Computer science | MATHEMATICAL ANALYSIS

cod. 0512100001

MATHEMATICAL ANALYSIS

	0512100001
	COMPUTER SCIENCE
	EQF6
	COMPUTER SCIENCE
	2021/2022

PERCORSO COMUNE

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

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/05	6	48	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS
MAT/05	3	24	EXERCISES	SUPPLEMENTARY COMPULSORY SUBJECTS

	Objectives
	KNOWLEDGE AND UNDERSTANDING: - BASIC NOTIONS OF CONTINUOUS MATHEMATICS. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: - TO KNOW HOW TO SOLVE EXERCISES CONNECTED TO THE STUDY OF THE FUNCTIONS OF A REAL VARIABLE (CALCULATION OF LIMITS OF FUNCTIONS, CALCULATION OF DERIVATIVES, STUDY OF THE TREND OF THE GRAPH OF A FUNCTION STARTING FROM ITS ALGEBRIC EXPRESSION, CALCULATION OF INTEGRALS). - TO KNOW HOW TO EXPLAIN IN A CLEAR AND STRICT WAY DEFINITIONS, PROBLEMS AND THEOREMS REGARDING THE TRAINING OBJECTIVES THE COURSE AIMS AT THE ACQUISITION OF ELEMENTS OF MATHEMATICAL ANALYSIS. THE EDUCATIONAL OBJECTIVES OF THE TEACHING CONSIST IN THE ACQUISITION OF RESULTS AND DEMONSTRATIVE TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELEVANT CALCULATION TOOLS. KNOWLEDGE AND UNDERSTANDING THE COURSE AIMS AT THE ACQUISITION OF THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS: NUMERICAL SETS, REAL FUNCTIONS OF A REAL VARIABLE, EQUATIONS AND INEQUATIONS, LIMITS, CONTINUOUS FUNCTIONS, DERIVATIVES, SERAPHIC OF A FUNCTION, INTEGRALS, REAL FUNCTIONS OF SEVERAL VARIABLES NUMERICAL. THE SPECIFIC EDUCATIONAL OBJECTIVES OF THE TEACHING CONSIST ESSENTIALLY IN THE ACQUISITION OF RESULTS AND DEMONSTRATIVE TECHNIQUES, AS WELL AS THE ABILITY TO SOLVE EXERCISES AND COMPARE IN A CONSTRUCTIVE WAY WITH TEXT BOOKS FOR A SUFFICIENT SELF APPROACH. STUDENTS WILL HAVE FUNDAMENTAL MATHEMATICAL TOOLS FOR AN APPROPRIATE QUANTITATIVE APPROACH TO THE COMPUTER TOPICS THAT WILL BE ADDRESSED DURING THE DEGREE COURSE. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING STUDENTS WILL BE ABLE TO APPLY THE QUANTITATIVE TOOLS LEARNED TO SOLVE SOME CLASSIC PROBLEMS IN COMPUTER SCIENCE. IN PARTICULAR, THEY WILL KNOW HOW TO APPLY THE THEOREMS AND RULES STUDIED TO THE SOLUTION OF PROBLEMS. PERFORM CALCULATIONS WITH LIMITS, DERIVATIVES, INTEGRALS. CARRY OUT THE STUDY OF THE GRAPH OF A FUNCTION. CALCULATE DERIVATIVES AND MAXIMUMS AND MINIMUMS OF FUNCTIONS OF MULTIPLE REAL VARIABLES. AUTONOMY OF JUDGMENT TO KNOW HOW TO IDENTIFY THE MOST APPROPRIATE METHODS TO SOLVE A MATHEMATICAL PROBLEM IN AN EFFICIENT WAY. COMMUNICATION SKILLS TO KNOW HOW TO ORALLY EXPLAIN A TOPIC RELATED TO MATHEMATICS. ABILITY TO LEARN TO KNOW HOW TO APPLY THE KNOWLEDGE ACQUIRED TO EXAMPLES OTHER THAN THOSE PRESENTED DURING THE LESSONS. TO KNOW HOW TO USE THE KNOWLEDGE ACQUIRED IN REASONING AND ALGORITHMS

KNOWLEDGE AND UNDERSTANDING:
- BASIC NOTIONS OF CONTINUOUS MATHEMATICS.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
- TO KNOW HOW TO SOLVE EXERCISES CONNECTED TO THE STUDY OF THE FUNCTIONS OF A REAL VARIABLE (CALCULATION OF LIMITS OF FUNCTIONS, CALCULATION OF DERIVATIVES, STUDY OF THE TREND OF THE GRAPH OF A FUNCTION STARTING FROM ITS ALGEBRIC EXPRESSION, CALCULATION OF INTEGRALS).
- TO KNOW HOW TO EXPLAIN IN A CLEAR AND STRICT WAY DEFINITIONS, PROBLEMS AND THEOREMS REGARDING THE TRAINING OBJECTIVES

THE COURSE AIMS AT THE ACQUISITION OF ELEMENTS OF MATHEMATICAL ANALYSIS.
THE EDUCATIONAL OBJECTIVES OF THE TEACHING CONSIST IN THE ACQUISITION OF RESULTS AND DEMONSTRATIVE TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELEVANT CALCULATION TOOLS.

