Rate of Change

Consider the function y = f(x) and consider two points on the x-axis "a" and "a + h", with "h" being a real number that corresponds to the increase of x (Δx).

Rate of Change

The rate of change of a function on the interval [a, a + h], denoted by Δy is the difference between the ordinates corresponding to points on the x-axis, a and a + h.

Δy = [f(a+h) − f(a)]

Average Rate of Change

Average rate of change in the interval [a, a + h] is represented by Rate of Change or Rate of Change, and is the quotient between rate of change and the amplitude of the interval considered on the horizontal axis, h or Δx. It can be written as follows:

Rate of Change

Geometric Interpretation

The previous expression coincides with the slope of the secant line to the function f(x), that passes through the points P and Q (represented on the graph above) which are represented on the x-axis as a and a + h.

Rate of Change

In the triangle PQR, we can see that:

Rate of Change

Examples

Calculate the average rate of change of the function f(x) = x2 − x in the interval [1,4].

Rate of Change Exercise

A stock market index increased from 1,350 to 1,510 points in one year. Find the average monthly rate of change.

Rate of Change Exercise