27991 - Mathematical Analysis T-1

Learning outcomes

At the end of the course the student will have the fundamental notions of differential and integral calculus for functions of a real variable, and the fundamentals of linear algebra in Euclidean space. The classical tools of this course find useful applications in other fields.

Course contents

Limits and continuity. Definition of sequences of convergent and divergent real numbers. Theorems about limit of a sequence. Definition of limit for real functions of a real variable. Definition of continuous function of a real variable. Weierstrass's theorems, zeros and intermediate values. Limits from the right and from the left. Monotone functions: definition and their limits.

Differential calculus. Definition of derivable and derivative function of a function. Geometric meaning. The theorems of the average value. Derivatives of higher order. Taylor's formula with the rest of Peano.

Local extremes: definitions, necessary conditions (Fermat's theorem), sufficient conditions. Convex functions.

Integral calculus: definition of Riemann integral. Properties of the integral: linearity, additivity, monotony, mean theorem. Classes of integrable functions. The fundamental theorems of integral calculus. Substitution integration theorem and Part integration theorem.

Generalized integrals. Definition of generalized integral for unrestricted functions or defined on unrestricted intervals. Definition of convergence of an integral. Basic examples: generalized integral of the functions of type 1 (x-c) α on limited and not limited intervals. Criterion of comparison and asymptotic comparison. The absolute integrability and the relationship with the simple integrability.

Linear algebra The vector space R^n: operations between vectors. Linear combinations of vectors and linear independence. Vector subspaces and basic definition of subspaces. Euclidean space R ^ n: scalar product of two vectors. Orthogonal vectors. A vector's standard and property. Distance between two points. The vector product.Matrix: determinant, invertible matrices, rank of a matrix. Linear systems: Rouché-Capelli's theorem, Cramer's theorem.

Readings/Bibliography

Elementi di Analisi matematica uno, Marcellini, Sbordone, Liguori Ed.

Elementi di Analisi matematica, vol 1, Giulio Cesare Barozzi , Giovanni Dore , Enrico Obrecht. Ed. Zanichelli

Esercitazioni di Analisi Matematica 1, Marco Bramanti. Ed. Esculapio.

Introduzione all'algebra lineare, Rita Fioresi , Marta Morigi. Ed. Zanichelli.

Teaching methods

Lectures and exercises in the classroom.

Assessment methods

The examination consists of a preliminary written test and an oral one.

The written test consists of 6/7 exercises related to the arguments of the course. In order to be admitted to the oral examination it is necessary to obtain a sufficient score in the written test.

The written test remains valid for the oral exam in the same examination period.

The oral test follows the written test; it mainly concerns the theoretical aspects of the course. The student must show to know the concepts explained during the course  and how to connect with each other.

Teaching tools

Exercises, notes and other online material on the page https://www.unibo.it/sitoweb/francesca.incensi3/didattica and at https://iol.unibo.it/

Office hours

See the website of Francesca Incensi