# 31586 - Integrational Elements of Mathematical Analysis and Elements of Probability Calculation T

## Learning outcomes

At the end of the course, after passing the final verification test, the student has the basic knowledge related to the calculation of functions of several real variables (properties, maxima and minima) curves, potentials, multiple integrals, their meaning, solution of some simple types of differential equations. He/She also possesses the elementary notions of probability, with particular reference to some distributions in the continuum (uniform and normal distributions).

## Course contents

General program

Integrational Elements of Mathematical Analysis (60h)

1. Differential equations
Separable differential equations; Linear equations of the first order; Linear equations with constant coefficients of the second order.
2. Curves
Vector-valued functions: limits and continuity; Regular curves and vector differential calculus; Length of a curve; Line integrals of the first kind.
3. Functions of multiple variables
Limits
: Graphs and level sets; Calculation of limits; Continuity and theorems on continuous functions.
Differential calculus: Partial derivatives; Tangent plane; Differential and differentiability conditions; Higher-order derivatives, Hessian matrix, and Taylor's formula.
Free optimization: Fermat's theorem; Study of the nature of critical points: sufficient conditions of the second order. Constrained optimization: Lagrange multiplier theorem.
4. Vector-valued functions of multiple variables
Limits, continuity and differentiability; E.g. coordinate transformations in the plane.
5. Double integrals
Simple domains; Reduction theorems; Change of variables.
6. Vector fields
Line integrals of the second kind, work and line integrals along a closed curve; Gradient and rotor; Irrotational fields and conservative fields.

Elements of Probability Calculation (30h)

1. Probability spaces
Probability spaces; Conditional probability and Independence, Partition Equation, Bayes' formula; Combinatorics.
2. Discrete models
Discrete random variables and main distributions: Bernoulli, binomial, geometric, and Poisson distribution; Cumulative distribution function; Expected value; Variance.
3. Continuous models
Absolutely continuous random variables; Density and Cumulative distribution function; Expected value; Variance. Examples: uniform, normal, and exponential random variables.

• Analisi Matematica 2. Teoria con esercizi svolti
Author: Francesca G. Alessio
Editor: Esculapio
Year: 2020

• Analisi matematica 2
Authors: Marco Bramanti, Carlo D. Pagani, Sandro Salsa
Editor: Zanichelli
Year: 2009

• Esercizi di Analisi matematica 2
Authors: Sandro Salsa, Annamaria Squellati
Editor: Zanichelli
Year: 2011

• Introduzione alla probabilità - con elementi di statistica, 2a edizione
Author: Paolo Baldi
Editor: McGrawHill
Year: 2012

## Teaching methods

Theoretical lessons and exercises (in mixed mode or online only).

## Assessment methods

The exam consists of a written test consisting of exercises and theory questions aimed at verifying the knowledge and understanding of the concepts explained during both modules of the course.

In particular, the student must demonstrate that

• she/he knows and knows how to apply to concrete cases the definitions, the statements of the theorems and the properties presented in class;
• she/he knows and have understood the proofs of some theorems and propositions that will be listed at the end of the course (refer to the file "Detailed program" on the Virtual platform).

SCORE:

• The maximum score achievable with the written test is 33; scores from 31 to 33 correspond to a mark of 30 laude;
• the 33 points are divided into 22 points for exercises and questions related to the Analysis module and 11 points for exercises and questions related to the Probability module;
• the exam is passed if you get an evaluation greater than or equal to 18, of which at least 4 points must have been obtained on the exercises and questions of Probability and at least 8 points on the exercises and questions of Analysis.

The exam may also include an oral test, to supplement the written one, in the following two cases:

• at the request of the student, if she/he has passed the written test (grade greater than or equal to 18);
• at the request of the teacher in case she needs to test further.

The oral exam consists of questions aimed at ascertaining the knowledge and understanding of definitions, properties, statements of theorems and proofs included in the list of those subject to examination (refer to the file "Detailed program"). Any oral test contributes to determining the final grade of the exam together with the written one. In particular, it may also lead to the lowering of the grade obtained in the written test. If the student fails the oral test, she/he has to repeat also the written test.

GENERAL INSTRUCTIONS:

• The exam can only be accessed by registering for the lists on AlmaEsami, in the dedicated time window;
• it is necessary to present yourself for the exam with a university card or other identification document;
• during the exam you can use pens, pencils, paper fr calculations, but it is not possible to use calculators (of any kind), books, notes, smartphones and other paper or electronic media.

Remote mode
(applicable only at the disposal of the University):The exam will be carried out remotely, using the EOL (OnLine Exams), Zoom, Teams platforms.
The general instructions written above apply. In addition,

• the student must have a computer equipped with a microphone and webcam and an internet connection that supports a good audio / video transmission.

## Teaching tools

Other material useful for the preparation of the exam will be made available on the online page of the course on the Virtuale platform.

## Office hours

See the website of Francesca Colasuonno