# MATHEMATICAL ANALYSIS I

**Civil and Environmental Engineering 8715 (coorte 2018/2019)**- GEOTECHNICS 99062
- STRUCTURAL ENGINEERING I 72543
- REPRESENTATION OF THE TERRITORY 80343
- INTRODUCTION TO BUILDING AND TERRITORY ENGINEERING 98962
- INTRODUCTION TO ENVIRONMENTAL ENGINEERING 98961
- URBAN PLANNING AND TRANSPORTATION ENGINEERING 84522
- STRUCTURAL MECHANICS II 66285
- MATHEMATICAL ANALYSIS II 60243
- HIDRAULIC CONSTRUCTIONS 80332
- HYDROLOGY & HYDRAULIC URBAN INFRASTRUCTURES 66097

OVERVIEW

The lectures are delivered in Italian.

This is a full year course. Between the first and the second semester there will be a break, during which no lectures will be delivered.

The first semester will be devoted to limits and differential calculus for functions of one real variable, while in the second semester integral calculus in one variable, functions of two variables and an introduction to differential equations will be dealt with.

## AIMS AND CONTENT

LEARNING OUTCOMES

Capability in following a line of logical reasoning; understanding of the essential properties of differential and integral calculus for functions of one variable, differential equations and functions of several variables. Acquisition of a sufficient calculation ability.

AIMS AND LEARNING OUTCOMES

The main objective is to supply the fundamentals of differential and integral calculus in one variable, differential equations and functions of several variables: these are some fundamental instruments in Mathematical Analysis, essential to get a well grounded knowledge in the basic branches of Mathematics and for the understanding of simultaneous and next courses.

**Expected learning outcomes:**

The students will become acquainted with the concepts and proofs carried out in class and how they are used in practice to solve exercises; moreover they will know how to produce easy examples on topics covered in this course.

In particular, students are expected to develop the following skills: operations on limits and derivatives, study of functions of one variable, integral calculus (indefinite, definite and improper integrals), study of integral functions, elementary study of level curves of functions of two varlables, resolution of simple differential equations of first order, resolution of linear differential equations of order n with constant coefficients.

Teaching methods

Lectures (120 hours).

Both theory and exercises are presented by the teachers in the classroom on the blackboard. Moreover some tutorial exercitations will be carried out during the semester.

SYLLABUS/CONTENT

Real numbers, cartesian coordinates in a plane, functions of one real variable, monotonic functions, composition and invertibility of functions, elementary functions, supremum and infimum, limits of functions, limits of sequences, infinitesimal and infinite functions, continuous functions and their properties, derivatives, derivation rules, derivatives of elementary functions, sign of derivatives in the study of monotonicity and convexity, maxima and minima, theorems of Fermat, Rolle, Lagrange, de l'Hospital, Taylor expansions and applications to critical points, definite and indefinite integrals, fundamental theorem of integral calculus, integral functions, improper integrals, real functions of two variables (domain, limits at a point and at infinity, continuity, partial and directional derivatives, differentiability and tangent plane, maxima and minima), linear first order differential equations with continuous coefficients, linear differential equations of order n with constant coefficients, separation of variables.

RECOMMENDED READING/BIBLIOGRAPHY

O. Caligaris, P. Oliva: Analisi matematica 1, ECIG (1990);

M. Bramanti, C. Pagani, S. Salsa: Analisi Matematica 1, Zanichelli (2008);

T. Zolezzi: Dispense di Analisi Matematica I, edizioni ERSU (anni 90);

P. Marcellini, C. Sbordone: Esercitazioni di matematica, Liguori (1988);

F. Buzzetti, E. Grassini Raffaglio, A. Vasconi: Esercizi di analisi matematica, Masson (1989);

M. Bertsch, R. Dal Passo: Elementi di analisi matematica, Aracne (2000).

M. Baronti, F. De Mari, R. Van der Putten, I. Venturi: Calculus problems, Springer (2016).

## TEACHERS AND EXAM BOARD

**Ricevimento:** At the end of lectures or by appointment.

Exam Board

MAURIZIO CHICCO (President)

ADA ARUFFO (President)

## LESSONS

Teaching methods

Lectures (120 hours).

Both theory and exercises are presented by the teachers in the classroom on the blackboard. Moreover some tutorial exercitations will be carried out during the semester.

LESSONS START

The class will start according to the academic calendar: September 17, 2018

ORARI

## EXAMS

Exam description

The exam consists of a written and an oral test. In order to access to the oral test, the student will be required of an evaluation of at least 10/30. Oral test has to be taken in the same exam appeal of written test.

Furthermore, during the year, there will be two partial written tests. The students that will pass them (that is, obtaining an average greater than or equal to 15/30 and, in each partial test, an evaluation of at least 10/30), will be enabled to access directly the oral exam; this is valid for all the exam appeals for the academic year, hence until February included.

Assessment methods

The written examination consists in some exercises about the topics covered in this course. In such a test is proven the ability to solve problems relating to calculation of limits, derivatives and integrals of one variable functions, study of properties of one or more variables functions and ordinary differential equations.

The test lasts two hours and it is possible to consult the notes and textbooks; the same procedures apply to each of the two partial written tests.

In the oral exam a discussion of the written examination is done; moreover some questions are asked on the course content and/or about the solution of some exercises about the topics covered in this course. In such way they are assessed the understanding and the knowledge of the concepts and reasoning skills acquired by the students.

## FURTHER INFORMATION

Attendance is recommended.

Students are suggested to enroll in AulaWeb, in order to be able to get further information by the teachers about the course.