#
Continuum mathematics

A.Y. 2020/2021

Learning objectives

The aim of the curse is twofold. First, to provide students with a basic mathematical language, allowing them to formulate a problem in a correct way and to understand a problem formulated by other people. Secondly, to provide the necessary instruments to solve some specific problems, ranging from the behaviour of sequences to that of series and functions of a single variable.

Expected learning outcomes

Students have to correctly express a selected number of basic mathematical notions and instruments. Moreover, they must know which instrument is the most suitable to solve

some classical problem in Mathematical Analysis. Finally, they must be able to use such instrument to solve the problem itself, or at least have the appropriate know-how to understand some helpful mathematical text.

some classical problem in Mathematical Analysis. Finally, they must be able to use such instrument to solve the problem itself, or at least have the appropriate know-how to understand some helpful mathematical text.

**Lesson period:** First semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

Lessons and exercises lectures in asynchronous/synchronous mode, recorded.

Review and clarification of the topics of the lessons and exercises lectures during the tutoring online in synchronous mode.

The program remains unchanged

All the written tests, the in itinere tests, the oral exams will be online.

Review and clarification of the topics of the lessons and exercises lectures during the tutoring online in synchronous mode.

The program remains unchanged

All the written tests, the in itinere tests, the oral exams will be online.

**Course syllabus**

Natural, rational and real numbers. Real numbers and operations.

Maximum and minimal element of subsets of the real numbers, greatest lower bound and least upper bound. The symbols +∞ e -∞. The real line.

Comparison between rational and irrational numbers.

Complex numbers Algebraic and trigonometric representations, algebraic operations and roots of unity. The Fundamental Theorem of Algebra, factoring of polynomials.

Natural numbers: Induction over the integers and properties that holds eventually.

Sequences of real numbers: basic properties, boundedness and monotonicity.

Limits of sequences: the notion of limit, uniqueness, boundedness of convergent sequences, comparison theorems. Algebraic operations with limits and forms of indecision, comparison between infinite/infinitesimal sequences, Landau's symbols and their use. Regularity of monotone sequences, the number e (of Napier).

Limits of functions. Continuous functions: the notion of continuity and its graphical interpretation, points of discontinuity. The concept of limit for functions and its relation to continuity. Continuity and discontinuity of elementary functions: rational, exponential, logarithmic functions, the absolute value and step functions, the integer and fractional part functions. Change of variables in limits and the limit of compositions of functions. The theorem on zeros and Weirstrass' theorem for continuous functions.

Differential calculus: the concept of the derivative: linear approximations and the tangent to a curve. Calculation of derivatives for elementary functions. Angular points and cusps. Algebraic operations and the derivative. The theorems of Fermat, Rolle, Lagrange and applications. Cauchy's theorem. De l'Hôpital's theorem. The Taylor's formula and its applications. Optimization problems (finding maxima and minima).

Integral calculus: computing areas, approximation and the method of exhaustion. The integral of Riemann: definition of the definite integral, classes of integrable functions, properties of the definite integral.

The mean value theorem for integrals, the fundamental theorem of integral calculus, the fundamental formula of integral calculus.

Indefinite integrals and integration methods: integration by substitution and by parts,

integration of rational functions. Improper integrals: definition and fundamental examples.

Finite sums. Fundamental examples: powers of integers and geometric progressions.

The concept of series: fundamental examples, the geometric series and telescopic series.

Necessary condition for convergence, regularity, comparison test, absolute convergence and simple convergence. Estimates and asymptotic estimates on the rate of conver-gence/divergence of a series: comparison test, limit comparison test. The generalized harmonic series (p-series).

Real power series and the radius of convergence. Taylor series and analytic functions. Algebraic operations on power series, derivative and integrals of series.

Maximum and minimal element of subsets of the real numbers, greatest lower bound and least upper bound. The symbols +∞ e -∞. The real line.

Comparison between rational and irrational numbers.

Complex numbers Algebraic and trigonometric representations, algebraic operations and roots of unity. The Fundamental Theorem of Algebra, factoring of polynomials.

Natural numbers: Induction over the integers and properties that holds eventually.

Sequences of real numbers: basic properties, boundedness and monotonicity.

