20410388 - AM120 - MATHEMATICAL ANALYSIS 2

To acquire a good knowledge of the main theorems of the Mathematical Analysis in R and of the corresponding methods of proof.

Curriculum

teacher profile | teaching materials

Programme

Part 1: Axiomatics of R and its main subsets Axiomatic definition of R.
Inductive assemblies; definition of N and induction principle. Definition of Z
and Q; Z is a ring, Q is a field.
Nth roots; rational powers.

Part 2: Theory of Limits
The extended line R*: intervals, neighbourhoods and accumulation points.
Limits of functions in R*.
Comparison theorems.
Lateral limits; limits of monotone functions.
Algebra of limits on R and R*.
Composition limit of functions.
Limits of inverse functions.
Notable limits. The number of Napier. Exponential and trigonometric functions.

Part 3: Continuous functions
Topology of R.
Theorem of existence of zeroes. Bolzano-Weierstrass theorems. Weierstrass’ Theorem.
Uniformly continuous functions.

Part 4: Differentiable functions
Rules of derivation. Derivatives of elementary functions.
Local minima and maxima and elementary theorems on derivatives (Fermat, Rolle, Cauchy,
Lagrange).
Bernoulli-Hopital theorem.
Convexity.
Taylor’s formulae.

Part 5: Riemann integral in R.
The Riemann integral and its fundamental properties.
Integration criteria. Integrability of continuous and monotone functions.
The fundamental theorem of calculus and its applications (integration by parts,
changes of variables in integration). Generalized ("improper") integrals and
related integrability criteria.

Core Documentation

Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su R
McGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte

Testi di esercizi:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010

Type of delivery of the course

Lectures and exercises in class. All the material of the program will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself. In the event of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) governing the methods of carrying out educational activities will be implemented. In particular, distance learning may be necessary.

Attendance

Attendance is optional and the understanding of the text adopted is sufficient for the full use of the course. Of course, attendance is desirable and STRONGLY recommended, as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.

Type of evaluation

The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher. In the case of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) that regulate the methods of student assessment will be implemented. In particular, distance assessments may be necessary and in this case the oral evaluation will be preceded by a preliminary written test which is an integral part of the oral examination.

teacher profile | teaching materials

Programme

Part 1: Axiomatics of R and its main subsets Axiomatic definition of R.
Inductive assemblies; definition of N and induction principle. Definition of Z
and Q; Z is a ring, Q is a field.
Nth roots; rational powers.

Part 2: Theory of Limits
The extended line R*: intervals, neighbourhoods and accumulation points.
Limits of functions in R*.
Comparison theorems.
Lateral limits; limits of monotone functions.
Algebra of limits on R and R*.
Composition limit of functions.
Limits of inverse functions.
Notable limits. The number of Napier. Exponential and trigonometric functions.

Part 3: Continuous functions
Topology of R.
Theorem of existence of zeroes. Bolzano-Weierstrass theorems. Weierstrass’ Theorem.
Uniformly continuous functions.

Part 4: Differentiable functions
Rules of derivation. Derivatives of elementary functions.
Local minima and maxima and elementary theorems on derivatives (Fermat, Rolle, Cauchy,
Lagrange).
Bernoulli-Hopital theorem.
Convexity.
Taylor’s formulae.

Part 5: Riemann integral in R.
The Riemann integral and its fundamental properties.
Integration criteria. Integrability of continuous and monotone functions.
The fundamental theorem of calculus and its applications (integration by parts,
changes of variables in integration). Generalized ("improper") integrals and
related integrability criteria.

Core Documentation

Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su R
McGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte

Testi di esercizi:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010

Type of delivery of the course

Lectures and exercises in class. All the material of the program will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself. In the event of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) governing the methods of carrying out educational activities will be implemented. In particular, distance learning may be necessary.

Attendance

Attendance is optional and the understanding of the text adopted is sufficient for the full use of the course. Of course, attendance is desirable and STRONGLY recommended, as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.

Type of evaluation

The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher. In the case of an extension of the health emergency from COVID-19, all the provisions (of the State and of the Roma Tre University) that regulate the methods of student assessment will be implemented. In particular, distance assessments may be necessary and in this case the oral evaluation will be preceded by a preliminary written test which is an integral part of the oral examination.