27991 - Mathematical Analysis T-1

Learning outcomes

After completing the course the student has a basic knowledge of the differential and integral calculus for functions of one variable, and the fundamentals of linear algebra in space euclideo. The mathematical instruments of this course  also formalize problems of other disciplines. 

Course contents

Limits and continuity. Definition of sequence of real numbers convergent and divergent. The theorems on limits of sequences: uniqueness of the limit theorem the two policemen. Monotone sequences: definition and their limits. The Euler number. Composition of functions, invertible functions. Definition of limit for functions.. Definition of continuous function of one variable. Continuity of the composition of two continuous functions. Weierstras'ss, the zeros and the intermediate value theorems. Limits from right and left. Monotone functions: definition and their limits. The circular functions.
Differential calculus. Definition of differentiable function and of derivative of a function. Geometric meaning. The theorems of the mean value. Higher order derivatives. Taylor's formula. Local extrema: definitions, necessary conditions (Fermat's theorem), sufficient conditions. Convex functions.
Integral calculus. Definition of Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem. Classes of integrable functions. The fundamental theorem of calculus. Integration by substitution and integration by parts.
Linear algebra The vector space R ^ n: vector operations. Linear combinations of vectors and linear independence. Subspaces. Euclidean space R ^ n: scalar product of two vectors. Orthogonal vectors. Norm of a vector and property. Distance between two points. The vector product.

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

The course provides theoretical lessons supported by exercises who aims to help the students to become familiar with and master the tools and mathematical methods introduced during the lectures.

Assessment methods

The exam consists of a written part and an oral part.

Achievements will be assessed by the means of a final exam. This is based on an analytical assessment of the "expected learning outcomes" described above.

In order to properly assess such achievement the examination is composed of  two different sections: a written session, which consist of  6/7 exercises and  an oral session (if the written test is sufficient).

The oral session, consists of three questions at least about the program (definition and theorem).

Higher grades will be awarded to students who demonstrate an organic understanding of the subject, a high ability for critical application, and a clear and concise presentation of the contents.

To obtain a passing grade, students are required to at least demonstrate a knowledge of the key concepts of the subject, some ability for critical application, and a comprehensible use of mathematical language.

A failing grade will be awarded if the student shows knowledge gaps in key-concepts of the subject, inappropriate use of language, and/or logic failures in the analysis of the subject.

Teaching tools

Exercises , notes and and other materials available online at: http://www.dm.unibo.it/~manfredi/didattica.html and at AMS campus https://campus.unibo.it/

Links to further information

http://www.dm.unibo.it/~manfredi/didattica.html

Office hours

See the website of Maria Manfredini