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.

C. Trapani: “Analisi Matematica, funzioni di una variabile reale”, Ed. McGraw-Hill.

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.

The exam consists of a written test and a subsequent oral test. The written test usually consists of 4-6 exercises on different subjects of the course. In each written test, most of the subjects contained in the course are covered.
The presence of questions on the theoretical part is not excluded. The oral test is made of preliminary questions about the basic notions of mathematics and about elementary functions, of a discussion on the exercises contained in the written test and of some final questions on statements and proofs of the main theoretical results. Admission to the oral test is subject to passing the written test with a mark greater than or equal to 16/30; the final mark will be obtained having as a starting point the mark of the written test to which points will be added or subtracted based on the progress of the oral test. Passing the written test, even with full marks, does not guarantee passing the exam in case of insufficient oral 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.