# MATHEMATICAL ANALYSIS 1

*Last update 14/06/2021 17:48*

OVERVIEW

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

## AIMS AND CONTENT

LEARNING OUTCOMES

The first part of the course covers differentiation and integration of functions of one variable. The second part of the course provides the basic concepts of infinite series, differentiation of functions of two variables and it concludes with a brief discussion of linear 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

PREREQUISITES

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

Teaching methods

Lecture classes and exercise classes (with possible streaming broadcast allowing to remotely attend the lectures).

The teaching methods may change according to restrictions motivated by the pandemic emergency.

As part of the innovation learning project (adopted by the Bachelor Degree Course in Mechanical Engineering), novel tools will be used for the active learning of students. The goal is to increase students' skills via interactive, experience-based, learning methodologies (e-learning, teamwork, etc.) for enhanced participation, using an advanced level of communication that makes the student more aware and autonomous

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. Schwarz’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

- C. Canuto, A. Tabacco, Analisi Matematica 1, 4a edizione, Springer-Verlag Italia, 2014,
- C. Canuto, A. Tabacco, Analisi Matematica 2, 2a edizione, Springer-Verlag Italia, 2014
- M. Baronti, M., F. De Mari, R. van der Putten, I. Venturi,
*Calculus Problems*, Springer International Publishing Switzerland, 2016

## TEACHERS AND EXAM BOARD

**Ricevimento:** By appointment, to be scheduled by e-mail

## LESSONS

Teaching methods

Lecture classes and exercise classes (with possible streaming broadcast allowing to remotely attend the lectures).

The teaching methods may change according to restrictions motivated by the pandemic emergency.

As part of the innovation learning project (adopted by the Bachelor Degree Course in Mechanical Engineering), novel tools will be used for the active learning of students. The goal is to increase students' skills via interactive, experience-based, learning methodologies (e-learning, teamwork, etc.) for enhanced participation, using an advanced level of communication that makes the student more aware and autonomous

LESSONS START

ORARI

## EXAMS

Exam description

The exam consists of

- Written exam with multiple choice test, and a second part with open questions and exercises
- Oral test (optional)

An intermediate exam (only written) will be held between the first and the second term.

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

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

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

Assessment methods

**Multiple choice test**. It is aimed to verift the student's ability to manage mathematical notation and to carry out simple computations and simple deductive reasoning. It consists of multiple-choice tests, each with one and only one correct answer.

**Open questions and exercises.** It is aimed to verify the knowledge of the main tools of differential and integral calculus. The test consists of exercises with several questions of different difficulty. The student must be able to solve the exercises correctly and to justify the necessary steps to obtain the final result, and to use the correct formalism.

**Optional oral test**. It is aimed at verifying the logical/deductive reasoning skills and consists of an oral test on the topics covered in the lectures, with particular focus on the correct statement of the theorems, the proofs of the results discussed during the lectures, and the solution to exercises. In particular, the student's logical/deductive ability and the degree of understanding of the concepts are assessed.