The average rate of change of a function can be calculated by dividing the difference between the dependent variable and the independent variable. In this post, we discuss how to compute the average rate of change of functions in different spaces.

#### The average rate of change in single variable calculus

Let \(y = f(x)\) be a real-valued single-variable function defined over the interval \([a,b]\). The average rate of change of the function \(f\) with respect to \(x\) over the interval \([a,b]\) is $$\frac{\Delta y}{\Delta x} = \frac{f(b) – f(a)}{b-a}.$$

**Example**. Find the average rate of change of the function $$ y = x^2 + x + 1 $$ over the interval \([1,3]\).

**Solution**. Let \(f(x) = x^2 + x + 1\). It is clear that \(f(3) = 13\) and \(f(1) = 3\), and so, \(f(3) – f(1) = 10\). Therefore, the average rate of change of the function \(f\) over the interval \([1,3]\) is $$\frac{\Delta y}{\Delta x} = \frac{10}{2} = 5.$$

#### The average rate of change of a linear function

The average rate of change of a linear function \(f(x) = mx + b\) with respect to \(x\) over any interval \([a,b]\) is the same as the slope of the straight line \(L\) in the plane \(\mathbb{R}^2\) with the equation \(y = mx+b\). From college mathematics, we know that if a straight line \(L\) passes through two different points \((x_1, y_1)\) and \((x_2, y_2)\), then the slope \(m\) of the line \(L\) is $$m = \frac{y_2 – y_1}{x_2 – x_1}.$$

By definition, a line passing through two different points \((a,f(a))\) and \((b,f(b))\) of a curve \(y=f(x)\) is a secant line. It is clear that the slope of the mentioned secant line is the average rate of change of \(f\) with respect to \(x\) over the interval \([a,b]\).

#### The average rate of change in multivariable calculus

Let \(y = f(t)\) be a function from the interval \([a,b]\) into the set of \(n\)-dimensional real vectors, i.e., \(\mathbb{R}^n\). The average rate of the function \(f\) with respect to \(t\) over the interval \([a,b]\) is $$\frac{\Delta y}{\Delta t} = \frac{1}{b-a}(f(b) – f(a)).$$

**Example**. Find the average rate of the function $$ y = (\cos t, \sin t, t) $$ over the interval \([-\pi, \pi]\).

**Solution**. Let \(f(t) = (\cos t, \sin t, t)\). Note that $$f(-\pi) = (\cos(-\pi), \sin(-\pi), -\pi) = (1,0,-\pi)$$ $$f(\pi) = (\cos(\pi), \sin(\pi), \pi) = (1,0,\pi),$$ and so, $$f(\pi) – f(-\pi) = (0,0,2\pi).$$ Consequently, the average rate of change of the function \(f\) over the interval \([-\pi, \pi]\) is $$\frac{\Delta y}{\Delta t} = \frac{1}{2\pi}(0,0,2\pi) = (0,0,1).$$

#### The average rate of change in linear algebra

Let \(v = f(t)\) be a function from the interval \([a,b]\) into the real vector space \(V\). Similar to multivariable calculus, the average rate of change of the function \(f\) with respect to \(t\) over the interval \([a,b]\) is $$\frac{\Delta v}{\Delta t} = \frac{1}{b-a}(f(b) – f(a)).$$

For an etymological discussion of the word average, please refer to the average entry on the Wiktionary website.