Teachings
IN/0186  MATHEMATICAL ANALYSIS 1
Academic Year 2020/2021
Free text for the University
 Professor

GIUSEPPE VIGLIALORO (Tit.)
 Period

First Semester
 Teaching style

Convenzionale
 Lingua Insegnamento

ITALIANO
Informazioni aggiuntive
Course  Curriculum  CFU  Length(h) 

[70/72] CIVIL ENGINEERING  [72/00  Ord. 2013] PERCORSO COMUNE  9  90 
[70/73] ENVIRONMENTAL AND LAND ENGINEERING  [73/00  Ord. 2020] PERCORSO COMUNE  9  90 
[70/77] CHEMICAL ENGINEERING  [77/00  Ord. 2020] PERCORSO COMUNE  9  90 
Objectives
1. Knowledge and understanding. At the end of the course the student will have knowledge of topics concerning real sequences, the infinitesimal calculus of real functions of a real variable and ordinary differential equations.
2. Applying knowledge and understanding. The student will be introduced to the main applications of the theoretical notions of the course, concerning both the solution of mathematical problems and the study of physical problems.
3. Autonomy of judgment. The student will learn how to classify single problems of differential and integral calculus, as well as differential equations, in the right class and to apply the most convenient method.
4. Communicative Skills. Students will acquire the ability to communicate what they learn and elaborate and also to express and argue the choice of methodology with respect to another to solve a mathematical problem.
5. Learning skills. Thanks to the notions acquired in this course, the student will be able to perfect his knowledge of higher mathematics and its applications to engineering.
Objectives
The objectives of the Mathematical Analysis Course 1 are defined in accordance with the information contained in the SUd CdS frame and more in detail can be described as follows:
Knowledge and understanding: At the end of the course the student will have knowledge of topics concerning real sequences, the infinitesimal calculus of real functions of a real variable and ordinary differential equations.
Ability to Apply Knowledge and Understanding: Students will be able to understand and interpret mathematical problems whose resolution is related to the knowledge of infinitesimal calculus as well as to real sequences and differential equations.
Autonomy of judgment: theoretical and applied notions will enable the student to understand which mathematical models and techniques are most appropriate for describing natural phenomena.
Communicative Skills: Students will acquire the ability to communicate what they learn and elaborate and also to express and argue the choice of methodology with respect to another to solve a mathematical problem.
Learning Abilities: Students will learn methodologies and tools such as: infinitesimal analysis and its direct applications to optimization problems and mathematical modeling in the general sense.
Prerequisites
Good knowledge of algebra, trigonometry and elementary analytic geometry
Contents
Introduction to set theory. Sets of Natural, Integer, Rational Numbers. Real Numbers: definitions, algebraic operations, distance. Subsets of real numbers. Least upper bound, greatest lower bound; maximum and minimum, accumulation, isolated, internal , external, boundary points; bounded, open, closed sets.
Real functions. Domain of definition, graph of elementary functions. Bounded, periodic, symmetric, monotonic, composite and inverse functions. Maxima and minima.
Limit theory. Basic limit theorems.
Continuous functions. The definition of continuity and basic theorems. Types of discontinuities. Weierstrass Theorem. The intermediate value theorem.
Differential calculus. Definition of derivative. The algebra of derivatives. The derivative of polynomials, of rational , exponential , logarithm, trigonometric functions. Geometric interpretation. Higher derivatives. The chain rule. Estreme values of a function. Increasing and decreasing functions, The mean value, Rolle Cauchy, DeL’Hopital theorems. Second derivative test for extrema. Convex and concave functions. Graph. Taylor and MacLaurin formula.
Integration. Antiderivative. Definite integral.Partitions of intervals. Definition of integral by upper and lower integrals. The area of a set. Theory and techniques of integrations. Fundamental theorem of integral calculus. Improper integrals.
Differential equations. Physical motivations. Theminology and notations. First order differential equation: separable equations, linear, Bernoulli and Clairaut equations. Cauchy theorem for the existence and uniqueness of the solution. Linear equations of order n. Wronskian determinant, Liouville theorem. Linear equations with constant coefficients: Lagrange method and special methods for determining a particular solution of the nonhomogeneous equation.
