CODE  52474 

ACADEMIC YEAR  2023/2024 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/05 
LANGUAGE  Italian 
TEACHING LOCATION 

PREREQUISITES 
Propedeuticità in uscita
Questo insegnamento è propedeutico per gli insegnamenti:

MODULES  Questo insegnamento è composto da: 
TEACHING MATERIALS  AULAWEB 
OVERVIEW
Language: Italian
AIMS AND CONTENT
LEARNING OUTCOMES
Introduction to the rigorous treatment of mathematical analysis, while developing at the same time the methods of differential and integral calculus in the context of real functions of a real variable.
PREREQUISITES
Elementary algebra; trigonometry
TEACHING METHODS
Traditional: blackboard.
SYLLABUS/CONTENT
1. Real numbers. The axioms of ordered fields. Absolute value. Natural and integer numbers. Rational numbers and their geometric representation. Completeness and its consequences. Real numbers and the straight line. Archimedean property. Decimal representations.
2. Functions. Relations, functions, domain, codomain, image and graph of a function. Composition of functions. Invertible functions. Operations on real functions. Monotone functions. Polynomials and rational functions. Trigonometric functions. The exponential function on rational numbers.
3. Limits. Metric and e topological properties of R. Continuity. Operations with continuous functions. Limits and their properties. The algebra of limits. Comparison theorems. Limits of monotone functions. Limits of compositions and change of variables. Sequences and their limits. Sunsequences. BolzanoWeierstrass' theorem. Cauchy sequences. Sequences defined by recurrence and their limits. Neper's number e.
4. Global properties of continuous functions. Weierstrass' theorem. Zeroes of continuous functions. Intermediate value theorem. Continuity and monotonicity. Continuity of the inverse function. Uniform continuity. HeineCantor's theorem. The exponential funcion on real numbers.
5. Differential calculus. The derivative: definition and elementary properties. Differentiability and the properties of the differential. Derivative of compositions and inverse functions. Derivatives of elementary functions. Higher order derivatives. The classical theorems by Rolle, Lagrange and Cauchy and their consequences. The theorem of de l'Hopital. Local comparison of functions. Vanishing and diverging functions. Taylor's formula. Convexity. Study of monotonicity and convexity by means of first and second derivatives. Newton's method. Iterative procedures for the solution of equations.
6.The indefinite integral. Integration techniques. Integration of elementary functions. Integration by parts and by substitution. Integration of rational functions.
7. The Riemann integral. Definition and properties of the definite integral. Integrability of continuous and monotonic functions. The oriented integral. The integral mean theorem. Relations between derivation and integration: integral functions, the fundamental theorem of calculus and its consequences. Improper integrals. Convergence criteria.
8. Series. Geometric and telescopic series. Convergence. Series with non negative terms: comparison, root and ratio criteria; condensation, order and integral tests. Alternating series and Leibniz' theorem.
9. Differential equations. Separation of variables. Linear first order equations Second order linear equations with constant coefficients.
RECOMMENDED READING/BIBLIOGRAPHY
A.Bacciotti, F.Ricci  Analisi Matematica I  Liguori Editore
M. Baronti, F. De Mari, R. van der Putten, I. Venturi  Calculus Problems, Springer, 2016
Further readings will be posted on the web page (AULAWEB)
TEACHERS AND EXAM BOARD
Ricevimento: The teacher is available for explanations one afternoon a week.
Ricevimento: On appointment
Ricevimento: Weekly office hours will be communicated. Meetings upon email requests will also be considered.
Ricevimento: Send a message to ernesto.devito@unige.it
Ricevimento: By appointment
LESSONS
LESSONS START
The class will start according to the academic calendar.
Class schedule
L'orario di tutti gli insegnamenti è consultabile all'indirizzo EasyAcademy.
EXAMS
EXAM DESCRIPTION
The exam consists of a written test and an oral test.
Students enrolled in the course of study in Physics are not required to study the proofs of "the 6 theorems" described in the file published on Aulaweb.
Students with a certified DSA, disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.
ASSESSMENT METHODS
1. Two intermediate written tests will be provided during the year. If a student obtains an average mark greater than or equal to 18/30 and if he scores at least 15/30 in both, the average of the two marks counts as a written test. 2. A written test with a score greater than or equal to 15/30 allows access to the oral test. 3. If a student submits a written test, the intermediate written tests will be considered canceled. Oral tests. During the oral exam, the commission asks questions about the entire program. In particular, the knowledge of the definitions of the main concepts, and of the statements and proofs of the most important results will be evaluated, and the ability to carry out exercises will be verified.
Exam schedule
Data  Ora  Luogo  Degree type  Note 

22/01/2024  09:00  GENOVA  Scritto  riservata agli studenti iscritti a.a.2022/23 e aa.aa. precedenti 
24/01/2024  09:00  GENOVA  Orale  riservata agli studenti iscritti a.a.2022/23 e aa.aa. precedenti 
12/02/2024  09:00  GENOVA  Scritto  riservata agli studenti iscritti a.a.2022/23 e aa.aa. precedenti 
15/02/2024  09:00  GENOVA  Orale  riservata agli studenti iscritti a.a.2022/23 e aa.aa. precedenti 
27/06/2024  09:00  GENOVA  Scritto  
04/07/2024  09:00  GENOVA  Orale  
19/07/2024  09:00  GENOVA  Scritto  
23/07/2024  09:00  GENOVA  Orale  
13/09/2024  09:00  GENOVA  Scritto  
16/09/2024  09:00  GENOVA  Orale  
22/01/2024  09:00  GENOVA  Scritto  riservata agli studenti iscritti a.a.2022/23 e aa.aa. precedenti 
24/01/2024  09:00  GENOVA  Orale  riservata agli studenti iscritti a.a.2022/23 e aa.aa. precedenti 
12/02/2024  09:00  GENOVA  Scritto  riservata agli studenti iscritti a.a.2022/23 e aa.aa. precedenti 
15/02/2024  09:00  GENOVA  Orale  riservata agli studenti iscritti a.a.2022/23 e aa.aa. precedenti 
27/06/2024  09:00  GENOVA  Scritto  
04/07/2024  09:00  GENOVA  Orale  
19/07/2024  09:00  GENOVA  Scritto  
23/07/2024  09:00  GENOVA  Orale  
13/09/2024  09:00  GENOVA  Scritto  
16/09/2024  09:00  GENOVA  Orale 
FURTHER INFORMATION
Teaching style: in presence.