MATHEMATICAL ANALYSIS 1
Learning outcomes of the course unit
Knowledge and understanding:
At the end of this course the student should know the essential definitions and results of the analysis in one variable, and he should be able to grasp how these enter in the solution to problems.
Applying knowledge and understanding:
The student should be able to apply the forementioned notions to solve medium level problems, and to understand how they will be used in a more applied context.
The student should be able to evaluate coherence and correctness of the results obtained by himself or offered him.
The student should be able to communicate in a clear and precise way, also in a context broader than mere calculus.
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.
Theory and basic examples:
E. ACERBI e G. BUTTAZZO: "Primo corso di Analisi matematica", Pitagora editore, Bologna, 1997
D. MUCCI: “Analisi matematica esercizi vol.1”, Pitagora editore, Bologna, 2004
Exercises for cross-examination:
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. Laboratory activities 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.