KNOWLEDGE AND UNDERSTANDING
THE COURSE AIMS AT THE ACQUISITION OF THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS: NUMERICAL SETS, REAL FUNCTIONS OF A REAL VARIABLE, EQUATIONS AND INEQUATIONS, LIMITS, CONTINUOUS FUNCTIONS, DERIVATIVES, SERAPHIC OF A FUNCTION, INTEGRALS, REAL FUNCTIONS OF SEVERAL VARIABLES NUMERICAL.

THE SPECIFIC EDUCATIONAL OBJECTIVES OF THE TEACHING CONSIST ESSENTIALLY IN THE ACQUISITION OF RESULTS AND DEMONSTRATIVE TECHNIQUES, AS WELL AS THE ABILITY TO SOLVE EXERCISES AND COMPARE IN A CONSTRUCTIVE WAY WITH TEXT BOOKS FOR A SUFFICIENT SELF APPROACH.
STUDENTS WILL HAVE FUNDAMENTAL MATHEMATICAL TOOLS FOR AN APPROPRIATE QUANTITATIVE APPROACH TO THE COMPUTER TOPICS THAT WILL BE ADDRESSED DURING THE DEGREE COURSE.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING
STUDENTS WILL BE ABLE TO APPLY THE QUANTITATIVE TOOLS LEARNED TO SOLVE SOME CLASSIC PROBLEMS IN COMPUTER SCIENCE.
IN PARTICULAR, THEY WILL KNOW HOW TO APPLY THE THEOREMS AND RULES STUDIED TO THE SOLUTION OF PROBLEMS.
PERFORM CALCULATIONS WITH LIMITS, DERIVATIVES, INTEGRALS.
CARRY OUT THE STUDY OF THE GRAPH OF A FUNCTION.
CALCULATE DERIVATIVES AND MAXIMUMS AND MINIMUMS OF FUNCTIONS OF MULTIPLE REAL VARIABLES.

AUTONOMY OF JUDGMENT
TO KNOW HOW TO IDENTIFY THE MOST APPROPRIATE METHODS TO SOLVE A MATHEMATICAL PROBLEM IN AN EFFICIENT WAY.

COMMUNICATION SKILLS
TO KNOW HOW TO ORALLY EXPLAIN A TOPIC RELATED TO MATHEMATICS.

ABILITY TO LEARN
TO KNOW HOW TO APPLY THE KNOWLEDGE ACQUIRED TO EXAMPLES OTHER THAN THOSE PRESENTED DURING THE LESSONS. TO KNOW HOW TO USE THE KNOWLEDGE ACQUIRED IN REASONING AND ALGORITHMS

	Prerequisites
	PREREQUISITES: POLYNOMIAL EQUATIONS AND DISEQUATIONS. IRRATIONAL, EXPONENTIAL, LOGARITHMIC AND GONIOMETRIC EQUATIONS AND DISEQUATIONS. PROPEDEUTICITY: NONE

	Contents
	• ELEMENTS OF SET THEORY. • NUMBERABLE SETS. PRINCIPLE OF INDUCTION AND BERNOUILLI'S INEQUALITY. • IRRATIONAL NUMBERS AND CONTINUOUS PROPERTIES. AXIOM OF DEDEKIND. • COMPLEX NUMBERS IN ALGEBRIC, TRIGONOMETRIC AND EXPONENTIAL FORM • REAL FUNCTIONS. DOMAIN AND LIMITS OF A FUNCTION. CONTINUOUS FUNCTIONS AND POINTS OF DISCONTINUITY. OPERATIONS AND THEOREMS ON LIMITS. INDETERMINATE FORMS. ASYMPTOTES. • DERIVATIVE OF A FUNCTION AND ITS GEOMETRIC MEANING. STRAIGHT LINE TANGENT. DERIVABILITY AND CONTINUITY. DERIVATION RULES. RELATIVE EXTREME. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS. HIGHER ORDER DERIVATIVES. CONCAVITY, CONVEXITY AND FLEXES. DE L’HOSPITAL THEOREM. • STUDY OF FUNCTIONS AND GRAPHIC REPRESENTATION • SUCCESSIONS AND NUMERICAL SERIES. • INDEFINITE INTEGRAL AND INTEGRATION RULES. RIEMANN INTEGRAL. PEANO-JORDAN MEASURE AND GEOMETRIC MEANING OF THE RIEMANN INTEGRAL. • MCLAURIN AND TAYLOR SERIES

