[list][*]Use the input box for [math]f(x)[/math] to define a function. Use the input boxes for [math]a[/math] and [math]b[/math] to define the interval over which you want to calculate the average rate of change of [math]f(x)[/math]. [/*][*]Use the slider tool for [math]\Delta x[/math] and then for [math]\Delta y[/math] to observe the visualization for the average rate of change of [math]f(x)[/math] between [math]x=a[/math] and [math]x=b[/math]. 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 [i]average[/i]. Use the slider tool for [math]n[/math] to subdivide the interval between [math]x=a[/math] and [math]x=b[/math] into [math]n[/math] pieces. The [math]n[/math] sub-intervals have their own slopes. The average of all these slopes is given by [math]m_{avg}[/math]. [/*][/list]

For linear functions, [b]slope [/b]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. [br][br]The [b]average rate of change[/b] of a function [math]f(x)[/math] between [math]x=a[/math] and [math]x=b[/math] is the [b]slope [/b]of the line that connects the points [math](a,f(a))[/math] and [math](b,f(b))[/math]. The slope formula gives: [br][br][math]ARoC=\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}[/math][br][br][b]Observation 1: [/b]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. [br][br][b]Observation 2: [/b]This rate of change is called an [i]average [/i]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.