Interactive mathlets with Sage

“Mathlets”?

Applets for math.

A “mathlet” is a small, interactive applet used to demonstrate or experiment with mathematical ideas. They are often web-based and written in Java.

You can easily make mathlets with Sage's “@interact” feature. And the embeddable Sage cell server means you can put @interacts anywhere — including right into this presentation!

Example: Taylor polynomials

Taylor polynomials for $$\sin(x^2 + 1)\sqrt{x+2}$$:

The sequence

• $$\frac{1}{4} \sqrt{2} \sin\left(1\right) x + \sqrt{2} \sin\left(1\right) + \mathcal{O}(x^2)$$
• $$-\frac{1}{32} \left(\sin\left(1\right) - 32 \cos\left(1\right)\right) \sqrt{2} x^{2} + \frac{1}{4} \sqrt{2} x \sin\left(1\right) + \sqrt{2} \sin\left(1\right) + \mathcal{O}(x^3)$$
• ...
doesn't say much to me.

Making a bunch of drawings is better, but can you make them look good? Make them correct? Convey the idea of a sequence of polynomials converging to the original function? If not...

Taylor polynomials, continued

This is the “canonical” example from the Sage wiki.

Try it yourself

1. Visit the Sage cell server: aleph.sagemath.org, or log into your account on any Sage server.
2. Cut and paste some code from wiki.sagemath.org/interact or these slides.
3. Enjoy!

Thank you

These slides online at goo.gl/Nx9U3

Credits

• Thanks to Jason Grout and his students for the Sage cell server.
• Thanks to Ira Hanson for CSS help.
• The Newton's method code is available from gist.github.com/2469090.
• Dyck/Schröder code (and my paper) is at arXiv:1006.1959.

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

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