Average Rate of Change
The average rate of change measures how much a function changes per unit of input over an interval, equal to the slope of the secant line.
Formula
\frac{f(b) - f(a)}{b - a}
Definition
The average rate of change tells you how much a quantity changed overall, divided by how long it took, the "big picture" rate over a whole interval. For $f$ on $[a, b]$, it equals the difference quotient $\frac{f(b) - f(a)}{b - a}$, the slope of the secant line through $(a, f(a))$ and $(b, f(b))$; as $b \to a$, it approaches the instantaneous rate of change. The Mean Value Theorem guarantees that for a function continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c$ in $(a,b)$ where $f'(c) = (f(b)-f(a))/(b-a)$, meaning the instantaneous rate equals the average rate somewhere in between.
Example
If you drove $120$ miles in $2$ hours, your average rate of change of distance is $60$ mph, even though you might have gone faster or slower at different moments. For $f(x) = x^2$ on $[1, 4]$: average rate $= (16 - 1)/(4 - 1) = 15/3 = 5$, the slope of the secant line from $(1,1)$ to $(4,16)$. In numerical differentiation, finite-difference approximations (forward, backward, central) all approximate the derivative using average rates of change over small intervals.
Key Insight
Average rate of change is just the slope formula (rise over run) applied to a function over an interval. The entire theory of numerical integration methods, like the trapezoidal rule and Simpson's rule, is built on piecewise average rates of change, connecting discrete arithmetic to continuous calculus.