The average rate of change of a function over an interval from x1 to x2 is calculated by taking the difference in the function’s values at these two points and dividing it by the difference in the x-values. Mathematically, it can be expressed as:
Average Rate of Change = (f(x2) – f(x1)) / (x2 – x1)
This formula tells us how much the function f(x) changes, on average, for each unit change in x, which is the main idea behind the average rate of change.
To graphically represent this, you would plot the points (x1, f(x1)) and (x2, f(x2)) on a coordinate plane. The average rate of change corresponds to the slope of the line that connects these two points. This line is often referred to as the secant line. The slope of the secant line gives a visual and numeric representation of the average rate of change over the specified interval.
Regarding the units, the average rate of change has units that are derived from the units of the function and the independent variable. For example, if f(x) represents distance in meters and x represents time in seconds, then:
- The unit for the change in f(x) would be meters (m).
- The unit for the change in x would be seconds (s).
Therefore, the average rate of change would have units of meters per second (m/s). This means that for every second that passes, the distance changes by a certain number of meters on average.