ANALISI MATEMATICA I

Module MODULO B

The aim of the course of Mathematical Analysis I - Module A is to give the basic skills on real and complex numbers and Differential Calculus for real functions of one real variable.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

• Knowledge and understanding: The student will learn some basic concepts of Mathematical Analysis and will develop both computing ability and the capacity of manipulating some common mathematical structures, as complex numbers, limits and derivatives.
• Applying knowledge and understanding: The student will be able to apply the acquired knowledge in the basic processes of mathematical modeling of classical problems arising from Engineering.
• Makin​g judgements: The student will be stimulated to autonomously deepen his/her knowledge and to carry out exercises on the topics covered by the course. Constructive discussion between students and constant discussion with the teacher will be strongly recommended so that the student will be able to critically monitor his/her own learning process.
• Communication skills: The attendance of the lessons and the reading of the recommended books will help the student to be familiar with the rigor of the mathematical language. Through constant interaction with the teacher, the student will learn to communicate the acquired knowledge with rigor and clarity, both in oral and written form. At the end of the course the student will have learned that mathematical language is useful for communicating clearly in the scientific field.
• Learning skills: The student will be guided in the process of perfecting his/her study method. In particular, through suitable guided exercises, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.

The course of Mathematical Analysis I (12 CFU) for the Bachelor Degree in Electronic Engineering is divided into two modules: Mathematical Analysis I – Module A (6 CFU) and Mathematical Analysis I – Module B (6 CFU).

Theory lectures and exercises related to the topics covered will be offered. Theory lectures and exercises will be carried out in frontal mode.

For the Module B of the course, 28 hours of theory and 30 hours of other activities (typically, these are exercises) are expected.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.

In-depth knowledge of the contents of Arithmetic, Algebra, Analytical Geometry, Trigonometry usually covered in High Schools and of the contents covered in Module A.

The proofs of the topics marked with an asterisk are not required during the exam.

1. Sets of numbers. 
• Natural numbers, integer numbers, rational numbers, real numbers. Basic notions on the set N of natural numbers, on the set Z of integer numbers and on the set Q of rational numbers*. The set R of real numbers. Bounds for numerical sets. Archimedean property*. Density of Q and R-Q in R*. Powers with real exponent*.
• Complex numbers. Basic definitions. Total order. Polar coordinates in the plane. Trigonometric form of a complex number. Product and power of complex numbers in trigonometric form. Exponential form of a complex number. Product and power of complex numbers in exponential form. nth roots of a complex number. Algebraic equations*.
2. Functions and limits. 
• Functions. Basic definitions.
• Limits. Topology in R. Definitions of limits. Algebra of limits. Indeterminate forms. Theorems on limits: uniqueness of the limit, sign permanence, comparison theorems. Theorem on limits of monotone functions*. Theorem on the limit of the composite function*. Sequences: basic definitions, limits, sequential characterization of the limit of a function*, subsequences.
3. Continuous functions and local comparison.
• Continuous functions. Definition of continuous function and basic results. Continuity of elementary functions and operations with continuous functions. Singularity points: removable singularity, singularity of the first kind and singularity of the second kind. Properties of continuous functions: local properties and global properties. Theorem on the existence of zeroes, Intermediate value theorem, Weierstrass Theorem. Injectivity and strict monotonicity for continuous functions. Theorem on the continuity of the inverse function*. Fundamental limits.
• Local comparison of functions. Bachmann-Landau symbols, comparison between infinitesimal and infinite functions. Asymptotes.
• Uniformly continuous functions*.
4. Differential Calculus.
• Definition of differentiable function and definition of derivative. Geometric and kinematic meaning of the first derivative. Relationship between continuity and differentiability. Derivatives of the elementary functions. One-sided derivatives. Point of non-differentiability. Definition of differential. Rules of differentiation. Theorem on the differentiability of the composite function. Theorem on the differentiability of the inverse function.
• Fundamental theorems of Differential Calculus and their consequences. Fermat's Theorem, Rolle's Theorem, Lagrange's Theorem and its consequences (functions with null derivative on an interval, monotonicity test and local extrema, characterization of differentiable strictly monotone funcitons). De L'Hôpital's Theorem*. Theorem on the limit of the derivative. Limit of the derivative and points of non-differentiability. Higher-order derivatives. Taylor formula with Peano’s remainder and Taylor formula with Lagrange’s remainder*. Concave functions, convex functions: definitions of concave function and convex function, inflection points, necessary condition for the inglection points, relationship between the concavity/convexity and the sign of the second derivative, test for the inflection points. Higher-order derivative test for the study of stationary points. Qualitative study of a function.

Teaching’s contribution to the Goals of the Agenda 2030 for Sustainable Development:

GOAL 4: Quality education. Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all.

Self-assessment tests

During the period of delivery of the lessons, some self-assessment tests will be administered. These self-assessment tests have the task of guiding the student in the gradual learning of the contents displayed during the lessons. In addition, the self-assessment tests allow the teacher to quickly implement any additional activities aimed at supporting students in view of the exams.

