f(b)).\) Restating that in more mathematical language:

\[f(a) + m(b - a) = f(b),\]

where we multiply \(m\) by the change in \(x\) from \(a\) to \(b\) to get the total change in \(f\) from \(a\) to \(b\) because

\[\frac{\text{Change in }f}{\text{Change in }x} \cdot \text{Change in }x = \text{Change in }f.\]

So, to arrive at our definition of average rate of change, all we have to do is solve for \(m\).

Definition: Average Rate of Change

Let \(f(x) \colon X \subseteq \mathbb{R} \longrightarrow \mathbb{R}.\) Let \(\exists a, b \in \mathbb{R} \mid a < b, [a,b] \subseteq X.\) Then the average rate of change (a.k.a. the average slope) of \(f\) on the interval \([a,b]\) is defined as

\[\frac{f(b) - f(a)}{b - a}.\]