Limits of sequences: the notion of limit, uniqueness, boundedness of convergent sequences, comparison theorems. Algebraic operations with limits and forms of indecision, comparison between infinite/infinitesimal sequences, Landau's symbols and their use. Regularity of monotone sequences, the number e (of Napier).

Limits of functions. Continuous functions: the notion of continuity and its graphical interpretation, points of discontinuity. The concept of limit for functions and its relation to continuity. Continuity and discontinuity of elementary functions: rational, exponential, logarithmic functions, the absolute value and step functions, the integer and fractional part functions. Change of variables in limits and the limit of compositions of functions. The theorem on zeros and Weirstrass' theorem for continuous functions.

Differential calculus: the concept of the derivative: linear approximations and the tangent to a curve. Calculation of derivatives for elementary functions. Angular points and cusps. Algebraic operations and the derivative. The theorems of Fermat, Rolle, Lagrange and applications. Cauchy's theorem. De l'Hôpital's theorem. The Taylor's formula and its applications. Optimization problems (finding maxima and minima).

Integral calculus: computing areas, approximation and the method of exhaustion. The integral of Riemann: definition of the definite integral, classes of integrable functions, properties of the definite integral.

The mean value theorem for integrals, the fundamental theorem of integral calculus, the fundamental formula of integral calculus.

Indefinite integrals and integration methods: integration by substitution and by parts,

integration of rational functions. Improper integrals: definition and fundamental examples.

Finite sums. Fundamental examples: powers of integers and geometric progressions.

The concept of series: fundamental examples, the geometric series and telescopic series.

Necessary condition for convergence, regularity, comparison test, absolute convergence and simple convergence. Estimates and asymptotic estimates on the rate of conver-gence/divergence of a series: comparison test, limit comparison test. The generalized harmonic series (p-series).

Real power series and the radius of convergence. Taylor series and analytic functions. Algebraic operations on power series, derivative and integrals of series.

**Prerequisites for admission**

- basic algebra

- solving basic equations and inequalities

- elementary functions and their graphs

- graphic interpretation of inequalities

- basic plane analytical geometry

- basic trigonometry

- basic set theory

- basic elements of logic

- solving basic equations and inequalities

- elementary functions and their graphs

- graphic interpretation of inequalities

- basic plane analytical geometry

- basic trigonometry

- basic set theory

- basic elements of logic

**Teaching methods**

Lessons and exercises lectures in asynchronous/synchronous mode, recorded.

Review and clarification of the topics of the lessons and exercises lectures during the tutoring online in synchronous mode.

Possible tutoring meetings with students in presence, in compliance with the university directives.

Further information on the video platforms and the methods of connection or participation in the possible face-to-face tutoring meetings will appear on the Ariel website of the course.

Review and clarification of the topics of the lessons and exercises lectures during the tutoring online in synchronous mode.

Possible tutoring meetings with students in presence, in compliance with the university directives.

Further information on the video platforms and the methods of connection or participation in the possible face-to-face tutoring meetings will appear on the Ariel website of the course.

**Teaching Resources**

Bibliography:

P. Marcellini and C. Sbordone, Calcolo, Liguori

Other material:

MiniMat course, available online on the Ariel website (prerequisites)

Matematica Assistita course, available online on the Ariel website

P. Marcellini and C. Sbordone, Calcolo, Liguori

Other material:

MiniMat course, available online on the Ariel website (prerequisites)

Matematica Assistita course, available online on the Ariel website

**Assessment methods and Criteria**

The exam consists of a written test and an oral test in the same exam session..

The written test lasts 2 hours and consists of two distinct parts:

Part 1: The student will have to answer some simple questions indicating only the result.

Part 2: The student will have to solve some exercises on the topics covered during the lessons, indicating for some exercises only the answer, for others the procedure and the answer. The student will also have to answer some theoretical questions related to the program (definitions, statements of theorems, proofs of theorems).

In addition to the answers, the procedure to solve the exercises and the correctness of the demonstrations will be taken into account.

Passing Part 1 is a necessary condition for the correction of Part 2 (i.e. if Part 1 is not passed, the test is insufficient and Part 2 is not corrected). The final mark is determined only by the mark of Part 2. The written test is passed if the mark is greater than or equal to 18/30.