Structure of the exam

The Mathematical Analysis I exam can be passed in two ways:

Mode 1: written mid-term tests and oral tests;

Mode 2: written test and oral test.

The examination procedures are described below.

Mode 1

Mode 1 includes two intermediate mid-term tests: the first at the end of the first teaching period and the second at the end of the second teaching period. It is possible to take the second written mid-term test only if the first has been previously passed. Once both written intermediate tests have been passed, the student will have to take an oral exam

Written mid-terms dates.

There is a date for the written mid-term test related to Module A at the end of the first teaching period and a date for the written mid-term test related to Module B at the end of the second teaching period. The dates of the intermediate test can be found on the degree course website.

Structure of the written mid-term tests

In the intermediate written test, three exercises will be proposed. The duration of the intermediate written test is 90 minutes.

Evaluation of the written mid-term tests.

The maximum grade obtainable in the first written mid-term test is 30/30. The first written mid-term test is passed if the student has achieved a score of at least 12/30. A score will be assigned to each exercise. The maximum score will be assigned if the exercise is carried out correctly, otherwise, a partial score will be assigned and it will be determined based on the errors made. If the total score is greater than or equal to 15 and less than 18, the Examination Commission may admit the student to the oral test with reservations and may require some preliminary exercises to the student.

Oral exam.

The oral exam covers all the contents of the course (see the section “Detailed Course Contents” in the Syllabus related to Module A and in the Syllabus related to Module B) and must be taken within the first round of Second Exam Session according to a calendar that will be prepared by the Examination Commission.

The final evaluation will take into account the outcome of the written exam. If the student does not pass the oral exam on Module A or decides not to show up for the convocation, it will be necessary to take the exam again according to Mode 2.

Mode 2: written test and oral test

In this mode, a single written test is proposed that focuses on the contents of Module A and the contents of Module B and, if passed, the student will have to take the oral test on the entire program. The schedule of the oral tests will be prepared by the Examination Commission.

Dates of the exam sessions. 

The dates of the exam sessions are available on the degree course website.

Structure of the written test.

Four exercises will be proposed in the written test and it lasts 120 minutes. 

Evaluation of the written test. 

The maximum score obtainable in the written test is 30/30. The written test is considered passed if the student has achieved a score of at least 18/30. A score will be assigned to each exercise. Each exercise will be assigned the maximum score expected if and only if it is carried out correctly. Otherwise, a partial score will be assigned that will be determined based on the errors made. In the event that the total score is greater than or equal to 15 and less than 18, the Examination Commission may admit the student to the oral exam with reservations and may require some preliminary exercises to the student. 

Oral exam. 

The oral exam covers all the contents of the course (see the section “Detailed Course Contents” in the Syllabus related to Module A and in the Syllabus related to Module B). The final evaluation will take into account the outcome of the written exam. If the student does not pass the oral exam or decides not to show up for the convocation, it will be necessary to take the exam again according to Mode 2.

Grading criteria.

Written and oral tests will assess students' understanding of the topics covered in the course and their ability to use the relevant language. A successful exam requires a complete and accurate presentation of definitions, statements, and examples. Assessment of content acquisition also includes theorem proofs, where applicable. The final grade is expressed in thirtieths according to the following table:

• NOT PASSED (<18): The student demonstrates a poor and fragmented knowledge of the subject matter, exhibits serious comprehension errors and do not present the contents in an acceptable manner;
• 18-21: The student demonstrates limited knowledge and a basic understanding of the subject matter, presents the contents unclearly and with little precision;
• 22-24: The student demonstrates an acceptable knowledge and a basic understanding of the subject matter, presents the contents correctly but not in a fully structured way;
• 25-27: The student demonstrates a broad knowledge and adequate understanding of the subject matter, presents the contents correctly but not in a complete way;
• 28-29: The student demonstrates an in-depth knowledge and a solid understanding of the subject matter, presents the contents clearly and in a fully structured way;
• 30-30 cum laude: The student demonstrates a complete and detailed knowledge and an excellent understanding of the subject matter, presents the contents clearly and in a fully structured way.

Note. Information for students with disabilities and / or SLD.

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of the Department of Electrical, Electronic and Computer Engineering.

Note. Verification of learning can also be carried out electronically, should the conditions require it. In this case, the duration of the written test may be subject to change.

All the topics mentioned in the program can be requested during the exam.

The attendance of the lessons, the study of the recommended texts and the study of the material provided by the teacher (handouts and collections of exercises carried out and proposed) allow the student to have a clear and detailed idea of ​​the questions that may be proposed during the exam.

An adequate exposition of the theory involves the use of the rigorous language characteristic of the discipline, the exposition of simple examples and counterexamples that clarify the exposed concepts (definitions, propositions, theorems, corollaries).

The main types of exercises related to the Module A of the course of Mathematical Analysis I are:

• Finding the bounds of a numerical set. Finding interior points, boundary points, accumulation points of an assigned numerical set.
• Exercises on complex numbers.
• Calculation of sequences limits.
• Calculation of limits of functions. Study of the continuity, boundedness and invertibility of real functions of a real variable.
• Qualitative study of a function and applications.