Ingegneria Gestionale | MATHEMATICS I

cod. 0612600001

MATHEMATICS I

	0612600001
	DIPARTIMENTO DI INGEGNERIA INDUSTRIALE

	INDUSTRIAL ENGINEERING AND MANAGEMENT
	2014/2015

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	9	90	LESSONS	BASIC COMPULSORY SUBJECTS

	Objectives
	THE COURSE WILL PROVIDE THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS, LINEAR ALGEBRA AND ANALYTICAL GEOMETRY IN THE EUCLIDEAN SPACE. ANOTHER COURSE'S AIM CONSISTS OF DEVELOPING A SCIENTIFIC AND LOGICAL APPROACH TO PROBLEMS AND PHENOMENA THAT STUDENTS WILL ENCOUNTER DURING THEIR FURTHER STUDIES. COURSE SUBJECTS WILL BE PRESENTED IN A RIGOROUS AND CONCISE WAY AND THEY WILL BE ACCOMPANIED BY EXERCISES DEVOTED TO PROMOTE UNDERSTANDING OF CONCEPTS ALSO IN ORDER TO PROMOTE THE INTERACTIONS WITH THE STUDENTS. KNOWLEDGE AND UNDERSTANDING: LEARNERS SHOULD BE ABLE TO UNDERSTAND THE TERMINOLOGY AND THE LOGICAL DEVELOPMENT OF A MATHEMATICAL REASONING, THEY SHOULD KNOW HOW TO PERFORM A CALCULATION PROCEDURE RELATED TO THE COURSE SUBJECTS. APPLYING KNOWLEDGE AND UNDERSTANDING: STUDENTS SHOULD BE ABLE TO APPLY LEARNED RESULTS AND RULES IN ORDER TO SOLVE BASIC PROBLEMS. THEY SHOULD ALSO ORGANIZE A LOGICAL REASONING AND A RIGOROUS PROCEDURE TO EXPOSE THE ARGUMENTS PROPER OF CALCULUS, LINEAR ALGEBRA AND ANALYTICAL GEOMETRY IN ORDER TO FACE AND TARGET ALSO QUESTIONS ARISING FROM OTHER CONTEXTS. MAKING JUDGEMENTS: STUDENTS SHOULD START RECOGNIZING HOW TO ADDRESS AND TO FORMULATE, IN TERMS OF BASIC MATHEMATICS, SIMPLE PROBLEMS DERIVING FROM APPLICATIONS. THEY SHOULD KNOW HOW TO INTERPRET A SINGLE VARIABLE REAL VALUED FUNCTION AND ITS GRAPH AND ANSWER BASIC QUESTIONS IN TERMS OF MATHEMATICS. COMMUNICATION SKILLS: IT IS EXPECTED THAT STUDENTS SHOULD KNOW HOW TO MAKE RIGOROUS COMPUTATIONS, AND TO STATE, IN A CONCISE AND RIGOROUS WAY, KEY-CONCEPTS REGARDING THE TOPICS OF THE COURSE, ALSO IN ORDER TO BE ABLE TO WORK IN GROUPS. LEARNING SKILLS: STUDENTS SHOULD BE ABLE TO APPLY THE BASIC CONCEPTS OF CALCULUS, LINEAR ALGEBRA AND ANALYTICAL GEOMETRY TO DIFFERENT CONTEXTS, ESSENTIALLY TO THOSE COMING FROM APPLIED SCIENCES. FURTHERMORE THEY HAVE TO BE ABLE TO ENLARGE THEIR KNOWLEDGE IN BASIC MATHEMATICS BY USING DIFFERENT SOURCES AND/OR TEXTBOOKS.

THE COURSE WILL PROVIDE THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS, LINEAR ALGEBRA AND ANALYTICAL GEOMETRY IN THE EUCLIDEAN SPACE. ANOTHER COURSE'S AIM CONSISTS OF DEVELOPING A SCIENTIFIC AND LOGICAL APPROACH TO PROBLEMS AND PHENOMENA THAT STUDENTS WILL ENCOUNTER DURING THEIR FURTHER STUDIES. COURSE SUBJECTS WILL BE PRESENTED IN A RIGOROUS AND CONCISE WAY AND THEY WILL BE ACCOMPANIED BY EXERCISES DEVOTED TO PROMOTE UNDERSTANDING OF CONCEPTS ALSO IN ORDER TO PROMOTE THE INTERACTIONS WITH THE STUDENTS.

