Vittorio ZAMPOLI | MATHEMATICS I

cod. 0612500001

MATHEMATICS I

	0612500001
	DEPARTMENT OF CIVIL ENGINEERING
	EQF6
	CIVIL AND ENVIRONMENTAL ENGINEERING
	2024/2025

PERCORSO CIVIL AND ENVIRONMENTAL ENGINEERING

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2022
	AUTUMN SEMESTER

TEACHING UNITS

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

PROFESSORS

	FRANCESCA PASSARELLA T Curriculum
	VITTORIO ZAMPOLI Curriculum

	Objectives
	EXPECTED LEARNING RESULTS AND COMPETENCE TO BE ACQUIRED: LEARNING THE BASIC CONCEPTS OF MATHEMATICAL ANALYSIS AND CALCULUS FOR FUNCTIONS OF A VARIABLE, WITH ELEMENTS OF ANALYTICAL GEOMETRY OF THE PLANE AND PHYSICAL APPLICATIONS. KNOWLEDGE AND UNDERSTANDING: ACQUISITION OF SKILLS RELATED TO BASIC MATHEMATICAL CONCEPTS AND THEIR GRAPHIC REPRESENTATION WITH PARTICULAR REGARD TO THE FOLLOWING TOPICS: ANALYTICAL AND CONICAL GEOMETRY, FUNCTIONS OF A VARIABLE, LIMITS, DIFFERENTIAL AND INTEGRAL CALCULUS, SEQUENCES AND NUMERICAL SERIES. ABILITY TO UNDERSTAND AND ACQUIRE MATHEMATICAL LANGUAGE. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: APPLICATION OF THE KNOWLEDGE ACQUIRED TO CALCULATE LIMITS, DERIVATIVES AND INTEGRALS; STUDY AND DRAW THE GRAPH OF A FUNCTION OF A VARIABLE AND A CONIC IN THE PLANE; SOLVE MAXIMUM AND MINIMUM PROBLEMS; CALCULATE AREAS; CALCULATE THE LIMIT OF A SEQUENCE AND ESTABLISH THE CONVERGENCE OF A SEQUENCE; PERFORM CALCULATIONS WITH COMPLEX NUMBERS. ABILITY TO FORMULATE IN MATHEMATICAL TERMS AND SOLVE SIMPLE PROBLEMS OF APPLIED SCIENCES AND IN PARTICULAR OF ENGINEERING. AUTONOMY OF JUDGMENT: ABILITY TO CHOOSE THE MOST SUITABLE MATHEMATICAL MODELS AND METHODS FOR THE VARIOUS SITUATIONS AND VERIFY THE VALIDITY OF THE RESULTS OBTAINED FROM A QUALITATIVE AND QUANTITATIVE POINT OF VIEW. COMMUNICATION SKILLS: ABILITY TO EXPOSE, WITH APPROPRIATE TECHNICAL LANGUAGE AND WITH ADEQUATE GRAPHIC REPRESENTATION, THE ACQUIRED MATHEMATICAL NOTIONS AND METHODS, ALSO INTEGRATING THE ACQUIRED KNOWLEDGE WITH THOSE TYPICAL OF OTHER DISCIPLINES. ABILITY TO LEARN: CONSOLIDATION OF THE KNOWLEDGE AND SKILLS ACQUIRED TO LEARN MORE ADVANCED MATHEMATICAL SUBJECTS AND CONTENTS OF OTHER SCIENTIFIC DISCIPLINES THAT USE MATHEMATICAL TOOLS WITHOUT DIFFICULTY.

