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, Giappichelli, 2020 (exercises)

- D'Ercole R., Precorso di Matematica, Pearson, 2019 (prerequisites)

Reaching of a good knowledge and understanding of the basic mathematical notions about infinitesimal, differential and integral calculus in one real variable.
Maturation of a satisfactory ability of choosing and using the right tools for the solution of simple problems concerning function of one real variable.
Development of the skill of adapting the tools choice to necessary variants and of the ability of communicating in a sufficiently clear and rigorous way the logic-deductive way of problem solving.

Elementary set theory, numerical sets, equations and inequalities, analytic geometry.

Lectures and tutorials.

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/servicesstudents- 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. When dealing with ESG aspects and the sustainability objective, the importance of good corporate governance will be highlighted in the various profiles, including gender equality.

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:
- complementary learning material: examples and/or extra material;
- recent past exams proofs with text and sketch of the solutions; samples of unfolded exercises which are more frequently assigned at exams.

A compulsory written exam with exercises and theoretical questions, and an optional oral exam.
Written exam:
- two theoretical questions to test the knowledge and the comprehension of the main topics of the course;
- four short exercises to test the ability of applying the knowledges reached to the solution of simple problems;
- two longer and more complex exercises, to test the autonomous ability of choosing the adequate tools for problem solving and the ability of communicating through an appropriate and rigorous language.
Oral exam:
- some questions on topics that can be analyzed through the tools and the knowledges 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.

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.

At the end of the course the student should be able to analyse, with the infintesimal and differntial calculus, a real function of real variable. The student should also be able to determine the primitive, the definite integral of a function and to study a simple integral function.