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 active 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.

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 written exam will test both knowledge of the theory (theorems, proofs, definitions) and the ability to solve exercises. The oral exam is at the discretion of the instructor if additional assessment elements are needed.

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.