Non-calculus stuff done in calc classes:
basic algebraic geometry/ring theory:
- conic sections
- polynomial long division
- partial fraction decomposition

....actually, I'm being unfair: these are important functions...

conic sections can lead to:
 - algebraic geometry
 (e.g., try classifying/talking about degree 3 forms)
 - quadratic forms
 ...and lattices, etc.
 (e.g., over the Reals, just need Sylvester's theorem)
   
...i.e., increase the dimension or degree, or change the field.
classic conic sections are:
- binary quadratic forms over R
(i.e., dim=2, deg=2, field=R)
