MATHEMATICAL ANALYSIS 1
Learning outcomes of the course unit
The course provides the basic mathematical instruments for a solid comprehension of other the courses
Course contents summary
Functions depending on one variable
Elementary algebraic properties of the real numbers (standard
types of equations and inequations); logic and set theory.
Numerical sets: natural numbers and induction principle;
combinatoric calculus; rational numbers; real numbers and supremum
of a set; complex numbers and n-roots.
Real functions: maximum and supremum; monotonicity; odd and even
functions; powers; irrational functions; absolute value;
trigonometric, exponential and hyperbolic functions; graphs of the
elementary functions and geometric transformations of the same.
Sequences: topology; limits and related theorems; monotonic
sequences; Bolzano-Weierstrass and Cauchy theorems; basic
examples; the Neper number “e”; recursive sequences; complex
Properties of continuous functions (including mean value,
existence of a maximum, Lipschitz continuity); limits of functions
and of sequences of real numbers; infinitesimals.
Properties of differentiable functions (including Rolle, Lagrange,
Hopital theorems); Taylor expansion (with Peano and Lagrange
remainder); graphing a function.
Indefinite and definite integral: definition and computation
(straightforward, by parts, by change of variables); integral mean
and fundamental theorems; Torricelli theorem; generalised
integrals: definition and comparison principles.
Numerical series: definition, convergence criteria, Leibniz and
All statements are rigorously proved.
As to the theory:
E. ACERBI e G. BUTTAZZO: "Primo corso di Analisi matematica", Pitagora editore, Bologna, 1997
As to the exercises
D. MUCCI: “Analisi matematica esercizi vol.1”, Pitagora editore, Bologna, 2004
E. ACERBI: "Esami di Analisi Matematica 1", Pitagora Editore, Bologna, 2012
A. COSCIA e A. DEFRANCESCHI: "Primo esame di Analisi matematica", Pitagora editore, Bologna, 1997
Lectures in classrom.
in smaller groups of students.
Assessment methods and criteria
The cross-examination consists in a written text divided into two parts followed by a colloquium.