# Computer science | MATHEMATICAL ANALYSIS

## Computer science MATHEMATICAL ANALYSIS

 0512100001 DIPARTIMENTO DI INFORMATICA EQF6 COMPUTER SCIENCE 2019/2020

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2017 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 48 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS MAT/05 3 24 EXERCISES SUPPLEMENTARY COMPULSORY SUBJECTS

 ELMO BENEDETTO T
Objectives
KNOWLEDGE AND UNDERSTANDING: TO PROVIDE STUDENTS WITH THE BASIC NOTIONS OF CALCULUS.
APPLYING KNOWLEDGE AND UNDERSTANDING: LEARN CALCULUS THEORY AND APPLICATIONS, SO AS TO BE ABLE TO SOLVE A RANGE OF SIMPLE EXERCISES. IN PARTICULAR, STUDENTS WILL BE ABLE TO DRAW THE GRAPHIC OF A FUNCTION GIVEN ITS ALGEBRAIC FORM AND TO CALCULATE SIMPLE INTEGRALS. TO BE ABLE TO PRESENT, IN A CLEAR AND RIGOROUS MANNER, THE THEOREMS, DEFINITIONS AND PROBLEMS STUDIED THROUGHOUT THE COURSE. TO BE ABLE TO USE THE ACQUIRED KNOWLEDGE IN REASONING AND ALGORITHMS
Prerequisites
ELEMENTARY ASPECTS OF ALGEBRA ARE MANDATORY AS WELL AS FAMILIARITY WITH SOLUTION METHODS FOR FIRST AND SECOND ORDER EQUALITIES AND INEQUALITIES. KNOWLEDGE OF SOME ELEMENTS OF TRIGONOMETRY ARE ALSO CONSIDERED A PREREQUISITE.

Contents
NUMERICAL SETS
THE THEORY SETS.NUMERICAL SETS. INDUCTION PRINCIPLE AND APPLICATIONS. PRIME NUMBERS AND COMPUTING APPLICATIONS. NUMERICAL SETS: UPPER AND LOWER BOUNDS, LIMITED SET, MAXIMUM AND MINIMUM, LOWER AND UPPER EXTREMITIES. INTERVALS. COMPLEX NUMBERS: ALGEBRAIC, TRIGONOMETRIC AND EXPONENTIAL REPRESENTATION. CALCULATION WITH COMPLEX NUMBERS

REAL FUNCTIONS OF A REAL VARIABLE AND ELEMENTARY FUNCTIONS

REAL FUNCTIONS OF A REAL VARIABLE. DEFINITION AND NOTATIONS. EXTREMES. CARTESIAN AND GRAPHICAL REPRESENTATION. EQUAL FUNCTIONS, ODD FUNCTIONS AND PERIODIC FUNCTIONS. ALGEBRA OF FUNCTIONS. COMPOSITION OF FUNCTIONS. INVERTIBLE FUNCTIONS. MONOTONOUS FUNCTIONS. THE INERTIA THEOREM ON THE MONOTONE FUNCTIONS. ELEMENTARY FUNCTIONS.

NUMERICAL SEQUENCES AND SERIES

DEFINITION AND NOTATIONS. CONVERGENT SUCCESSIONS. DIVERGENT SUCCESSIONS. LIMIT CHECKS. SUCCESSIONS NOT REGULAR. LIMITED SUCCESSIONS. THE NUMBER E. SOME REMARKABLE LIMITS. EXTRACTED SEQUENCES. INTRODUCTION TO SERIES, TELESCOPIC SERIES, POSITIVE SERIES, CONVERGENCE CRITERIA, VARIABLE SIGNAL SERIES.

LIMITS OF FUNCTIONS

DEFINITION AND NOTATIONS. UNIQUENESS OF THE LIMIT (WITH PROOF ). LIMIT CHECKS. OPERATIONS WITH FUNCTION LIMITS (PROOF OF SUM AND PRODUCT). INDEFINITE FORMS. THEOREM OF THE SIGN A PROOF) LIMIT OPERATIONS INDEFINITE FORMS COMPOSITE FUNCTIONS LIMITS. LIMITS OF ELEMENTARY FUNCTIONS. ASYMPTOTES OF THE FIRST ORDER AND OF ORDER HIGHER THAN THE FIRST.

