#
Mathematical Analysis 1

A.Y. 2021/2022

Learning objectives

The aim of the course is to introduce the basic concepts of Mathematical Analysis, in particular those related to the study of differential calculus in one real variable.

Expected learning outcomes

At the end of the course students should prove to have a sufficient knowledge of basic differential calculus concepts concerning functions of one real variable; they should also be able to apply the fundamental calculus techniques to solve exercise, including those of some complexity.

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Analisi Matematica 1 (ediz.1)

Responsible

Lesson period

First semester

The didactic activity will be performed on the basis of what is stated by the university. As to what is stated at the moment (July 2020), classes will be given on-line as scheduled in the time-table, as well as the tutoring. More tutoring could be available in presence. Exams will be performed following what will be stated by the university. Any on-line activity will be performed by using Zoom. The content of the course as well as the rules and the level of the exams will be independent on the employed methods.

**Course syllabus**

The real and complex numbers.

Review of elementary set theory and of functions between sets. The set of real numbers R and its characterisation as an ordered field with the existence of supremum property. Existence of n-th roots of positive real numbers. The extended real line. Euclidean spaces. The complex field C. Algebraic and trigonometric forms. De Moivre's formula, n-th roots. The fundamental theorem of algebra. Sets of the same cardinality. Finite and infinite sets. Countable sets. Sets of the continuum cardinality. Uncountability of R.

Metric spaces and limits of sequences.

Definition of metric space, examples and metric balls. Bounded, open, closed, compact sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences and complete metric spaces. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. Definition of the Nepero's number e and applications.

Series of real numbers.

Series in R. Convergent, divergent and irregular series. Cauchy's criterion for convergence. Series with positive terms and convergence criteria: comparison, ratio and root, condensation. Absolute convergence of a series. Alternating sign series and Leibniz's criterion for convergence.

Limits and continuity.

Limits of functions. Equivalent definition using sequences. Continuity of functions between metric spaces. Preimages of open sets. Relationship between continuity and compactness. Continuity of the composition of functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities. Weierstrass Theorem. Intermediate value theorem and applications. Continuity of the inverse function.

Differential calculus for real functions of one real variable.

Differentiability and the definition of derivative. Derivatives of elementary functions. Computation of derivatives: algebraic operations, composition of functions and inverse function. The theorems of Rolle, Lagrange, Cauchy. Higher order derivatives.

Applications of differential calculus to the study of functions: monotonicity, local and global optimisation. De L'Hospital Theorem. Taylor formulas and applications.

The final program will we published at the end of classes on the Ariel website https://mtaralloam1.ariel.ctu.unimi.it

Review of elementary set theory and of functions between sets. The set of real numbers R and its characterisation as an ordered field with the existence of supremum property. Existence of n-th roots of positive real numbers. The extended real line. Euclidean spaces. The complex field C. Algebraic and trigonometric forms. De Moivre's formula, n-th roots. The fundamental theorem of algebra. Sets of the same cardinality. Finite and infinite sets. Countable sets. Sets of the continuum cardinality. Uncountability of R.

Metric spaces and limits of sequences.

Definition of metric space, examples and metric balls. Bounded, open, closed, compact sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences and complete metric spaces. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. Definition of the Nepero's number e and applications.

Series of real numbers.

Series in R. Convergent, divergent and irregular series. Cauchy's criterion for convergence. Series with positive terms and convergence criteria: comparison, ratio and root, condensation. Absolute convergence of a series. Alternating sign series and Leibniz's criterion for convergence.

Limits and continuity.

Limits of functions. Equivalent definition using sequences. Continuity of functions between metric spaces. Preimages of open sets. Relationship between continuity and compactness. Continuity of the composition of functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities. Weierstrass Theorem. Intermediate value theorem and applications. Continuity of the inverse function.

Differential calculus for real functions of one real variable.

Differentiability and the definition of derivative. Derivatives of elementary functions. Computation of derivatives: algebraic operations, composition of functions and inverse function. The theorems of Rolle, Lagrange, Cauchy. Higher order derivatives.

