# Ingegneria Edile-Architettura | ANALISI MATEMATICA I

## Ingegneria Edile-Architettura ANALISI MATEMATICA I

 0660100001 DIPARTIMENTO DI INGEGNERIA CIVILE CORSO DI LAUREA MAGISTRALE A CICLO UNICO DI 5 ANNI BUILDING ENGINEERING - ARCHITECTURE 2015/2016

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2012 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 60 LESSONS BASIC COMPULSORY SUBJECTS
 VITTORIO ZAMPOLI T
Objectives
THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS.
THE LEARNING OBJECTIVES OF THE COURSE ARE THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE RELATED CALCULATION TOOLS.
THE COURSE HAS AS ITS MAIN AIM TO CONSOLIDATE BASIC MATHEMATICAL KNOWLEDGE AND TO PROVIDE AND DEVELOP USEFUL TOOLS FOR A SCIENTIFIC APPROACH TO PROBLEMS AND PHENOMENA THAT STUDENTS WILL ENCOUNTER IN THE FOLLOWING OF THEIR STUDIES.
THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS BUT CONCISE WAY AND ACCOMPANIED BY A PARALLEL PRACTICAL ACTIVITY AIMED AT PROMOTING THE UNDERSTANDING OF THE CONCEPTS.
Prerequisites
FOR A SUCCESSFUL ACHIEVEMENT OF THE OBJECTIVES, SKILLS ARE REQUIRED WITH RESPECT TO:
- ALGEBRA, WITH PARTICULAR REGARD TO ALGEBRAIC, LOGARITHMIC, EXPONENTIAL, TRIGONOMETRIC AND TRANSCENDENTAL EQUATIONS AND INEQUALITIES;
- TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC TRIGONOMETRIC FUNCTIONS.
Contents
NUMERICAL SETS.
INTRODUCTION TO THE SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO THE REAL NUMBERS. EXTREMES OF A NUMERICAL SET. REAL INTERVALS. NEIGHBOURHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO THE COMPLEX NUMBERS. IMAGINARY UNIT. COMPLEX NUMBERS OPERATIONS. GEOMETRIC AND TRIGONOMETRIC FORMS. DE MOIVRE FORMULA FOR POWERS AND FOR N-TH ROOTS. (LESSON HOURS: 4, EXERCISE HOURS: 3).

REAL FUNCTIONS.
DEFINITION. DOMAIN CODOMAIN AND FUNCTION GRAPH. EXTREMES OF A REAL FUNCTION. MONOTONE FUNCTIONS. COMPOSITE FUNCTIONS. INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER FUNCTION, N-TH ROOT FUNCTION, EXPONENTIAL AND LOGARITHMIC FUNCTIONS, POWER FUNCTION, TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES. (LESSON HOURS: 3, EXERCISE HOURS: 2).

SEQUENCES AND NUMERICAL SERIES.
DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER NUMBER. CAUCHY'S CONVERGENCE CRITERION.
INTRODUCTION TO NUMERICAL SERIES. CONVERGENT, DIVERGENT AND INDETERMINATE SERIES. GEOMETRIC AND HARMONIC SERIES. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: COMPARISON, RATIO AND ROOT CRITERIA. (LESSON HOURS: 3, EXERCISE HOURS: 3).

LIMITS OF A FUNCTION.
DEFINITION. RIGHT AND LEFT LIMITS. UNIQUENESS THEOREM. COMPARISON THEOREMS. OPERATIONS ON LIMITS AND INDETERMINATE FORMS. SIGNIFICANT LIMITS. (LESSON HOURS: 5, EXERCISE HOURS: 3).

CONTINUOUS FUNCTIONS.
DEFINITION. CONTINUITY AND DISCONTINUITY. EXTREME VALUE THEOREM. BOLZANO'S THEOREM, INTERMEDIATE VALUE THEOREM. UNIFORM CONTINUITY. (LESSON HOURS: 4, EXERCISE HOURS: -).

DERIVATIVE OF A FUNCTION.
DEFINITION. RIGHT AND LEFT DERIVATIVES. GEOMETRIC MEANING, 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. (LESSON HOURS: 4, EXERCISE HOURS: 2).

FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS.
ROLLE'S THEOREM. LAGRANGE'S THEOREM AND COROLLARIES. CHAUCHY THEOREM. THEOREM OF DE L'HOSPITAL. CONDITIONS FOR MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS. (LESSON HOURS: 4, EXERCISE HOURS: 2).

STUDY OF THE GRAPH OF A FUNCTION.
ASYMPTOTES OF A GRAPH. MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS, INFLEXION POINTS. GRAPH OF A FUNCTION THROUGH ITS CHARACTERISTIC ELEMENTS. (LESSON HOURS: 4, EXERCISE HOURS: 6).

INTEGRATION OF ONE VARIABLE FUNCTIONS.
DEFINITION OF ANTIDERIVATIVE AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. INTEGRATION RULES AND METHODS. INTEGRATION OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF CALCULUS. (LESSON HOURS: 5, EXERCISE HOURS: 3).
Teaching Methods
THE COURSE COVERS THEORETICAL LESSONS, DURING WHICH THE COURSE TOPICS WILL BE PRESENTED THROUGH LECTURES AND CLASSROOM EXERCISES DURING WHICH THE MAIN TOOLS NECESSARY FOR THE RESOLUTION OF EXERCISES RELATED TO THE CONTENT OF TEACHING WILL BE PROVIDED.
Verification of learning
THE ASSESSMENT OF THE ACHIEVEMENT OF 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, MAGGIOLI EDITORE (2015).
TEACHING MATERIAL AVAILABLE ON THE E-LEARNING IWT PLATFORM.
NOTES FROM LESSONS.