OVERVIEW

The course is directed to first year students which are familiar with basic notions in elementary Mathematics

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to provide the basic knowledge preparatory to other courses that require mathematical methods and tools.

AIMS AND LEARNING OUTCOMES

The knowledge of mathematical basic tools useful in physical problems modelling.  The skill of setting up and solving problems by using intuitive and deductive reasoning as well as recognizing and using the suitable mathematical tools in solving problems in a physical setting. At the end of the course the student will be able

1. to state the concepts ( theorems and definitions ) introduced during the course ( f.i. Infimum and Supremum of a set, derivative, integral, line integral, Existence and Uniqueness theorems for differential problems );

2. to give physical and geometric interpretation of the basic concepts of Mathematical Analysis;

3. to select the suitable mathematical tools in problem solving;

4. to solve problems with deductive reasoning.

PREREQUISITES

Basic notions in elementary mathematics.

TEACHING METHODS

The course consists of 90 hours of lectures and practices. In the lectures the topics of the syllabus are explained with definitions and theorems and some proofs which can be useful for the comprehension of the topics and to develop the logical and deductive skills. Every theoretical topic is explained with easy examples and some exercises. In the practices, many exercises are solved with the aim of going into the knowledge of theoretical topics treated in the lectures and preparing the student for the exam. Some guided practices will be held to help the student to valuate one's preparation. Several intermediate tests are provided.

Students  have several exercises at their disposal on Aulaweb.

SYLLABUS/CONTENT

1) Real numbers and the real line. Cartesian coordinates in the plane. Functions and their graphs. Limits and continuity. Theorems about continuous functions. The derivative. Differentiation rules: product, reciprocal, quotient and chain rule. Monotone functions: the inverse function theorem. Derivatives of some elementary functions. Theorems about differentiable functions: Rolle, Lagrange, Cauchy. Higher order derivatives. Extreme values, convexity and inflection. Sketching the graph of a function.

2) Integration. Sums and sigma notation. Areas as limits of sums. The definite integral. Properties of the definite integral. The fundamental Theorem of Calculus. Changes of variable: the method of substitution. Areas of plane regions. Integration by parts. Integrals of rational functions.

3) L’Hopital’s rule.  Taylor’ formula and its applications.

3) Ordinary differential equations. First order equations: separable equations, linear equations. Cauchy problems: existence and uniqueness theorem. Linear ODE’s with constant coefficients.

RECOMMENDED READING/BIBLIOGRAPHY

Main books

T. Zolezzi : Dispense di analisi matematica I e II.
C. Canuto – A. Tabacco : Analisi Matematica 1. Teoria ed esercizi. Unitext, Springer – Verlag. 2014
F. Parodi – T. Zolezzi : Appunti di analisi matematica. ECIG, 2002
R. Adams : Calcolo differenziale I. Funzioni di una variabile reale. Casa ed. Ambrosiana, 1992.
P. Marcellini – C. Sbordone : Analisi Matematica II. Liguori Editori
R. Adams : Calcolo differenziale II. Funzioni di più variabili. Casa ed. Ambrosiana, 1993.

Practices

M. Baronti – F. De Mari – R. van der Putten – I. Venturi : Calculus Problems. Springer 2016
M. Pavone: Temi svolti di analisi matematica I.
Marcellini-Sbordone : Esercitazioni di matematica, I volume

S. Salsa – A. Squellati : Esercizi di Matematica, volume 1.

TEACHERS AND EXAM BOARD

Exam Board

ROBERTUS VAN DER PUTTEN (President)

VALENTINA BERTELLA

MARCO BARONTI (President Substitute)

MARIA VIRGINIA CATALISANO (President Substitute)

ALBERTO PERELLI (President Substitute)

LESSONS

LESSONS START

https://corsi.unige.it/8721/p/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists in a written and oral examination. The written examination consists in two problems concerning the topics treated. The students have two hours at their disposal. After the written examination, the students who obtained a grade higher than 13/30 may take the oral examination. Two intermediate examinations will be held which may substitute the written examination.

Students with SLD, disability or other regularly certified special educational needs are advised to contact the instructor at the beginning of the course to agree on teaching and examination methods that, in compliance with the course objectives, take into account the individual learning requirements.

ASSESSMENT METHODS

The aim of the examination is verifying the skills acquired by the student. The problems proposed in the examination call for the choice and the application of suitable mathematical tools, besides their  solution needs the skill of constructing a logical connection applying theoretical topics treated. The student must solve the exercises justifying the most important passages recalling  theorems and definitions and underlying the physical and geometric interpretation of the problem.

The final evaluation depends also on the quality of the written exposition and on the ability of reasoning.

Exam schedule

FURTHER INFORMATION

The course requires knowledge of elementary Algebra and some Trigonometric tools.