[my thoughts in 2006-dec]
What's the point of calculus & linear algebra for most folk?
It's to learn underlying basic math of several variables,
which is hugely useful.
(Stats also, but the math there is drier and subtler.)

key ideas of calculus:
- increment/cumulate
  simple but underlies much: can and should be done discretely at
  first, and seen as a "change of coordinates" (from position domain to
  velocity + initial position domain, etc.)
- infintesimal/infinity: measure, derivative, integral, limits

practical: approximation and optimization
- infintesimal polynomial approximation
  Taylor POV & polys

- global polynomial approximation
  linear/quadratic etc. regressions

- optimization
  Reduce 'optimize function' to 'solve system of equations'
  1-d (1st/2nd deriv test)
  n-d: lagrange, simplex

functions:
- visualize functions R^n -> R^m
- elementary transcendental functions
  (which feature compact-open convergence of Taylor series;
   however, Taylor series should be about infintesimal approximation
   -first-, about analytic functions -second-)

underlying math: real topology:
- ivt: roots exist, continuity
- max principal (boundedness too): extrema exist, compactness
- limits exist: completeness

Lastly, the real reason most people learn about eigenvalues is
-not- because they have operators that they want to diagonalize,
but b/c they want to do Principal Component Analysis,
and eigenvalues are a very, very confusing way of thinking about it.