Ch 0 Introduction
1) Some history, the exhaustion method of the Greeks,
the area of a disc and the length of a circle.
2) Tangents by example: a circle, y=x^2 and y=x^3 and how to
draw them using a sraight edge and a compass. Generilize to y=x^d.
3) Area under y=x, y=x^2 (induction and recursion as a side line),
the geometric series and the area under y=x^d (after Fermat).
4) Area under y=1/x and natural logarithms.
5) Newton-Leibniz by example (as an emerging pattern).
6) Approximation as the most important idea behind Calculus.
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.
* 3) Antiderivatives and integration, acceleration, free falls,
conservation of energy, the leaky bucket problem.
* 4) A heuristic treatment of sin', cos', log and exp.
Newton's 2-nd law, harmonic oscillations, a rope sliding off
a table, gravitation and Kepler's laws for a circular orbit.
* 5) Differentiation as approximation: why a tangent looks
like a tangent, the Increasing Function Theorem, f'=0 --> f=const,
uniform Lipschitz continuity (ULC) of the derivatives,
f'>0 --> f increases, Fermat, differentiation of sin and cos:
a rigorous proof.
6) Higher derivatives, convexity, concavity, max, min, graphs.
information and max area of a polygon inscribed in a circle.
Taylor formula and series, relation of exp, sin and cos, linear DE's,
damped oscillations, resonance.
* 7) Newton's method: an example (x^2=2), f'>0 & f''>0 --> convergence.
8**) A rigorous treatment of definite integrals and area (via IFT),
integrability of ULC functions via approximation.
9*) 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.