Sequences of real number. Limits of real sequences and theorems. Subsequences and convergence criteria
Contents
Introduction to set theory. Sets of Natural, Integer, Rational Numbers. Real Numbers: definitions, algebraic operations, distance. Subsets of real numbers. Least upper bound, greatest lower bound; maximum and minimum, accumulation, isolated, internal , external, boundary points; bounded, open, closed sets.
Real functions. Domain of definition, graph of elementary functions. Bounded, periodic, symmetric, monotonic, composite and inverse functions. Maxima and minima.
Limit theory. Basic limit theorems.
Continuous functions. The definition of continuity and basic theorems. Types of discontinuities. Weierstrass Theorem. The intermediate value theorem.
Differential calculus. Definition of derivative. The algebra of derivatives. The derivative of polynomials, of rational , exponential , logarithm, trigonometric functions. Geometric interpretation. Higher derivatives. The chain rule. Estreme values of a function. Increasing and decreasing functions, The mean value, Rolle Cauchy, DeL’Hopital theorems. Second derivative test for extrema. Convex and concave functions. Graph. Taylor and MacLaurin formula.
Integration. Antiderivative. Definite integral.Partitions of intervals. Definition of integral by upper and lower integrals. The area of a set. Theory and techniques of integrations. Fundamental theorem of integral calculus. Improper integrals.
Differential equations. Physical motivations. Theminology and notations. First order differential equation: separable equations, linear, Bernoulli and Clairaut equations. Cauchy theorem for the existence and uniqueness of the solution. Linear equations of order n. Wronskian determinant, Liouville theorem. Linear equations with constant coefficients: Lagrange method and special methods for determining a particular solution of the nonhomogeneous equation.
Sequences of real number
Limits of sequence. Theorems.
Teaching Methods
Compatibly with the teaching method established in the Manifesto Accademico 202021, as consequence of the COVID19 emergency, the tools used for the lectures will be both the blackboard and tablet with projection system via classroom screen and via internet streaming.
Frontal lectures (theory): 72 hours
Frontal lectures (exercises): 18 hours
Teaching Methods
Compatibly with the teaching method established in the Manifesto Accademico 202021, as consequence of the COVID19 emergency, the tools used for the lectures will be both the blackboard and tablet with projection system via classroom screen and via internet streaming.
Frontal lectures (theory): 72 hours
Frontal lectures (exercises): 18 hours
Verification of learning
Compatibly with the modality of exams established in the Manifesto degli Studi 202021, as consequence of the the COVID19 emergency, the exams will be held either in the presence or on the Teams platform or on an alternative telematic platform previously agreed between the teacher and the student.
The exam consists of a written test in which the following topics are proposed: general analysis of a real function of a real variable, integral calculus and applications, differential equations and numeric sequences. The student will have to demonstrate that he has understood and learnt the techniques for handling and exposing each of the topics discussed and to know how to apply the various methodologies linked to the techniques of resolution. The exam score is awarded by a vote between 1 and 30. The test consists of 5 exercises, 4 of which are mandatory (with score 7.5 each) and 1 optional (with score 3) and the score is determined according to the following rule: sum of the scores obtained in the individual exercises. In evaluating the examination, the final vote determination takes into account the logic followed by the student for each proposed exercise, the calculation strategy chosen in terms of the hypothesis of the problem, the clarity of the exposition and the reasoning.
Texts
Marco Bramanti, Carlo D. Pagani, Sandro Salsa: Analisi matematica 1. Zanichelli,
Sandro Salsa, Annamaria Squellati: Esercizi di Analisi matematica 1, Zanichelli.
Texts
P. Marcellini, C. Sbordone.
Elementi di Analisi Matematica 1
Liguori Editore
P. Marcellini, C. Sbordone
Esercitazioni di matematica.
vol 1, parte 1 e 2.
Liguori Editore
More Information
More information is available in the personal web page of the Prefessor
More Information
More information are available in the personal web page of the Prefessor