Higher order Calc project

[Abelard]

I was wondering if there are any topics for calculus based or advanced high school math project that I can devote my whole semester to at school.

 

[micromass]

It depends of what you know about calculus. Some ideas are:

- polar coordinates and integration of polar curves
- fractional derivatives
- series and Taylor series

All of these should be quite ok for an interested high-school student...

 

[Abelard]

Hmm. Fractional calculus sounds pretty cool. But any cool applications associated with that?

 

[micromass]

Hmm, I don't know any applications of fractional calculus. There might be some, but I think they're very complicated...

If you're looking for applications, then the other two topics sure have a lot of applications. But they might be less cool than fractional calculus

 

[Gib Z]

Fractional calculus is possible for an advanced high school student to "discover", but probably not to "develop" the theory very much. Doing so, at the very least, requires some complex analytic properties of the Gamma function which we use as the meromorphic continuation of factorials onto the complex plane.

If you already know basic calculus, perhaps your project could be to write an essay that places some of those calculus concepts on a firmer, more rigorous groundwork (ie, introduce yourself to basic real analysis). Apostol's Calculus Vol.1 does this quite well.

 

[Abelard]

Is it possible to interpret fractional calculus in a physical sense like the first derivative is the rate of change, but if it's fractional, what does that represent?

 

[Abelard]

http://www.icp.uni-stuttgart.de/Jahresberichte/01/node2.html [Broken]

provides a good detailed explanation of the applications. It doesn't really look too hard though.