Applications of differential calculus to the study of functions: monotonicity, local and global optimisation. De L'Hospital Theorem. Taylor formulas and applications.

The final program will we published at the end of classes on the Ariel website https://mtaralloam1.ariel.ctu.unimi.it

**Prerequisites for admission**

Part of the ministerial program for secondary high schools, namely:

- basic algebra: monomials, polynomials, rational functions, powers, roots, exponentials and logarithms

- solving basic equations and inequalities

- basic theory of functions, elementary functions and their graphs, graphic interpretation of inequalities

- basic analytic geometry on the plane: lines, circles and parabolas

- basic trigonometry: sine, cosine and tangent, addition formulas

- basic set theory

- basic elements of logic

- basic algebra: monomials, polynomials, rational functions, powers, roots, exponentials and logarithms

- solving basic equations and inequalities

- basic theory of functions, elementary functions and their graphs, graphic interpretation of inequalities

- basic analytic geometry on the plane: lines, circles and parabolas

- basic trigonometry: sine, cosine and tangent, addition formulas

- basic set theory

- basic elements of logic

**Teaching methods**

Before the beginning of classes of the course of Mathematical Analysis 1, the course Elements of Basic Mathematics is given. Such a course aims to help students to acquire missing prerequisites and to deepen some aspects of basic mathematics. Attending classes of Elements of Basic Mathematics and passing its exam is strongly suggested.

The course of Mathematical Analysis 1 consists of lectures and exercise sessions, that alternate according to the timetable published on the Ariel website: attending both lectures and exercise sessions is strongly suggested.

A list of exercises about the topics already discussed during classes is published every week on the Ariel website.

Two supplementary activities are offered to the students. In the first one, each week a tutor gives a special two-hours exercise session, solving some of the proposed exercises and answering questions posed by the students. The second activity is in preparation to the first midterm exam. In the second half of October, two tests formed by exercises similar to those of the written exams are published on the Ariel website. Students have about a week to write down the solution and, on a voluntary basis, turn the test in to be corrected by two tutors. The duty of the tutors is twofold: they need to correct the tests and give them back to each student with an explanation on the correction, so that students can realise the mistakes they made and what is expected from them in the written exams.

The course of Mathematical Analysis 1 consists of lectures and exercise sessions, that alternate according to the timetable published on the Ariel website: attending both lectures and exercise sessions is strongly suggested.

A list of exercises about the topics already discussed during classes is published every week on the Ariel website.

Two supplementary activities are offered to the students. In the first one, each week a tutor gives a special two-hours exercise session, solving some of the proposed exercises and answering questions posed by the students. The second activity is in preparation to the first midterm exam. In the second half of October, two tests formed by exercises similar to those of the written exams are published on the Ariel website. Students have about a week to write down the solution and, on a voluntary basis, turn the test in to be corrected by two tutors. The duty of the tutors is twofold: they need to correct the tests and give them back to each student with an explanation on the correction, so that students can realise the mistakes they made and what is expected from them in the written exams.

**Teaching Resources**

Ariel website: Mathematical Analysis 1 https://mtaralloam1.ariel.ctu.unimi.it

Textbook: P. M. Soardi, Analisi Matematica, II edition, Città Studi, 2010.

Other suggested textbooks:

- W. Rudin, Principi di Analisi Matematica, Mc Graw Hill, 1997.

- G. Gilardi, Analisi Matematica di Base, II edition, McGraw-Hill, 2011

- E. Giusti, Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000.

- L. De Michele, G. L. Forti: Analisi Matematica: problemi ed esercizi, CLUP, 2000.

Some material is available on the Ariel website of the course, namely:

- detailed Course Syllabus

- one or more lists of exercises for every topic

- notes by the teacher on some specific topics

- texts of all the written exams done in the last years

Textbook: P. M. Soardi, Analisi Matematica, II edition, Città Studi, 2010.

Other suggested textbooks:

- W. Rudin, Principi di Analisi Matematica, Mc Graw Hill, 1997.

- G. Gilardi, Analisi Matematica di Base, II edition, McGraw-Hill, 2011

- E. Giusti, Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000.

