Academic Year 2020/2021 - 1° Year

MAT/07 - 6 CFU - 1° Semester

**Knowledge and understanding:** the student will learn some basic concepts of mathematics. He will develop computational abilities of the most common mathematical objects among which sets, relations, differential and integral calculus, differential equations.

**Application of knowledge and understanding:** through the investigation of simple mathematical models, the student will understand the importance of mathematical modelization in science.

**Judgment authonomy:** to develop logical abilities, the student will approach with rigor some simple proofs. Mani statements and proofs will be given in an intuitive way, to adapt them to the inclination of the audience and the time constraints. The student will learn to choose among different techniques the ones that will allow him to solve problems.

**Communication abilities:** the study of mathematics, with its axiomatic-deductive scheme, will make the student capable to communicate with rigor and clarity his ideas. He will learn that the use of a proper language is the most important tool in science in particular, and in life in general.

**Learning abilities:** the students will be stimulated to deepn their investigation on some arguments. They will be invited to work in groups and to learn from different sources.

Lectures on theoretical arguments and exercises will be given in class at the blackboard. Teaching may also be carried out on line, should the conditions require it.

Axioms related to operations, ordering and completenes of the reals. Sets; definition, representation, operations. Relations: definition, ordering relations, equivalence relations, quotients, numerical sets, completeness. Upper and lower bounds, inf and sup, max and min. Functions: definition, domain, codomain, injective surjective, and bijective functions. Composition of functions, inverse functions, montone functions. Global and local max and min. Cartesian plane, representation of elementary functions (polynomial exponential, logarithmic, goniometric). Limits of sequences and functions: the definition of limit, limit of a sequence, theorems on limits of sequences; limits of functions; continuous functions and fundamental theorems; discontinuous functions. Derivatives: definition and fundamental theorems (Lagrange, Rolle, Cauchy, de l'Hopital); graphic of a function.Integration: definite integrals and area; indefinite intregrals; integration techniques; definite integral. Differential equations: separation of variables, first and second order linear equations; models for biology, chemistry, and physics.

- Elementi di Calcolo, Paolo Marcellini, Carlo Sbordone, Liguori Editore

- Notes from the teacher