Mathematical Analysis I R - Z

The course in Mathematical Analysis I aims to provide students with basic knowledge of the set of real numbers, the set of complex numbers, sequences and numerical series, the concept of a real-valued function of a real variable and its related properties, the notions of limit and continuity, and differential calculus for real-valued functions of a real variable. The course also covers integral calculus for real-valued functions of a real variable and some types of ordinary differential equations.

In particular, the learning objectives of the course, defined according to the Dublin Descriptors, are as follows:

Knowledge and understanding:
Students will acquire some fundamental mathematical concepts and develop computational and manipulation skills concerning the most common objects of basic Mathematical Analysis. These include complex numbers, limits and derivatives of real-valued functions of a real variable, integral calculus, and differential equations.

Applying knowledge and understanding:
Students will be able to apply the knowledge acquired to the basic mathematical modelling of classical problems in Engineering.

Making judgements:
Students will be encouraged to independently deepen their knowledge and practice exercises on the topics covered in the course. Constructive interaction among students and continuous discussion with the instructor will be strongly encouraged, enabling students to critically monitor their own learning process.

Communication skills:
Attending lectures and reading the recommended textbooks will help students become familiar with the rigor of mathematical language. Through continuous interaction with the instructor, students will learn to communicate the knowledge they have acquired with rigor and clarity, both orally and in writing. By the end of the course, students will have learned that mathematical language is a valuable tool for communicating clearly in scientific contexts.

Learning skills:
Students will be guided in improving and refining their study methods. In particular, through appropriate guided exercises, they will acquire the ability to independently approach new topics by identifying the prerequisites necessary for their understanding.

The course includes theoretical lectures and exercise sessions related to the topics covered. Both the theoretical lectures and the exercise sessions will be conducted in a traditional, in-person format.

The course consists of 35 hours of theoretical lectures and 60 hours of other activities (typically, exercise sessions or practical explorations of selected theoretical topics).

If the course is delivered in a blended or distance-learning format, the necessary adjustments may be made to the arrangements described above in order to ensure that the planned syllabus is fully covered.

Proofs of the topics marked with an asterisk (*) are not required in the examination.

1. Number Systems

Natural numbers, integers, rational numbers, and real numbers. General properties of the sets N of natural numbers, Z of integers, and Q of rational numbers. The set R of real numbers and its properties. Density of Q in R. Completeness of R*. Upper and lower bounds of a numerical set.

Complex numbers. Basic definitions. Polar coordinates in the plane. Algebraic and trigonometric forms of a complex number. Product and powers of complex numbers in trigonometric form. Exponential form of a complex number. Product and powers of complex numbers in exponential form. nth roots of a complex number. Algebraic equations*.

2. Sequences

Numerical sequences: basic definitions and limits.

Theorems on limits: uniqueness, preservation of sign, and comparison. Algebra of limits.

Monotone sequences and subsequences. Bolzano–Weierstrass theorem. Cauchy convergence criterion.

3. Numerical Series

Basic definitions. Standard series. Convergence criteria for numerical series: addition, removal, and modification of a finite number of terms of a numerical series*; algebra of numerical series; necessary condition for the convergence of a numerical series; properties of numerical series with non-negative terms.

Convergence criteria for series with non-negative terms: ratio test, root test, comparison test, limit comparison test, Cauchy condensation test* and its applications.

Convergence criteria for alternating series. Leibniz's test and its corollaries.

Absolute convergence criterion.

4. Functions and Their Properties

Functions. Basic definitions and properties of functions. Elementary functions.

Limits. Topology of R. Definitions of limits. Algebra of limits. Indeterminate forms. Theorems on limits: sequential characterization of the limit of a function (sequential criterion or bridge theorem), uniqueness of the limit, preservation of sign, and comparison theorems. Theorem on the limit of monotone functions*. Theorem on the limit of a composite function.

5. Continuous Functions and Local Comparison

Continuous functions. Definition of continuity. Continuity of elementary functions and operations on continuous functions. Singular points: removable singularities, first-kind and second-kind singularities. Properties of continuous functions: Intermediate Value Theorem, Intermediate Value Theorem for zeros, and Weierstrass theorem. Injectivity and strict monotonicity of continuous functions*. Continuity theorem for the inverse function*. Standard limits.

Local comparison of functions. Bachmann–Landau notation, comparison of infinitesimals and infinities. Asymptotes.

Uniformly continuous functions. Cantor's theorem.

6. Differential Calculus

Definition of the incremental ratio, differentiability at a point, and derivative. Geometric interpretation of the first derivative. Relationship between continuity and differentiability. Derivatives of elementary functions. Points of non-differentiability. Definition of the differential. Algebra of derivatives. Differentiation rule for composite functions. Differentiation rule for inverse functions.

Fundamental theorems of Differential Calculus and their consequences. Fermat's theorem, Rolle's theorem, Cauchy's mean value theorem, Lagrange's mean value theorem and its consequences. L'Hôpital's theorem. Theorem on the limit of the derivative. Limits of derivatives and points of non-differentiability. Higher-order derivatives. Taylor's formula*. Concave and convex functions and inflection points: definitions and theorems. Higher-derivative criteria for identifying stationary points. Qualitative analysis of the graph of a function.

7. Integral Calculus

Indefinite integrals. Antiderivatives of a function on an interval and indefinite integrals. Rules of indefinite integration: linearity, integration by parts, and integration by substitution. Integration of rational functions. Various techniques of indefinite integration.

Definite integrals. Riemann integral: lower and upper sums, definition of a Riemann-integrable function, Riemann integrability condition*. Classes of Riemann-integrable functions. Properties of the Riemann integral*. Integral over an oriented interval. Definition of the integral mean value and its geometric interpretation. Mean Value Theorem for integrals. Fundamental Theorem of Integral Calculus (Torricelli's theorem).

