Antonio SELLITTO | CALCULUS I

cod. 0612300001

CALCULUS I

	0612300001
	DEPARTMENT OF INDUSTRIAL ENGINEERING
	EQF6
	MECHANICAL ENGINEERING
	2024/2025

PERCORSO IN COMMON

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

TEACHING UNITS

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

	Objectives
	THE COURSE OF MATHEMATICS I AIMS AT THE KNOWLEDGE OF BASIC CONCEPTS OF MATHEMATICAL ANALYSIS AND OF CALCULUS, AND TO THEIR PHYSICAL AND GEOMETRICAL INTERPRETATIONS. IT HAS THE FOLLOWING EDUCATIONAL OBJECTIVES: -) KNOWLEDGE AND UNDERSTANDING THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF MATHEMATICAL ANALYSIS, 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, THE INTEGRALS OF THE FUNCTIONS OF A VARIABLE AND NUMERICAL SERIES OF REAL NUMBERS, THE CANONICAL FORM OF CONICAL SECTIONS. -) APPLYING KNOWLEDGE AND UNDERSTANDING 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). -) LEARNING 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.

THE COURSE OF MATHEMATICS I AIMS AT THE KNOWLEDGE OF BASIC CONCEPTS OF MATHEMATICAL ANALYSIS AND OF CALCULUS, AND TO THEIR PHYSICAL AND GEOMETRICAL INTERPRETATIONS. IT HAS THE FOLLOWING EDUCATIONAL OBJECTIVES:

-) KNOWLEDGE AND UNDERSTANDING

THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF MATHEMATICAL ANALYSIS, 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, THE INTEGRALS OF THE FUNCTIONS OF A VARIABLE AND NUMERICAL SERIES OF REAL NUMBERS, THE CANONICAL FORM OF CONICAL SECTIONS.

-) APPLYING KNOWLEDGE AND UNDERSTANDING

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).

-) LEARNING 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 (2 HOURS OF THEORY): INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS. COMPLEX NUMBERS (3 HOURS OF THEORY AND 4 HOURS OF EXERCISE). OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE MOIVRE’S FORMULA. N-TH ROOTS. REAL FUNCTIONS (4 HOURS OF THEORY AND 6 OF EXERCISE): DOMAIN AND CODOMAIN. EXTREMA. MONOTONE, COMPOSITE AND INVERTIBLE FUNCTIONS. MAIN ELEMENTARY FUNCTIONS. NUMERICAL SEQUENCES (6 HOURS OF THEORY): BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE. LIMITS OF A FUNCTION (4 HOURS OF THEORY AND 6 HOURS OF EXERCISE): UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS. CONTINUOUS FUNCTIONS (7 HOURS OF THEORY AND 3 HOURS OF EXERCISE): CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM, ZEROS THEOREMS). DERIVATIVE OF A FUNCTION (5 HOURS OF THEORY AND 3 HOURS OF EXERCISE): DERIVABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER-ORDER DERIVATIVES. DERIVATIVE OF A FUNCTION AND ITS GEOMETRIC MEANING. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, AND LAGRANGE THEOREMS, THEIR COROLLARIES. DE L'HOSPITALTHEOREM. MAXIMA AND MINIMA. TAYLOR AND MACLAURIN FORMULAS. THE STUDY OF A FUNCTION (3 HOURS OF THEORY AND 5 OF EXERCISE): ASYMPTOTES, MAXIMA AND MINIMA. CONCAVE AND CONVEX BEHAVIOR, INFLECTION POINTS. THE GRAPH. INTEGRATION OF REAL FUNCTIONS (6 HOURS OF THEORY AND 9 HOURS OF EXERCISE): 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. NUMERICAL SERIES (3 HOURS OF THEORY AND 3 HOURS OF EXERCISE): INTRODUCTION TO MAIN NUMERICAL SERIES. CONIC SECTIONS (4 HOURS OF THEORY AND 4 HOURS OF EXERCISE): CANONICAL FORMS OF CIRCLES, ELLIPSES, HYPERBOLAS, AND PARABOLAS

Contents

NUMERICAL SETS (2 HOURS OF THEORY): INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBORHOODS, ACCUMULATION POINTS.

COMPLEX NUMBERS (3 HOURS OF THEORY AND 4 HOURS OF EXERCISE). OPERATIONS ON COMPLEX NUMBERS. POWERS AND DE MOIVRE’S FORMULA. N-TH ROOTS.

REAL FUNCTIONS (4 HOURS OF THEORY AND 6 OF EXERCISE): DOMAIN AND CODOMAIN. EXTREMA. MONOTONE, COMPOSITE AND INVERTIBLE FUNCTIONS. MAIN ELEMENTARY FUNCTIONS.

NUMERICAL SEQUENCES (6 HOURS OF THEORY): BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. EULER’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE.

LIMITS OF A FUNCTION (4 HOURS OF THEORY AND 6 HOURS OF EXERCISE): UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS.

CONTINUOUS FUNCTIONS (7 HOURS OF THEORY AND 3 HOURS OF EXERCISE): CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM, ZEROS THEOREMS).

DERIVATIVE OF A FUNCTION (5 HOURS OF THEORY AND 3 HOURS OF EXERCISE): DERIVABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER-ORDER DERIVATIVES. DERIVATIVE OF A FUNCTION AND ITS GEOMETRIC MEANING.
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, AND LAGRANGE THEOREMS, THEIR COROLLARIES. DE L'HOSPITALTHEOREM. MAXIMA AND MINIMA. TAYLOR AND MACLAURIN FORMULAS.