KNOWLEDGE AND UNDERSTANDING:
LEARNERS SHOULD BE ABLE TO UNDERSTAND THE TERMINOLOGY AND THE LOGICAL DEVELOPMENT OF A MATHEMATICAL REASONING, THEY SHOULD KNOW HOW TO PERFORM A CALCULATION PROCEDURE RELATED TO THE COURSE SUBJECTS.

APPLYING KNOWLEDGE AND UNDERSTANDING:

STUDENTS SHOULD BE ABLE TO APPLY LEARNED RESULTS AND RULES IN ORDER TO SOLVE BASIC PROBLEMS. THEY SHOULD ALSO ORGANIZE A LOGICAL REASONING AND A RIGOROUS PROCEDURE TO EXPOSE THE ARGUMENTS PROPER OF CALCULUS, LINEAR ALGEBRA AND ANALYTICAL GEOMETRY IN ORDER TO FACE AND TARGET ALSO QUESTIONS ARISING FROM OTHER CONTEXTS.

MAKING JUDGEMENTS:
STUDENTS SHOULD START RECOGNIZING HOW TO ADDRESS AND TO FORMULATE, IN TERMS OF BASIC MATHEMATICS, SIMPLE PROBLEMS DERIVING FROM APPLICATIONS. THEY SHOULD KNOW HOW TO INTERPRET A SINGLE VARIABLE REAL VALUED FUNCTION AND ITS GRAPH AND ANSWER BASIC QUESTIONS IN TERMS OF MATHEMATICS.

COMMUNICATION SKILLS:
IT IS EXPECTED THAT STUDENTS SHOULD KNOW HOW TO MAKE RIGOROUS COMPUTATIONS, AND TO STATE, IN A CONCISE AND RIGOROUS WAY, KEY-CONCEPTS REGARDING THE TOPICS OF THE COURSE, ALSO IN ORDER TO BE ABLE TO WORK IN GROUPS.

LEARNING SKILLS:
STUDENTS SHOULD BE ABLE TO APPLY THE BASIC CONCEPTS OF CALCULUS, LINEAR ALGEBRA AND ANALYTICAL GEOMETRY TO DIFFERENT CONTEXTS, ESSENTIALLY TO THOSE COMING FROM APPLIED SCIENCES. FURTHERMORE THEY HAVE TO BE ABLE TO ENLARGE THEIR KNOWLEDGE IN BASIC MATHEMATICS BY USING DIFFERENT SOURCES AND/OR TEXTBOOKS.

	Prerequisites
	BASIC ALGEBRA: EQUATIONS, INEQUALITIES, RATIONAL AND IRRATIONAL ONES, EQUATIONS AND INEQUALITIES INVOLVING LOGARITHMS, EXPONENTIALS, TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES. BASIC TRIGONOMETRY. BASIC CARTESIAN GEOMETRY IN THE PLANE.

