-short- overview of calc

ABC: diff/int/taylor
underlying theme: limits
big thread: "elementary transcendental functions",
  IE, exp/log, trig (sin/cos/tan, etc.)
small thread: number systems

In a quip:
calculus class:
- discusses infinity: infinitly big and infinitly small
- introduces two operations on functions,
  differentiation and integration (D \& I)
- proves properties of quite -general- functions,
  mostly but not exclusively using D \& I
- expands your notion of what a "function" is
  + polynomials
  + rational functions
  + elementary transcendentals
  + integrals
  + power series


Why -don't- you get this feeling (of coherence) from a usual calculus
class?

2 main reasons:
1: "diamond shards"
 high school textbooks are designed by commitee to please the textbook
 requirements of Texas/California, as these are big states
 and have -statewide- (instead of local) textbook requirements.
 
 They hence read like a checklist of topics b/c -that's how they're chosen-:
 commitees come up with a checklist of topics, and textbooks have to cover them.
 (this is true not just for math textbooks)

 Also if a new topic is introduced, it's a huge amount of work to
 weave it through the text, and instead it's just dumped into a new
 section.

 Thus instead of covering a few unified theories (``diamonds''), it
 covers \emph{shards}: scraps from various theories.
 Each theory is beautiful in itself, but if you only see a shard, you
 can't appreciate it.

2: stupid problems
 They're designed for low-paid and unqualified calc teachers, so they
 can just assign lists of problems from them, and so that students can
 do them w/o thinking.

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Some nice EGs of applications of hard-to-apply stuff:
- fundamental theorem:
  illustrate by:
  - $x^n$: the cube cut into pieces
  - $\sin$ and $\cos$: integrate velocity and get position
    (coordinates on circle)
    (indeed, the velocity/position (aka, displacement) POV
     makes the Fund. Thm -obvious-)
    (also give the baby stoke's theorem POV: it's flow)
- mean value theorem: liouville's approximation!
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A historical perspective

By this I don't just mean "well Weierstrass introduced this concept":
I mean give the historical -context- and -motivation-.

EGs
- Uniform convergence was motivated by Cauchy proving -wrong- theorems,
  like that the pointwise limit of continuous functions was continuous.
- sup/inf, as distinct from max/min,
  was advocated by Weierstrass in the context of variational problems:
  how do you know that the inf is actually achieved?
  (by a function with the characteristics you want)
[also logs, obviously]

BTW, the historical POV helps explain why certain topics have such
prominence -- they're often arbitrary and boring in themselves, but
arise as important concepts -in a more sophisticated context-.

Now:
- without this context, they really are boring
- you can study them, but who cares?

...and this problem is -ubiquitous- in math instruction.
It happens because simple notions are important in fancy areas.
This does -not- mean that you should bother teaching these simple
notions out of context!

EGs include:
- compactness, sup/inf, uniform convergence,
  pointwise definitions of continuity
  (this is interesting once you learn about nowhere differentiable
   functions, which are early fractals)

More fancily:
- Sobolev spaces, functional analysis generally
  (very interesting in context of PDEs, meaningless abstraction
   outside it)