THE STUDY OF A FUNCTION (3 HOURS OF THEORY AND 5 OF EXERCISE): ASYMPTOTES, MAXIMA AND MINIMA. CONCAVE AND CONVEX BEHAVIOR, INFLECTION POINTS. THE GRAPH.

INTEGRATION OF REAL FUNCTIONS (6 HOURS OF THEORY AND 9 HOURS OF EXERCISE): 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.

NUMERICAL SERIES (3 HOURS OF THEORY AND 3 HOURS OF EXERCISE): INTRODUCTION TO MAIN NUMERICAL SERIES.

CONIC SECTIONS (4 HOURS OF THEORY AND 4 HOURS OF EXERCISE): CANONICAL FORMS OF CIRCLES, ELLIPSES, HYPERBOLAS, AND PARABOLAS

	Teaching Methods
	COMPULSORY ATTENDANCE. LECTURES ARE IN ITALIAN. THE COURSE CONSISTS OF THEORETICAL LECTURES, DEVOTED TO THE EXPLANATION OF ALL THE COURSE CONTENTS AND CLASSROOM PRACTICE, PROVIDING THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES.

	Verification of learning
	THE EXAM IS COMPOSED BY A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT HAS TO PASS BOTH THE WRITTEN AND ORAL TEST. THERE WILL BE IN-COURSE WRITTEN TESTS, ON THE LATEST TOPICS SEEN DURING THE LECTURES. THE STUDENTS WHO PASS THOSE TESTS WILL BE EXONERATED FROM THE WRITTEN TEST. WRITTEN TEST: IT CONSISTS IN SOLVING PROBLEMS SIMILAR TO THE ONES WHICH HAVE BEEN STUDIED DURING THE COURSE. IN MORE DETAILS, THERE WILL BE THE FOLLOWING TYPES OF EXERCISES: DETERMINATION OF THE DOMAIN OF A SCALAR FUNCTION (AT LEAST 6 POINTS); SOLUTION OF AN EQUATION IN THE COMPLEX DOMAIN (AT LEAST 3 POINTS); SOLUTION OF A LIMIT DISPLAYING AN INDETERMINATE FORM (AT LEAST 6 POINTS); STUDY OF A SCALAR FUNCTION (AT LEAST 6 POINT); COMPUTATION OF (DEFINITE/INDEFINITE) INTEGRALS (AT LEAST 6 POINTS); CHECK OF A NUMERICAL-SERIES' BEHAVIOR (AT LEAST 3 POINTS). SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30 AND IS THE SUM OF THE SCORES OBTAINED IN EACH EXERCISES ABOVE. THOSE SCORES DEPEND ON THE STUDENT'S ABILITY IN APPLYING OWN KNOWLEDGE. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED. ORAL INTERVIEW: IT IS SUBJECTED TO BEING SELECTED IN THE PREVIOUS WRITTEN TEST. IT NORMALLY TAKES 20 MINUTES. IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING. IT WILL BE PASSED PROVIDED A SUITABLE KNOWLEDGE OF THE FOLLOWING TOPICS HAS BEEN ACHIEVED: LIMITS OF FUNCTIONS, DERIVATIVES OF FUNCTIONS, INTEGRATION OF FUNCTIONS. 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.

Verification of learning

THE EXAM IS COMPOSED BY A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT HAS TO PASS BOTH THE WRITTEN AND ORAL TEST.

THERE WILL BE IN-COURSE WRITTEN TESTS, ON THE LATEST TOPICS SEEN DURING THE LECTURES. THE STUDENTS WHO PASS THOSE TESTS WILL BE EXONERATED FROM THE WRITTEN TEST.

WRITTEN TEST: IT CONSISTS IN SOLVING PROBLEMS SIMILAR TO THE ONES WHICH HAVE BEEN STUDIED DURING THE COURSE. IN MORE DETAILS, THERE WILL BE THE FOLLOWING TYPES OF EXERCISES: DETERMINATION OF THE DOMAIN OF A SCALAR FUNCTION (AT LEAST 6 POINTS); SOLUTION OF AN EQUATION IN THE COMPLEX DOMAIN (AT LEAST 3 POINTS); SOLUTION OF A LIMIT DISPLAYING AN INDETERMINATE FORM (AT LEAST 6 POINTS); STUDY OF A SCALAR FUNCTION (AT LEAST 6 POINT); COMPUTATION OF (DEFINITE/INDEFINITE) INTEGRALS (AT LEAST 6 POINTS); CHECK OF A NUMERICAL-SERIES' BEHAVIOR (AT LEAST 3 POINTS). SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30 AND IS THE SUM OF THE SCORES OBTAINED IN EACH EXERCISES ABOVE. THOSE SCORES DEPEND ON THE STUDENT'S ABILITY IN APPLYING OWN KNOWLEDGE. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED.

ORAL INTERVIEW: IT IS SUBJECTED TO BEING SELECTED IN THE PREVIOUS WRITTEN TEST. IT NORMALLY TAKES 20 MINUTES. IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING. IT WILL BE PASSED PROVIDED A SUITABLE KNOWLEDGE OF THE FOLLOWING TOPICS HAS BEEN ACHIEVED: LIMITS OF FUNCTIONS, DERIVATIVES OF FUNCTIONS, INTEGRATION OF FUNCTIONS.

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.

	Texts
	-) THEORY - P. MARCELLINI - C. SBORDONE, “ANALISI MATEMATICA UNO “, LIGUORI EDITORE (1998) -) EXERCISES - P. MARCELLINI - C. SBORDONE, “ESERCITAZIONI DI MATEMATICA - VOL. 1 E 2“, LIGUORI EDITORE (2015)

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	SUBJECT DELIVERED IN ITALIAN.

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