Ateneo - UPO

Numerical sets. Real functions of one real variable: definition and properties, operations, differential calculus, antiderivatives and their applications.

- Salinelli E., Appunti di Matematica, Giappichelli, 2021 (theory)
- Salinelli E., Esercizi svolti di Matematica, III edizione, Giappichelli, 2020 (exercises)
- D'Ercole R., Precorso di Matematica, Pearson, 2011 (math refresher)

The course aims at helping the students to acquire the knowledge and skills of mathematics, related to the differential and integral calculus in a variable, necessary for the study and analysis of economic-business models and for the understanding of the tools of basic statistics.

Elementary set theory, numerical sets, equations and inequalities (of 1st and 2nd degree, rational, irrational, logarithmic and exponential ones), analytic geometry.

Lectures aimed at presenting the topics in the program (knowledge) and their critical discussion; exercises integrated in the lessons aimed at applying the concepts introduced; interactive discussion of some examples, both theoretical and relating to some significant economic problems. Use of the DIR platform to support lessons (interactive graphs, tests) and individual work (self-assessment test).

On the web page of the course at the URL: www.dir.uniupo.it any useful information on the course and learning resources can be found. More precisely:
- self-assessments: self-assessment tests are available divided by topics, useful for verifying the level of learning achieved;
- complementary learning material: examples and/or extra material; review test for each macro-topic;
- samples of unfolded exercises which are more frequently assigned at exams.

There will be 20 additional hours of laboratory, during the period of the lessons, to allow the active verification of one's level of preparation.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/services-
students-physical-or-learning-disabilities.

Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The examination consists of a written test and an oral test.
The written test is divided into two parts:
- First part: compulsory, it consists of 12 multiple-choice questions, one of which is theoretical, aimed at verifying knowledge of the course contents, and eleven exercises aimed at verifying the skills achieved in the application of the knowledge acquired to the resolution of not excessively articulated problems; the maximum mark achievable is 24/30.
- Second part: optional, it consists of an open-ended theoretical question to test understanding of the course content and two open-ended exercises or, alternatively, depending on the session, a function study, aimed at testing the ability to choose autonomously the appropriate tools for solving problems and the ability to communicate in appropriate and rigorous language; the maximum mark achievable in this part is 8/30. Access to the second part is permitted only to those who obtain a mark of at least 18/30 in the first part.
The oral test, which is optional, includes
- questions on issues that can be analysed using the tools and knowledge acquired during the course, in order to further test learning and communication skills;
- analysis of situations that deviate slightly from those seen during the course, in order to assess the autonomous reasoning ability acquired.
The maximum mark achievable is 4/30.
Access to the oral test is only permitted to those who obtain a score of at least 22/30 by adding together the first and second part scores.

Seven sessions are scheduled in a calendar year, but a maximum of three may be attended. It is compulsory to register for the exam via the student portal: once three exam appearances have been reached, including withdrawals and refusals, the system will not allow further
On the didactic portal (DIR), a path is proposed divided into self-assessments at the end of which it is possible to use a simulator of the first part of the exam.

Details and specifications on the content, participation modes and assessment of the examinations are published on the DIR page of the course.

The set of the real numbers, the real line, intervals. Introduction to the topology of the real line. Ordered sets on the real line, g.l.b., l.u.b., maximum and minimum of a set. Real functions of one real variable: definition, elementary functions, operations among functions, injective, surjective and bijective functions, invertibility, monotonicity, concavity and convexity, extrema of a function. Limits and continuity: definitions and basic theorems. Differential calculus in one variable. Definitions: difference quotient, derivative, tangent line. Derivatives of elementary functions, differentiation rules. Higher-order derivatives. Some theorems of differential calculus. Applications of differential calculus to computation of limits, monotonicity and convexity study, and determination of extrema. Introduction to the antiderivatives and their applications: definition, elementary integrals, linearity of integration, integration by parts, integration by substitution. Application to the computation of definite integrals. Functions defined by integrals. Introduction to generalized integrals.

The expected results related to a minimum level of sufficiency are: - Basic knowledge of the main mathematical notions related to the theory of function of real variable, to differential and integral calculus in a real variable. - Sufficient ability to apply these concepts in solving simple problems. - Maturation of a minimum capacity of judgment in the choice and use of calculation tools suitable for the resolution of the aforementioned problems. The expected results related to a medium / advanced level are: - Good knowledge and understanding of the main mathematical notions related to the theory of function of real variable, to the differential and integral calculus in a real variable. - Good ability to apply these concepts in solving sufficiently complex problems. - Maturation of a satisfactory judgment capacity in the choice and use of the appropriate tools to solve the aforementioned problems. - Discrete ability to adapt the choice of instruments to the necessary variations. - Discrete ability to communicate in a clear and rigorous way the logical-deductive paths followed in dealing with the proposed problems.