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## Subject: MATHEMATICAL ANALYSIS A (A.A. 2020/2021)

### degree course in MATHEMATICS

Course year 1 15 Unit Analisi matematica A - mod 1 Basic Mathematics (lesson) TAF: Basic compulsory subjects SSD: MAT/05 CFU: 9 Teachers: Carlo 6/8/1962 BENASSI Unit Analisi matematica A - mod 2 Basic Mathematics (lesson) TAF: Basic compulsory subjects SSD: MAT/05 CFU: 6 Teachers: Michela ELEUTERI oral final vote Italiano

### Contents for i semestre

Contents download ### Overview

This course gives the basic knowledge on numerical sequences and series, on the differential calculus for real functions of one real variable.

- Sets and main operations on sets.
- Integer, rational and real numbers and their main properties
- Algebraic equations and inequalities.
- Exponential and logarhitmic equations
- Trigonometric functions and trigonometric identities.
- Analytic geometry (equations of lines and conics).

### Course contents

Real numbers. Bounded sets. Sup and Inf, axiom of completeness. Numerical sequences. Limit of a sequence. Monotone sequences. Bolzano-Weiestrass theorem. Numerical sequence: simple and absolute convergence, convergence criteria. Metric and normed vector spaces. Limit of functions of one real variable. Continuous functions. Bolzano's theorem, Weierstrass theorem. Derivatives, Rolle's and Lagrange's theorems. Higher order derivatives and applications. De l'Hopital's rule. Taylor's polynomials and formulas.

### Teaching methods

By written and oral examination.

### Assessment methods

By written and oral examination.

### Learning outcomes

Knowledge and understanding: Through lectures, material and educational activities carried out during the course, the student will acquire the basic knowledge differential and integral calculus.

Applying knowledge and understanding: Through classroom exercises, support activities and individual work,
the student will be able to model and solve mathematical problems by analytical tools.

Making judgments: At the end of the course, the student will be able to verify the contents presented through rigorous arguments and to recognize independently resolution methods appropriate to different types of problems.
Communication skills: Thanks to discussions with the teacher and the final interview at the end of the course the student will be able to report orally on the arguments presented in the course in a appropriate technical language and a correct mathematical formalism .
Learning skills: Development of skills of independent learning and study of side arguments to those presented in the course.

E.Acerbi, G.Buttazzo - Primo corso di analisi matematica - ed. Pitagora

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

### Contents for ii semestre

Contents download ### Overview

ANALISI MATEMATICA A/ANALISI MATEMATICA (CDL MATEMATICA E FISICA)

This course gives the basic knowledge on differential calculus for real functions of two or more variables, on the integral calculus for real functions of one real variable and for ordinary differential equations of the first order.

COMPLEMENTI DI ANALISI MATEMATICA (CDL IN INFORMATICA)

This course gives the basic knowledge on differential calculus for real functions of two or more variables, on the integral calculus for real functions of two real variable and for ordinary differential equations of the first and second order.

ANALISI MATEMATICA A/ANALISI MATEMATICA (CDL MATEMATICA E FISICA)

Differential calculus for functions of one variable

COMPLEMENTI DI ANALISI MATEMATICA (CDL INFORMATICA)

Differential and integral calculus for functions of one variable

### Course contents

ANALISI MATEMATICA A/ANALISI MATEMATICA (CDL MATEMATICA E FISICA)

Riemann integral. The fundamental theorem of the calculus. Generalized integrals. Ordinary differential equations of first order: generalities, linear and separable equations. Optional: linear ordinary differential equations of second order.
Real and vector valued functions of two or more variables. Limits, continuity, partial derivatives. Differentiability and directional derivatives. Examples and counterexamples. Taylor formula. Hessian matrix. Free and constrained optimization. Dini's Theorem (implicit function theorem). Optional: constrained optimization: Lagrange multipliers method; metric spaces.

COMPLEMENTI DI ANALISI MATEMATICA (CdL INFORMATICA)

Ordinary differential equations of first order: generalities, linear and separable equations. Linear ordinary differential equations of second order.
Real and vector valued functions of two or more variables. Limits, continuity, partial derivatives. Differentiability and directional derivatives. Examples and counterexamples. Taylor formula. Hessian matrix. Free optimization. Optional: Dini's Theorem (implicit function theorem); constrained optimization.
Basic results about integration in two variables

### Teaching methods

More details will be given according to the evolution of the sanitary emergency. For sure lessons will be given also remotely

### Assessment methods

ANALISI MATEMATICA A/ANALISI MATEMATCA (CDL MATEMATICA E FISICA) Two written tests: a first test is concerned with integral calculus and ordinary differential equations and the second written test is concerned with differential calculus (in several variables) and optimization. It follows then an oral test. Students can get to the oral test only if to both the written tests have obtained a mark equal or greater than 18/30 and if they already passed successfully the oral test of the first modulus. COMPLEMENTI DI ANALISI MATEMATICA (CDL INFORMATICA) Written test. The oral test is optional.

### Learning outcomes

Knowledge and understanding: Through lectures, material and educational activities carried out during the course, the student will acquire the basic knowledge concerning differential and integral calculus and ordinary differential equations.

Applying knowledge and understanding: Through classroom exercises, support activities and individual work,
the student will be able to model and solve mathematical problems by analytical tools.

Making judgments: At the end of the course, the student will be able to verify the contents presented through rigorous arguments and to recognize independently resolution methods appropriate to different types of problems.

Communication skills: Thanks to discussions with the teacher and the final interview at the end of the course the student will be able to report orally on the arguments presented in the course in a appropriate technical language and a correct mathematical formalism.

Learning skills: Development of skills of independent learning and study of side arguments to those presented in the course.