# MATHEMATICAL ANALYSIS 1

OVERVIEW

The aim of tje course is to provide the basic elements of differential and integral calculus for the functions of one variable, and a short introduction to the theory of diferential equations and differential calculus for multivariate functions.

## AIMS AND CONTENT

LEARNING OUTCOMES

The course aims at providing the student with basic operative knowledge on differential and integral calculus for functions of one and two real variables, with some attention to mathematical rigour. Some of the founding elements of mathematical modeling are developed in the second half of the course, such as the elementary theory of ordinary differential equations.

AIMS AND LEARNING OUTCOMES

The main expected learning outcomes are

- to master the mathematical notation
- the knowledge of the properties of the main elementary functions
- the ability to follow the logical concatenation of arguments
- to master simple demonstration techniques
- the ability to solve exercises, discussing the reasonableness of the results obtained

PREREQUISITES

Numerical sets, equations and inequalities, analytical geometry, trigonometry.

Teaching methods

Lectures and practice

SYLLABUS/CONTENT

*Functions of one real variable*. Real numbers, the oriented real line. The Cartesian plane, graphs of elementary functions. Operations on functions and their graphical interpretation. Monotonicity. Composition and inversion. Powers, exponentials and logarithms. Supremum and infimunm. Sequences and series: the basic notions and examples. Limits of functions. Infinitesimal and infinite functions. Continuous functions and their local and global, derivative, derivation rules. Derivatives of elementary functions. Sign of derivatives in the study of monotonicity and convexity. The classical theorems of Rolle, Cauchy, Lagrange and de l'Hôpital. Taylor expansions and applications to critical points. Definite and indefinite integrals.

* Functions of two (or more) real variables*. Continuity, directional and partial derivatives, gradient. Differentiability and tangent plane. Level sets. Local minima and maxima: second order derivatives and the Hessian. Schwartz’s theorem.

*Differential equations*. Separation of variables. The existence and uniqueness theorem for the Cauchy problem. Linear first and second order differential equations: solving methods. General solution for the linear equation.

RECOMMENDED READING/BIBLIOGRAPHY

A. Bacciotti, F. Ricci, Lezioni di Analisi Matematica 1 e 2, Levrotto & Bella, 1991.

C. Canuto, A. Tabacco, Analisi Matematica 1 e 2, Springer-Verlag Italia, 2003.

F.De Mari, Dispense di Analisi Matematica 1, http://www.dima.unige.it/~demari/DIDA.html

## TEACHERS AND EXAM BOARD

**Ricevimento:** Friday 14-16

Exam Board

ERNESTO DE VITO (President)

FILIPPO DE MARI CASARETO DAL VERME

EMANUELA SASSO

CESARE MOLINARI (President Substitute)

## LESSONS

Teaching methods

Lectures and practice

LESSONS START

According to the academic rule

## EXAMS

Exam description

The exam consists of three parts

- Multiple choice test
- Written test with open questions
- Oral test (optional)

To enroll the exam you must register by the deadline on the website

https://servizionline.unige.it/studenti/esami/prenotazione

The overall assessment of the first two tests is the weighted average (1/3 test and 2/3 written test).

If the average is between 18 and 26 the student can choose not to take the oral test and to confirm the written grade. If the written test has a grade greater than 26, the student can decide whether to take the oral test or, if he or she renounces the oral test, accept the grade of 26.

In the months of February and June two intermediate written tests are proposed, to be considered, if passed, as substitutes for the written examination.

The examination modalities may change according to the evolution of the health emergency.

Assessment methods

**First part (multiple choice test)**. It is aimed to verift the student's ability to manage mathematical notation and to carry out simple deductive reasoning. It consists of 10 multiple-choice tests, only one of which is correct. The questions are aimed to test the basic knowledge of subjects already studied by the student in high school and reviewed in the first part of the teaching. The correct answers are worth 3 points, the wrong one -1, while unanswered questions are worth 0. To access the second part you must take a grade greater than or equal to 18. Duration of the test: 1 hour. It is not possible to consult notes or books. The use of a calculator, computer or mobile phone is not permitted.

**Second part (open questions).** It is aimed to verify the knowledge of the main tools of differential and integral calculus. The test consists of three exercises divided into several questions of different difficulty. The student must be able to solve the exercises correctly and be able to justify the necessary steps to obtain the final result and to use the correct formalism.

The duration of the test is 2.30 hours. You can consult the notes, textbooks and use the calculator. The use of a computer or mobile phone is not permitted.

The final grade of the two written tests is given by

(test grade)*1/3 + (exercise grade)*2/3

rounded off to the largest integer.

**Third part (oral test)**. It is aimed at verifying the logical/deductive reasoning skills and consists of an oral test on the topics covered in the lesson, with particular attention to the correct wording of the theorems and the demonstrations of the results seen in the lesson. In particular, the student's logical/deductive ability and the degree of understanding of the concepts seen in class are assessed. The test is optional.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

11/06/2021 | 09:00 | GENOVA | Compitino con accettazione online | |

28/06/2021 | 15:00 | GENOVA | Scritto | |

14/07/2021 | 15:00 | GENOVA | Scritto | |

17/09/2021 | 15:00 | GENOVA | Scritto |