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).
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 or , 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:
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.
In the triangle PQR, we can see that:
Calculate the average rate of change of the function f(x) = x2 − x in the interval [1,4].
A stock market index increased from 1,350 to 1,510 points in one year. Find the average monthly rate of change.