27210 - Mathematical Analysis 1

Academic Year 2017/2018

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Astronomy (cod. 8004)

Course contents

REAL NUMBERS
Sets and numbers. The real line and the cartesian plane. Completeness axiom for real numbers. Maximum and minimum value, superior and infimum value.

FUNCTIONS
Definitions. Properties of functions: boundedness, monotonicity, injectivity, surjectivity, invertibility. Even, odd, periodic functions. Elementary functions.

SEQUENCES

Definition. Limit of a sequence. Convergence theorems.

LIMITS OF FUNCTIONS AND CONTINUOUS FUNCTIONS
Limits: definitions and properties. Continuity. Theorems for continuous functions.

DIFFERENTIAL CALCULUS
Derivatives: definition and properties. Fundamental theorems of differential calculus: Fermat's, Rolle's, Lagrange's Theorems. Motonoticity. Derivatives of hygher orders and Taylor formula of order k. Convexity and concavity. Graph of a function.

INTEGRAL CALCULUS

One dimensional integral calculus. Integration by parts and by change of variables.

SERIES AND IMPROPER INTEGRALS

Numerical series: definition, convergence and absolute convergence; criteria for the convergence of series.
Improper integrals: definition, convergence and absolute convergence; criteria for the convergence of integrals.

COMPLEX NUMBERS

The field of complex numbers; algebraic, trigonometric and exponential form of a complex number; n-th power and root.

Readings/Bibliography

Theory:
Marcellini P.-Sbordone C.: Analisi Matematica 1 - Liguori Editore 
Exercises:
Bramanti M.: Esercitazioni di Analisi Matematica 1 , Ed. Esculapio
M.Amar-M.Bersani: Analisi Matematica. Esercizi e richiami di teoria vol.1, La Dotta

Teaching methods

The course consists of lessons describing the fundamental concepts of real numbers, sequences and numerical series, and, especially, of real functions of one real variable. Lessons are completed with examples and counterexamples illuminating the theoretical content. Futhermore a lot of exercises are solved in the classroom.

Assessment methods

The examination consists of a preliminary written test (exercises) and a test about the theoretical part.
The preliminary written test consists of exercises related "to the arguments of the course. In order to sustain it the student must register at the test through AlmaEsami [https://almaesami.unibo.it/] . If this written test is passed, the student can sit for the test concerning the theoretical aspects of the course. In this part, the student must show to know the concepts explained during the course (in particular definitions and theorems) and how to connect them. The theoretical part of the exam must be passed in the same "sessione" of the preliminary written test (exercises): in the same "appello"  or in the subsequent one.

Teaching tools

Tutorship (if appointed)

Office hours

See the website of Giovanni Cupini