CONTINUOUS FUNCTIONS

DEFINITION. CONTINUITY OF ELEMENTARY FUNCTIONS. OPERATION WITH CONTINUOUS FUNCTIONS. THEOREM OF SIGN PERMANENCE. ZEROS THEOREM. CONTINUITY AND DISCONTINUITY PATHOLOGIES: CLASSIFICATION AND EXAMPLES. WEIERSTRASS THEOREM. THEOREMS OF THE EXISTENCE OF INTERMEDIATE VALUES (ONLY THE FIRST PROOF). SIGNIFICANT LIMITS.
DERIVED
DEFINITION AND GEOMETRIC MEANING. CONTINUITY OF THE DERIVATIVE FUNCTIONS OF A VARIABLE (WITH PROOF). OPERATIONS WITH DERIVATIVES. DERIVED OF A COMPOSITE FUNCTION. REVERSE FUNCTION DERIVATIVE. DERIVATIVES OF ELEMENTARY FUNCTIONS. HIGHER ORDER DERIVATIVES. DERIVATIVE PATHOLOGY: CUSPID, ANGULAR POINTS AND VERTICAL TANGENT FLANKS. DE L'HOPITAL THEOREMS. TAYLOR'S FORMULA.

APPLICATIONS OF DERIVATIVES. STUDY OF FUNCTIONS

RELATIVE MAXIMUM AND MINIMUM. ROLLE THEOREM (WITH PROOF). LAGRANGE THEOREM (WITH PROOF AND GRAPHIC INTERPRETATION). CAUCHY THEOREM (WITH PROOF). CONVEX AND CONCAVE FUNCTIONS. GRAPHING A FUNCTION GRAPH.

RIEMANN INTEGRATION AND INDEFINITE INTEGRALS

DEFINITIONS AND PROPERTIES. GEOMETRIC INTERPRETATION. INTEGRABILITY. THEOREM OF THE MEAN (WITH PROOF).PRIMITIVE. THEOREM OF INTEGRAL CALCULUS (WITH PROOF). INDEFINITE INTEGRAL AND ITS PROPERTIES. INTEGRATION SUM DECOMPOSITION. INTEGRATION OF RATIONAL FUNCTIONS. INTEGRATION FOR PARTS. SOME TYPES OF INTEGRATION BY SUBSTITUTION.

DIFFERENTIAL EQUATIONS

INTRODUCTION TO DIFFERENTIAL PROBLEMS. FIRST ORDER ELEMENTARY DIFFERENTIAL EQUATIONS AND APPLICATIONS. LINEAR EQUATIONS. LINEAR EQUATIONS.

FUNCTIONS OF TWO AND MORE VARIABLES.
VIEWS ON FUNCTIONS IN MULTIPLE VARIABLES, WITH EXAMPLES OF APPLICATION INTEREST IN THE IT FIELD.
COMPARISON OF CONTINUITY, DERIVABILITY AND INTEGRABILITY IN ONE AND MORE VARIABLES.

FOR ALL CHAPTERS: EXAMPLES, APPLICATIONS AND DESIGN OF ELEMENTAL ALGORITHM
Teaching Methods
LESSONS
PRACTICE LESSON
Verification of learning
THE KNOWLEDGE AND UNDERSTANDING OF THE TOPICS DESCRIBED WITHIN THE COURSE WILL BE TESTED BY MEANS OF A FINAL WRITTEN EXAMINATION, FOLLOWED BY AN ORAL EXAM.
THE WRITTEN TEST HELP TO ASSESS THE ABILITY OF STUDENT OF APPLIED MATHEMATICS CONCEPTS FOR THE RESOLUTION OF EXERCISES ON THE STUDY OF FUNCTION AND CALCULATING INTEGRTALE .

THE ORAL EXAM HELP TO ASSESS THE ABILITY OF STUDENT OF EXHIBIT CLEARLY AND RIGOROUSLY MATHEMATICAL CONCEPTS AND THEOREMS DEMONSTRATED DURING THE LESSONS.
EXEMPTIONS OF THE EXAMINATIONS (WRITTEN AND ORAL) WILL BE FORWARDED DURING THE COURSE
Texts
E. GIUSTI ANALISI MATEMATICA I, BOLLATI BORINGHIERI
P. MARCELLINI - C. SBORDONE, ANALISI MATEMATICA UNO , LIGUORI EDITORE
P. MARCELLINI - C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA UNO , LIGUORI EDITORE

TEXTS DEPTH

M. TROISI ANALISI MATEMATICA I, LIGUORI EDITORE

M.BRAMANTI-C.PAGANI-S. SALSA, ANALISI MATEMATICA I, LIGUORI ZANICHELLI

P. MARCELLINI - C. SBORDONE, ESERCITAZIONI DI MATEMATICA I , LIGUORI EDITORE
A. ALVINO - L. CARBONE- G. TROMBETTI, ESERCITAZIONI DI MATEMATICA I , LIGUORI EDITORE
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