Tiziana DURANTE | CALCULUS 1

cod. 0612700112

CALCULUS 1

	0612700112
	DIPARTIMENTO DI INGEGNERIA DELL'INFORMAZIONE ED ELETTRICA E MATEMATICA APPLICATA
	EQF6
	COMPUTER ENGINEERING
	2021/2022

CURRICULUM NETWORKS

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2017
	AUTUMN SEMESTER

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/05	5	40	LESSONS	BASIC COMPULSORY SUBJECTS
MAT/05	4	32	EXERCISES	BASIC COMPULSORY SUBJECTS

	Objectives
	"THE COURSE PRESENTS THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS. THE TRAINING OBJECTIVES THE TEACHING CONSISTS IN THE ACQUISITION OF RESULTS AND DEMONSTRATIVE TECHNIQUES, AS WELL AS IN THE ABILITY TO USE THE RELATED CALCULATION TOOLS. KNOWLEDGE AND UNDERSTANDING NUMERICAL SETS. REAL FUNCTIONS. REFERENCES ON EQUATIONS AND INEQUATIONS. NUMERICAL SUCCESSIONS. LIMITS OF A FUNCTION. CONTINUOUS FUNCTIONS. DERIVATIVE OF A FUNCTION. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULATION. STUDY OF THE GRAPH OF A FUNCTION. INTEGRATION OF FUNCTIONS OF A VARIABLE. SOFTWARE TOOLS FOR MATHEMATICS. APPLYING KNOWLEDGE AND UNDERSTANDING APPLY THE THEOREMS AND RULES STUDIED TO TROUBLESHOOTING. PERFORM CALCULATIONS WITH LIMITS, DERIVATIVES, INTEGRAL. CONDUCT THE STUDY OF THE GRAPH OF A FUNCTION."

"THE COURSE PRESENTS THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS. THE TRAINING OBJECTIVES THE TEACHING CONSISTS IN THE ACQUISITION OF RESULTS AND DEMONSTRATIVE TECHNIQUES, AS WELL AS IN THE ABILITY TO USE THE RELATED CALCULATION TOOLS.

KNOWLEDGE AND UNDERSTANDING
NUMERICAL SETS. REAL FUNCTIONS. REFERENCES ON EQUATIONS AND INEQUATIONS. NUMERICAL SUCCESSIONS. LIMITS OF A FUNCTION. CONTINUOUS FUNCTIONS. DERIVATIVE OF A FUNCTION. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULATION. STUDY OF THE GRAPH OF A FUNCTION. INTEGRATION OF FUNCTIONS OF A VARIABLE. SOFTWARE TOOLS FOR MATHEMATICS.

APPLYING KNOWLEDGE AND UNDERSTANDING
APPLY THE THEOREMS AND RULES STUDIED TO TROUBLESHOOTING. PERFORM CALCULATIONS WITH LIMITS, DERIVATIVES, INTEGRAL. CONDUCT THE STUDY OF THE GRAPH OF A FUNCTION."

	Prerequisites
	IN ORDER TO REACH THE OBJECTIVES AND, IN PARTICULAR, FOR AN ADEQUATE UNDERSTANDING OF THE CONTENTS PROVIDED BY THE TEACHING, THE STUDENTS SHOULD HAVE A BACKGROUND RELATED TO THE ALGEBRA, IN PARTICULAR TO ALGEBRAIC EQUATIONS AND INEQUALITIES, LOGARITHMIC, EXPONENTIAL, TRIGONOMETRIC AND TRANSCENDENTAL INEQUALITIES, AND THEY SHOULD UNDERSTAND TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC THEORY OF TRIGONOMETRIC FUNCTIONS.

Prerequisites

IN ORDER TO REACH THE OBJECTIVES AND, IN PARTICULAR, FOR AN ADEQUATE UNDERSTANDING OF THE CONTENTS PROVIDED BY THE TEACHING, THE STUDENTS SHOULD HAVE A BACKGROUND RELATED TO THE ALGEBRA, IN PARTICULAR TO ALGEBRAIC EQUATIONS AND INEQUALITIES, LOGARITHMIC, EXPONENTIAL, TRIGONOMETRIC AND TRANSCENDENTAL INEQUALITIES, AND THEY SHOULD UNDERSTAND TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC THEORY OF TRIGONOMETRIC FUNCTIONS.

