Teaching code

4S009864

Academic staff

Mauro Bonafini, Bogdan Mihai Maris

Coordinator

Mauro Bonafini

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Period

Semester 1 dal Oct 3, 2022 al Jan 27, 2023.

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 3-hour written test and covers all the topics covered during the course, theory and exercises. Regarding the theoretical part, all definitions / statements are subject to examination. As for the proofs, at the end of the course a list of proofs to know will be identified and the exam will include at least one question such as "State and prove theorem X".

Cheat sheet.
It is allowed to bring cheat sheet, that is
- single A4 page
- handwritten (by yourself)
- name-surname-matriculation number written at the top
- on the space available you can write whatever you want.
The cheat sheet must be delivered together with the exam.

Structure.
The exam is divided into two parts.
The first part, the preliminary test, includes 8 short exercises. The preliminary test is considered passed by solving at least 6 out of 8 exercises, otherwise the candidate is automatically rejected.
The second part, the complete test, it includes exercises and theoretical questions.

Score.
Passing the preliminary test assigns 3 points, the various exercises of the complete test assign 30 points, for a total of 33 points.

Registration of the exam.
The final grade is given by the minimum between the score obtained and 30. Each sufficient grade, i.e. from 18/30 upwards, can be accepted and consequently recorded.

Oral exam (optional).
The oral exam focuses on theory, is optional and can be taken by all candidates who have achieved at least 17/30. Attention: with the oral test, even if you start with a sufficient grade, you may be rejected and forced to repeat the written test. The oral exam is a necessary condition for "30 e lode''.

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