Ciro D'APICE | Mathematics I

cod. 0612100001

MATHEMATICS I

	0612100001
	DIPARTIMENTO DI INGEGNERIA CIVILE

	BSC DEGREE IN CIVIL ENGINEERING
	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

PROFESSORS

	CIRO D'APICE T Curriculum
	GIOVANNINA ALBANO Curriculum

	Objectives
	THE COURSE AIMS AT THE ACQUIRING OF THE BASIC ELEMENTS OF CALCULUS AND LINEAR ALGEBRA. LEARNING OUTCOMES OF COURSE CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELATED COMPUTATIONAL TOOLS. THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS BUT CONCISE WAY AND IT WILL BE SUPPORTED BY PARALLEL EXERCISE SESSIONS DESIGNED TO PROMOTE MEANINGFUL UNDERSTANDING OF CONCEPTS. KNOWLEDGE AND UNDERSTANDING ELEMENTS OF VECTOR ALGEBRA. NUMERICAL SETS. REAL FUNCTIONS. BASIC NOTIONS ON EQUATIONS AND INEQUALITIES. NUMERICAL SEQUENCES. LIMITS OF A FUNCTION. CONTINUOUS FUNCTIONS. DERIVATIVE OF A FUNCTION. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS. GRAPH OF A FUNCTION. MATRICES AND LINEAR SYSTEMS. VECTOR SPACES. LINEAR OPERATORS AND DIAGONALIZATION. ANALYTICAL GEOMETRY. APPLYING KNOWLEDGE AND UNDERSTANDING BEING ABLE TO APPLY THEOREMS AND RULES IN PROBLEM SOLVING. BEING ABLE TO CONSISTENTLY BUILD PROOFS. BEING ABLE TO PERFORM CALCULATIONS WITH LIMITS, DERIVATIVES. MAKING JUDGEMENTS TO IDENTIFY THE BEST AND EFFICIENT METHOD TO SOLVE A MATHEMATICAL PROBLEM. COMMUNICATION SKILLS ABILITY TO WORK IN GROUPS. ABILITY TO ORALLY PRESENT A TOPIC OF THE COURSE. LEARNING SKILLS BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE. SKILL TO DEEPEN THE TOPICS DEALT WITH BY USING MATERIALS DIFFERENT FROM THOSE PRESENTED DURING THE COURSE.

THE COURSE AIMS AT THE ACQUIRING OF THE BASIC ELEMENTS OF CALCULUS AND LINEAR ALGEBRA. LEARNING OUTCOMES OF COURSE CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELATED COMPUTATIONAL TOOLS.
THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS BUT CONCISE WAY AND IT WILL BE SUPPORTED BY PARALLEL EXERCISE SESSIONS DESIGNED TO PROMOTE MEANINGFUL UNDERSTANDING OF CONCEPTS.
KNOWLEDGE AND UNDERSTANDING
ELEMENTS OF VECTOR ALGEBRA. NUMERICAL SETS. REAL FUNCTIONS. BASIC NOTIONS ON EQUATIONS AND INEQUALITIES. NUMERICAL SEQUENCES. LIMITS OF A FUNCTION. CONTINUOUS FUNCTIONS. DERIVATIVE OF A FUNCTION. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS. GRAPH OF A FUNCTION. MATRICES AND LINEAR SYSTEMS. VECTOR SPACES. LINEAR OPERATORS AND DIAGONALIZATION. ANALYTICAL GEOMETRY.
APPLYING KNOWLEDGE AND UNDERSTANDING
BEING ABLE TO APPLY THEOREMS AND RULES IN PROBLEM SOLVING. BEING ABLE TO CONSISTENTLY BUILD PROOFS. BEING ABLE TO PERFORM CALCULATIONS WITH LIMITS, DERIVATIVES.
MAKING JUDGEMENTS
TO IDENTIFY THE BEST AND EFFICIENT METHOD TO SOLVE A MATHEMATICAL PROBLEM.
COMMUNICATION SKILLS
ABILITY TO WORK IN GROUPS. ABILITY TO ORALLY PRESENT A TOPIC OF THE COURSE.
LEARNING SKILLS
BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE. SKILL TO DEEPEN THE TOPICS DEALT WITH BY USING MATERIALS DIFFERENT FROM THOSE PRESENTED DURING THE COURSE.

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

	Contents
	VECTOR ALGEBRA: INTRODUCTION TO VECTOR ALGEBRA AND OPERATIONS WITH VECTORS. (HOURS LECTURE/PRACTICE/LABORATORY 1/2/-) 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’SFORMULA. N-THROOTS.(HOURS 5/3/-) REAL FUNCTIONS:DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE INVERTIBLE FUNCTIONS.ELEMENTARY FUNCTIONS: N-THPOWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS.(HOURS 4/2/-) 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.(HOURS 2/3/-) NUMERICAL SEQUENCES:DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.(HOURS 2/2/-) LIMITS OF A FUNCTION: DEFINITION. RIGHTAND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.(HOURS 5/3/-) CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY.(HOURS 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.(HOURS 5/3/-) FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITALTHEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.(HOURS 4/3/-) GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. FINDING LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAQWING GRAPH.(HOURS 6/8/-) MATRICES AND LINEAR SYSTEMS: MATRICES AND DETERMINANTS. LINEAR SYSTEMS: ROUCHÉ-CAPELLIAND CRAMER THEOREMS.(HOURS 2/2/-) 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. DOTPRODUCT. REAL EUCLIDEAN VECTOR SPACE. NORM. CAUCHY–SCHWARZ INEQUALITY. ANGLE. ORTHOGONAL VECTORS. ORTHONORMAL BASES. COMPONENTS IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT PROCEDURE.(HOURS 3/2/-) 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. SPECTRALTHEOREM.(HOURS 5/3/-) 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.(HOURS 3/2/-) TOTAL HOURS 52/38/-