Contents

• ELEMENTS OF SET THEORY.
• NUMBERABLE SETS. PRINCIPLE OF INDUCTION AND BERNOUILLI'S INEQUALITY.
• IRRATIONAL NUMBERS AND CONTINUOUS PROPERTIES. AXIOM OF DEDEKIND.
• COMPLEX NUMBERS IN ALGEBRIC, TRIGONOMETRIC AND EXPONENTIAL FORM
• REAL FUNCTIONS. DOMAIN AND LIMITS OF A FUNCTION. CONTINUOUS FUNCTIONS AND POINTS OF DISCONTINUITY. OPERATIONS AND THEOREMS ON LIMITS. INDETERMINATE FORMS. ASYMPTOTES.
• DERIVATIVE OF A FUNCTION AND ITS GEOMETRIC MEANING. STRAIGHT LINE TANGENT. DERIVABILITY AND CONTINUITY. DERIVATION RULES. RELATIVE EXTREME. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS. HIGHER ORDER DERIVATIVES. CONCAVITY, CONVEXITY AND FLEXES. DE L’HOSPITAL THEOREM.
• STUDY OF FUNCTIONS AND GRAPHIC REPRESENTATION
• SUCCESSIONS AND NUMERICAL SERIES.
• INDEFINITE INTEGRAL AND INTEGRATION RULES. RIEMANN INTEGRAL. PEANO-JORDAN MEASURE AND GEOMETRIC MEANING OF THE RIEMANN INTEGRAL.
• MCLAURIN AND TAYLOR SERIES

	Teaching Methods
	72 HOURS OF LECTURES. 48 HOURS OF THEORY AND 24 HOURS OF EXERCISE. THE LESSONS WILL ALLOW THE STUDENT TO ACQUIRE BOTH THEORETICAL KNOWLEDGE AND APPLICATIONS FROM THE POINT OF VIEW OF THE CALCULATION. COURSE ATTENDANCE IS RECOMMENDED BUT NOT OBLIGATORY. INTERACTION WITH THE TEACHER WILL TAKE PLACE IN THE CLASSROOM, BY EMAIL AND DURING RECEPTION HOURS.

	Verification of learning
	EXERCISES BY THE TEACHER, SIMULATIONS, WRITTEN TEST AND FINAL ORAL TEST. STUDENTS, ON THE BASIS OF THE TECHNIQUES ACQUIRED, SHOULD BE ABLE TO CARRY OUT EXERCISES, TO STATE AND DEMONSTRATE THEOREMS AND TO SUPPORT, IN A CLEAR AND EFFECTIVE WAY, AN ORAL DISCUSSION WITH APPROPRIATE REFERENCES TO THE CONTENTS OF THE COURSES. IN PARTICULAR, THE ACHIEVEMENT OF THE OBJECTIVES OF THE COURSE IS CERTIFIED BY PASSING AN EXAM WITH ASSESSMENT IN THIRTIES. THE EXAM INCLUDES A WRITTEN TEST AND AN ORAL TEST THAT MAY TAKE PLACE ON DIFFERENT DAYS. THE WRITTEN TEST IS PREPARED TO THE ORAL TEST. THE DATE OF THE WRITTEN TEST IS THAT PROVIDED BY THE DEPARTMENT'S CALENDAR WHILE THE DAY OF THE ORAL TEST IS AGREED WITH THE STUDENTS ON THE DAY OF THE WRITTEN TEST. EACH TEST IS ASSESSED IN THIRTIES AND IS INTENDED TO BE PASSED WITH A MINIMUM MARK OF 18/30. THE FINAL VOTE IS GIVEN BY THE AVERAGE OF THE VOTES REPORTED IN EACH TEST. THE WRITTEN TEST, LASTING ABOUT 120 MINUTES, IS AIMED TO ASSESS THE LEVEL OF KNOWLEDGE AND UNDERSTANDING OF THE TOPICS INDICATED IN THE PROGRAM AND PROPOSED DURING THE COURSE, THE KNOWLEDGE OF THE ANALYTICAL TOOLS AND THE ABILITY TO APPLY THE KNOWLEDGE FINALLY, THE ABILITY TO COMMUNICATE, IN AN EFFECTIVE AND RELEVANT WAY, IN WRITTEN FORM. IT CONSISTS IN THE SOLUTION OF A CERTAIN NUMBER OF EXERCISES CONCERNING THE TOPICS OF THE COURSE, EACH OF WHICH IS ASSIGNED A SCORE WHICH VARY ACCORDING TO THE COMPLEXITY OF THE CALCULATIONS REQUIRED AND WHICH IS NOTIFIED TO THE STUDENT BY THE TEACHER. THE PROPOSED EXERCISES ARE SIMILAR TO THOSE SOLVED DURING THE HOURS OF LESSON. THE WRITTEN TEST IS INTENDED TO BE PASSED IF THE STUDENT HAS CORRECTLY SOLVED THE PROPOSED EXERCISES (WITH A TOTAL VOTE NOT LESS THAN 18 THIRTY). THE SCORE OF THE WRITTEN TEST IS EQUAL TO THE SUM OF POINTS ASSIGNED TO THE INDIVIDUAL QUESTIONS CARRIED OUT BY THE STUDENT. DURING THE WRITTEN TEST IT IS NOT ALLOWED TO CONSULT TEXTS, USE PCS AND MOBILE PHONES; THE USE OF THE CALCULATOR IS PERMITTED ONLY IN CERTAIN CASES. THE ORAL EXAM, LASTING ABOUT 30 MINUTES, CONSISTS OF A DISCUSSION ON DEMONSTRATION OF THEOREMS PROPOSED DURING THE LESSONS AND ON THE THEORETICAL AND METHODOLOGICAL CONTENT INDICATED IN THE PROGRAM, IN ORDER TO ASK FOREST NOT ONLY THE LEVEL OF KNOWLEDGE AND SKILLS OF UNDERSTANDING STUDENT, BUT ALSO THE ABILITY TO EXPOSE THE TOPICS WITH THE APPROPRIATE TERMINOLOGY. NO INTERCURSE TESTS ARE PROVIDED. THE ORAL EXAM WILL BE USED TO ASSESS THE STUDENT'S ABILITY TO EXPOSE THE MATHEMATICAL CONCEPTS AND THEOREMS PROVEN DURING THE LESSONS IN A CLEAR AND RIGOROUS WAY.