Improper integrals. Improper integrals over unbounded intervals. Improper integrals over bounded intervals. Convergence criteria: algebra of improper integrals, integrability of non-negative functions*, comparison test*, absolute convergence criterion*, limit comparison test*.

8. Ordinary Differential Equations

Basic definitions.

Solution methods for selected types of differential equations. First-order separable differential equations, first-order linear differential equations, second-order linear differential equations with constant coefficients, and nth-order linear differential equations with constant coefficients.

Contribution of the Course to the Objectives of the UN 2030 Agenda for Sustainable Development

Goal 4: QUALITY EDUCATION

Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all.

Goal 5: GENDER EQUALITY

Achieve gender equality and empower female students by promoting greater confidence, self-esteem, and awareness.

Goal 8: DECENT WORK AND ECONOMIC GROWTH

Promote sustained, inclusive, and sustainable economic growth, full and productive employment, and decent work for all.

Self-Assessment Tests

During the periods of teaching activities, students will be given a number of self-assessment tests. These tests are intended to guide students through the gradual learning of the topics covered during the lectures. In addition, self-assessment tests enable the instructor to promptly implement any targeted supplementary activities to support students in preparation for the examinations.

Examination Structure

The Mathematical Analysis I examination can be completed through one of two examination formats:

• Format 1: two written mid-term tests and one oral examination;
• Format 2: one comprehensive written examination and one oral examination.

A description of the two examination formats is provided below.

Format 1

Format 1 consists of two written mid-term tests: the first is held during the teaching suspension period and the second at the end of the course. The second written test may be taken only if the first written test has been passed. After passing both written mid-term tests, the student must take an oral examination.

Structure of the Written Mid-Term Tests

Each mid-term test consists of four exercises and lasts 120 minutes.

Assessment of the Written Mid-Term Tests

The maximum grade obtainable in each written mid-term test is 30/30. A written mid-term test is considered passed if the student obtains a score of at least 18/30.

Each exercise is assigned a specific score. The maximum score is awarded if the solution is correct; otherwise, a partial score is awarded according to the errors made.

If the total score is equal to or greater than 15/30 but lower than 18/30, the Examination Board may provisionally admit the student to the oral examination and may require the student to solve one or more additional exercises beforehand.

Oral Examination

The oral examination covers all the topics included in the course (see the section “Course Contents”) and must be taken by the end of the First Examination Session, according to a schedule established by the Examination Board.

The final grade takes into account both the grades obtained in the written tests and the assessment of the oral examination. If the student does not pass the oral examination or decides not to attend the scheduled oral examination, the examination must be taken again from the beginning, following Format 2.

Format 2: Comprehensive Written Examination and Oral Examination

Under this format, students take a single written examination covering the course contents. If the written examination is passed, the student must then take an oral examination. The schedule of oral examinations will be established by the Examination Board.

Examination Dates

Examination dates are available on the website of the degree programme.

Structure of the Written Examination

The written examination consists of four exercises and lasts 120 minutes.

Assessment of the Written Examination

The maximum grade obtainable in the written examination is 30/30. The written examination is considered passed if the student obtains a score of at least 18/30.

Each exercise is assigned a specific score. The maximum score assigned to each exercise is awarded if and only if the solution is correct. Otherwise, a partial score is awarded according to the errors made.

If the total score is equal to or greater than 15/30 but lower than 18/30, the Examination Board may provisionally admit the student to the oral examination and may require the student to solve one or more additional exercises beforehand.

Oral Examination and Final Grade

The oral examination covers all the topics included in the course.

The final grade takes into account both the grade obtained in the written examination and the assessment of the oral examination. If the student does not pass the oral examination or decides not to attend the scheduled oral examination, the examination must be taken again from the beginning, including a new written examination.

Note: Information for Students with Disabilities and/or Specific Learning Disorders (SLD)

To ensure equal opportunities and in compliance with current legislation, students concerned may request an individual meeting in order to plan any appropriate compensatory and/or exemption measures, taking into account the educational objectives and the student's specific needs.

Students may also contact the CInAP faculty coordinator (Centre for Active and Participatory Integration – Services for Students with Disabilities and/or Specific Learning Disorders) of the Department of Electrical, Electronic and Computer Engineering.

Note: Remote Assessment

The assessment of learning outcomes may also be conducted remotely, should circumstances require it. In such cases, the duration of the written examination may be subject to change.

Examination Registration

In order to take the final examination, students must register through the SmartEdu portal. For any technical issues concerning examination registration, students should contact the Teaching Office.

All the topics mentioned in the syllabus may be included in the examination.

Attending lectures, studying the recommended textbooks, and studying the material provided by the instructor (lecture notes and collections of solved and proposed exercises) enable students to gain a clear and detailed understanding of the types of questions that may be included in the examination.

An adequate presentation of the theory requires the use of the rigorous language characteristic of the discipline, together with the presentation of simple examples and counterexamples that clarify the concepts discussed (definitions, propositions, theorems, and corollaries).

The main types of exercises related to the contents of the Mathematical Analysis I course are the following:

• Finding the extrema of a numerical set. Finding the interior points, boundary points, and accumulation points of a given numerical set.
• Exercises involving complex numbers.
• Computing limits of sequences.
• Determining the convergence or divergence of numerical series.
• Computing limits of real functions of a real variable. Studying the continuity, boundedness, and invertibility of a function.
• Qualitative analysis of the graph of a function.
• Computing indefinite and definite integrals.
• Determining an antiderivative of a function satisfying a given condition.
• Studying the convergence of improper integrals and computing improper integrals.
• Determining the general solution of an ordinary differential equation.
• Determining a solution of an ordinary differential equation satisfying a given condition.