I think the geometric principles of
global/local/infintesimal
should be stressed;
it's a good organizing principle for understanding many principles

The other main calc concept is the relationship between a function $f$
and its derivative $f'$.

What is -precisely- meant by ``infintesimal''?
The infintesimal data -means- the jet, i.e.,
  $f(a),f'(a),f''(a),\dotsc$
The local data about $a$ -means- the germ, i.e.,
  the behaviour on an arbitrarily small neighborhood
  (germs are elements of the stalk of the relevant sheaf of functions)
The global data is the whole function $f$

key EGs --------------
infintesimal -----
continuity/differentiability are -infintesimal- properties;
roughly, if it has $\lim_{x\to a}$ in the definition, it's probably an
infintesimal property.

local ---------
being a local max/min is a local (but not infintesimal) property

global --------
being constant is a global property

-------------------------------

EGs of theorems:
- 1st, 2nd derivative tests
  1st: local $f'$ -> local $f$
  2nd: infintesimal $f',f''$ -> local $f$

- if your derivative vanishes everywhere (infintesimal info about $f'$
at each point), then the function is constant (global info about $f$)

IVT/MVT is global -> infintesimal
  (if something is true over an interval, then at some point
   it must be true)
NB: IVT is not true for maps $\bR^2 \to \bR^2$, as it's not in the
order topology/deleting a point doesn't make the space disconnected.

- increasing/decreasing
  these are $f$ -> $f'$ and $f'$ -> $f$

-------------------------------------
Generalizations:
brain-dead calculus is: single-variable, all linear
Essentially:
- a single
- linear polynomial
- in one variable

The natural generalizations (as is always the case with polynomials) are:
- more variables
  (more dimensions in domain)
- more polynomials
  (more dimensions in target)
- higher degree
  (Taylor series, and curvature)

The first two are easiest:
- derivative of a function $\bR^2 \to \bR$
- derivative of a path: $\bR \to \bR^2$
...and really clarify things.

The third is trickier, and involves asymptotics, but clarifies what's
really meant by ``(best) local linear approximation''
