Maria TOTA | MATHEMATICS I

cod. 0612200001

MATHEMATICS I

	0612200001
	DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
	EQF6
	CHEMICAL ENGINEERING
	2021/2022

CURRICULUM FOOD ENGINEERING

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2016
	AUTUMN SEMESTER

TEACHING UNITS

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

PROFESSORS

	MARIA TOTA T Curriculum
	CARMINE MONETTA Curriculum

	Objectives
	KNOWLEDGE AND UNDERSTANDING: THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF MATHEMATICAL ANALYSIS AND LINEAR ALGEBRA, IN PARTICULAR SETS OF NUMBERS, REAL-VALUED FUNCTIONS, SEQUENCES OF REAL NUMBERS, LIMITS OF REAL-VALUED FUNCTIONS, CONTINUOUS FUNCTIONS, DERIVATIVES OF REAL-VALUED FUNCTIONS, THE FUNDAMENTAL THEORY OF DIFFERENTIAL CALCULATION, THE GRAPHIC STUDY OF A FUNCTION, MATRICES AND LINEAR SYSTEMS. VECTOR SPACES. EIGENVALUES AND DIAGONALIZATION. ANALYTIC GEOMETRY. APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS: THE STUDENT HAS TO DEVELOP IN A RIGOROUS AND COHERENT WAY A MATHEMATICAL ARGUMENT, APPLY THE THEOREMES AND THE STUDIED RULES TO SOLVING PROBLEMS. HE HAS TO BE ABLE TO MAKE CALCULATIONS WITH LIMITS, DERIVATIVES AND INTEGRALS (BOTH UNDEFINED AND DEFINED). APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING DESIGN: THE STUDENT HAS TO PRESENT A DEMONSTRATION WITH MATHEMATICAL RIGOR. HE HAS TO KNOW HOW TO BUILD METHODS AND PROCEDURES TO SOLVE PROBLEMS. MAKING JUDGMENTS - ENGINEERING PRACTICE: THE STUDENT HAS TO KNOW HOW TO APPLY KNOWLEDGE ACQUIRED TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE. COMMUNICATION SKILLS – TRANSVERSAL SKILLS: THE STUDENT HAS TO KNOW HOW TO WORK IN THE GROUP. HE HAS TO KNOW HOW TO ORGANIZE THE ARGUMENTS OBJECT OF THE COURSE. LEARNING SKILLS – TRANSVERSAL SKILLS: THE STUDENT HAS TO DEVELOP THE LEARNING SKILLS THAT WILL BE NECESSARY FOR INSERTING HIM IN THE FOLLOWING STUDIES WITH A HIGH AUTONOMY OF STUDY, AND CRITICALLY FACE MORE GENERAL PROBLEMS.

KNOWLEDGE AND UNDERSTANDING:

THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF MATHEMATICAL ANALYSIS AND LINEAR ALGEBRA, IN PARTICULAR SETS OF NUMBERS, REAL-VALUED FUNCTIONS, SEQUENCES OF REAL NUMBERS, LIMITS OF REAL-VALUED FUNCTIONS, CONTINUOUS FUNCTIONS, DERIVATIVES OF REAL-VALUED FUNCTIONS, THE FUNDAMENTAL THEORY OF DIFFERENTIAL CALCULATION, THE GRAPHIC STUDY OF A FUNCTION, MATRICES AND LINEAR SYSTEMS. VECTOR SPACES. EIGENVALUES AND DIAGONALIZATION. ANALYTIC GEOMETRY.

APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS:

THE STUDENT HAS TO DEVELOP IN A RIGOROUS AND COHERENT WAY A MATHEMATICAL ARGUMENT, APPLY THE THEOREMES AND THE STUDIED RULES TO SOLVING PROBLEMS. HE HAS TO BE ABLE TO MAKE CALCULATIONS WITH LIMITS, DERIVATIVES AND INTEGRALS (BOTH UNDEFINED AND DEFINED).

APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING DESIGN:

THE STUDENT HAS TO PRESENT A DEMONSTRATION WITH MATHEMATICAL RIGOR. HE HAS TO KNOW HOW TO BUILD METHODS AND PROCEDURES TO SOLVE PROBLEMS.

MAKING JUDGMENTS - ENGINEERING PRACTICE:

THE STUDENT HAS TO KNOW HOW TO APPLY KNOWLEDGE ACQUIRED TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE.

COMMUNICATION SKILLS – TRANSVERSAL SKILLS:

THE STUDENT HAS TO KNOW HOW TO WORK IN THE GROUP. HE HAS TO KNOW HOW TO ORGANIZE THE ARGUMENTS OBJECT OF THE COURSE.

LEARNING SKILLS – TRANSVERSAL SKILLS:

THE STUDENT HAS TO DEVELOP THE LEARNING SKILLS THAT WILL BE NECESSARY FOR INSERTING HIM IN THE FOLLOWING STUDIES WITH A HIGH AUTONOMY OF STUDY, AND CRITICALLY FACE MORE GENERAL PROBLEMS.

