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Continuum mathematics

A.Y. 2019/2020

Learning objectives

The aim of the course is to provide the basic mathematical tools, both from a conceptual and from a calculus point of view, which are essential to successfully attend a university undregraduate program in a scientific area. The course should also provide the required mathematics prerequisites for the other courses of the program.

Expected learning outcomes

At the end of the course, students should prove to have a sufficient knowledge of basic mathematics, that includes the main properties of sets, of the main number sets, in particular of real numbers, of functions between sets, of elementary functions, of combinatorics and of complex numbers. Also, she/he should know the basic results in the theory of differential and integral calculus for functions of one real variable. Finally, at the end of the course students should be able to apply the theoretical results to solve elementary problems and exercises and in

particular they should be able to tackle the following kinds of problems: computation of limits of sequences or functions, analysis of the continuity of a function, computation of derivates, study of the qualitative graph of a function, computation of the Taylor polynomial and expansion, computation of definite and indefinite integrals.

particular they should be able to tackle the following kinds of problems: computation of limits of sequences or functions, analysis of the continuity of a function, computation of derivates, study of the qualitative graph of a function, computation of the Taylor polynomial and expansion, computation of definite and indefinite integrals.

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Lesson period

First semester

**Course syllabus**

Real numbers and real functions.

The set of real numbers. Maximum, minimum, supremum, infimum. Elementary properties of functions. Elementary functions. Basics of combinatorics. Complex numbers.

Limits of sequences.

Definitions and first properties. Bounded sequences. Operations with limits. Comparison theorems. Monotone sequences. Undetermined forms. Special limits.

Limits of functions and continuous functions.

Definition and first properties of limits of functions and of continuous functions. Types of discontinuities. Limits and continuity of the composition of functions. Some important theorems on continuous functions.

Derivatives and study of functions.

Definition of derivatives. Computation of derivatives. Theorems of Fermat, Rolle, Lagrange and Cauchy and their consequences. Second and higher order derivatives. Applications to the study of functions. L'Hopital theorem and Taylor formula.

Integration

Definite integrals and method of exhaustion. Definition of integrable functions and classes of integrable functions. Properties of the definite integrals. Indefinite integrals. Fundamental theorem of integral calculus. Integration methods. Integration by parts and by substitution. Integration of rational functions.

The final program will we published at the end of classes on the course web page https://lrondimc.ariel.ctu.unimi.it

The set of real numbers. Maximum, minimum, supremum, infimum. Elementary properties of functions. Elementary functions. Basics of combinatorics. Complex numbers.

Limits of sequences.

Definitions and first properties. Bounded sequences. Operations with limits. Comparison theorems. Monotone sequences. Undetermined forms. Special limits.

Limits of functions and continuous functions.

Definition and first properties of limits of functions and of continuous functions. Types of discontinuities. Limits and continuity of the composition of functions. Some important theorems on continuous functions.

Derivatives and study of functions.

Definition of derivatives. Computation of derivatives. Theorems of Fermat, Rolle, Lagrange and Cauchy and their consequences. Second and higher order derivatives. Applications to the study of functions. L'Hopital theorem and Taylor formula.

Integration

Definite integrals and method of exhaustion. Definition of integrable functions and classes of integrable functions. Properties of the definite integrals. Indefinite integrals. Fundamental theorem of integral calculus. Integration methods. Integration by parts and by substitution. Integration of rational functions.

The final program will we published at the end of classes on the course web page https://lrondimc.ariel.ctu.unimi.it

**Prerequisites for admission**

There no particular prerequisites except the basic notions of mathematics that can be acquired in any secondary high school. All the topics of the course are developed from the very beginning and students are not required to know anything about them in advance.

**Teaching methods**

Lectures and classwork.

**Teaching Resources**

Textbook: P. Marcellini and C. Sbordone, Elementi di Analisi Matematica uno, Liguori, 2002.

Suggested exercise textbooks: P. Marcellini and C. Sbordone, Esercitazioni di Matematica, primo volume, parte prima and parte seconda, Liguori, 2013 and 2017.

