I'm not able to change the 3-semester sequence of calculus in the US;
so I'll make 3 parts:
- derivatives
  ...taking derivatives (diff. rules)
  ...applying derivatives to understanding functions:
    - max/min, curve sketching
    - growth/decay (what you -can't- see on a calculator)
- integrals
- series: Taylor and Fourier
  (note: point of "seq. and series" is really -taylor- series)

...together with "appendices":
- basic logic/proof techniques
  (intro to proofs, including stuff like induction, contraposition,
   contradiction, etc.)
  (there exists, for all)
- limits
  fundamental result: if $\epsilon \geq 0$, and it's less than every
  positive number (in particular, $1/n$: clever sequence going to $0$),
  then it's $0$
- continuity
  continuity preserved under algebraic operations, -uniform- limits
  continuity at a point is -weird-
  "uniform -blah- (convergence/continuity) means: forall epsilon exist
  (something: delta, N) s.t. forall x, -splat- is true"
- important functions
  - trig
  - exp/log
  - statistical?
  - others
- foundations
  - set theory
  - numbers:
   - naturals, negative, rationals, reals, complex, quaternionic
     (note how -each- step is essentially -different-)
   - p-adics
   - polynomials, formal power series, laurent series

...an instructor should intersperse the appendices with the main text
as desired/necessary: this can last 3 quarters/semesters, or can be
squished into 2, however you want.
----------------
How to organize the -parts-:

- Motivation
(start with picture: tangent line, area of a parabola, taylor/fourier
series: some finite approximation)
- calculations
(brief: how-to worksheet, no understanding)
(e.g., derivatives table; integration tricks; convergence tests)
- understanding/elaboration/proofs
- applications, more chat...