	Contents
	VECTOR ALGEBRA: INTRODUCTION TO VECTOR ALGEBRA AND OPERATIONS WITH VECTORS.(HRS: LECT. 1, PRACT. 2) NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE MOIVRE’S FORMULA. N-TH ROOTS.(HRS: LECT. 5, PRACT. 3) REAL FUNCTIONS:DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS.(HRS: LECT 4, PRACT. 2) BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST ORDER, QUADRATIC, BINOMIAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC EQUATIONS. SYSTEMS. FIRST ORDER, QUADRATIC, RATIONAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC INEQUALITIES. SYSTEMS.( HRS: LECT. 2, PRACT. 3) NUMERICAL SEQUENCES: DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.(HRS: LECT. 2, PRACT. 2) LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.(HRS: LECT. 5, PRACT. 3) CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY.(LECT. HRS 5) DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND ITS GEOMETRIC MEANING.( HRS: LECT. 5, PRACT. 3) FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.(HRS: LECT. 4, PRACT. 3) GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. FINDING LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAQWING GRAPH.(HRS: LECT. 6, PRACT. 8) MATRICES AND LINEAR SYSTEMS: MATRICES AND DETERMINANTS. LINEAR SYSTEMS: ROUCHÉ-CAPELLI AND CRAMER THEOREMS.(HRS: LECT. 2, PRACT. 2) VECTOR SPACES:THE STRUCTURE OF THE VECTOR SPACE. LINEAR DEPENDENCE AND INDEPENDENCE. VECTOR SPACES, FINITE DIMENSION AND RELATED THEOREM. SUBSPACES. INTERSECTION AND SUM OF SUBSPACES , DIRECT SUM. DOT PRODUCT. REAL EUCLIDEAN VECTOR SPACE. NORM. CAUCHY–SCHWARZ INEQUALITY. ANGLE. ORTHOGONAL VECTORS. ORTHONORMAL BASES. COMPONENTS IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT PROCEDURE.(HRS: LECT. 3, PRACT. 2) LINEAR OPERATORS AND DIAGONALIZATION: DEFINITIONS OF LINEAR OPERATOR. KERNEL AND IMAGE. PROPERTIES AND CHARACTERIZATIONS. RANK-NULLITY THEOREM. MATRIX REPRESENTATION. CHARACTERISTIC POLYNOMIAL. EIGENSPACE. ALGEBRAIC AND GEOMETRIC MULTIPLICITIES. DIAGONALIZATION: DEFINITIONS AND CHARACTERIZATIONS. SUFFICIENT CONDITION FOR THE DIAGONALIZATION. ORTHOGONAL DIAGONALIZATION. SPECTRAL THEOREM.(HRS: LECT. 5, PRACT. 3) ANALYTICALGEOMETRY: PLANE CARTESIAN COORDINATE SYSTEM. IMPLICIT, EXPLICIT, SEGMENTAL EQUATION OF A LINE. PARALLELISM OF LINES. IMPROPER AND PROPER BUNDLE OF LINES. LINE THROUGH A POINT. LINE PASSING THROUGH A POINT AND PARALLEL TO A GIVEN LINE. CONDITIONS FOR PERPENDICULAR LINES. CONICS. ALGORITHM FOR REDUCING CONICS TO CANONICAL FORM. SPATIAL CARTESIAN COORDINATE SYSTEM. PLANE EQUATION: CARTESIAN AND PARAMETRIC. LINE EQUATION: PARAMETRIC, CARTESIAN, SYMMETRIC. BUNDLES AND STARS OF PLANS. CONDITIONS FOR PARALLEL AND PERPENDICULAR LINES, LINE AND PLAN, PLANS.(HRS: LECT. 3,PRACT. 2) TOTAL HOURS LECT.: 52, PRACT.:38

