Ch 0 Introduction
1) The exhaustion method of the Greeks and integration:
the length of a circle, the area of a disc, the volume of a ball,
the Archimedes discovery (why he cried "eureca!").
2) The area under the curve y=x^n.
3) Tangents and differentiation, the derivative of x^n.
4) Geometric series and the area under y=x^d (after Fermat).
5) The area under y=1/x and the natural logarithm.
6) Integration and differentiation: Newton-Leibniz theorem.
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: an algebraic approach
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, integration rules,
acceleration, free falls, conservation of energy, the leaky bucket.
4) A heuristic treatment of sin', cos', ln and exp, examples
of exp growth and decay, harmonic and damped oscillations,
a rope sliding off a table etc.
5) More integration techniques.
6) Newton's 2-nd law, gravitation and Kepler's laws
for a circular orbit (maybe an elliptic orbit too)
Ch 3 Theory of Calculus via uniform Lipschitz estimates
1) Differentiation as approximation: why a tangent looks
like a tangent, the Increasing Function Theorem
2) Uniform Lipschitz continuity (ULC) of the derivatives,
f'>0 --> f increases, f'=0 --> f=const. etc.
3) Differentiation of sin and cos: a rigorous proof.
4) Higher derivatives, convexity, concavity, max, min, graphs,
information and max area of a polygon inscribed in a circle.
5) Newton's method: an example (x^2=2), f'>0 & f''>0 --> convergence,
dealing with deterioration of convergence for multiple roots.
6) Definite integral via antiderivatives, IFT and area, existence
of antiderivatives via approximation of area under the graph of a ULC
function.
7) Taylor formula and Taylor series, relation of exp, sin and cos.
8) Numerical integration and numerical solution of ODEs.
9) ULC differentiation as division of ULC functions. Using Holder
estimates to get a theory with a wider sweep.
Ch 4 Sequences, series, convergence, continuity, limits
1) Zeno paradox and the geometric series, the Neumann series
and another look at 3.7. Fibonacci numbers, their asymptotic and
an application to the Euclidean algorithm.
2) A closer look at the series for sin, cos and exp, convergence of
the series, the uniform convergence, convergence of power series.
3) Convergence of numerical methods (a simple example treated in
detail and some hints about the general theory).
4) Convergence of sequences and the Cauchy criterion, completeness.
5) Continuity and limits. Classical differentiation as division of
continuous functions, derivatives as limits. Continuity in 2 variables
and smoothness.
Ch 5 Real numbers, what are they, really?
0) Decimals and a minimalistic theory of reals.
1) Rational approximations of real numbers.
2) Real numbers as Cauchy sequences of rationals.
3) Completeness of the reals.
Ch 6 Infinitesimals made easy: the ideas of Leibniz salvaged.
0) Infinitesimals: the idea and how it was used.
1) An informal view of infinitely small sequences.
2) The arithmetic of the sequences: the natural extension of the real
functions.
3) How to get rid of the zero divisors (the ultrafilter construction).
4) The standard part of the finite hyperreals.
5) Continuity and differentiation through hyperreals.
6) Differentiability on a hyperreal interval and smoothness.