Rate (mathematics) - Rate of Change

Rate of Change

A rate of change can be formally defined in two ways:

\begin{align}\mbox{Average rate of change}&=\frac{f(a+h)-f(a)}{h}\\ \mbox{Instantaneous rate of change}&=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\end{align}

where f(x) is the function with respect to x over the interval from a to a+h. An instantaneous rate of change is equivalent to a derivative.

An example to contrast the differences between the average and instantaneous definitions: the speed of a car can be calculated:

  1. An average rate can be calculated using the total distance travelled between a and b, divided by the travel time
  2. An instantaneous rate can be determined by viewing a speedometer.

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