Ingegneria Edile-Architettura | ANALISI MATEMATICA I

cod. 0660100001

ANALISI MATEMATICA I

	0660100001
	DIPARTIMENTO DI INGEGNERIA CIVILE
	CORSO DI LAUREA MAGISTRALE A CICLO UNICO DI 5 ANNI
	BUILDING ENGINEERING - ARCHITECTURE
	2013/2014

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2012
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	6	60	LESSONS	BASIC COMPULSORY SUBJECTS

	Objectives
	THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS. THE AIM OF THE COURSE IS THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AND THE ABILITY TO USE COMPUTATIONAL TOOLS. THE COURSE HAS AS ITS MAIN PURPOSE THE CONSOLIDATION OF BASIC MATHEMATICAL KNOWLEDGE AND THE DEVELOPMENT OF USEFUL TOOLS FOR A SCIENTIFIC APPROACH TO THE PROBLEMS AND PHENOMENA THAT STUDENTS ENCOUNTER IN THE PURSUIT OF HIS STUDIES. THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS YET CONCISE WAY AND ACCOMPANIED BY A PARALLEL EXERCISE ACTIVITIES DESIGNED TO FOSTER THE UNDERSTANDING OF CONCEPTS.

THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS.
THE AIM OF THE COURSE IS THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AND THE ABILITY TO USE COMPUTATIONAL TOOLS.
THE COURSE HAS AS ITS MAIN PURPOSE THE CONSOLIDATION OF BASIC MATHEMATICAL KNOWLEDGE AND THE DEVELOPMENT OF USEFUL TOOLS FOR A SCIENTIFIC APPROACH TO THE PROBLEMS AND PHENOMENA THAT STUDENTS ENCOUNTER IN THE PURSUIT OF HIS STUDIES.
THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS YET CONCISE WAY AND ACCOMPANIED BY A PARALLEL EXERCISE ACTIVITIES DESIGNED TO FOSTER THE UNDERSTANDING OF CONCEPTS.

	Prerequisites
	FOR THE SUCCESSFUL ACHIEVEMENT OF OBJECTIVES, THE STUDENTS ARE REQUIRED TO HAVE KNOWLEDGE ABOUT: - ALGEBRA, WITH PARTICULAR REFERENCE TO, ALGEBRAIC EQUATIONS AND INEQUALITIES, LOGARITHMIC, EXPONENTIAL, TRIGONOMETRIC AND TRANSCENDENTAL; - TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC TRIGONOMETRIC FUNCTIONS.

	Contents
	SET THEORY. INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS OF A SET. INTRODUCTION TO REAL NUMBERS. EXTREMES OF A NUMERICAL SET. INTERVALS OF R. NEIGHBORHOODS, LIMIT POINTS. CLOSED SETS AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. CARTESIAN AND TRIGONOMETRIC FORM. DE MOIVRE'S FORMULA FOR POWER AND FOR ROOTS. REAL FUNCTIONS. DEFINITION. DOMAIN, CODOMAIN AND GRAPH OF THE FUNCTION. EXTREMES OF A REAL FUNCTION. MONOTONE FUNCTIONS. FUNCTION COMPOSITION. INVERSE FUNCTIONS. ELEMENTARY FUNCTIONS: NTH POWER AND NTH ROOT FUNCTIONS, EXPONENTIAL FUNCTION, LOGARITHMIC FUNCTION, REAL POWER FUNCTION, TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES. SEQUENCES AND SERIES. DEFINITIONS. BOUNDED SEQUENCES, CONVERGENT, DIVERGENT AND OSCILLATING. MONOTONE SEQUENCES. EULER NUMBER. CAUCHY CONVERGENCE CRITERION. INTRODUCTION TO NUMERICAL SERIES. CONVERGING SERIES, DIVERGING, AND INDETERMINATE. GEOMETRIC SERIES, HARMONIC SERIES. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: THE CRITERIA OF COMPARISON, RATIO CRITERION, ROOT CRITERION. LIMITS OF A FUNCTION. DEFINITION. RIGHT AND LEFT LIMIT. UNIQUENESS THEOREM. COMPARISON THEOREMS. OPERATIONS AND INDETERMINATE FORMS. PARTICULAR LIMITS. CONTINUOUS FUNCTIONS. DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM. BOLZANO THEOREM. INTERMEDIATE VALUES THEOREM. UNIFORM CONTINUITY. DERIVATIVE OF A FUNCTION. DEFINITION. LEFT AND RIGHT DERIVATIVE. GEOMETRIC MEANING, THE TANGENT LINE TO THE GRAPH OF A FUNCTION. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY FUNCTIONS. DERIVATIVES OF COMPOSITE FUNCTION AND INVERSE FUNCTION. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND GEOMETRIC MEANING. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS. ROLLE'S THEOREM. CAUCHY'S THEOREM. LAGRANGE'S THEOREM AND COROLLARIES. THEOREM OF DE L'HOSPITAL. CONDITIONS FOR MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS. STUDY OF THE GRAPH OF A FUNCTION. ASYMPTOTES OF A GRAPH. SEARCH OF MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. GRAPH OF A FUNCTION BY ITS CHARACTERISTIC ELEMENTS. INTEGRATION OF FUNCTIONS OF ONE VARIABLE. DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF CALCULUS.