During the exam the student cannot consult notes or books, nor use calculators or other calculation tools.

The written tests will be held in the 5 exam sessions distributed in the months of January, February, June, July, September.

The written test can be replaced by two "in itinere" tests reserved for freshmen only (matricole). The first in itinere test will take place in November, the second immediately after the end of the lessons in January.

The structure and rules of the in itinere tests are the same as those of the written tests, except that they last 1 hour and 30 minutes. To pass the written part with the in itinere tests it is necessary to obtain at least 16/30 in each test with an average of at least 18/30. The final mark of the written part will be the average of the two marks (rounded up).

Only freshmen (matricole) are admitted to participate in the in itinere tests. The in itinere tests will be online.

The oral exam will take place in two ways:

1) if the student wants to confirm the grade obtained in the written test, the oral test is limited to the discussion of the written or in itinere tests;

2) if the student wants to improve the grade obtained in the written test or in the in itinere tests or obtain honors, then he/she must also pass an oral test on the topic of the program in the same session (definitions, statements of theorems, proofs of theorems).

For the in itinere tests the oral exam will be held in January.

In case of negative evaluation of the oral test (in both modalities) the mark obtained in the written test could be modified accordingly or even the written test will have to be repeated.

In case of negative evaluation of the oral test, the mark obtained in the written test could be modified accordingly or even the written test will have to be repeated.

To participate in a written test or in an in itinere test it is necessary to register through the system provided by the university, by the deadline indicated. The candidate is required to show a personal identification document with a photograph.

Further and updated information will appear on the Ariel page of the course.

The written test lasts 2 hours and consists of two distinct parts:

Part 1: The student will have to answer some simple questions indicating only the result.

Part 2: The student will have to solve some exercises on the topics covered during the lessons, indicating for some exercises only the answer, for others the procedure and the answer. The student will also have to answer some theoretical questions related to the program (definitions, statements of theorems, proofs of theorems).

In addition to the answers, the procedure to solve the exercises and the correctness of the demonstrations will be taken into account.

Passing Part 1 is a necessary condition for the correction of Part 2 (i.e. if Part 1 is not passed, the test is insufficient and Part 2 is not corrected). The final mark is determined only by the mark of Part 2. The written test is passed if the mark is greater than or equal to 18/30.

During the exam the student cannot consult notes or books, nor use calculators or other calculation tools.

The written tests will be held in the 5 exam sessions distributed in the months of January, February, June, July, September.

The written test can be replaced by two "in itinere" tests reserved for freshmen only (matricole). The first in itinere test will take place in November, the second immediately after the end of the lessons in January.

The structure and rules of the in itinere tests are the same as those of the written tests, except that they last 1 hour and 30 minutes. To pass the written part with the in itinere tests it is necessary to obtain at least 16/30 in each test with an average of at least 18/30. The final mark of the written part will be the average of the two marks (rounded up).

Only freshmen (matricole) are admitted to participate in the in itinere tests. The in itinere tests will be online.

The oral exam will take place in two ways:

1) if the student wants to confirm the grade obtained in the written test, the oral test is limited to the discussion of the written or in itinere tests;

2) if the student wants to improve the grade obtained in the written test or in the in itinere tests or obtain honors, then he/she must also pass an oral test on the topic of the program in the same session (definitions, statements of theorems, proofs of theorems).

For the in itinere tests the oral exam will be held in January.

In case of negative evaluation of the oral test (in both modalities) the mark obtained in the written test could be modified accordingly or even the written test will have to be repeated.

In case of negative evaluation of the oral test, the mark obtained in the written test could be modified accordingly or even the written test will have to be repeated.

To participate in a written test or in an in itinere test it is necessary to register through the system provided by the university, by the deadline indicated. The candidate is required to show a personal identification document with a photograph.

Further and updated information will appear on the Ariel page of the course.

MAT/01 - MATHEMATICAL LOGIC - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

Practicals: 48 hours

Lessons: 64 hours

Lessons: 64 hours

Professors:
Cavaterra Cecilia, Gori Anna

Educational website(s)

Professor(s)