Cantor set applet

I made this Cantor set applet for my Real Analysis class. It is nothing fancy, but it saves me from drawing it on the board. Advertisements

Applet to illustrate the epsilon-delta definition of limit

Here’s a GeoGebra applet that I made for my Real Analysis class. It can be used to explore the definition of limit: Definition. The limit of as approaches is , or equivalently if for any there exists such that whenever , it follows that .

The Japanese theorem for nonconvex polygons

A couple of years ago I wrote blog posts about two beautiful theorems from geometry: the so-called Japanese theorem and Carnot’s theorem. Today I finished a draft of a web article that looks at both of these theorems in more detail. It contains all that you could want—connections between these theorems, generalizations of them, and consequence of…

An Euler line geogebra applet

Lately I’ve been thinking a lot about Euler and his many contributions to mathematics. So, just for fun I decided to make a Geogeba applet showing the “Euler Line.” In 1763 Euler proved that three different centers of a triangle—the centroid, the orthocenter, and the circumcenter—are collinear. This line is called the Euler line. The centroid…

Three applets for linear algebra or multivariable calculus

This semester I’m teaching two sections of Calculus III (multivariable calculus) and I happen to be teaching the first four weeks of Linear Algebra. The first couple weeks of both courses cover properties of vectors in Rn. (Of course, just to confuse the instructor and the students who happen to be in both classes, the…

The relative sizes of the stars and planets

My colleague sent me this link which shows the relative sizes of the planets in our solar system and some of the brightest stars in the sky. Not only does it make the Earth look small, it makes our sun look small. Pretty amazing! Just for fun I decided to create an interactive GeoGebra applet…

Geogebra applet for families of discrete dynamical systems

As I mentioned recently, I taught the last two weeks of my colleague’s differential equations course. The topic was discrete dynamical systems. I posted links to a few Geogebra applets that I made, namely, applets for illustrating one-dimensional dynamical systems and an applet to generate cobweb plots for the logistic map. This is my third and…

Cobweb plots for the logistic map: a Geogebra applet

A few days ago I posted Geogebra applets illustrating discrete dynamical systems. I was using these in a differential equations lecture that I gave. In the next lecture I showed the students how to draw cobweb plots for 1-dimensional discrete dynamical systems. A discrete dynamical system is a function in which the range is a…

An applet illustrating a continuous, nowhere differentiable function

Open any calculus book and you will find a discussion about how differentiability implies continuity, but continuity does not imply differentiability. The absolute value function is the standard example of a continuous function that is not differentiable (at ). The inquisitive student may ask: how bad can continuous, nondifferentiable functions get? Can we make a…

Applet for discrete dynamical systems

I’m teaching the last two weeks of one of my colleagues’ Differential Equations course. I’m leading the class through a chapter on discrete dynamical systems. In preparation for the first lecture I created a couple of java applets using Geogebra. I thought others might be interested in them, so I’m linking to them here. The…

An applet for teaching the limit of a sequence

I’m currently teaching real analysis. Right now we’re discussing limits of sequences. The definition is: The limit of a sequence is (or converges to ) if, given any , there exists a natural number such that for all . I used GeoGebra to create the following applet, which illustrates the definition of a limit. (Clicking…

Carnot’s Theorem

Here’s a neat theorem from geometry. Begin with any triangle. Let R be the radius of its circumscribed circle and r be the radius of its inscribed circle. Let a, b, and c be the signed distances from the center of the circumscribed circle to the three sides. The sign of a, b, and c…