Mythic Logos

GeoGebra is an awesome program.

I plan on doing something with stuff I am working on, but just to give a taste: Using GeoGebra, one can setup very easily a quadratic approximation to a function at a point. Then drag the point on the function showing how the quadratic approximation changes. In particular, through an inflection point, one changes concavity and you can see it beautifully as well as seeing the quadratic become increasingly linear locally.

And then one can implement a quadratic version of Newton’s method for finding maxs/mins. Do a quadratic approximation, get the max/min of that parabola, descend to the function, and iterate. Works beautifully locally, but one can easily see where and why bad stuff starts to happen. Never seen it in a textbook, probably because computations are laborious. But with GeoGebra, it is trivial, particularly with their really neat tool creator. You go through the steps once, and then you make a tool that will mimic that, producing just the output that you want. So one can do several iterations easily. And then one can drag the original point around and see how the iterations change. Absolutely amazing. And yes, I am under a deadline to produce a lecture and this is seriously distracting me.

Newton’s Method I found was always terrible for a demonstration because of how quickly it converged. Or if it failed, how crazy it went. Getting a good view is exceedingly difficult. But the quadratic seems very tame and easy in comparison.

Brilliant.

Here is an attempt to display it: Quadratic Newton

And here is one on integration and the Fundamental Theorem of Calculus Integration