# Ingegneria Gestionale | CALCULUS I

## Ingegneria Gestionale CALCULUS I

 0612600001 DIPARTIMENTO DI INGEGNERIA INDUSTRIALE EQF6 INDUSTRIAL ENGINEERING AND MANAGEMENT 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

 TIZIANA DURANTE T MARIA PIA D'ARIENZO
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 THE ABILITY TO USE THE CALCULATION TOOLS STUDIED.

KNOWLEDGE AND UNDERSTANDING

THE AIM IS TO REACH A GOOD LEVEL OF UNDERSTANDING AND KNOWLEDGE OF THE TOPIC:
NUMERICAL SETS. REAL FUNCTIONS. 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 WHICH ARE THE POSSIBLE METHODS OF RESOLUTION TO THE MATHEMATICAL PROBLEMS AND TO STUDY IN A CRITICAL AND INDEPENDENT WAY.

COMMUNICATIVE ABILITIES

THE STUDENT WILL BE ABLE TO DESCRIBE IN A CLEAR AND EXAUSTIVE WAY THE TOPICS COVERED SHOWING MASTERY OF THE MATHEMATICAL LANGUAGE.
Prerequisites
IN ORDER TO REACH THE OBJECTIVES STUDENTS SHOULD HAVE A BACKGROUND RELATING TO ALGEBRA, IN PARTICULAR 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 THE THEORY OF THE SETS. INTRODUCTION TO REAL NUMBERS. EXTREME OF A NUMERICAL SET. INTERVALS. INTORNI, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. ALGECRICA FORM AND TRIGONOMETRIC FORM. DE MOIVRE FORMULA. N-ESIME ROOTS.

LESSON / EXERCISES : HOURS 4/3

REAL FUNCTIONS: DEFINITION. EXTREME OF A REAL FUNCTION. MONOTONE FUNCTIONS. COMPOSED FUNCTIONS. INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-MA POWER FUNCTION AND N-MA ROOT, EXPONENTIAL FUNCTION, LOGARITHMIC FUNCTION, POWER FUNCTION, TRIGONOMETRIC FUNCTIONS AND THEIR INVERSE.

LESSON / EXERCISES : HOURS 3/3

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.
LESSON / EXERCISES : HOURS 2/5

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.
LESSON /EXERCISE HOURS: 5/4

LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.
LESSON /EXERCISE HOURS: 5/5

CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY.
LESSON / EXERCISES : HOURS 6/0

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.

LESSON / EXERCISES : HOURS 5/3

FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.
LESSON / EXERCISES : HOURS 5/3

GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH.
LESSON / EXERCISES : HOURS 6/9

INTEGRATION OF ONE VARIABLE FUNCTIONS: PRIMITIVE FUNCTIONS AND INDEFINITE INTEGRAL. DEFINITE INTEGRAL AND GEOMETRICAL MEANING. BASIC INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. FUNDAMENTAL THEOREM OF CALCULUS
LESSON / EXERCISES : HOURS 7/7
Teaching Methods
THE COURSE CONSISTS IN THEORETICAL LECTURES AND EXERCISES. 90 HOURS IN TOTAL: 48 HOURS FOR THE THEORY AND 42 HOURS FOR THE EXERCISES.
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 USE THE ACQUIRED KNOWLEDGE.
THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL EXAMINATION.
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 AND MASTERY IN ALL THE TOPICS OF THE COURSE, AS DEFINITIONS, AS PROOFS OF THEOREMS AND IN SOLVING EXERCISES.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL VALUTATION OF THE STUDENT.
Texts
BASIC TEXT FOR THEORY:
P.MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA UNO, LIGUORI 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.