Some historical approaches to calculus:
- symbolic (formal) (Leibniz)
  This is what's actually taught in US high schools these days:
  it's formal manipulations.
  This lets you do stuff like $\int \sin x$ very easily, but you are
  completely lost when facing $\int e^{x^2}$
  Only useful for elementary functions, and falls apart even then.

- infintesimal (17th century vagueness -> 20th century model theory)
  "fuzzy nonsense", which can be made to make sense using logic, and
  really captures the intuition.

- power series (Newton -> Weierstrass)
  Very powerful, and leads into complex analysis, but often ignores
  delicate questions of convergence and is only useful for analytic
  functions.

- geometric/physical (Newton; intuitive)

19th century:
- real analysis, Bolzano/Cauchy
  Hard ass: where all the proofs come from.
  powerful, and leads to 20th century analysis and deeper
  understanding of function spaces (e.g., nowhere-differentiable
  functions)
  Unfortunately, the -worst- exposition for beginners.