- L. De Michele, G. L. Forti: Analisi Matematica: problemi ed esercizi, CLUP, 2000.

Some material is available on the Ariel website of the course, namely:

- detailed Course Syllabus

- one or more lists of exercises for every topic

- notes by the teacher on some specific topics

- texts of all the written exams done in the last years

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam, both of which contribute to the final score.

During the written exam, students must solve some exercises, with the aim of assessing their ability to solve problems in the field of Mathematical Analysis. The duration of the written exam is adequate to the number and difficulty of the assigned exercises, in any case it does exceed three hours. The written exam of the first exam session might be replaced by two midterm tests, the first one in November and the second one in January. The outcome of the written exams and of the midterm tests will be readily made available on the online system.

The oral exam can be taken only if the written exam (or the midterm tests) of the same exam session has been successfully passed.

In the oral exam students are required to illustrate some of the main topics and results developed during the course, describing them in specific situations and using them to solve suitable problems in Mathematical Analysis, in order to evaluate their knowledge and actual comprehension of the arguments covered as well as their capability to apply them.

The exam is passed if both parts, written and oral, are successfully passed, with a combined score greater than or equal to 18/30.

Final marks are given using the numerical range 0-30 and will be immediately communicated at the end of the oral exam.

During the written exam, students must solve some exercises, with the aim of assessing their ability to solve problems in the field of Mathematical Analysis. The duration of the written exam is adequate to the number and difficulty of the assigned exercises, in any case it does exceed three hours. The written exam of the first exam session might be replaced by two midterm tests, the first one in November and the second one in January. The outcome of the written exams and of the midterm tests will be readily made available on the online system.

The oral exam can be taken only if the written exam (or the midterm tests) of the same exam session has been successfully passed.

In the oral exam students are required to illustrate some of the main topics and results developed during the course, describing them in specific situations and using them to solve suitable problems in Mathematical Analysis, in order to evaluate their knowledge and actual comprehension of the arguments covered as well as their capability to apply them.

The exam is passed if both parts, written and oral, are successfully passed, with a combined score greater than or equal to 18/30.

Final marks are given using the numerical range 0-30 and will be immediately communicated at the end of the oral exam.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 9

Practicals: 48 hours

Lessons: 45 hours

Lessons: 45 hours

Professors:
Calanchi Marta, Stuvard Salvatore

### Analisi Matematica 1 (ediz.2)

Responsible

Lesson period

First semester

**Course syllabus**

The real and complex numbers.

Review of elementary set theory and of functions between sets. The set of real numbers R and its characterisation as an ordered field with the existence of supremum property. Existence of n-th roots of positive real numbers. The extended real line. Euclidean spaces. The complex field C. Algebraic and trigonometric forms. De Moivre's formula, n-th roots. The fundamental theorem of algebra. Sets of the same cardinality. Finite and infinite sets. Countable sets. Sets of the continuum cardinality. Uncountability of R.

Metric spaces and limits of sequences.

Definition of metric space, examples and metric balls. Bounded, open, closed, compact sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences and complete metric spaces. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. Definition of the Nepero's number e and applications.

Series of real numbers.

Series in R. Convergent, divergent and irregular series. Cauchy's criterion for convergence. Series with positive terms and convergence criteria: comparison, ratio and root, condensation. Absolute convergence of a series. Alternating sign series and Leibniz's criterion for convergence.

Limits and continuity.

Limits of functions. Equivalent definition using sequences. Continuity of functions between metric spaces. Preimages of open sets. Relationship between continuity and compactness. Continuity of the composition of functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities. Weierstrass Theorem. Intermediate value theorem and applications. Continuity of the inverse function.

Differential calculus for real functions of one real variable.

Differentiability and the definition of derivative. Derivatives of elementary functions. Computation of derivatives: algebraic operations, composition of functions and inverse function. The theorems of Rolle, Lagrange, Cauchy. Higher order derivatives.

Applications of differential calculus to the study of functions: monotonicity, local and global optimisation. De L'Hospital Theorem. Taylor formulas and applications.