	Prerequisites
	FOR THE SUCCESSFUL ACHIEVEMENT OF THE GOALS OF THE COURSE, STUDENTS ARE REQUIRED TO HAVE THE FOLLOWING PREREQUISITES: -KNOWLEDGE OF ALGEBRA, WITH PARTICULAR REFERENCE TO: ALGEBRAIC, LOGARITHMIC AND EXPONENTIAL EQUATIONS AND INEQUALITIES. -KNOWLEDGE RELATED TO TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC TRIGONOMETRIC FUNCTIONS.

	Contents
	NUMERICAL SETS (THEORY HOURS AND EXERCISES HOURS 4/4): 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. REAL FUNCTIONS (THEORY HOURS AND EXERCISES HOURS 6/4): 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. NUMERICAL SEQUENCES (THEORY HOURS AND EXERCISES HOURS 4/2): DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. NEPERO’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE. LIMITS OF A FUNCTION (THEORY HOURS AND EXERCISES HOURS 6/4): DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS. CONTINUOUS FUNCTIONS (THEORY HOURS AND EXERCISES HOURS 6/4): DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY. DERIVATIVE OF A FUNCTION (THEORY HOURS AND EXERCISES HOURS 4/4): 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. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS. GRAPH OF A FUNCTION (THEORY HOURS AND EXERCISES HOURS 4/4): ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH. MATRICES (THEORY HOURS AND EXERCISES HOURS 2/2): DEFINITIONS AND PROPERTIES. DETERMINANT AND ITS COMPUTATION: LAPLACE’S FORMULA. RANK OF A MATRIX. INVERSE OF A MATRIX. LINEAR SYSTEMS OF EQUATIONS (THEORY HOURS AND EXERCISES HOURS 2/2): DEFINITIONS, MATRIX EQUATION, CONSISTENCY AND INCONSISTENCY, NUMBER OF SOLUTIONS. ROUCHÉ-CAPELLI THEOREM. CRAMER THEOREM AND RULE. GAUSSIAN ELIMINATION ALGORITHM. DISCUSSION OF LINEAR SYSTEMS WITH A PARAMETER. VECTOR SPACES (THEORY HOURS AND EXERCISES HOURS 2/2): THE VECTOR SPACE STRUCTURE. LINEAR DEPENDENCE AND INDEPENDENCE OF VECTORS. GENERATORS. BASES. DIMENSION OF A VECTOR SPACE AND RELATED RESULTS. VECTOR SUBSPACES. DIAGONALIZATION (THEORY HOURS AND EXERCISES HOURS 2/2): EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND EQUATION. ALGEBRAIC AND GEOMETRIC MULTEPLICITIES. DEFINITION OF DIAGONALIZATION AND ITS CHARACTERIZATIONS (FOR MATRICES). SUFFICIENT CONDITION FOR THE DIAGONALIZATION. ANALYTICAL GEOMETRY (THEORY HOURS AND EXERCISES HOURS 4/4): PLANE AND SPACE CARTESIAN COORDINATE SYSTEM. EQUATION OF A LINE (CARTESIAN, PARAMETRIC) AND OF A PLANE. PARALLELISM OF LINES AND PLANES. PROPER AND IMPROPER BUNDLE OF STRAIGHT LINES. NUMERICAL SERIES (THEORY HOURS AND EXERCISES HOURS 4/2): INTRODUCTION TO MAIN NUMERICAL SERIES.

Contents

NUMERICAL SETS (THEORY HOURS AND EXERCISES HOURS 4/4): 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.

REAL FUNCTIONS (THEORY HOURS AND EXERCISES HOURS 6/4): 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.

NUMERICAL SEQUENCES (THEORY HOURS AND EXERCISES HOURS 4/2): DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. NEPERO’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.

LIMITS OF A FUNCTION (THEORY HOURS AND EXERCISES HOURS 6/4): DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.

CONTINUOUS FUNCTIONS (THEORY HOURS AND EXERCISES HOURS 6/4): DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY.

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

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

MATRICES (THEORY HOURS AND EXERCISES HOURS 2/2): DEFINITIONS AND PROPERTIES. DETERMINANT AND ITS COMPUTATION: LAPLACE’S FORMULA. RANK OF A MATRIX. INVERSE OF A MATRIX.

LINEAR SYSTEMS OF EQUATIONS (THEORY HOURS AND EXERCISES HOURS 2/2): DEFINITIONS, MATRIX EQUATION, CONSISTENCY AND INCONSISTENCY, NUMBER OF SOLUTIONS. ROUCHÉ-CAPELLI THEOREM. CRAMER THEOREM AND RULE. GAUSSIAN ELIMINATION ALGORITHM. DISCUSSION OF LINEAR SYSTEMS WITH A PARAMETER.

VECTOR SPACES (THEORY HOURS AND EXERCISES HOURS 2/2): THE VECTOR SPACE STRUCTURE. LINEAR DEPENDENCE AND INDEPENDENCE OF VECTORS. GENERATORS. BASES. DIMENSION OF A VECTOR SPACE AND RELATED RESULTS. VECTOR SUBSPACES.

