1. The average rate of change of a function $f(x)$ over an interval $[a,b]$ measures how much the function's output changes per unit change in input between $a$ and $b$. 2. It is calculated using the formula: $$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$$ 3. Here, $f(a)$ and $f(b)$ are the values of the function at points $a$ and $b$, respectively. 4. This formula essentially finds the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$ on the graph of $f$. 5. For example, if $f(x) = x^2$, and we want the average rate of change from $x=1$ to $x=3$, then: $$f(1) = 1^2 = 1$$ $$f(3) = 3^2 = 9$$ So, $$\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = \frac{8}{2} = 4$$ This means the average rate of change of $f(x) = x^2$ from $1$ to $3$ is $4$.