The final program will we published at the end of classes on the Ariel website https://mtaralloam1.ariel.ctu.unimi.it

Review of elementary set theory and of functions between sets. The set of real numbers R and its characterisation as an ordered field with the existence of supremum property. Existence of n-th roots of positive real numbers. The extended real line. Euclidean spaces. The complex field C. Algebraic and trigonometric forms. De Moivre's formula, n-th roots. The fundamental theorem of algebra. Sets of the same cardinality. Finite and infinite sets. Countable sets. Sets of the continuum cardinality. Uncountability of R.

Metric spaces and limits of sequences.

Definition of metric space, examples and metric balls. Bounded, open, closed, compact sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences and complete metric spaces. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. Definition of the Nepero's number e and applications.

Series of real numbers.

Series in R. Convergent, divergent and irregular series. Cauchy's criterion for convergence. Series with positive terms and convergence criteria: comparison, ratio and root, condensation. Absolute convergence of a series. Alternating sign series and Leibniz's criterion for convergence.

Limits and continuity.

Limits of functions. Equivalent definition using sequences. Continuity of functions between metric spaces. Preimages of open sets. Relationship between continuity and compactness. Continuity of the composition of functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities. Weierstrass Theorem. Intermediate value theorem and applications. Continuity of the inverse function.

Differential calculus for real functions of one real variable.

Differentiability and the definition of derivative. Derivatives of elementary functions. Computation of derivatives: algebraic operations, composition of functions and inverse function. The theorems of Rolle, Lagrange, Cauchy. Higher order derivatives.

Applications of differential calculus to the study of functions: monotonicity, local and global optimisation. De L'Hospital Theorem. Taylor formulas and applications.

The final program will we published at the end of classes on the Ariel website https://mtaralloam1.ariel.ctu.unimi.it

**Prerequisites for admission**

Part of the ministerial program for secondary high schools, namely:

- basic algebra: monomials, polynomials, rational functions, powers, roots, exponentials and logarithms

- solving basic equations and inequalities

- basic theory of functions, elementary functions and their graphs, graphic interpretation of inequalities

- basic analytic geometry on the plane: lines, circles and parabolas

- basic trigonometry: sine, cosine and tangent, addition formulas

- basic set theory

- basic elements of logic

- basic algebra: monomials, polynomials, rational functions, powers, roots, exponentials and logarithms

- solving basic equations and inequalities

- basic theory of functions, elementary functions and their graphs, graphic interpretation of inequalities

- basic analytic geometry on the plane: lines, circles and parabolas

- basic trigonometry: sine, cosine and tangent, addition formulas

- basic set theory

- basic elements of logic

**Teaching methods**

Before the beginning of classes of the course of Mathematical Analysis 1, the course Elements of Basic Mathematics is given. Such a course aims to help students to acquire missing prerequisites and to deepen some aspects of basic mathematics. Attending classes of Elements of Basic Mathematics and passing its exam is strongly suggested.

The course of Mathematical Analysis 1 consists of lectures and exercise sessions, that alternate according to the timetable published on the Ariel website: attending both lectures and exercise sessions is strongly suggested.

A list of exercises about the topics already discussed during classes is published every week on the Ariel website.

Two supplementary activities are offered to the students. In the first one, each week a tutor gives a special two-hours exercise session, solving some of the proposed exercises and answering questions posed by the students. The second activity is in preparation to the first midterm exam. In the second half of October, two tests formed by exercises similar to those of the written exams are published on the Ariel website. Students have about a week to write down the solution and, on a voluntary basis, turn the test in to be corrected by two tutors. The duty of the tutors is twofold: they need to correct the tests and give them back to each student with an explanation on the correction, so that students can realise the mistakes they made and what is expected from them in the written exams.

The course of Mathematical Analysis 1 consists of lectures and exercise sessions, that alternate according to the timetable published on the Ariel website: attending both lectures and exercise sessions is strongly suggested.

A list of exercises about the topics already discussed during classes is published every week on the Ariel website.

