Programme

Part 1: School Skills Review.
Real numbers and their subsets (N, Z, Q).
Roots and properties of rational powers.
Inequalities (also graphic resolution).
Fundamental properties of exponential, logarithmic, trigonometric and inverse trigonometric functions.

Part 2: Introduction to the concept of limit, continuity and differentiability through definitions, examples and exercises
Definition of limit for functions from R to R.
Calculation of delta as a function of epsilon in simple cases.
Fundamental properties of limits: algebra of limits and computation of finite limits.
Infinite limits, limit of sequences.
Extended limits algebra: extension of the calculus of limits.
Continuous functions and points of discontinuity.
Derivative: definition and rules of derivation (statements). Calculation of derivatives.
Relation between derivative and monotony.
Convexity: definition and criteria for C^1 and C^2 functions.
Applications to the qualitative study of function graphs.

Part 3: Introduction to the concept of integral and series through definitions, examples and exercises.
Definition of Riemann integral and its fundamental properties (linearity, invariance by translation, positivity). Calculation of simple integrals using the definition.
Illustration of the fundamental theorem of integral calculus.
Calculation of Primitives: main methods (substitution, integration by parts); integration of rational functions and other special classes.
Numerical series. Convergence criteria: statements and applications.
Improper integrals. Convergence criteria: statements and applications.

Part 4: Elementary solution methods of ordinary differential equations
Solution methods for special classes of ordinal differential equations (EDO) including:
linear first order, separation of variables, second order with constant coefficients, etc.

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.

Programme

Part 1: School Skills Review.
Real numbers and their subsets (N, Z, Q).
Roots and properties of rational powers.
Inequalities (also graphic resolution).
Fundamental properties of exponential, logarithmic, trigonometric and inverse trigonometric functions.

Part 2: Introduction to the concept of limit, continuity and differentiability through definitions, examples and exercises
Definition of limit for functions from R to R.
Calculation of delta as a function of epsilon in simple cases.
Fundamental properties of limits: algebra of limits and computation of finite limits.
Infinite limits, limit of sequences.
Extended limits algebra: extension of the calculus of limits.
Continuous functions and points of discontinuity.
Derivative: definition and rules of derivation (statements). Calculation of derivatives.
Relation between derivative and monotony.
Convexity: definition and criteria for C^1 and C^2 functions.
Applications to the qualitative study of function graphs.

Part 3: Introduction to the concept of integral and series through definitions, examples and exercises.
Definition of Riemann integral and its fundamental properties (linearity, invariance by translation, positivity). Calculation of simple integrals using the definition.
Illustration of the fundamental theorem of integral calculus.
Calculation of Primitives: main methods (substitution, integration by parts); integration of rational functions and other special classes.
Numerical series. Convergence criteria: statements and applications.
Improper integrals. Convergence criteria: statements and applications.

Part 4: Elementary solution methods of ordinary differential equations
Solution methods for special classes of ordinal differential equations (EDO) including:
linear first order, separation of variables, second order with constant coefficients, etc.

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.