Teaching code

4S00006

Credits

12

Coordinator

Bogdan Mihai Maris

Language

Italian

Also offered in courses:

• Mathematical analysis [Matricole pari] - Analisi matematica I - teoria of the course Bachelor's degree in Computer Engineering for Robotic and Intelligent Systems
• Mathematical analysis [Matricole pari] - Analisi matematica I - esercitazioni of the course Bachelor's degree in Computer Engineering for Robotic and Intelligent Systems
• Mathematical analysis [Matricole pari] - Analisi matematica II - teoria of the course Bachelor's degree in Computer Engineering for Robotic and Intelligent Systems
• Mathematical analysis [Matricole pari] - Analisi matematica II - esercitazioni of the course Bachelor's degree in Computer Engineering for Robotic and Intelligent Systems

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Courses Single

Authorized

Moodle Seminars 0

The teaching is organized as follows:

Learning objectives

The course is divided into two modules. The Analysis 1 module aims to provide the fundamental concepts of single-variable mathematical analysis. The goal is to develop an understanding of the methods used, with a view to their applications in analysis. By the end of the course, students are expected to be able to apply basic techniques of single-variable mathematical analysis to problem-solving; apply these techniques to problems involving the concepts of function, derivative, integral, and series in various situations, including those outside strictly mathematical contexts; choose the most appropriate technique for the problem at hand; present the solution to a problem using correct terminology; and develop the skills necessary to expand their knowledge starting from the concepts learned.
In the Analysis 2 module, the concepts and techniques of differential and integral calculus for real functions of several real variables are developed. By the end of the course, students are expected to have knowledge and understanding of multivariable mathematical analysis techniques and the ability to use them to solve problems; to be able to choose which tools or theoretical results may be useful in solving a given problem; and to appropriately use the language and formalism of multivariable mathematical analysis in various engineering and modeling applications.

Prerequisites and basic notions

Basic knowledge required for enrollment in the degree program.

Program

First semester program (analysis 1): 1. Introduction. Intuitive notion of a set, notions of logic, elementary set operations, numerical sets, least and greatest terms, least infimum and least upper bound, minimum and maximum, axiom of continuity of real numbers. Definition of function, injective function, surjective function, composite function, and inverse function. 2. Sequences. Definition of sequence, monotone sequences, limited sequences, definition of limit for a sequence, uniqueness of the limit, algebra of limits, comparisons and asymptotic estimates. Indeterminate forms and calculation of limits. 3. Real functions of one real variable. Bounded functions, maximums and minimums, periodic functions, monotone functions. Graph of a function, elementary operations on the graph. 4. Limits. Accumulation points, limits for real functions of one real variable, algebra of limits. Special limits, comparisons, asymptotic estimates, change of variable in limits, and calculation of limits. 5. Continuity. Continuity of a real function of one variable. Continuous functions on a closed and bounded interval: zero theorem, Weierstrass's theorem, intermediate value theorem. 6. Derivative. Definition of the derivative of a real function of one variable, tangent line. Calculating derivatives for elementary functions, algebra of derivatives, derivative of the composite function, derivative of the inverse function. Corner points, cusps, inflection points with a vertical tangent, continuity and differentiability. Stationary points, local maxima and minima. Differentiable functions on an interval: Fermat's theorem, Lagrange's (or mean value) theorem, monotonicity test. L'Hospital's theorem. Second derivative: geometric meaning, concavity, convexity. Studying the graph of a function. Asymptotic expansions. Notion of "small o", MacLaurin/Taylor polynomial, MacLaurin-Taylor formula with Peano remainder and Lagrange remainder. Application to the calculation of limits using asymptotic expansion. 7. Integral calculus. Definition of integral and various interpretations, classes of integrable functions, properties of integrals, mean value theorem. Notion of primitive and fundamental theorem of integral calculus. Methods for finding a primitive: immediate integrals, integrals by substitution, integrals by parts, integrals of rational functions. Integral functions and the second fundamental theorem of integral calculus. Applications of integral calculus. Generalized integrals: integrals of unbounded functions and criteria for integrability at finite intervals; integration over unbounded intervals and criteria for integrability at infinity. 8. Differential equations. Notion of differential equation and solution of a differential equation. First-order differential equations with separable variables, solution method and existence and uniqueness theorem for the problem. Cauchy's differential equations. First- and second-order linear differential equations: existence of solutions, structure of the general integral, solution methods.
Recommended books: Mathematical Analysis 1, authors: Bramanti, Marco, Bologna, Zanichelli.
Exercises in Mathematical Analysis 1, authors: Salsa Sandro, Squellati Annamaria.
Second semester program (analysis 2):
Differential calculus in several variables. Limits and continuity for functions of several variables, level curves, directional derivatives and differential of functions of several variables, total differential theorem, gradient of scalar functions. Higher-order derivatives, Hessian matrix, Schwarz's theorem, Taylor expansion. Optimization problems for functions of several variables. Critical points, free optimization, study of the Hessian matrix for the determination of free relative maxima and minima. Constrained optimization, Lagrange multipliers, Dini's theorem, inverse function theorem. Multiple integrals for continuous functions defined on n-dimensional rectangles. Fubini's theorem. Formula of the Change of variables in double and triple integrals. Vector fields, curl, divergence. Regular curves, differential calculus for curves. Curvilinear integrals of the first and second kind. Regular surfaces and surface integrals of the first and second kind. Gauss-Green formula in the plane. Flux, divergence theorem, and curl theorem.

Bibliography

Didactic methods

Lectures and exercises in the classroom.

Learning assessment procedures

The partial exam consists of a multiple-choice quiz followed by a written test.
Quiz.
The quiz lasts 30 minutes and consists of multiple-choice questions. All candidates who obtain at least 60% correct answers are admitted to the written test; all others are failed. The quiz result does not affect the final grade or contributes to honors as specified below.
Written test.
The written test lasts 2-3 hours (depending on the complexity of the exercises) and consists of open-ended theory questions and exercises. The final grade out of 30 is determined based on the results of the written test.
During the full exam, candidates may choose one of the two modules. For those who choose only one module, the exam follows the partial exam format (quiz + written test) for a total duration of 2 hours and 30 minutes. The full exam includes a multiple-choice quiz (30 minutes) and a written test lasting 4 hours.
Oral exam.
All candidates who pass the written exam are admitted to an optional oral exam. The oral exam focuses on theory and simple exercises and allows for unlimited increases/decreases in grades.
Students who achieve the highest score in the quiz (100%) and the written exam (30%) are awarded a grade of 30 cum laude. Students who do not achieve a grade of 30 cum laude may still obtain it by choosing the oral exam, but only after passing the written exam.

Evaluation criteria

Understanding of the topics covered during the course and the ability to apply the acquired notions in solving exercises and in the construction of new statements.

Criteria for the composition of the final grade

Written exam grade (and oral exam grade, if applicable). If separate partial exams are taken, the final grade is the average of the partial grades.

Exam language

Italiano