Ciro D'APICE | MATHEMATICS I

cod. 0612500001

MATHEMATICS I

	0612500001
	DIPARTIMENTO DI INGEGNERIA CIVILE
	EQF6
	CIVIL AND ENVIRONMENTAL ENGINEERING
	2016/2017

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
	MARIA PIA D'ARIENZO Curriculum

	Objectives
	KNOWLEDGE AND UNDERSTANDING THE TEACHING AIMS AT THE ACQUIRING OF THE BASIC ELEMENTS OF CALCULUS AND LINEAR ALGEBRA: NUMBER 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, STUDY OF THE GRAPH OF A FUNCTION, INDEFINITE AND DEFINITE INTEGRALS, NUMERICAL SERIES. LEARNING OUTCOMES OF THE TEACHING CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELATED COMPUTATIONAL TOOLS. THE TEACHING'S MAIN AIM IS TO STRENGTHEN BASIC MATHEMATICAL KNOWLEDGE AND TO PROVIDE AND DEVELOP A SCIENTIFIC APPROACH TO PROBLEMS AND PHENOMENA THAT STUDENTS ARE GOING TO ENCOUNTER DURING THE REST OF THEIR STUDIES. THE THEORETICAL PART OF THE TEACHING 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. 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. TO IDENTIFY THE MOST APPROPRIATE METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM. TO BE ABLE TO FIND SOME OPTIMIZATIONS TO THE SOLVING PROCESS OF A MATHEMATICAL PROBLEM. BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE TEACHING. SKILL TO DEEPEN THE TOPICS DEALT WITH BY USING MATERIALS DIFFERENT FROM PRESENTED DURING THE TEACHING.

KNOWLEDGE AND UNDERSTANDING
THE TEACHING AIMS AT THE ACQUIRING OF THE BASIC ELEMENTS OF CALCULUS AND LINEAR ALGEBRA: NUMBER 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, STUDY OF THE GRAPH OF A FUNCTION, INDEFINITE AND DEFINITE INTEGRALS, NUMERICAL SERIES.
LEARNING OUTCOMES OF THE TEACHING CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELATED COMPUTATIONAL TOOLS.
THE TEACHING'S MAIN AIM IS TO STRENGTHEN BASIC MATHEMATICAL KNOWLEDGE AND TO PROVIDE AND DEVELOP A SCIENTIFIC APPROACH TO PROBLEMS AND PHENOMENA THAT STUDENTS ARE GOING TO ENCOUNTER DURING THE REST OF THEIR STUDIES.
THE THEORETICAL PART OF THE TEACHING 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.
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.
TO IDENTIFY THE MOST APPROPRIATE METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM. TO BE ABLE TO FIND SOME OPTIMIZATIONS TO THE SOLVING PROCESS OF A MATHEMATICAL PROBLEM.
BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE TEACHING. SKILL TO DEEPEN THE TOPICS DEALT WITH BY USING MATERIALS DIFFERENT FROM PRESENTED DURING THE TEACHING.

	Prerequisites
	PREREQUISITES FOR 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 PREPARATORY TEACHINGS NONE

