# Ingegneria Elettronica | Mathematics I

## Ingegneria Elettronica Mathematics I

 0612400001 DIPARTIMENTO DI INGEGNERIA INDUSTRIALE EQF6 ELECTRONIC ENGINEERING 2020/2021

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 9 90 LESSONS BASIC COMPULSORY SUBJECTS
 ANNA CANALE T ABDELAZIZ RHANDI FEDERICA GREGORIO
Objectives
THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS. THE OBJECTIVES OF THE COURSE ARE THE ACQUISITION OF RESULTS, PROOF'S TECHNIQUES AND ABILITY TO USE THE CALCULATION TOOLS STUDIED.

KNOWLEDGE AND UNDERSTANDING.
THE AIM IS TO REACH A GOOD LEVEL OF UNDERSTANDING AND KNOWLEDGE ABOUT THE FOLLOWING TOPICS:
NUMERICAL SETS. REAL FUNCTIONS. BASIC NOTIONS OF EQUATIONS AND INEQUALITIES. NUMERIC SUCCESSIONS. LIMITS. CONTINUOUS FUNCTIONS. DERIVATIVES. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS. STUDY OF A FUNCTION. INTEGRATION OF A SINGLE VARIABLE FUNCTION.
APPLYING KNOWLEDGE AND UNDERSTANDING.
APPLYING THE THEOREMS AND THE RULES TO THE SOLUTION OF PROBLEMS. COMPUTING LIMITS, DERIVATIVES AND INTEGRALS. PERFORMING THE GRAPHICAL STUDY OF A FUNCTION.
JUDGMENT AUTONOMY.
THE STUDENT WILL BE ABLE TO EVALUATE IN AN INDEPENDENT MANNER WHICH ARE THE POSSIBLE RESOLUTION APPROACHES TO THE QUESTIONS PROPOSED TO HIM AND/OR AUTONOMELY SELECTED, SUCCESSFULLY IDENTIFYING THE BEST POSSIBLE CHOICE.
COMMUNICATION SKILLS.
THE STUDENT WILL BE ABLE TO EXPOSE IN A CLEAR, RIGOROUS AND CRITICAL MANNER DEFINITIONS AND PROOFS OF THE PROPOSED THEOREMS, SHOWING AT THE SAME TIME A SUITABLE AND COMPLETE CONTROL OF THE MATHEMATICAL LANGUAGE.
Prerequisites
IN ORDER TO REACH THE OBJECTIVES AND, IN PARTICULAR, FOR AN ADEQUATE UNDERSTANDING OF THE CONTENTS PROVIDED BY THE TEACHING, THE STUDENTS SHOULD HAVE A BACKGROUND RELATED TO THE ALGEBRA, IN PARTICULAR TO ALGEBRAIC EQUATIONS AND INEQUALITIES, LOGARITHMIC, EXPONENTIAL, TRIGONOMETRIC AND TRANSCENDENTAL INEQUALITIES, AND THEY SHOULD UNDERSTAND TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC THEORY OF TRIGONOMETRIC FUNCTIONS.
Contents
NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INDUCTION PRINCIPLE. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBOURHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. GEOMETRIC AND TRIGONOMETRIC FORM. POWERS AND DE MOIVRE'S FORMULA. N-TH ROOTS (HOURS OF THEORY/EXERCISES: 6/6).
REAL FUNCTIONS: DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE AND INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS (HOURS OF THEORY/EXERCISES: 4/4).
BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST ORDER, QUADRATIC, BINOMIAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. FIRST ORDER, SECOND ORDER, RATIONAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC INEQUALITIES. SYSTEMS (HOURS OF THEORY/EXERCISES: 0/4).
NUMERICAL SEQUENCES: DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. NEPERO'S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE. NUMERICAL SERIES: INTRODUCTION. CONVERGENCE. ARMONIC AND GEOMETRIC SERIES. POSITIVE SERIES AND CONVERGENCE CRITERIA (HOURS OF THEORY/EXERCISES: 4/6).
LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS (HOURS OF THEORY/EXERCISES: 4/5).
CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY (HOURS OF THEORY/EXERCISES: 4/0).
DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING. TANGENTIAL LINE. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND ITS GEOMETRIC MEANING (HOURS OF THEORY/EXERCISES: 4/5).
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS (HOURS OF THEORY/EXERCISES: 4/4).
GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH (HOURS OF THEORY/EXERCISES: 4/8).
INTEGRATION OF ONE VARIABLE FUNCTIONS: DEFINITION OF ANTIDERIVATIVE AND INDEFINITE INTEGRAL. BASIC INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND GEOMETRICAL MEANING. FUNDAMENTAL THEOREM OF CALCULUS (HOURS OF THEORY/EXERCISES: 8/6).
Teaching Methods
THE COURSE CONSISTS IN THEORETICAL LECTURES AND EXERCISES. 90 HOURS (9 CFU) IN TOTAL: 42 HOURS FOR THE THEORY AND 48 HOURS FOR THE EXERCISES.
THE TEACHING IS PROVIDED IN PRESENCE WITH MANDATORY FREQUENCY, CERTIFIED BY THE STUDENT THROUGH THE USE OF THE PERSONAL BADGE. IN ORDER TO BE ADMITTED TO THE FINAL VERIFICATION OF PROFIT AND TO ACHIEVE THE RELATED NUMBER OF CFU, THE STUDENT WILL HAVE TO ATTENDED AT LEAST THE 70% OF THE SCHEDULED HOURS OF DIDACTIC ACTIVITY.
Verification of learning
IN RELATION TO THE TEACHING OBJECTIVES, THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE, THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL TEST, THE SKILL OF PROVING THEOREMS, THE SKILL OF SOLVING EXERCISES, THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING, THE ABILITY TO USE THE ACQUIRED KNOWLEDGE.
THE EXAM CONSISTS OF A WRITTEN TEST, THAT THE STUDENT WILL BE HELD TO ADDRESS IN TOTAL AUTONOMY, AND AN ORAL EXAMINATION. THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE (THERE ARE SAMPLES AVAILABLE ON THE DEPARTMENT WEBSITE). THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION.
THERE WILL BE A MID-TERM TEST CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE WHICH, IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENTS ON THESE TOPICS AT THE FINAL WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS.
THE ORAL INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE AND MASTERY IN ALL THE TOPICS OF THE COURSE, AS DEFINITIONS AND PROOFS OF THEOREMS, AS WELL AS IN SOLVING EXERCISES.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL EVALUATION OF THE STUDENT.
Texts
BASIC TEXT FOR THEORY:
P. MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA UNO, LIGUORE EDITORE.
BASIC TEXT FOR THE EXERCISES:
P. MARCELLINI, C. SBORDONE, ESERCITAZIONI DI MATEMATICA I, VOL.I, PARTE I, II, LIGUORI EDITORE.
DIDACTIC SUPPORTS ON THE PLATFORM OF THE DEPARTMENT.