Contents

VECTOR ALGEBRA: INTRODUCTION TO VECTOR ALGEBRA AND OPERATIONS WITH VECTORS. (HOURS LECTURE/PRACTICE/LABORATORY 1/2/-)
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’SFORMULA. N-THROOTS.(HOURS 5/3/-)
REAL FUNCTIONS:DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE INVERTIBLE FUNCTIONS.ELEMENTARY FUNCTIONS: N-THPOWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS.(HOURS 4/2/-)
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.(HOURS 2/3/-)
NUMERICAL SEQUENCES:DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.(HOURS 2/2/-)
LIMITS OF A FUNCTION: DEFINITION. RIGHTAND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.(HOURS 5/3/-)
CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY.(HOURS 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.(HOURS 5/3/-)
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITALTHEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.(HOURS 4/3/-)
GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. FINDING LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAQWING GRAPH.(HOURS 6/8/-)
MATRICES AND LINEAR SYSTEMS: MATRICES AND DETERMINANTS. LINEAR SYSTEMS: ROUCHÉ-CAPELLIAND CRAMER THEOREMS.(HOURS 2/2/-)
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. DOTPRODUCT. REAL EUCLIDEAN VECTOR SPACE. NORM. CAUCHY–SCHWARZ INEQUALITY. ANGLE. ORTHOGONAL VECTORS. ORTHONORMAL BASES. COMPONENTS IN AN ORTHONORMAL BASIS. ORTHOGONAL PROJECTIONS. GRAM-SCHMIDT PROCEDURE.(HOURS 3/2/-)
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. SPECTRALTHEOREM.(HOURS 5/3/-)
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.(HOURS 3/2/-)
TOTAL HOURS 52/38/-

	Teaching Methods
	THE COURSE COVERS THEORETICAL LECTURES, DEVOTED TO THE FACE-TO-FACE DELIVERY OF ALL THE COURSE CONTENTS , AND CLASSROOM PRACTICE DEVOTED TO PROVIDE THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES.

	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 PROOFS; •THE SKILL OF PROVING THEOREMS; •THE SKILL OF SOLVING EXERCISES; •THE SKILL TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHOD IN EXERCISES SOLVING; •THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE. THE EXAM CONSISTS OF A WRITTEN PROOF AND AN ORAL INTERVIEW. WRITTEN PROOF: THE WRITTEN PROOF CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE. IN THE CASE OF A SUFFICIENT PROOF, IT WILL BE EVALUATED BY THREE SCALES. ORAL INTERVIEW: THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING. FINAL EVALUATION: THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE MARK OF THE WRITTEN PROOF, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW. MASTERING ABILITY OF THE COURSE CONTENTS, TAKING INTO ACCOUNT THE QUALITY OF THE WRITTEN AND ORAL ELABORATION AND THE SELF-EVALUATION CAPABILITY SHOWN.

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 PROOFS;
•THE SKILL OF PROVING THEOREMS;
•THE SKILL OF SOLVING EXERCISES;
•THE SKILL TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHOD IN EXERCISES SOLVING;
•THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE.
THE EXAM CONSISTS OF A WRITTEN PROOF AND AN ORAL INTERVIEW.
WRITTEN PROOF: THE WRITTEN PROOF CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE. IN THE CASE OF A SUFFICIENT PROOF, IT WILL BE EVALUATED BY THREE SCALES.
ORAL INTERVIEW: THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
FINAL EVALUATION: THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE MARK OF THE WRITTEN PROOF, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.
MASTERING ABILITY OF THE COURSE CONTENTS, TAKING INTO ACCOUNT THE QUALITY OF THE WRITTEN AND ORAL ELABORATION AND THE SELF-EVALUATION CAPABILITY SHOWN.

	Texts
	G. ALBANO, C. D’APICE, S. SALERNO, LIMITI E DERIVATE, CUES (2002). G. ALBANO, C. D’APICE, S. SALERNO, ALGEBRA LINEARE, CUES (2002). C. D’APICE, R. MANZO, VERSO L’ESAME DI MATEMATICA I, CUES (2007). G. ALBANO, LA PROVA SCRITTA DI GEOMETRIA:TRA TEORIA E PRATICA, CUES (2011). EDUCATIONAL CONTENTS ON E-LEARNING PLATFORM IWT. LECTURE NOTES.

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	COMPULSORY ATTENDANCE. TEACHING IN ITALIAN.

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Ciro D'APICE | Mathematics I