Local Linearization
What is "local linearization?"
As the term states, many functions are "linear" in a very local setting. By that, we mean, that if we zoom in on a particular part of a function, the graph will appear to be linear. This may not happen at every point of the function (for example, the absolute value function will not appear linear at the origin no matter how much we zoom in). By exploring this feature of functions, moving the focal point along the function, and examining a range of functions, we can gain a better understanding of the nature of derivatives.
To use this applet, position your cursor to the right of the two coordinate planes and scroll down. By clicking the ZoomIn button, you can zoom in on the function, with the zoom centered on point B. As you move point B in the left-hand graph, the function in the right-hand graph will remain centered on B. To change the position of point B, slide point A along the x-axis in the left-hand screen. The Reset button will restore the original zoom setting/ The Axes button will toggle the axes on and off.
In the left-hand screen, the Function slider changes the graphed function. A list of those functions with their numbers, is located on the bottom of the right-hand panel. The slope-trace checkbox will turn on and off a point (x,y) where is is the x-value of point B and y is the slope of the function at that point. As you move A along the x-axis to move point B, this slope-trace button will leave traces of its movement so you can see its path.