OVERVIEW

The course aims to provide basic knowledge of Mathematical Analysis, such as differential and integral calculus for functions of one variable, which are essential for understanding the topics covered in subsequent courses.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide knowledge of basic Mathematical Analysis tools useful for modelling physical phenomena, the ability to formulate and solve problems using both intuitive and deductive methods, and to recognize and apply appropriate mathematical tools in solving problems in the field of physics. By the end of the course, students will be able to: 1. State the concepts (theorems, definitions) covered in the course (e.g., the supremum and infimum of a set, the concepts of continuity and derivative of a function). 2. Physically and geometrically interpret the fundamental concepts of mathematical analysis. 3. Approach problem-solving using an intuitive method. 4. Select the appropriate mathematical tools for solving problems. 5. Solve problems using a deductive approach. Additionally, the course provides students with fundamental knowledge of mathematical analysis related to the theory of real functions of a real variable.

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 set up problem solving with an intuitive approach;

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

5. to solve problems with deductive reasoning.

PREREQUISITES

Elementary Algebra, Trigonometry and Analytic Geometry in the plane.
 

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. Several intermediate tests are provided.

Students  have several exercises at their disposal on Aulaweb.

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

SYLLABUS/CONTENT

Real numbers and the real line. Cartesian coordinates in the plane. Sequences: properties and elementary examples. 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. L’Hopital’s rule.  Taylor’ formula and its applications. Antiderivative. Riemann integrals. Mean value integrals. Fundamental theorem and formula of integrals calculus. Integration formulas. Integral functions. Improper integrals. Ordinary differential equations. Cauchy problem : existence and unicity theorems and resolution methods in some special cases : separation variable equation. Linear Differential Equations.  Structure of the set of solutions of a linear differential equations in the homogeneous and non homogeneous.  First orderl inear differential equatios with continuous coefficient. Metods  for solving linear differential equations with constant coefficients.                                                                                    

RECOMMENDED READING/BIBLIOGRAPHY

Main books

T. Zolezzi: Dispense di analisi matematica I.
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.
 

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

LESSONS

LESSONS START

Lessons start on Thursday, 26 september 202. One can find the lesson timetable on https://corsi.unige.it/en/corsi/8784/studenti-orario

Class schedule

MATHEMATICAL ANALYSIS 1

EXAMS

EXAM DESCRIPTION

The exam consists of a written examination. . The written examination consistsin  open-ended exercises concerning the topics treated. The students have two/three hours at their disposal and during the written exam the student can consult notes and texts, use calculators but cannot use laptopas or smartphone. Two intermediate examinations will be held which may substitute the written examination.

To enroll the exam you must register by the deadline on the website https://servizionline.unige.it/studenti/esami/prenotazione

Students with SLD, disability or other regularly certified special educational needs are advised to contact the professor and the University referent at least 15 days before each examination to agree on exhamination methods that, in compliance with the course objectives, take into account the individual learning requirements.

ASSESSMENT METHODS

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The aim of the written 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.