Other Topics (Type 7)

Any topic in the Course and Exam Description may be the subject of a free-response question. The two topics listed here have been the subject of full free-response questions or major parts of them. Implicitly defined relations and implicit differentiation These questions may ask students to find the first or second derivative of an implicitly…

From One Side or the Other.

Recently, a reader wrote and suggested my post on continuity would be improved if I discussed one-sided continuity. This, along with one-sided differentiability, are today’s topic. The definition of continuity requires that for a function to be continuous at a value x = a in its domain  and that both value are finite. That is, the…

Seeing Difference Quotients

Third in the graphing calculator series.  In working up to the definition of the derivative you probably mention difference quotients. They are The forward difference quotient (FDQ):  The backwards difference quotient (BDQ): , and The symmetric difference quotient (SDQ):  Each of these is the slope of a (different) secant line and the limit of each as…

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…

Tangents and Slopes

Using the function to learn about its derivative.  In this post we will look at a way of helping students discover the numerical and graphical properties of the derivative and how they can be determined from the graph of the function. These ideas can be used very early, when you are first relating the function…

Discovering the Derivative

Discovering the Derivative with a Graphing Calculator This is an outline of how to introduce the idea that the slope of the line tangent to a graph can be found, or at least approximated, by finding the slope of a line through two very close points in the graph.  It is a set of graphing calculator…