Local Linearity

If you use your calculator or graphing program and zoom-in of the graph of a function (with equal zoom factors in both directions), the graph eventually looks like a line: the graph appears to be straight. This property is called Local Linearity. The slope of this line is the number called the derivative. (There are exceptions:…

Difference Quotients

Difference quotients are the path to the definition of the derivative. Here are three posts exploring difference quotients. Difference Quotients I  The forward and backward difference quotients Difference Quotients II      The symmetric difference quotient and seeing the three difference quotients in action.  Showing that the three difference quotients converge to the same value. Seeing Difference…

Tangent Lines

Second in the Graphing Calculator/Technology series This graphing calculator activity is a way to introduce the idea if the slope of the tangent line as the limit of the slope of a secant line. In it, students will write the equation of a secant line through two very close points. They will then compare their…

Concepts Related to Graphs

This and the next several posts will be about graphing and specifically how the function and its first and second derivatives are related. Since I do not intend this to be a textbook I will not be doing textbook stuff. Rather, I hope to add some big picture things on the concepts involved. Hope you find it…

Local Linearity II

Using Local Linearity to introduce difference quotient and the derivative. A good way to introduce difference quotients and derivatives is to write the equation of the “line” you see when you zoom-in on a locally linear function. First: Ask your class to use their calculator or computer grapher to graph a function, say y =…