Rate of Change

A rigorous guide to average and instantaneous rates, slope, units, signs, models, and verification.

Slope compares vertical change with horizontal change. A line contains two marked points joined by a right-angled rise-and-run triangle. The vertical and horizontal legs make the signed ratio behind slope visible.

Precise definition

The average rate of change of $f$ from $a$ to $b$ is $[f(b)-f(a)]/(b-a)$ for $a\ne b$. It is the secant slope and has output units per input unit. The instantaneous rate at $a$ is $f'(a)=\lim_{h\to0}[f(a+h)-f(a)]/h$ when that limit exists.

Notation and mathematical language

A positive rate means output increases as input increases locally or on the interval; negative means decrease. Constant rate corresponds to a linear function. Difference quotients require consistent order in numerator and denominator.

Conceptual picture

Average rate compresses net change over an interval and can hide variation within it. Instantaneous rate is the slope of the best local linear approximation. In data, a fitted slope is estimated and carries uncertainty; in an exact formula, symbolic rates are exact relative to the model.

Fully worked example

Interpretation and application

Rates model speed, marginal cost, population growth, temperature change, and concentration. A statistical association slope does not alone show that changing the explanatory variable causes the response to change.