MATHEMATICAL ANALYSIS 1
Learning outcomes of the course unit
Knowledge of arguments introduced in the course
Elementary algebra, elementary equation and inequality, elementary logic
Course contents summary
Integral and differential calculus
PREREQUISITES: elementary algebra, trigonometry, analytic geometry, rational powers, exponentials and logarithms; elementary functions.
LOGIC: propositions and predicates, sets, functions, order relations and equivalence.
NUMERICAL SETS: natural numbers and the principle of induction, combinatorics and elementary probability, integers, rational, real numbers, complex numbers and n-th roots.
REAL FUNCTIONS: extrema of real functions, monotone functions, even and odd functions, powers, absolute value, trigonometric functions, hyperbolic functions, graphs of real functions.
SEQUENCES: overview of topology, sequences and their limits; comparison theorems and algebraic theorems, continuity, monotone sequences, theorems of Bolzano-Weierstrass and Cauchy, key examples, the number of Napier; recursively defined sequences.
CONTINUOUS FUNCTIONS: limits of functions, continuity, first properties of continuous functions, continuous functions on an interval (zeros, intermediate values); Weierstrass theorem, uniformly continuous functions, theorem of Heine-Cantor; lipschitz functions; infinitesimals.
DERIVATIVES: definition of the derivative, the first properties; algebraic operations on derivatives, derivatives and local properties of functions; theorems of Rolle, Lagrange, Cauchy; indeterminate forms, de l'Hôpital theorem, Taylor's formula and various remains, asymptotic developments; functions convex qualitative study of functions.
INTEGRATION: construction and first properties of the integral, primitive, fundamental theorem of integral calculus, methods of integration, improper integrals, integration of rational functions.
SERIES: standard definition and first properties; convergence criteria set in terms of non-negative; series in terms of alternating sign.
• Tom APOSTOL “Calcolo vol. 1 - Analisi 1” Boringhieri
• Enrico GIUSTI “Analisi matematica vol.1” Boringhieri
• Walter RUDIN “Principi di Analisi Matematica” Mc Graw-Hill
• Emilio ACERBI, Giuseppe BUTTAZZO “Analisi matematica ABC.
1-Funzioni di una variabile” Pitagora
• Emilio ACERBI, Giuseppe BUTTAZZO “Primo corso di Analisi
Testi di esercizi
• V. DEMIDOVICH “Esercizi e problemi di Analisi Matematica”
• Enrico GIUSTI “Esercizi e complementi di analisi matematica vol.1”
• BUTTAZZO, GAMBINI e SANTI “Esercizi di analisi matematica
• Domenico MUCCI “Analisi matematica - Esercizi 1. Funzioni di
una variabile” Pitagora
Frontal lesson, exercise to little groups, use of tablet PC
Assessment methods and criteria
A written examination (in two parts) and then, for those who obtained an vote adequate, an oral exam
The lessons, in pdf format, can be downloaded from my webpage.