Contents

VECTOR ALGEBRA: INTRODUCTION TO VECTOR ALGEBRA AND OPERATIONS WITH VECTORS.(HRS: LECT. 1, PRACT. 2)
NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE MOIVRE’S FORMULA. N-TH ROOTS.(HRS: LECT. 5, PRACT. 3)
REAL FUNCTIONS:DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS.(HRS: LECT 4, PRACT. 2)
BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST ORDER, QUADRATIC, BINOMIAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC EQUATIONS. SYSTEMS. FIRST ORDER, QUADRATIC, RATIONAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC INEQUALITIES. SYSTEMS.( HRS: LECT. 2, PRACT. 3)
NUMERICAL SEQUENCES: DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.(HRS: LECT. 2, PRACT. 2)
LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.(HRS: LECT. 5, PRACT. 3)
CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY.(LECT. HRS 5)
DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND ITS GEOMETRIC MEANING.( HRS: LECT. 5, PRACT. 3)
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.(HRS: LECT. 4, PRACT. 3)
GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. FINDING LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAQWING GRAPH.(HRS: LECT. 6, PRACT. 8)
MATRICES AND LINEAR SYSTEMS: MATRICES AND DETERMINANTS. LINEAR SYSTEMS: ROUCHÉ-CAPELLI AND CRAMER THEOREMS.(HRS: LECT. 2, PRACT. 2)
VECTOR SPACES:THE STRUCTURE OF THE VECTOR SPACE. LINEAR DEPENDENCE AND INDEPENDENCE. VECTOR SPACES, FINITE DIMENSION AND RELATED THEOREM. SUBSPACES. INTERSECTION AND SUM OF SUBSPACES , DIRECT SUM. DOT PRODUCT. REAL EUCLIDEAN VECTOR SPACE. NORM. CAUCHY–SCHWARZ INEQUALITY. ANGLE. ORTHOGONAL VECTORS. ORTHONORMAL BASES. COMPONENTS IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT PROCEDURE.(HRS: LECT. 3, PRACT. 2)
LINEAR OPERATORS AND DIAGONALIZATION: DEFINITIONS OF LINEAR OPERATOR. KERNEL AND IMAGE. PROPERTIES AND CHARACTERIZATIONS. RANK-NULLITY THEOREM. MATRIX REPRESENTATION. CHARACTERISTIC POLYNOMIAL. EIGENSPACE. ALGEBRAIC AND GEOMETRIC MULTIPLICITIES. DIAGONALIZATION: DEFINITIONS AND CHARACTERIZATIONS. SUFFICIENT CONDITION FOR THE DIAGONALIZATION. ORTHOGONAL DIAGONALIZATION. SPECTRAL THEOREM.(HRS: LECT. 5, PRACT. 3)
ANALYTICALGEOMETRY: PLANE CARTESIAN COORDINATE SYSTEM. IMPLICIT, EXPLICIT, SEGMENTAL EQUATION OF A LINE. PARALLELISM OF LINES. IMPROPER AND PROPER BUNDLE OF LINES. LINE THROUGH A POINT. LINE PASSING THROUGH A POINT AND PARALLEL TO A GIVEN LINE. CONDITIONS FOR PERPENDICULAR LINES. CONICS. ALGORITHM FOR REDUCING CONICS TO CANONICAL FORM. SPATIAL CARTESIAN COORDINATE SYSTEM. PLANE EQUATION: CARTESIAN AND PARAMETRIC. LINE EQUATION: PARAMETRIC, CARTESIAN, SYMMETRIC. BUNDLES AND STARS OF PLANS. CONDITIONS FOR PARALLEL AND PERPENDICULAR LINES, LINE AND PLAN, PLANS.(HRS: LECT. 3,PRACT. 2)
TOTAL HOURS LECT.: 52, PRACT.:38

	Teaching Methods
	1)FRONTAL LECTURES DEVOTED TO SHOW THE MOTIVATIONS BEHIND THE THEORY, THE APPLICATIONS TO OTHER SCIENCES, THE METHODS USEFUL TO ADDRESS A PROBLEM. 2) EXERCISES INVOLVING THE STUDENTS AS A REALLY ACTIVE PART.

	Verification of learning
	THE FINAL EXAM WILL CONSIST OF TWO PARTS: A WRITTEN AND AN ORAL EXAMINATION. STUDENTS WILL HAVE TO SHOW A COMPREHENSIVE UNDERSTANDING OF THE BASIC TOPICS OF THE COURSE. THEY WILL BE ABLE TO APPLY THE LEARNED TECHNIQUES TO SOLVE SIMPLE PROBLEMS. POSITIVE GRADES ARE AMONG 18 AND 30 POINTS OVER 30. EXCELLENT STUDENTS MIGHT GET 30/30 CUM LAUDE.

	Texts
	SEE THE ITALIAN VERSION FOR SUGGESTED READINGS IN ITALIAN. FOR A COMPREHENSIVE TREATMENT I ALSO RECOMMEND ADVANCED CALCULUS BY ANGUS E. TAYLOR AND W. ROBERT MANN.