Ateneo - UPO

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

Suggested book for the theoretical part:
Bramanti, Pagani, Salsa, “Analisi Matematica 1 con elementi di geometria e algebra lineare”, Ed. Zanichelli.
As an alternative: Bramanti, Pagani, Salsa, “Analisi Matematica 1”, Ed. Zanichelli.
Suggested book for exercises:
Salsa, Squellati: "Esercizi di Analisi Matematica 1", Ed. Zanichelli.

Acquisition of the basic notions of differential and integral calculus for real functions of one real variable. Ability to interlink these notions and use them to solve easy problems.

Basic notions of algebra and trigonometry.

The class is organized in specific lectures, that will cover both theory and excercises. In both situations we will look for the activ participation and the interaction of the students and among them, also through focused questions on the current arguments.

In addition to the suggested books for the theoretical part and the exercises, further material for the preparation of the exam will be provided during the development of the course.

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 for 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.

At the end of the class the student must know the basic notions of Calculus (functions in one variable, limits, derivatives, integrals) and master to be able to autonomously interlink them and use them jointly to solve simple theoretic problems. Moreover, the student must be able to explain in a clear manner both the notions, and their use in a general context or in a specific problem.