DIAGONALIZATION (THEORY HOURS AND EXERCISES HOURS 2/2): EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND EQUATION. ALGEBRAIC AND GEOMETRIC MULTEPLICITIES. DEFINITION OF DIAGONALIZATION AND ITS CHARACTERIZATIONS (FOR MATRICES). SUFFICIENT CONDITION FOR THE DIAGONALIZATION.

ANALYTICAL GEOMETRY (THEORY HOURS AND EXERCISES HOURS 4/4): PLANE AND SPACE CARTESIAN COORDINATE SYSTEM. EQUATION OF A LINE (CARTESIAN, PARAMETRIC) AND OF A PLANE. PARALLELISM OF LINES AND PLANES. PROPER AND IMPROPER BUNDLE OF STRAIGHT LINES.

NUMERICAL SERIES (THEORY HOURS AND EXERCISES HOURS 4/2): INTRODUCTION TO MAIN NUMERICAL SERIES.

	Teaching Methods
	ATTENDANCE AT THE LECTURES IS STRONGLY RECOMMENDED.

	Verification of learning
	THERE WILL BE WRITTEN EXERCISES AND AN ORAL INTERVIEW. WRITTEN EXERCISES: IT CONSISTS IN SOLVING TYPICAL PROBLEMS. SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED. ORAL INTERVIEW: IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING. FINAL EVALUATION: THE FINAL MARK, EXPRESSED ON A SCALE FROM 18 TO 30 (POSSIBLY WITH LAUDEM), DEPENDS ON THE MARK OF THE WRITTEN EXAM, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW. CRITERION CORRESPONDING TO THE ACHIEVEMENT OF A MINIMUM THRESHOLD TO PASS THE EXAM: THE KNOWLEDGE BASIS IS ACCEPTABLE IN RELATION TO SOME OF THE PROGRAM CONTENT. THE ABILITY TO SOLVE PROBLEMS EXTENDS TO SIMPLE "STANDARD" PROBLEMS, FOLLOWING ROUTINE PROCEDURES. TRANSFERABLE SKILLS ARE RUDIMENTAL. CRITERION CORRESPONDING TO THE ACHIEVEMENT OF THE EXCELLENCE: THE KNOWLEDGE BASE COVERS THE ENTIRE PROGRAM. THE CONCEPTUAL UNDERSTANDING IS FULL AND DEEP. TROUBLESHOOTING PROCEDURES ARE ADEQUATE TO THE NATURE OF THE PROBLEM. PERFORMANCE IN TRANSFERABLE SKILLS IS GENERALLY VERY GOOD.

Verification of learning

THERE WILL BE WRITTEN EXERCISES AND AN ORAL INTERVIEW.

WRITTEN EXERCISES: IT CONSISTS IN SOLVING TYPICAL PROBLEMS. SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED.

ORAL INTERVIEW: IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING.

FINAL EVALUATION: THE FINAL MARK, EXPRESSED ON A SCALE FROM 18 TO 30 (POSSIBLY WITH LAUDEM), DEPENDS ON THE MARK OF THE WRITTEN EXAM, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

CRITERION CORRESPONDING TO THE ACHIEVEMENT OF A MINIMUM THRESHOLD TO PASS THE EXAM:
THE KNOWLEDGE BASIS IS ACCEPTABLE IN RELATION TO SOME OF THE PROGRAM CONTENT.
THE ABILITY TO SOLVE PROBLEMS EXTENDS TO SIMPLE "STANDARD" PROBLEMS, FOLLOWING ROUTINE PROCEDURES.
TRANSFERABLE SKILLS ARE RUDIMENTAL.

CRITERION CORRESPONDING TO THE ACHIEVEMENT OF THE EXCELLENCE:
THE KNOWLEDGE BASE COVERS THE ENTIRE PROGRAM. THE CONCEPTUAL UNDERSTANDING IS FULL AND DEEP.
TROUBLESHOOTING PROCEDURES ARE ADEQUATE TO THE NATURE OF THE PROBLEM.
PERFORMANCE IN TRANSFERABLE SKILLS IS GENERALLY VERY GOOD.

	Texts
	THEORY - P. MARCELLINI - C. SBORDONE, "ANALISI MATEMATICA UNO", LIGUORI EDITORE (1996). - C. DELIZIA, P. LONGOBARDI, M. MAJ, C. NICOTERA, "MATEMATICA DISCRETA", MCGRAW-HILL (2009). EXERCISES - P. MARCELLINI - C. SBORDONE, "ESERCITAZIONI DI MATEMATICA I - VOL. 1 E 2", LIGUORI EDITORE (2016). - C. DELIZIA, P. LONGOBARDI, M. MAJ, C. NICOTERA, "MATEMATICA DISCRETA", MCGRAW-HILL (2009).

	More Information
	CONTACT THE LECTURER AT MTOTA@UNISA.IT

BETA VERSION Data source ESSE3 [Ultima Sincronizzazione: 2022-11-21]

Maria TOTA | MATHEMATICS I