Logic and set theory

Propositions and predicates, logical connectives, set operations and analogies with the logical connectives. Cartesian product, relations, functions and their graphs.

10 hours

The real number system

Axiomatic presentation of operations, ordering, completeness and consequences. Infimum and supremum, absolute value and metric, bounded sets, intervals.

10 hours

Number set theory and combinatorics

Natural numbers and mathematical induction, factorial, binomial coefficients, the Newton formula, combinatorics. Infinite sets, the integer numbers, the rationals, the axiom of Archimedes, density.

10 hours

The complex number system

Algebraic and metric structures, conjugate, modulus and argument, cartesian and polar forms, the De Moivre formula, roots, polynomials and usual complex functions.

10 hours

One variable functions

The elementary functions, infimum, supremum, maxima and minima, bounded, monotone, even, odd, periodic, injective, surjective and one-to-one functions, inverse map, convex, Lipschitz, Hölder functions, graphs.

12 hours

Sequences

Examples, general properties of limits, subsequences, algebraic and order properties of limits, monotonicity and comparison, Cauchy sequences, the Bolzano-Weierstraß Theorem, limit points, limits in metric spaces, neighborhoods, open and closed sets, completeness and compactness.

10 hours

Numerical series

Examples, computation of the sum when possible, convergence tests, absolute convergence, the Leibnitz and the Abel-Dirichlet tests. Real and complex power series and their behavior in the circle and on its boundary, the Abel Lemma.

12 hours

Limits of functions

Definitions, theorems on passing to limit, comparison, limits of common use, existence of the limit for monotone functions, use of sequences, asymptotes, estimates of infinitesimal and infinite order, local expansions and applications.

12 hours

Continuous functions

Properties, connection with monotonicity and convexity, continuous functions on intervals, zeros and continuity of the inverse, continuous functions on compact sets, the Weiersrass theorem, uniform continuity.

12 hours

Differential calculus

Derivative and differential, applications, rules, higher order derivatives, local maxima and minima, differentiable functions on intervals, monotonicity, convexity, Taylor expansion and applications, Taylor series and analytic functions, graph drawing.

12 hours

Integration

The Riemann integral, properties, class of integrable functions, the integral function, the fundamental theorem, primitives, rules of integration, improper integrals, absolute integration.

10 hours