	Contents
	NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INDUCTION PRINCIPLE. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBOURHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. GEOMETRIC AND TRIGONOMETRIC FORM. POWERS AND DE MOIVRE'S FORMULA. N-TH ROOTS (HOURS OF THEORY/EXERCISES: 4/2). REAL FUNCTIONS: DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE AND INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS (HOURS OF THEORY/EXERCISES: 2/2). BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST ORDER, QUADRATIC, BINOMIAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. FIRST ORDER, SECOND ORDER, RATIONAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC INEQUALITIES. SYSTEMS (HOURS OF THEORY/EXERCISES: 2/4). NUMERICAL SEQUENCES: DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. NEPERO'S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE. NUMERICAL SERIES: INTRODUCTION. CONVERGENCE. ARMONIC AND GEOMETRIC SERIES. POSITIVE SERIES AND CONVERGENCE CRITERIA (HOURS OF THEORY/EXERCISES: 4/4). LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS (HOURS OF THEORY/EXERCISES: 4/4). CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY (HOURS OF THEORY/EXERCISES: 4/0). DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING. TANGENTIAL LINE. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND ITS GEOMETRIC MEANING (HOURS OF THEORY/EXERCISES: 4/2). FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS (HOURS OF THEORY/EXERCISES: 4/2). GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH (HOURS OF THEORY/EXERCISES: 4/8). INTEGRATION OF ONE VARIABLE FUNCTIONS: DEFINITION OF ANTIDERIVATIVE AND INDEFINITE INTEGRAL. BASIC INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND GEOMETRICAL MEANING. FUNDAMENTAL THEOREM OF CALCULUS (HOURS OF THEORY/EXERCISES: 6/6).

Contents

NUMERICAL SETS: INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. INDUCTION PRINCIPLE. INTRODUCTION TO REAL NUMBERS. EXTREME VALUES. INTERVALS OF REAL NUMBERS. NEIGHBOURHOODS, ACCUMULATION POINTS. CLOSED AND OPEN SETS. INTRODUCTION TO COMPLEX NUMBERS. IMAGINARY UNIT. OPERATIONS ON COMPLEX NUMBERS. GEOMETRIC AND TRIGONOMETRIC FORM. POWERS AND DE MOIVRE'S FORMULA. N-TH ROOTS (HOURS OF THEORY/EXERCISES: 4/2).

REAL FUNCTIONS: DEFINITION. DOMAIN, CODOMAIN AND GRAPH. EXTREMA. MONOTONE, COMPOSITE AND INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER AND ROOT, EXPONENTIAL, LOGARITHMIC, POWER, TRIGONOMETRIC AND INVERSE FUNCTIONS (HOURS OF THEORY/EXERCISES: 2/2).

BASIC NOTIONS OF EQUATIONS AND INEQUALITIES: FIRST ORDER, QUADRATIC, BINOMIAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. FIRST ORDER, SECOND ORDER, RATIONAL, IRRATIONAL, TRIGONOMETRIC, EXPONENTIAL, LOGARITHMIC INEQUALITIES. SYSTEMS (HOURS OF THEORY/EXERCISES: 2/4).

NUMERICAL SEQUENCES: DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. NEPERO'S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE. NUMERICAL SERIES: INTRODUCTION. CONVERGENCE. ARMONIC AND GEOMETRIC SERIES. POSITIVE SERIES AND CONVERGENCE CRITERIA (HOURS OF THEORY/EXERCISES: 4/4).

LIMITS OF A FUNCTION: DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS (HOURS OF THEORY/EXERCISES: 4/4).

CONTINUOUS FUNCTIONS: DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY (HOURS OF THEORY/EXERCISES: 4/0).

DERIVATIVE OF A FUNCTION: DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING. TANGENTIAL LINE. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY, COMPOSITE, INVERSE FUNCTIONS. HIGHER ORDER DERIVATIVES. DIFFERENTIAL OF A FUNCTION AND ITS GEOMETRIC MEANING (HOURS OF THEORY/EXERCISES: 4/2).

FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS (HOURS OF THEORY/EXERCISES: 4/2).

GRAPH OF A FUNCTION: ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH (HOURS OF THEORY/EXERCISES: 4/8).

INTEGRATION OF ONE VARIABLE FUNCTIONS: DEFINITION OF ANTIDERIVATIVE AND INDEFINITE INTEGRAL. BASIC INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND GEOMETRICAL MEANING. FUNDAMENTAL THEOREM OF CALCULUS (HOURS OF THEORY/EXERCISES: 6/6).

