Ateneo - UPO

Sets, functions, limit for real functions of one real variable, continuity, differential and integral calculus for real functions of one real variable.

Suggested books (in alphabetic order):

M. Bramanti, C. Pagani, S. Salsa, "Analisi Matematica 1", Ed. Zanichelli.

Bramanti, Pagani, Salsa, ``Analisi Matematica 1 con elementi di geometria e algebra lineare'', Ed. Zanichelli.
This second volume is completely equivalent to the first one concerning the part of Mathematical Analysis 1 but it contains an additional part on the Geometry topics that are covered in the Discrete Mathematics course. The same volume is, among other things, recommended for the Discrete Mathematics course.

Acquire the basic notions of differential and integral calculus for real functions of one real variable.

Elementary notions of algebra and trigonometry.

The course is organized with frontal lessons with theoretical sessions and exercises. Each subject of the course is introduced by mean of a general discussion which has the purpose of making it comprehensible to the students as much as possible. In a second time the basic notions of each subject are introduced; they are successively followed by examples with the purpose of clarifying their meaning. The third step is devoted to the statements of the main theorems and their proofs. The last part is devoted to the exercises.
The active participation to the lectures is encouraged through questions addressed to students which have also the purpose to make possible understanding which are the difficulties met by them in attending these lectures; students are also encouraged to suggest possible exercises on topics that from their point of view need more clarifications.

In addition to the suggested books, further material for the preparation of the exam will be provided during the development of the course. It is made up of the slides used during the course of the last year and of the written exams of the previous years.

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 exam consists of a written test. The written test usually consists of 4-6 exercises on different topics of the course and one or more questions on the theoretical part. In each written test, most of the topics contained in the course are covered. The oral test is optional and you are admitted only if a sufficient evaluation is achieved in the written test. The oral exam consists of preliminary questions on the basic notions of Mathematical Analysis and elementary functions, a discussion on the exercises contained in the written test and some final questions on the statements and proofs of the main theoretical results. If the oral test is satisfactory, it will therefore be possible to obtain a final grade higher than that of the written test.

Sets and operations between sets; definition of function, composition between functions, injectivity, surjectivity, bijectivity, invertibility, cardinality of a set.
Limit of a real function of one real variable; right limit and left limit; limits and algebraic operations; Comparison Theorem; existence of the limit of a monotone function; change of variable.
Continuity for real functions of one real variable; continuity and algebraic operations; continuity of a composition ; Sign Permanence Theorem; Weierstrass Theorem; Intermediate Values Theorem; continuity of the inverse function.
Differential Calculus for real functions of one real variable; derivatives and algebraic operations; derivative of a composition; differentiability of the inverse function.
Relative maxima and minima; monotonicity of a function and sign of the first order derivative; de l'Hopital Theorems; convex and concave functions, flex points and sign of the second order derivative.
Riemann integral; integrability of sums and products of integrable functions; Mean Value Theorem; integral functions; Fundamental Theorem of Integral Calculus; integration by parts and change of variable formula.

- Knowledge and understanding: acquisition of the main notions of differential and integral calculus for real functions of one real variable.

- Applying knowledge and understanding: to be able to deduce the main qualitative and quantitative properties for real functions of one real variable.

- Learning skills: the student must learn a proper skill in using in the logical thinking and in the application of the methodological rigor essential for facing problmes based on mathematical models.