AIMS: Part I – Basics
Basics of set theory. Real numbers. Powers and logarithms. Sets of real numbers, intervals, maximum, minimum, infimum and supremum. Euclidean distance on the real line, neighborhoods, interior and boundary points, limit points and isolated points, open and closed sets.
Part II – Real functions
The concept of function. Elementary functions. Domain and image. Graph of a function. The inverse function and the composite function. The concept of limit of a function, calculus of limits, computation of simple limits. Theorem of the sign permanence.Continuous functions and related theorems (theorems and intermediate values and zeros). Derivarive of functions, relation between differentiability and continuity. Derivation rules. Maximum and minimum of a function, sufficient conditions. Rolle and Lagrange theorems. Monotone and convex functions.
Part III – Sequences and series
Convergencent sequences and series. Calculus of limits. Theorem of sign permanence. The Cauchy criterion. Basic examples of sequences and series.
Part IV – Integrals
Primitives. Rules for indefinite integration. Riemann integration and related properties. The fundamental theorem of integral calculus. Improper integrals. Riemann-Stieltjes integrals.
EXAMINATION METHOD: The exam consists of a written and an oral test. The written test, lasting 2 hours, consists in carrying out exercises on the topics of the course (that is one-variable functions, series, sequences and integrals). The written test is sufficient if the student reaches a total score of 18, which is necessary to be admitted to the oral examination. The oral exam must be held in the same round as the written test. The oral exam is considered sufficient only if the student has mastered the contents of the course, enunciates and correctly demonstrates a theorem chosen by the instructure, among those scheduled and explicitly demonstrated during the lessons.
PRE-REQUISITES: Equations and inequalities of I and II degree. Generalities on polynomials, algebric fractions of polynomials.
FURTHER INFORMATION: Portale Valutami