## MATHEMATICAL ANALYSIS

Italian

### Learning outcomes of the course unit

Knowledge and understanding: At the end of this course, the student should know the essential definitions and results, and tools of Calculus (limits, differential calculus and integral calculus for real functions of one real variable, series, ODE ), and he should be able to grasp how these enter in the solution of problems.

Applying knowledge and understanding: The student should be able to apply the aforementioned notions to solve mathematical problems not identical but strictly related to those already encountered, as well as to extrapolate the main results to be analyzed from a collection of data.

Making judgements: The student should be able to evaluate coherence and correctness of the solutions given during the written test, by constructing and developing logical arguments with a clear distinction of assumptions and conclusions; the student should be able to check correct proofs and spot wrong reasonings.

Communication skills: The student should be able to communicate in a clear and precise way, via a correct mathematical language, also through group work.

### Prerequisites

Abilty to handle basic mathematical expressions and to deal with mathematical equalities and inequalities. Elementary functions (powers, polynomials, exponentials, logarithms and trigonometric functions).

### Course contents summary

Basic notions of set theory and mathematical logic. Real numbers. Real functions of a real variable and their properties. Limit, continuity, differentiability, and Riemann integral. ODE.

### Course contents

1. Basic notions of mathematical logic. Basic notions of set theory.

2. Integers, rational numbers, irrational numbers. Upper bound, maximum, least upper bound (supremum). Functions and terminology concerning functions. Composite functions. One-to-one functions and inverses. Elementary functions (absolute value, rationals, exponentials, logarithms, powers, trigonometric functions) and their diagrams.

3. Limit of a function. One-side limit of a function. Properties of the limits of functions. Continuous functions. Theorems concerning continuous functions on an interval.

4. Series: definition and basic properties; convergence criteria for series with non-negative terms; alternating series.

5. Definition of derivative. Derivatives and continuity. Algebra of derivatives. The chain rule. One-sided derivatives and infinite derivatives. Zero derivatives and local extrema. Rolle's theorem. The mean-value theorem for derivatives. Higher order derivatives. Taylor's Formula with remainder. Convexity of a function. Diagram of a function.

6. Definition of the Riemann integral. Linear properties. Integration by parts. Change of variable in a Riemann integral. The integral as a function of the interval. Fundamental theorems of integral calculus.

6. Differential equations. Cauchy problems. Linear first order equations. Separable variables equations. Second order linear equations with constant coefficients. Some applications to concrete models.

### Recommended readings

One may use any good book on Calculus as e. g.,

E. Acerbi, G. Buttazzo: Analisi Matematica ABC vol. 1, Ed. Pitagora, 2003.

M. Bramanti, C. D. Pagani, S. Salsa, "Analisi Matematica 1", Ed. Zanichelli.

P. Marcellini, C. Sbordone, “Elementi di Analisi Matematica 1”, Liguori Ed.

G. Catino, F. Punzo, "Esercizi Svolti di Analisi Matematica e Geometria 1", Esculapio, 2020

### Teaching methods

The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are selected so that the student will be able to solve independently many related problems arising from the theoretical lessons. During the course, the weekly student reception is encouraged for any discussion on mathematical topics and for any individual in-depth analysis. In the case of pandemic emergency, the teaching will go online via Teams streaming.

Further teaching support documents will be shared via the related Elly blog.

### Assessment methods and criteria

Final written test(2h), and (possibly) in an oral discussion.

### Other informations

None