87295 - Mathematical Analysis T-A

Learning outcomes

Methodological and operational aspects of differential and integral calculus for functions of one real variable.

Course contents

1. LOGIC. Propositions, operations, sets, relations, functions, quantifiers, induction.
2. REAL NUMBERS. Intervals, topology, construction of R, inf and sup.
3. COMPLEX NUMBERS. Definitions, operations, algebraic and exponential forms, modulus and argument, roots, algebraic equations.
   
4. FUNCTIONS. Definition, properties, composition, inversion, sequences, real functions of one variable.
5. LIMITS. Convergent sequences, thereoms about limits of sequences (uniqueness, comparison,...) monotone sequences, limits of real functions of one variable, theorems about limits of real functions.
6. CONTINUITY. Definition, theorems (Weierstrass, Bolzano, intermediate values,...), continuity of the composition, change of variables, discontinuities.
7. DIFFERENTIAL CALCULUS. Derivatives and their calculus, mean value theorem, monotonicity test, critical points, higher order derivatives, convex functions, Taylor formula and its applications.
8. INTEGRAL CALCULUS. Riemann integral, preperties (linearity, additivity, monotonicity, mean value,...), primitives, the fundamental theorem, theorems on calculus (by parts, change of variables), generalized integrals, convergence criteria.
9. LINEAR DIFFERENTIAL EQUATIONS. Solutions to first and second order equations, Cauchy problems.

Readings/Bibliography

Lecture notes on Virtuale.

Suggested readings:

• G.C. Barozzi, G. Dore, E. Obrecht. Elementi di Analisi Matematica - Volume 1, Zanichelli (2009).
• M. Bramanti. Esercitazioni di Analisi Matematica 1, Esculapio (2011).

Teaching methods

Frontal lectures to explain the basic notions, examples, and counterexamples.

Exercises solved by the teacher.

Additional exercise sheets for personal study.

Office hours.

Assessment methods

The final examination of the course consists of a written test and an oral test, both compulsory and to be taken in that order.

The written test is aimed at testing the ability to apply theory to solving exercises of the type assigned during the course. Passages must be reported and justified. The use of books, notes or calculators is not permitted; pen and paper only. It lasts three hours.

The written test is passed with a grade greater or equal to 18/30.

If the written test is passed, you can proceed to the oral test. This aims to test your knowledge and understanding of the theory developed during the course. You will be asked to give definitions and examples of concepts and to give statements and demonstrations of theorems. It lasts approximately 30 minutes.

The oral test is to be taken within the same session (winter or summer) as the passed written test. Rejection at the oral examination results in the forfeiture of the validity of the score obtained in the written examination.

Teaching tools

Tutoring (TBA) and office hours.

Additional material will be made available on the Virtuale page of the class.

Office hours

See the website of Nicola Arcozzi

See the website of Mattia Francesco Galeotti