Asymptotes of Rational Functions.

Asymptotes of Rational Functions is a GeoGebra construction designed to help people explore the relationship between the equation of a rational function and its graph. You choose the rational function you'd like to explore. Here's a story of something neat that happened when I used it in class.

Intuitive Notion of the Limit - One-Sided Limits

This applet, which is part of Marc Renault's GeoGebra Calculus Applets collection, provides an intuitive introduction to the idea of a limit of a function as x approaches c from the right, left, or both.

The red, green, and blue lines along the x-axis, y-axis and the function which shrink as you decrease δ help with the visualization of motion along the function toward the point where x = c and the corresponding motion toward the limiting values on the axes. Choose from the good library of example functions that are included or enter your own. (I like to start with my own rational function which looks linear but actually has an unseen hole to emphasize from the very beginning the difference between the value of a function and its limit at a point.)

When I used this with a class for the first time, my students asked to see the examples with "Exploding" in the title and they reacted audibly to the motion of the green lines on the y-axis as the function approached its asymptote. My experience has been that when introduced to the idea of limits on a graph there are always a few students who have a lot of difficulty visualizing the various ways in which certain things approach approach other things, and starting with this applet seemed to really help.

Indiana Puzzle Quilt

This applet was inspired by a quilt square my mother-in-law made. It provides a nice way of visualizing the  the sum of an infinite geometric series.
Check out the pattern in some real quilts, too!

Update 1/5/2017: See the Snail's Trail Quilt Square

Constructing the Area of a Circle

A beautifully designed applet that walks you through finding the area of a circle by dividing it up into sectors and rearranging them to form a shape which becomes a rectangle as the number of sectors goes to infinity. I recommend using it in conjunction with Steven Strogatz's fabulous article, "Take it to the Limit". (If you like this, you might also like this applet which approaches the same problem in a slightly different way.)

Turning a Circle into a Rectangle

After reading Steven Strogatz's column Take It to the Limit, I developed this animation (using GeoGebra) of Strogatz's diagram which shows why the formula for the area of circle is what it is, and also provides an approximation for π.