	Teaching Methods
	THE COURSE CONSISTS IN THEORETICAL LECTURES AND EXERCISES. 72 HOURS (9 CFU) IN TOTAL: 38 HOURS FOR THE THEORY AND 34 HOURS FOR THE EXERCISES. THE TEACHING IS PROVIDED IN PRESENCE WITH MANDATORY FREQUENCY, CERTIFIED BY THE STUDENT THROUGH THE USE OF THE PERSONAL BADGE. IN ORDER TO BE ADMITTED TO THE FINAL VERIFICATION OF PROFIT AND TO ACHIEVE THE RELATED NUMBER OF CFU, THE STUDENT WILL HAVE TO ATTENDED AT LEAST THE 70% OF THE SCHEDULED HOURS OF DIDACTIC ACTIVITY.

Teaching Methods

THE COURSE CONSISTS IN THEORETICAL LECTURES AND EXERCISES. 72 HOURS (9 CFU) IN TOTAL: 38 HOURS FOR THE THEORY AND 34 HOURS FOR THE EXERCISES.
THE TEACHING IS PROVIDED IN PRESENCE WITH MANDATORY FREQUENCY, CERTIFIED BY THE STUDENT THROUGH THE USE OF THE PERSONAL BADGE. IN ORDER TO BE ADMITTED TO THE FINAL VERIFICATION OF PROFIT AND TO ACHIEVE THE RELATED NUMBER OF CFU, THE STUDENT WILL HAVE TO ATTENDED AT LEAST THE 70% OF THE SCHEDULED HOURS OF DIDACTIC ACTIVITY.

	Verification of learning
	IN RELATION TO THE TEACHING OBJECTIVES, 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 TEST, THE SKILL OF PROVING THEOREMS, THE SKILL OF SOLVING EXERCISES, THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING, THE ABILITY TO USE THE ACQUIRED KNOWLEDGE. THE EXAM CONSISTS OF A WRITTEN TEST, THAT THE STUDENT WILL BE HELD TO ADDRESS IN TOTAL AUTONOMY, AND AN ORAL EXAMINATION. THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE (THERE ARE SAMPLES AVAILABLE ON THE DEPARTMENT WEBSITE). THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION. THERE WILL BE A MID-TERM TEST CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE WHICH, IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENTS ON THESE TOPICS AT THE FINAL WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS. THE ORAL INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE AND MASTERY IN ALL THE TOPICS OF THE COURSE, AS DEFINITIONS AND PROOFS OF THEOREMS, AS WELL AS IN SOLVING EXERCISES. THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL EVALUATION OF THE STUDENT.

Verification of learning

IN RELATION TO THE TEACHING OBJECTIVES, 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 TEST, THE SKILL OF PROVING THEOREMS, THE SKILL OF SOLVING EXERCISES, THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING, THE ABILITY TO USE THE ACQUIRED KNOWLEDGE.
THE EXAM CONSISTS OF A WRITTEN TEST, THAT THE STUDENT WILL BE HELD TO ADDRESS IN TOTAL AUTONOMY, AND AN ORAL EXAMINATION. THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE (THERE ARE SAMPLES AVAILABLE ON THE DEPARTMENT WEBSITE). THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION.
THERE WILL BE A MID-TERM TEST CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE WHICH, IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENTS ON THESE TOPICS AT THE FINAL WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS.
THE ORAL INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE AND MASTERY IN ALL THE TOPICS OF THE COURSE, AS DEFINITIONS AND PROOFS OF THEOREMS, AS WELL AS IN SOLVING EXERCISES.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL EVALUATION OF THE STUDENT.

	Texts
	BASIC TEXT FOR THEORY: P. MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA UNO, LIGUORE EDITORE. BASIC TEXT FOR THE EXERCISES: P. MARCELLINI, C. SBORDONE, ESERCITAZIONI DI MATEMATICA I, VOL.I, PARTE I, II, LIGUORI EDITORE. SUPPLEMENTARY TEACHING MATERIAL WILL BE AVAILABLE ON THE UNIVERSITY E-LEARNING PLATFORM (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS USING THEIR OWN UNIVERSITY CREDENTIALS.

Texts

BASIC TEXT FOR THEORY:
P. MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA UNO, LIGUORE EDITORE.
BASIC TEXT FOR THE EXERCISES:
P. MARCELLINI, C. SBORDONE, ESERCITAZIONI DI MATEMATICA I, VOL.I, PARTE I, II, LIGUORI EDITORE.

SUPPLEMENTARY TEACHING MATERIAL WILL BE AVAILABLE ON THE UNIVERSITY E-LEARNING PLATFORM (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS USING THEIR OWN UNIVERSITY CREDENTIALS.

	More Information
	THE COURSE IS HELD IN ITALIAN

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Tiziana DURANTE | CALCULUS 1

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Tiziana DURANTE | CALCULUS 1