How is average rate of change related to slope?
The slope is the average rate of change about a point as the interval over which the average is being taken is reduced to zero.
In the set of graphs below,
- f(x) is shown in red
- the average rate of change between two points on either side of x is shown in orange (the average rate of change is the change in f(x) divided by the change in x between the 2 points i.e. it is the slope of the orange line.
- the slope of f(x) at x is indicated by the blue line.
Notice how the rate of change between the two points (the orange line) on either side of x becomes closer to the slope of x (the blue line) as the distance between the two points is reduced.