Ch 1 Functions

	1) Examples of functions: x, |x|, constants, domain and range. 
     Operations on functions: +, -, *, /, |.|; polynomials and rational
     functions.
	2) Long division of polynomials by linear functions, roots and
     their number, the vanishing theorem for polynomials,
     extension of functions beyond their original definition domains,
     uniqueness of extension for rational functions, the natural 
     definition domain.

Ch 2 Differentiation and Integration

	1) The derivative: some examples (polynomials, sqrt, 1/x),
     differentiation as making sense out of the undefined expressions,
     uniqueness of differentiation.
	2) The rules of differentiation: linearity, product rule,
     the derivative of a ratio, chain rule, implicit differentiation. 
	2.3**) Antiderivatives and integration, acceleration, free falls,
     conservation of energy, the leaky bucket problem. 
	2.6) A heuristic treatment of sin', cos', log and exp. 
	3) Differentiation as approximation: why a tangent looks
     like a tangent, the Increasing Function Theorem, f'=0 --> f=const.
     	4**) Uniform Lipschitz continuity (ULC) of the derivatives,
     f'>0 --> f increases, Fermat etc. 
	4.5*) Differentiation of sin and cos: a rigorous proof.
	5) Newton's 2-nd law, harmonic oscillations, gravitation and 
     Kepler's laws for a circular orbit.
	6) Higher derivatives, convexity, concavity, max, min, graphs.
	7) Newton's method: an example (x^2=2), f'>0 & f''>0 --> convergence.
	8**) Definite integrals and area (via IFT), 
     integrability of ULC functions. 
	9**) Integrating dx/x, log and exp, e, exp growth and decay,
     information and max area of a polygon inscribed in a circle.
	10*) Taylor formula and Taylor series (these can be done as soon as
     IFT is avalable), relation of exp, sin and cos,
     damped oscillations, a rope sliding off a table etc.
	11*) ULC differentiation as division of ULC functions. Using Holder
     estimates to get a theory with a wider sweep.

Ch 3 Sequences, series, convergence, reals, continuity, limits

	1) Examples: the sequence of roundings of a real, approximations
     of sqrt(2) by Newton's iterations, Maclaurin series for exp and sic,
     the geometric series and the Mercator series for ln, an elementary
     formulation of the Riemann Hypothesis using Liouville and Mobius
     functions. 
	2) Convergence and Cauchy criterion.
	3) Reals as Cauchy sequences of rationals, completeness.
	4) Approximation and continuity.
	5) Continuity and limits. Derivatives and integrals as limits.