EXPECTED LEARNING RESULTS AND COMPETENCE TO BE ACQUIRED:
LEARNING THE BASIC CONCEPTS OF MATHEMATICAL ANALYSIS AND CALCULUS FOR FUNCTIONS OF A VARIABLE, WITH ELEMENTS OF ANALYTICAL GEOMETRY OF THE PLANE AND PHYSICAL APPLICATIONS.
KNOWLEDGE AND UNDERSTANDING:
ACQUISITION OF SKILLS RELATED TO BASIC MATHEMATICAL CONCEPTS AND THEIR GRAPHIC REPRESENTATION WITH PARTICULAR REGARD TO THE FOLLOWING TOPICS: ANALYTICAL AND CONICAL GEOMETRY, FUNCTIONS OF A VARIABLE, LIMITS, DIFFERENTIAL AND INTEGRAL CALCULUS, SEQUENCES AND NUMERICAL SERIES.
ABILITY TO UNDERSTAND AND ACQUIRE MATHEMATICAL LANGUAGE.
ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
APPLICATION OF THE KNOWLEDGE ACQUIRED TO CALCULATE LIMITS, DERIVATIVES AND INTEGRALS; STUDY AND DRAW THE GRAPH OF A FUNCTION OF A VARIABLE AND A CONIC IN THE PLANE; SOLVE MAXIMUM AND MINIMUM PROBLEMS; CALCULATE AREAS; CALCULATE THE LIMIT OF A SEQUENCE AND ESTABLISH THE CONVERGENCE OF A SEQUENCE; PERFORM CALCULATIONS WITH COMPLEX NUMBERS.
ABILITY TO FORMULATE IN MATHEMATICAL TERMS AND SOLVE SIMPLE PROBLEMS OF APPLIED SCIENCES AND IN PARTICULAR OF ENGINEERING.
AUTONOMY OF JUDGMENT:
ABILITY TO CHOOSE THE MOST SUITABLE MATHEMATICAL MODELS AND METHODS FOR THE VARIOUS SITUATIONS AND VERIFY THE VALIDITY OF THE RESULTS OBTAINED FROM A QUALITATIVE AND QUANTITATIVE POINT OF VIEW.
COMMUNICATION SKILLS:
ABILITY TO EXPOSE, WITH APPROPRIATE TECHNICAL LANGUAGE AND WITH ADEQUATE GRAPHIC REPRESENTATION, THE ACQUIRED MATHEMATICAL NOTIONS AND METHODS, ALSO INTEGRATING THE ACQUIRED KNOWLEDGE WITH THOSE TYPICAL OF OTHER DISCIPLINES.
ABILITY TO LEARN:
CONSOLIDATION OF THE KNOWLEDGE AND SKILLS ACQUIRED TO LEARN MORE ADVANCED MATHEMATICAL SUBJECTS AND CONTENTS OF OTHER SCIENTIFIC DISCIPLINES THAT USE MATHEMATICAL TOOLS WITHOUT DIFFICULTY.

	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. NO PROPEDEUTIC COURSES ARE NEEDED.

	Contents
	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. (ORE 3/0/-) 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 5/4/-) 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 6/4/-) 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 5/6/-) FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.(HOURS 4/5/-) 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 6/7/-). 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 INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS.(HOURS 6/9-) INTRODUCTION TO NUMERICAL SERIES.(ORE 2/0/-) COMPLEX NUMBERS. OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE 3+MOIVRE’S FORMULA. N-THROOTS.(HOURS 3/1/-)

Contents

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. (ORE 3/0/-)
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 5/4/-)
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 6/4/-)
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 5/6/-)
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS.(HOURS 4/5/-)
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 6/7/-).
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 INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS.(HOURS 6/9-)
INTRODUCTION TO NUMERICAL SERIES.(ORE 2/0/-)
COMPLEX NUMBERS. OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE 3+MOIVRE’S FORMULA. N-THROOTS.(HOURS 3/1/-)

	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 CARRIED OUT AT THE END OF THE COURSE AND IT 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 LASTS 2,5 HOURS AND 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 LASTS ABOUT 30 MINUTES AND 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. LAUDE FOLLOWS FROM BRILLIANT WRITTEN PROOF AND ORAL INTERVIEW.

Verification of learning

THE FINAL EXAM CARRIED OUT AT THE END OF THE COURSE AND IT 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 LASTS 2,5 HOURS AND 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 LASTS ABOUT 30 MINUTES AND 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.
LAUDE FOLLOWS FROM BRILLIANT WRITTEN PROOF AND ORAL INTERVIEW.

	Texts
	P. MARCELLINI - C. SBORDONE, “ELEMENTI DI MATEMATICA “, LIGUORI EDITORE P. MARCELLINI - C. SBORDONE, “ESERCITAZIONI DI MATEMATICA (VOL. 1/1)“, LIGUORI EDITORE P. MARCELLINI - C. SBORDONE, “ESERCITAZIONI DI MATEMATICA (VOL. 1/2)“, LIGUORI EDITORE

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Vittorio ZAMPOLI | MATHEMATICS I