Two supplementary activities are offered to the students. In the first one, each week a tutor gives a special two-hours exercise session, solving some of the proposed exercises and answering questions posed by the students. The second activity is in preparation to the first midterm exam. In the second half of October, two tests formed by exercises similar to those of the written exams are published on the Ariel website. Students have about a week to write down the solution and, on a voluntary basis, turn the test in to be corrected by two tutors. The duty of the tutors is twofold: they need to correct the tests and give them back to each student with an explanation on the correction, so that students can realise the mistakes they made and what is expected from them in the written exams.

**Teaching Resources**

Ariel website: Mathematical Analysis 1 https://mtaralloam1.ariel.ctu.unimi.it

Textbook: P. M. Soardi, Analisi Matematica, II edition, Città Studi, 2010.

Other suggested textbooks:

- W. Rudin, Principi di Analisi Matematica, Mc Graw Hill, 1997.

- G. Gilardi, Analisi Matematica di Base, II edition, McGraw-Hill, 2011

- E. Giusti, Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000.

- L. De Michele, G. L. Forti: Analisi Matematica: problemi ed esercizi, CLUP, 2000.

Some material is available on the Ariel website of the course, namely:

- detailed Course Syllabus

- one or more lists of exercises for every topic

- notes by the teacher on some specific topics

- texts of all the written exams done in the last years

Textbook: P. M. Soardi, Analisi Matematica, II edition, Città Studi, 2010.

Other suggested textbooks:

- W. Rudin, Principi di Analisi Matematica, Mc Graw Hill, 1997.

- G. Gilardi, Analisi Matematica di Base, II edition, McGraw-Hill, 2011

- E. Giusti, Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000.

- L. De Michele, G. L. Forti: Analisi Matematica: problemi ed esercizi, CLUP, 2000.

Some material is available on the Ariel website of the course, namely:

- detailed Course Syllabus

- one or more lists of exercises for every topic

- notes by the teacher on some specific topics

- texts of all the written exams done in the last years

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam, both of which contribute to the final score.

During the written exam, students must solve some exercises, with the aim of assessing their ability to solve problems in the field of Mathematical Analysis. The duration of the written exam is adequate to the number and difficulty of the assigned exercises, in any case it does exceed three hours. The written exam of the first exam session may be replaced by two midterm tests, the first one in November and the second one in January. The outcome of the written exams and of the midterm tests will be readily made available on the online system.

The oral exam can be taken only if the written exam (or the midterm tests) of the same exam session has been successfully passed, with a score of at least 15/30.

In the oral exam students are required to illustrate some of the main topics and results developed during the course, describing them in specific situations and using them to solve suitable problems in Mathematical Analysis, in order to evaluate their knowledge and actual comprehension of the arguments covered as well as their capability to apply them.

The exam is passed if both parts, written and oral, are successfully passed, with a combined score greater than or equal to 18/30.

Final marks are given using the numerical range 0-30 and will be immediately communicated at the end of the oral exam.

During the written exam, students must solve some exercises, with the aim of assessing their ability to solve problems in the field of Mathematical Analysis. The duration of the written exam is adequate to the number and difficulty of the assigned exercises, in any case it does exceed three hours. The written exam of the first exam session may be replaced by two midterm tests, the first one in November and the second one in January. The outcome of the written exams and of the midterm tests will be readily made available on the online system.

The oral exam can be taken only if the written exam (or the midterm tests) of the same exam session has been successfully passed, with a score of at least 15/30.

In the oral exam students are required to illustrate some of the main topics and results developed during the course, describing them in specific situations and using them to solve suitable problems in Mathematical Analysis, in order to evaluate their knowledge and actual comprehension of the arguments covered as well as their capability to apply them.

The exam is passed if both parts, written and oral, are successfully passed, with a combined score greater than or equal to 18/30.

Final marks are given using the numerical range 0-30 and will be immediately communicated at the end of the oral exam.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 9

Practicals: 48 hours

Lessons: 45 hours

Lessons: 45 hours

Professors:
Ciraolo Giulio, Messina Francesca

Professor(s)

Reception:

Tuesday 15:30 - 17:30 and Wednesday 13:30 - 15:30

Room 1005, Department of Mathematics, Via Cesare Saldini 50, 20133 Milano