	Contents
	NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS OF A SET. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES OF A NUMERICAL SET. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. GEOMETRIC AND TRIGONOMETRIC FORM. POWERS AND DE MOIVRE’SFORMULA. N-THROOTS. (LECTUR/EXERCISES HOURS 4/3) REAL FUNCTIONS:DEFINITION. DOMAIN, CODOMAIN AND GRAPH OF A FUNCTION. EXTREMA OF A REAL FUNCTION. MONOTONE FUNCTIONS. COMPOSITE FUNCTIONS. INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-THPOWER AND N-TH ROOT FUNCTIONS, EXPONENTIAL AND LOGARITHMIC FUNCTIONS, POWER FUNCTIONS, TRIGONOMETRIC FUNCTIONS AND THEIR INVERSE FUNCTIONS. (3/5) BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST DEGREE EQUATIONS. QUADRATIC EQUATIONS. BINOMIAL EQUATIONS. IRRATIONAL EQUATIONS. TRIGONOMETRIC EQUATIONS. EXPONENTIAL AND LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. FIRST DEGREEINEQUALITIES. QUADRATIC INEQUALITIES. RATIONALINEQUALITIES. IRRATIONAL INEQUALITIES. TRIGONOMETRIC INEQUALITIES. EXPONENTIAL AND LOGARITHMIC INEQUALITIES. SYSTEMS OF INEQUALITIES. (3/4) NUMERICAL SEQUENCES: DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE. (2/2) LIMITS OF A FUNCTION: DEFINITION. RIGHT- AND LEFT-HAND LIMITS. UNIQUENESS THEOREM. LIMIT COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS. (5/6) CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS’ THEOREM. BOLZANO’S THEOREM. INTERMEDIATE VALUE THEOREM. UNIFORM CONTINUITY. ( 4/-) DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING, THE 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 ITS GEOMETRIC MEANING. (4/5) FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE'S THEOREM. CAUCHY'S THEOREM. LAGRANGE'S THEOREM AND COROLLARIES. DE L'HOSPITALTHEOREM. CONDITIONS FOR MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS. (4/3) GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. FINDING LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH OF A FUNCTION BY MEANS OF ITS CHARACTERISTIC ELEMENTS. (3/10) INTEGRATION OF FUNCTIONS OF ONE VARIABLE: DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRALS AND GEOMETRIC MEANING, MEAN VALUE THEOREM. INTEGRAL FUNCTION AND FUNDAMENTAL THEOREM OF THE INTEGRAL CALCULUS. (5/10) NUMERICAL SERIES: INTRODUCTION TO NUMERICAL SERIES. CONVERGENT SERIES. DIVERGENT AND INDETERMINATE SERIES. GEOMETRIC AND HARMONIC SERIES. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: CRITERION OF THE COMPARISON, OF THE RATIO, OF THE ROOT. (3/2) TOTAL HOURS 40/50

Contents

NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS OF A SET. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES OF A NUMERICAL SET. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. GEOMETRIC AND TRIGONOMETRIC FORM. POWERS AND DE MOIVRE’SFORMULA. N-THROOTS.
(LECTUR/EXERCISES HOURS 4/3)
REAL FUNCTIONS:DEFINITION. DOMAIN, CODOMAIN AND GRAPH OF A FUNCTION. EXTREMA OF A REAL FUNCTION. MONOTONE FUNCTIONS. COMPOSITE FUNCTIONS. INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-THPOWER AND N-TH ROOT FUNCTIONS, EXPONENTIAL AND LOGARITHMIC FUNCTIONS, POWER FUNCTIONS, TRIGONOMETRIC FUNCTIONS AND THEIR INVERSE FUNCTIONS.
(3/5)
BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST DEGREE EQUATIONS. QUADRATIC EQUATIONS. BINOMIAL EQUATIONS. IRRATIONAL EQUATIONS. TRIGONOMETRIC EQUATIONS. EXPONENTIAL AND LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. FIRST DEGREEINEQUALITIES. QUADRATIC INEQUALITIES. RATIONALINEQUALITIES. IRRATIONAL INEQUALITIES. TRIGONOMETRIC INEQUALITIES. EXPONENTIAL AND LOGARITHMIC INEQUALITIES. SYSTEMS OF INEQUALITIES.
(3/4)
NUMERICAL SEQUENCES: DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.
(2/2)
LIMITS OF A FUNCTION: DEFINITION. RIGHT- AND LEFT-HAND LIMITS. UNIQUENESS THEOREM. LIMIT COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.
(5/6)
CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS’ THEOREM. BOLZANO’S THEOREM. INTERMEDIATE VALUE THEOREM. UNIFORM CONTINUITY.
( 4/-)
DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING, THE 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 ITS GEOMETRIC MEANING.
(4/5)
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE'S THEOREM. CAUCHY'S THEOREM. LAGRANGE'S THEOREM AND COROLLARIES. DE L'HOSPITALTHEOREM. CONDITIONS FOR MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.
(4/3)
GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. FINDING LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH OF A FUNCTION BY MEANS OF ITS CHARACTERISTIC ELEMENTS.
(3/10)
INTEGRATION OF FUNCTIONS OF ONE VARIABLE: DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRALS AND GEOMETRIC MEANING, MEAN VALUE THEOREM. INTEGRAL FUNCTION AND FUNDAMENTAL THEOREM OF THE INTEGRAL CALCULUS.
(5/10)
NUMERICAL SERIES: INTRODUCTION TO NUMERICAL SERIES. CONVERGENT SERIES. DIVERGENT AND INDETERMINATE SERIES. GEOMETRIC AND HARMONIC SERIES. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: CRITERION OF THE COMPARISON, OF THE RATIO, OF THE ROOT.
(3/2)
TOTAL HOURS 40/50

