Ingegneria Elettronica | Mathematics I

cod. 0612400001

MATHEMATICS I

	0612400001
	DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
	EQF6
	ELECTRONIC ENGINEERING
	2016/2017

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2016
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	9	90	LESSONS	BASIC COMPULSORY SUBJECTS

PROFESSORS

	ABDELAZIZ RHANDI T Curriculum
	ANNA PIERRI Curriculum

	Objectives
	THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS. THE OBJECTIVES OF THE COURSE ARE THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AND THE ABILITY TO USE ITS CALCULATION TOOLS. THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS BUT CONCISE WAY AND WILL BE ACCOMPANIED BY A PARALLEL EXERCISE ACTIVITY AIMED AT PROMOTING THE UNDERSTANDING OF THE CONCEPTS.

THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS. THE OBJECTIVES OF THE COURSE ARE THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AND THE ABILITY TO USE ITS CALCULATION TOOLS. THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS BUT CONCISE WAY AND WILL BE ACCOMPANIED BY A PARALLEL EXERCISE ACTIVITY AIMED AT PROMOTING THE UNDERSTANDING OF THE CONCEPTS.

	Prerequisites
	IN ORDER TO REACH THE OBJECTIVES STUDENTS SHOULD HAVE A BACKGROUND RELATING TO ALGEBRA, IN PARTICULAR ALGEBRAIC EQUATIONS AND ALGEBRAIC, LOGARITHMIC, EXPONENTIAL, TRIGONOMETRIC AND TRANSCENDENTAL INEQUALITIES, AND UNDERSTANDING OF TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC THEORY OF TRIGONOMETRIC FUNCTIONS.

	Contents
	NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. 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 4/2) 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. (2/2) 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. (4/4) 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.(4/2) LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS. (4/4) CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY. (4/-) 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. (4/2) FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS. (4/2) GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH. (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. (8/4)

Contents

NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. 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 4/2)

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. (2/2)

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. (4/4)

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.(4/2)

LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS. (4/4)

CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY. (4/-)

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. (4/2)

FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS. (4/2)

GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH. (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. (8/4)

	Teaching Methods
	THE COURSE CONSISTS IN THEORETICAL LECTURES AND CLASSROOM PRACTICEDEVOTED LECTURES.

	Verification of learning
	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 APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE. THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL INTERVIEW. WRITTEN TEST: THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE (THERE ARE SAMPLES AVAILABLE ON THE WEBSITE) AND 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 THIS TOPICS AT THE WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS. THE ORAL INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING. THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE MARK OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

Verification of learning

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 APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE.
THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL INTERVIEW.
WRITTEN TEST: THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE (THERE ARE SAMPLES AVAILABLE ON THE WEBSITE) AND 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 THIS TOPICS AT THE WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS.
THE ORAL INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE MARK OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

	Texts
	P.MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA UNO, VERSIONE SEMPLIFICATA PER I NUOVI CORSI DI LAUREA, LIGUORE EDITORE 2002. C. D’APICE, R. MANZO, , VERSO L’ESAME DI MATEMATICA I, CUES (2007). EDUCATIONAL CONTENTS ON E-LEARNING PLATFORM IWT. LECTURE NOTES.