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…

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:…

Derivative Applications – Graphing

Graphing and the analysis of graphs given (1) the equation, (2) a graph, or (3) a table of values of a function and its derivative(s) makes up the largest group of questions on the AP exams. Most of the other applications of the derivative depend on understanding the relationship between a function and its derivatives.…

Graph Analysis (Type 3)

The long name is “Here’s the graph of the derivative, tell me things about the function.” Most often students are given the graph identified as the derivative of a function. There is no equation given and it is not expected that students will write the equation (although this may be possible); rather, students are expected…

How to Tell your Asymptote from a Hole in the Graph.

The fifth in the Graphing Calculator / Technology series (The MPAC discussion will continue next week) Seeing discontinuities on a graphing calculator is possible; but you need to know how a calculator graphs to do it. Here’s the story: The number you choose for XMIN becomes the x-coordinate of the (center of) the pixels in the…

Determining the Indeterminate

The other day someone asked me a question about the implicit relation . They had been asked to find where the tangent line to this relation is vertical. They began by finding the derivative using implicit differentiation: The derivative will be undefined when its denominator is zero. Substituting y = 0 this into the original…