1C. Average Rate of Change
Instructions:
- Use the input box for to define a function. Use the input boxes for and to define the interval over which you want to calculate the average rate of change of .
- Use the slider tool for and then for to observe the visualization for the average rate of change of between and . Notice the average rate of change (ARoC) calculation displayed on the left.
- Use the checkbox for "Why Average?" to visualize why this rate of change is called an average. Use the slider tool for to subdivide the interval between and into pieces. The sub-intervals have their own slopes. The average of all these slopes is given by .
Average Rate of Change
For linear functions, slope is used to quantify how fast the function changes. This slope is the same between any two points on the line. For nonlinear functions, the slope between any two points may be different. The function can change slowly at some points and more quickly at others. It can increase in some places and decrease in other places.
The average rate of change of a function between and is the slope of the line that connects the points and . The slope formula gives:
Observation 1: Notice that an average rate of change calculation requires two points. If those points are too far apart, then the average rate of change doesn't do a good job of approximating how the function is changing. But, if the two points are close enough together, then average rate of change can give you a pretty good idea about how steep the graph is at that point.
Observation 2: This rate of change is called an average for a reason. If you subdivide the interval, and calculate these average rates of change over smaller intervals, each average rate of change is a better approximation (because the two points are closer together). If you take all of these rates of change over smaller intervals, their average is equal to the average rate of change over the larger interval.