Rate of Change

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.

Conditions and key results

Derivative existence requires matching finite one-sided slopes at an interior point. At endpoints, one-sided rates may be appropriate. Units and domain matter; dividing total distance by time gives average speed, while displacement over time gives average velocity.

A reliable strategy

1. Define input and output with units and choose the interval or point.
2. For average rate, compute output change and input change in the same direction.
3. For instantaneous rate, use a derivative rule or justified difference-quotient limit.
4. Interpret sign and units and compare with graph, secants, or context.

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.

Common mistakes

Verification and reasonableness

• Estimate slope from a graph or nearby values.
• Check units by dividing output units by input units.
• For an exact derivative, compare with shrinking secant slopes.

Practice

1. Find average rate of $x^2$ from 2 to 5.
2. What is the rate unit for kilometres over hours?
3. Does average rate give every instantaneous rate?

Further deduction

For equally spaced data, first differences approximate rates over each interval. Constant first differences support a linear model; constant second differences support a quadratic model under equal spacing. These patterns suggest structure but measured noise requires residual analysis rather than exact classification from near-constant values.

The Mean Value Theorem links average and instantaneous rates under explicit hypotheses. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, some $c$ satisfies $f'(c)=[f(b)-f(a)]/(b-a)$. For $f(t)=t^2$ on $[1,5]$, the average rate is $(25-1)/4=6$, and $f'(t)=2t$ reaches 6 at $c=3$. The theorem guarantees at least one such interior point but does not say every instantaneous rate equals the average or that the point is the midpoint in general. A jump discontinuity or corner can invalidate the theorem's assumptions even when an endpoint average is calculable. In applications, keep units distinct: total distance divided by elapsed time is an average speed, while a derivative of position is signed instantaneous velocity; total distance and net displacement can differ.

Related topics

• Tangent Line
• Derivative
• Graphs of Functions