Teaching code

4S009864

Academic staff

Mauro Bonafini, Franco Zivcovich

Coordinator

Mauro Bonafini

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Period

Semester 1 dal Oct 2, 2023 al Jan 26, 2024.

Courses Single

Authorized

Lessons timetable Moodle Seminars 0

Learning objectives

The course will treat the fundamental concepts of mathematical analysis: the aim is to provide a bet- ter consciousness of the analytic methods in view of applications of analysis. At the end of the course, the students shall prove of being able: to apply mathematical analysis techniques to the solution of problems about functions, derivatives, integrals and series also in different contexts even not strictly mathematical; to apply mathematical analysis techniques to solution of problems; to choose among the various techniques the one better suited to the problem at hand; to describe the solution of a problem employing correct terminology; to widen their knowledge starting from what they learned.

Prerequisites and basic notions

Basic knowledge required for enrollment in the course of study.

Program

Introduction.
Intuitive notion of set, notions of logic, elementary set operations, numerical sets, lower and upper bound, infimum and supremum, minimum and maximum, continuity axiom for real numbers, complex numbers. Definition of a function, injective function, surjective function, function composition and inverse function.
Sequences.
Definition of a sequence, monotone sequences, bounded sequences, definition of limit for a sequence, uniqueness of the limit, algebra of limits, comparisons and asymptotic estimates. Indeterminate forms and calculation of limits.
Real functions of a real variable.
Bounded functions, maximum and minimum, periodic functions, monotone functions. Graph of a function, elementary operations on the graph.
Continuity.
Notion of continuity for a real valued function of a real variable. Continuous functions on a closed and bounded interval: existence of zeros, Weierstrass' theorem, intermediate values theorem.
Limits.
Accumulation points, limits for real valued functions of a real variable, algebra of limits. Simple limits, comparisons, asymptotic estimates, change of variable within limits and calculation of limits.
Derivative.
Definition of derivative of a real valued function of a real variable, tangent line. Calculus of derivatives for elementary functions, algebra of derivatives, derivative of the composition, derivative of the inverse function. Angular points, cusps, inflections with vertical tangents, continuity and derivability. Stationary points, local maximums and minimums. Differentiable functions on an interval: Fermat's theorem, Lagrange's (or mean value) theorem, monotony test. De l'Hospital's theorem. Second derivative: geometric meaning, concavity, convexity. Study of the graph of a function. Approximations: notion of ``small o'', MacLaurin / Taylor polynomial, MacLaurin-Taylor formula with Peano and Lagrange remainder. Application to the calculation of limits by asymptotic expansion.
Series.
Numerical series and convergence criteria (comparison, asymptotic comparison, ratio and root criteria, Leibniz criterion). Taylor series.
Integral calculus.
Definition of integral and various interpretations, classes of integrable functions, properties of the integral, 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 second fundamental theorem of integral calculus. Applications of integral calculus. Generalized integrals: integrals of unbounded functions and integrability criteria; integration over unlimited intervals and integrability criteria.
Differential equations.
Concept of differential equation and its solution. First order differential equations with separable variables, how to solve them, existence and uniqueness theorem for the Cauchy problem. Linear differential equations of the first and second order: existence of solutions, structure of the general integral, how to solve them.

Bibliography

Didactic methods

Lecture.

Learning assessment procedures

The exam consists of a multiple-choice quiz followed by a 3-hour written test.
Quiz.
The quiz lasts 30 minutes and consists of 10 multiple choice questions. All candidates who obtain at least 6/10 are admitted to the written test, the others are rejected. The result of the quiz does not affect the final grade.
Written test.
The written test lasts 3 hours and is made up of exercises and open theory questions.
Oral exam.
All candidates who have passed the written test have access to an optional oral exam. The oral is focused on theory and allows you to increase/decrease the grade of the written without limits. The oral exam is a necessary condition for praise.

Evaluation criteria

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

Criteria for the composition of the final grade

Mark of the written exam (plus an optional oral if requested).

Exam language

Italiano