	Teaching Methods
	THE TEACHING COVERS THEORETICAL LECTURES FOR A TOTAL OF 40 HOURS (AMOUNTING TO 4 CFU), DEVOTED TO THE FACE-TO-FACE DELIVERY OF ALL THE TEACHING CONTENTS, AND CLASSROOM PRACTICE FOR A TOTAL OF 50 HOURS (AMOUNTING TO 5 CFU), DEVOTED TO PROVIDE THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES. ATTENDANCE TO THE COURSE IS MANDATORY, AND IT IS CERTIFIABLE EXCLUSIVELY BY USING THE PERSONAL BADGE. THE MINIMUM PERCENTAGE OF ATTENDANCE, WITHOUT DISTINCTION BETWEEN THEORETICAL LECTURES AND EXERCISE SESSIONS, NECESSARY TO GUARANTEE THE ACCESS TO THE EXAMINATION IS EQUAL TO 70%.

Teaching Methods

THE TEACHING COVERS THEORETICAL LECTURES FOR A TOTAL OF 40 HOURS (AMOUNTING TO 4 CFU), DEVOTED TO THE FACE-TO-FACE DELIVERY OF ALL THE TEACHING CONTENTS, AND CLASSROOM PRACTICE FOR A TOTAL OF 50 HOURS (AMOUNTING TO 5 CFU), DEVOTED TO PROVIDE THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES. ATTENDANCE TO THE COURSE IS MANDATORY, AND IT IS CERTIFIABLE EXCLUSIVELY BY USING THE PERSONAL BADGE. THE MINIMUM PERCENTAGE OF ATTENDANCE, WITHOUT DISTINCTION BETWEEN THEORETICAL LECTURES AND EXERCISE SESSIONS, NECESSARY TO GUARANTEE THE ACCESS TO THE EXAMINATION IS EQUAL TO 70%.

	Verification of learning
	THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE: •THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE TEACHING •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 TEACHING. THE EXAM CONSISTS OF A WRITTEN PROOF AND AN ORAL INTERVIEW. WRITTEN PROOF: THE WRITTEN PROOF CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE LESSONS. IN THE CASE OF A SUFFICIENT PROOF, IT WILL BE EVALUATED BY THREE SCALES. THE PROOF WILL LAST 3 HOURS. ORAL INTERVIEW: THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING. FINAL EVALUATION: THE FINAL MARK, EXPRESSED IN THIRTIETHS, DEPENDS ON THE MARK OF THE WRITTEN PROOF, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

Verification of learning

THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE:
•THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE TEACHING
•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 TEACHING.
THE EXAM CONSISTS OF A WRITTEN PROOF AND AN ORAL INTERVIEW.
WRITTEN PROOF: THE WRITTEN PROOF CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE LESSONS. IN THE CASE OF A SUFFICIENT PROOF, IT WILL BE EVALUATED BY THREE SCALES. THE PROOF WILL LAST 3 HOURS.
ORAL INTERVIEW: THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND COVERS DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
FINAL EVALUATION: THE FINAL MARK, EXPRESSED IN THIRTIETHS, DEPENDS ON THE MARK OF THE WRITTEN PROOF, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE 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 (2015). EDUCATIONAL CONTENTS ON E-LEARNING PLATFORM IWT. LECTURE NOTES.

	More Information
	COMPULSORY ATTENDANCE. TEACHING IN ITALIAN.

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Ciro D'APICE | MATHEMATICS I