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.

The exam consists of a written test and an oral test. The written test is divided into two parts: - First Part: it is compulsory and consists of 14 multiple choice questions, three of which are theoretical questions aimed at verifying the knowledge and understanding of the main topics of the course, eleven exercises aimed at verifying the skills achieved in applying the knowledge acquired to the resolution of not excessively complex problems; the maximum score achievable is 25/30. - Second Part: it is optional and is made up of two open longer and more complex exercises or, alternately, a function study to test the autonomous ability of choosing the adequate tools for problem solving and the ability of communicating through an appropriate and rigorous language. The oral test is optional and includes: - some questions on topics that can be analyzed through the tools and the knowledge acquired during the course, in order to test both the learning and the communicative skills; - analysis of cases slightly different from the ones seen during the course, in order to test autonomous reasoning skills. There are 7 exam sessions in each solar year but students can enroll no more than three times, at their own choice. It is mandatory to register for the exam through the students portal: once the three exam participations have been reached, including withdrawals and refusals, the system will not allow further registrations. On the educational portal (DIR) a path divided into self-assessments and assessments available during the period of the lessons is proposed. Obtaining a score greater than or equal to 10 (out of 25), with the consequent conversion into exam points (from 1 to 4), allows the access to a first part of the written exam consisting of 10 questions, which will be held in the first session of the winter session. The exam points obtained with the individual work will be added to the result of this test in proportion to the one of the first part of the exam. Details and clarifications on the contents, methods of participation and the evaluation of the exams 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.