A: derivatives
B: integrals
C: Taylor series

themes:
- limits
  + continuity
  + defn of derivative
  + Darboux defn of integral :: nets, and limits over weird indexing sets
  + sequences/series: summation
- functions
  + black boxes
  + something you can calculate with
  + particular functions
    - exp/log
    - trig
    (- inverse trig, hyperbolic...)
  + classes of functions
    - diff, $C^1, C^\infty, C^\omega$, poly, rational...
- numbers
  (real and complex)
- proof/rigor/math expression (oral and written: hear/speak,
  read/write: math qua language/discourse)
- misc
  countable/uncountable: higher infinities

--------------
A: derivatives
- computations
- definition
- theory
  - MVT, etc.
- applications
  + motion
  + graph sketching
  + optimization
  + growth \& vanishing of functions/L'Hopital's rule
- examples
  + $\chi_\bQ$: nowhere continuous
    (mention nowhere differentiable function, but for later)
  + $x^2 \sin (1/x^2)$: diff but not $C^1$

B: integrals
- computations
  + antiderivatives
  + parts/substitution
  + tricks: partial fractions, trig subs, etc.
  + numerical
- definition
  ...and deeper understanding?
- theory
  + fundamental theorem
    ...theoretical importance
- applications
  + motion again
  + prove L'Hopital's (using Fundamental Theorem)
  + volume (discs/washers/shells)
- examples
  + $\chi_\bQ$: not Riemann integrable

C: taylor series
(2 divergent tracks:
 - real analysis, and detailed study of function spaces
 - complex analysis, and analytic functions)

- computations
  + convergence tests for series
  + radius of convergence of power series
  + formal power series manipulations
  + norms of function
  + convergence in: pointwise/norm topologies
- theory
  + analytic functions over $\bC$ and radius of convergence
  + Euler's formula and $\exp$ over $\bC$
  + analytic continuation (changing point of expansion),
    monodromy/residues, and covering spaces
  + convergence of Taylor series
    ...and formal manipulations are legit
  + Taylor series is infintesimal approximation
- applications
  ??? in physics we disregard higher order terms...
    (first few Taylor as ``better than tangent'')
- examples
  + geometric series, $\exp, \sin, \cos$ series, $\log, \arctan$
  + nowhere diff function
  + failure of Taylor series:
    + $e^{-1/x^2}$
    + bump functions
  + $\tan$ Taylor series

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The `C' quarter often feels a bit out of place:
`A' and `B' are clear: derivatives and integrals, two sides of a coin.

`C' is more subtle:
Taylor polynomials give an infintesimal meaning to higher order
derivatives (as quadratic etc. approximations), and are the first
example of functions as limits.
They also allow another look at limits (since they're rarely covered
  well in the `A' quarter).
In retrospect (which is rarely given), they explain continuity,
differentiability ($0$th and $1$st order approximation) and lead into
higher dimensions (local approximation by higher dimensional linear
functions, and curvature) and real analysis (function spaces),
and complex analysis (analytic functions).