Exercises provided on the course web page https://lrondimc.ariel.ctu.unimi.it

Further exercise textbook: M. Amar and A.M. Bersani, Esercizi di Analisi Matematica I - Esercizi e richiami di teoria, Edizioni La Dotta, 2014 (or any previous edition).

Suggested exercise textbooks: P. Marcellini and C. Sbordone, Esercitazioni di Matematica, primo volume, parte prima and parte seconda, Liguori, 2013 and 2017.

Exercises provided on the course web page https://lrondimc.ariel.ctu.unimi.it

Further exercise textbook: M. Amar and A.M. Bersani, Esercizi di Analisi Matematica I - Esercizi e richiami di teoria, Edizioni La Dotta, 2014 (or any previous edition).

**Assessment methods and Criteria**

The exam consists of a written test only. The written test lasts 2 hours and is divided in two parts.

Part 1: students should answer to some elementary questions on basic mathematics providing only the answer without any detailed explanation.

Part 2: students should solve some exercises on the topics of the course and answer to some questions of theoretical character on the program of the course. About exercises, both the correctness of the answer and the justification of it are evaluated.

Both parts are given to the students at the same time. Passing Part 1 is necessary for Part 2 to be corrected (that is, if Part 1 is failed the written test is failed as well and Part 2 is not corrected). The final score is determined by the score of Part 2 only. The exam is passed if the score is greater than or equal to 18/30.

The written test is a closed books test: no notes, books, calculators or similar instruments, items with a photocamera or able to connect to the internet are allowed.

Students are required to provide an ID card with a photograph.

The written test may be replaced by two midterm tests. The first midterm test usually takes place in the second half of November, the second at the same time with the first exam session of the January-February exam period. Structure and rules of the midterm tests are equal to those of the written tests, except the fact that they last 1 hour and 30 minutes. To pass the exam with the midterm tests students have to obtain a score of at least 15/30 in each of them with a mean of at least 18/30. The mean of the midterm tests will be the final score of the exam.

Any student, including those not attending classes or of previous academic years, are allowed to take the midterm tests.

Students are allowed, within a few days, to reject a score, obtained with the midterm tests or with any written test, and to retake the written test to improve their scores.

To sit at a written test or a midterm test, students are required to register through the university online system, within the deadline provided.

There will be 5 exam sessions: 2 or 3 in the January-February exam period, 2 or 1 in the June-July exam period and 1 in the September exam period. The actual distribution will be decided in accord with the students.

Part 1: students should answer to some elementary questions on basic mathematics providing only the answer without any detailed explanation.

Part 2: students should solve some exercises on the topics of the course and answer to some questions of theoretical character on the program of the course. About exercises, both the correctness of the answer and the justification of it are evaluated.

Both parts are given to the students at the same time. Passing Part 1 is necessary for Part 2 to be corrected (that is, if Part 1 is failed the written test is failed as well and Part 2 is not corrected). The final score is determined by the score of Part 2 only. The exam is passed if the score is greater than or equal to 18/30.

The written test is a closed books test: no notes, books, calculators or similar instruments, items with a photocamera or able to connect to the internet are allowed.

Students are required to provide an ID card with a photograph.

The written test may be replaced by two midterm tests. The first midterm test usually takes place in the second half of November, the second at the same time with the first exam session of the January-February exam period. Structure and rules of the midterm tests are equal to those of the written tests, except the fact that they last 1 hour and 30 minutes. To pass the exam with the midterm tests students have to obtain a score of at least 15/30 in each of them with a mean of at least 18/30. The mean of the midterm tests will be the final score of the exam.

Any student, including those not attending classes or of previous academic years, are allowed to take the midterm tests.

Students are allowed, within a few days, to reject a score, obtained with the midterm tests or with any written test, and to retake the written test to improve their scores.

To sit at a written test or a midterm test, students are required to register through the university online system, within the deadline provided.

There will be 5 exam sessions: 2 or 3 in the January-February exam period, 2 or 1 in the June-July exam period and 1 in the September exam period. The actual distribution will be decided in accord with the students.

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:
Camere Chiara, Rondi Luca

Professor(s)