Università degli studi di Modena e Reggio Emilia

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Teachers

Andres MANZINI
Luisa MALAGUTI

Overview

The course aims to provide the basic knowledge of algebra, geometry and calculus for the functions of a real variable. It also intends to provide the ability to analyze simple problems of an algebraic or geometric nature or deriving from calculus for functions of one real variable; the approach also includes the use of simple computer programs. More specifically, at the end of the course the student will be able to know: (1) the main elements of geometry in the plane and in the space; (2) the main elements of the calculus for functions of one real variable; (3) the ordinary differential equations of the first order. For a more complete understanding of the training objectives, please refer to the reading of the learning outcomes expected at the end of this training course.

Admission requirements

Main operations between sets. Sets of natural, integer, rational, real numbers and their main properties. Polynomial algebra. Algebraic equations and inequalities. Powers, roots, exponentials and logarithms. Trigonometric functions. Equations of lines and conics as geometric loci. To access the exam it is necessary not to have Additional Educational Obligations (OFA). In this regard, see the information contained in the web page of the Department of Engineering Sciences and Methods.
https://www.dismi.unimore.it/site/home/servizi/futuro-studente/obblighi-formativi-aggiuntivi-ofa.html

Course contents

a) First part - 2 CFU, 18 hours

a1) Review of set theory. N-ple: sum and linear combinations of N-ple.
a2) Matrix theory. Transposed matrix, symmetric, triangular, diagonal matrices. Operations on matrices. Regular matrices. Reduced matrices and elementary transformations. Determinant of a square matrix. Properties and methods of calculating the determinant. Inverse matrix.
a3) Linear systems and their solutions. Search for solutions using the complete matrix step reduction algorithm (Gauss method). Cramer systems and their resolution. Rouchè-Capelli theorem. Resolution of a linear system by reduction to normal form.

b) Second part - 4 CFU, 36 hours

b1) Limited, symmetric, monotone, periodic functions. Limits and continuity. Asymptotes.
b2) Derivative and tangent line. Rules of derivation and fundamental derivatives. Primitives. Monotony test. Search for relative and absolute maximums and minimums. Second derivative, concavity and convexity. Flex.
b3) Definite integral. Integration by replacement and by parts.
b4) Ordinary differential equations with linear and separable variables.

Teaching methods

The course is delivered in Italian through face-to-face lectures and laboratory activities carried out on the computer. The lessons include a theoretical part, a part of exercises and a part of laboratory activities. The theoretical part consolidates the understanding and knowledge of the topics presented. The exercises aim to refine the student's application skills and are dedicated to solving exercises on all the topics on the program. The simplest assisted calculation techniques on the main topics on the program are discussed during the laboratory activities. Attendance at lessons is not compulsory but strongly recommended. Collaborative activities are also proposed using the Team Based Learning strategy. Participation in these activities is optional.

Assessment methods

The exam will take place at the end of the course according to the official exam session calendar. It consists of a written test aimed at verifying the achievement by the student of the ability to independently analyze and solve problems of algebra, geometry and calculus. It is allowed to participate in more than one consecutive written exam. The written test is divided into three parts and lasts a total of 120 minutes. The first part includes some questions with multiple and open answers; they have the purpose of verifying the level of theoretical knowledge. The second part verifies the ability to apply theoretical notions in concrete contexts and provides the student with the solution, complete and articulated, of three exercises. The third and last part concerns the resolution of a problem by using some simple computer calculation program. The first part is awarded a maximum of 10 points, the second is awarded a maximum of 15 points, while the last part is a maximum of 5. Consultation of books and notes is not allowed. The results of the test are communicated through ESSE3 on average within one week from the date of the test itself. Students enrolled in the first year of the course can take two partial written tests which, if passed, allow them to pass the exam. The first partial test will be scheduled in the period of suspension of the lessons of the first semester; the second, after the end of the lessons of the first semester indicated in the calendar of didactic activities. The partial tests last 120 minutes and have a structure very similar to the final test. Even during their performance, the use of books and notes is not allowed. The results of the partial tests are communicated through ESSE3, on average within one week from the date of the test. You can access the second partial test only if the score of the first test is at least 18/30. For those who pass both partial tests, the final grade will be determined by their arithmetic average. Additional points (up to a maximum of three) can be assigned to students who have participated in team based learning meetings, depending on the results achieved during the collaborative activities. The rules illustrated apply to face-to-face exams. Times and methods may vary as a result of contingent situations that force and deliver online exams.

Learning outcomes

Knowledge and understanding.
Learn the main concepts of algebra, geometry and calculus.

Applied knowledge and understanding:
Knowing how to apply the knowledge acquired to model and solve simple mathematical problems using the techniques of algebra, geometry and calculus.

Learning ability:
Acquire the methodological tools to continue their studies and to be able to independently provide for their ability to learn.

Autonomy of judgment.
To be able to independently choose the methods of analysis and solution of problems related to the program of this Course.

Communication skills:
Knowing how to clearly explain the topics covered in the course.

Readings

Slide e video dei docenti relativi agli argomenti del corso messi a disposizione durante lo svolgimento del Corso.

A. Barani, L. Grasselli, C. Landi, "Algebra Lineare e Geometria: quiz ed esercizi commentati e risolti", Ed. Esculapio, seconda edizione (2014).
M.R. Casali, C. Gagliardi, L. Grasselli, "Geometria", Ed. Esculapio, terza edizione (2016).
M. Bramanti - C.D. Pagani - S. Salsa, ANALISI MATEMAICA 1, Zanichelli, 2008.
M. Bramanti: Esercitazioni di Analisi 1 Ed. Esculapio, Bologna, 2011.


N. Fusco- P.Marcellini-C.Sbordone, Elementi di Analisi Matematica 2, Liguori.