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…

Curves with Extrema?

We spend a lot of time in calculus studying curves. We look for maximums, minimums, asymptotes, end behavior, and on and on, but what about in “real life”? For some time I’ve been trying to find a real situation determined or modeled by a non-trigonometric curve with more than one extreme value. I’ve not been…

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…