Average Rate of Change
Calculus & Advanced MathThe 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. It is the "big picture" rate over a whole interval.
Example
If you drove $120$ miles in $2$ hours, your average rate of change of distance is $60$ mph. You might have gone faster or slower at different moments, but $60$ is the average.
Key Insight
Average rate of change is just the slope formula (rise over run) applied to a function over an interval.
Definition
For $f$ on $[a, b]$, the average rate of change is $\frac{f(b) - f(a)}{b - a}$. This equals 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.
Example
$f(x) = x^2$ on $[1, 4]$: average rate = $(16 - 1)/(4 - 1) = 15/3 = 5$. The secant line from $(1,1)$ to $(4,16)$ has slope $5$.
Key Insight
The Mean Value Theorem guarantees that for a differentiable function, there is at least one point in $(a, b)$ where the instantaneous rate equals the average rate.
Definition
The average rate of change is the difference quotient $(f(b) - f(a))/(b - a)$, the slope of the secant line. As the interval shrinks, its limiting value is the derivative. The Mean Value Theorem: if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c$ in $(a,b)$ with $f'(c) = (f(b)-f(a))/(b-a)$.
Example
In numerical differentiation, finite-difference approximations (forward, backward, central) all approximate the derivative using average rates of change over small intervals.
Key Insight
The entire theory of numerical integration methods (trapezoidal rule, Simpson's rule) is built on piecewise average rates of change, connecting discrete arithmetic to continuous calculus.