Contents

SET THEORY.
INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS OF A SET. INTRODUCTION TO REAL NUMBERS. EXTREMES OF A NUMERICAL SET. INTERVALS OF R. NEIGHBORHOODS, LIMIT POINTS. CLOSED SETS AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. CARTESIAN AND TRIGONOMETRIC FORM. DE MOIVRE'S FORMULA FOR POWER AND FOR ROOTS.

REAL FUNCTIONS.
DEFINITION. DOMAIN, CODOMAIN AND GRAPH OF THE FUNCTION. EXTREMES OF A REAL FUNCTION. MONOTONE FUNCTIONS. FUNCTION COMPOSITION. INVERSE FUNCTIONS. ELEMENTARY FUNCTIONS: NTH POWER AND NTH ROOT FUNCTIONS, EXPONENTIAL FUNCTION, LOGARITHMIC FUNCTION, REAL POWER FUNCTION, TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES.

SEQUENCES AND SERIES.
DEFINITIONS. BOUNDED SEQUENCES, CONVERGENT, DIVERGENT AND OSCILLATING. MONOTONE SEQUENCES. EULER NUMBER. CAUCHY CONVERGENCE CRITERION. INTRODUCTION TO NUMERICAL SERIES. CONVERGING SERIES, DIVERGING, AND INDETERMINATE. GEOMETRIC SERIES, HARMONIC SERIES. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: THE CRITERIA OF COMPARISON, RATIO CRITERION, ROOT CRITERION.

LIMITS OF A FUNCTION.
DEFINITION. RIGHT AND LEFT LIMIT. UNIQUENESS THEOREM. COMPARISON THEOREMS. OPERATIONS AND INDETERMINATE FORMS. PARTICULAR LIMITS.

CONTINUOUS FUNCTIONS.
DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM. BOLZANO THEOREM. INTERMEDIATE VALUES THEOREM. UNIFORM CONTINUITY.

DERIVATIVE OF A FUNCTION.
DEFINITION. LEFT AND RIGHT DERIVATIVE. GEOMETRIC MEANING, THE TANGENT LINE TO THE GRAPH OF A FUNCTION. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY FUNCTIONS. DERIVATIVES OF COMPOSITE FUNCTION AND INVERSE FUNCTION. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND GEOMETRIC MEANING.

FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS.
ROLLE'S THEOREM. CAUCHY'S THEOREM. LAGRANGE'S THEOREM AND COROLLARIES. THEOREM OF DE L'HOSPITAL. CONDITIONS FOR MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.

STUDY OF THE GRAPH OF A FUNCTION.
ASYMPTOTES OF A GRAPH. SEARCH OF MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. GRAPH OF A FUNCTION BY ITS CHARACTERISTIC ELEMENTS.

INTEGRATION OF FUNCTIONS OF ONE VARIABLE.
DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF CALCULUS.

	Teaching Methods
	THE COURSE COVERS THEORETICAL LESSONS, DURING WHICH THE COURSE TOPICS WILL BE PRESENTED THROUGH LECTURES, AND CLASSROOM EXERCISES DURING WHICH TO PROVIDE THE MAIN TOOLS NEEDED FOR SOLVING EXERCISES RELATED TO THE CONTENT OF TEACHING.

	Verification of learning
	THE ASSESSMENT OF THE ACHIEVEMENT OF THE OBJECTIVES WILL BE DONE THROUGH A WRITTEN TEST AND AN ORAL INTERVIEW.

	Texts
	G. ALBANO, C. D’APICE, S. SALERNO, LIMITI E DERIVATE, CUES (2002). C. D’APICE, R. MANZO, VERSO L’ESAME DI MATEMATICA I, CUES (2007). TEACHING MATERIALS OF E-LEARNING PLATFORM IWT. LECTURE NOTES.

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