CODE  52474 

ACADEMIC YEAR  2021/2022 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/05 
LANGUAGE  Italian 
TEACHING LOCATION 

PREREQUISITES 
Prerequisites (for future units)
This unit is a prerequisite for:

MODULES  This unit is composed by: 
TEACHING MATERIALS  AULAWEB 
Language: Italian
Rigorous treatment of Mathematical Analysis, focusing on differential and integral calculus of functions of one real variable.
Traditional: blackboard.
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.
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)
Office hours: The teacher is available for explanations one afternoon a week.
Office hours: On appointment
Office hours: By appointment via email.
Office hours: Friday 1416
MARCO BARONTI (President)
ERNESTO DE VITO
TOMMASO BRUNO (President Substitute)
CRISTINA CAMPI (President Substitute)
MATTEO SANTACESARIA (President Substitute)
The class will start according to the academic calendar.
All class schedules are posted on the EasyAcademy portal.
The exam consists of a written test and an oral test.
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 12/30 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.
Date  Time  Location  Type  Notes 

10/01/2022  09:00  GENOVA  Scritto  riservato agli studenti iscritti a.a. 2020/2021 e anni accademici precedenti 
11/01/2022  09:00  GENOVA  Orale  riservato agli studenti iscritti a.a. 2020/2021 e anni accademici precedenti 
31/01/2022  09:00  GENOVA  Scritto  riservato agli studenti iscritti a.a.2020/2021 e anni accademici precedenti 
01/02/2022  09:00  GENOVA  Orale  riservato agli studenti iscritti a.a. 2020/2021 e anni accademici precedenti 
27/06/2022  09:00  GENOVA  Scritto  
29/06/2022  09:00  GENOVA  Orale  
18/07/2022  09:00  GENOVA  Scritto  
20/07/2022  09:00  GENOVA  Orale  
20/09/2022  09:00  GENOVA  Scritto  
21/09/2022  09:00  GENOVA  Orale  
10/01/2022  09:00  GENOVA  Scritto  riservato agli studenti iscritti a.a. 2020/2021 e anni accademici precedenti 
11/01/2022  09:00  GENOVA  Orale  riservato agli studenti iscritti a.a. 2020/2021 e anni accademici precedenti 
31/01/2022  09:00  GENOVA  Scritto  riservato agli studenti iscritti a.a.2020/2021 e anni accademici precedenti 
01/02/2022  09:00  GENOVA  Orale  riservato agli studenti iscritti a.a. 2020/2021 e anni accademici precedenti 
27/06/2022  09:00  GENOVA  Scritto  
29/06/2022  09:00  GENOVA  Orale  
18/07/2022  09:00  GENOVA  Scritto  
20/07/2022  09:00  GENOVA  Orale  
20/09/2022  09:00  GENOVA  Scritto  
21/09/2022  09:00  GENOVA  Orale 
Teaching style: in presence.