Verification of learning

EXERCISES BY THE TEACHER, SIMULATIONS, WRITTEN TEST AND FINAL ORAL TEST.
STUDENTS, ON THE BASIS OF THE TECHNIQUES ACQUIRED, SHOULD BE ABLE TO CARRY OUT EXERCISES, TO STATE AND DEMONSTRATE THEOREMS AND TO SUPPORT, IN A CLEAR AND EFFECTIVE WAY, AN ORAL DISCUSSION WITH APPROPRIATE REFERENCES TO THE CONTENTS OF THE COURSES.
IN PARTICULAR, THE ACHIEVEMENT OF THE OBJECTIVES OF THE COURSE IS CERTIFIED BY PASSING AN EXAM WITH ASSESSMENT IN THIRTIES. THE EXAM INCLUDES A WRITTEN TEST AND AN ORAL TEST THAT MAY TAKE PLACE ON DIFFERENT DAYS. THE WRITTEN TEST IS PREPARED TO THE ORAL TEST. THE DATE OF THE WRITTEN TEST IS THAT PROVIDED BY THE DEPARTMENT'S CALENDAR WHILE THE DAY OF THE ORAL TEST IS AGREED WITH THE STUDENTS ON THE DAY OF THE WRITTEN TEST. EACH TEST IS ASSESSED IN THIRTIES AND IS INTENDED TO BE PASSED WITH A MINIMUM MARK OF 18/30. THE FINAL VOTE IS GIVEN BY THE AVERAGE OF THE VOTES REPORTED IN EACH TEST.
THE WRITTEN TEST, LASTING ABOUT 120 MINUTES, IS AIMED TO ASSESS THE LEVEL OF KNOWLEDGE AND UNDERSTANDING OF THE TOPICS INDICATED IN THE PROGRAM AND PROPOSED DURING THE COURSE, THE KNOWLEDGE OF THE ANALYTICAL TOOLS AND THE ABILITY TO APPLY THE KNOWLEDGE FINALLY, THE ABILITY TO COMMUNICATE, IN AN EFFECTIVE AND RELEVANT WAY, IN WRITTEN FORM. IT CONSISTS IN THE SOLUTION OF A CERTAIN NUMBER OF EXERCISES CONCERNING THE TOPICS OF THE COURSE, EACH OF WHICH IS ASSIGNED A SCORE WHICH VARY ACCORDING TO THE COMPLEXITY OF THE CALCULATIONS REQUIRED AND WHICH IS NOTIFIED TO THE STUDENT BY THE TEACHER. THE PROPOSED EXERCISES ARE SIMILAR TO THOSE SOLVED DURING THE HOURS OF LESSON.
THE WRITTEN TEST IS INTENDED TO BE PASSED IF THE STUDENT HAS CORRECTLY SOLVED THE PROPOSED EXERCISES (WITH A TOTAL VOTE NOT LESS THAN 18 THIRTY). THE SCORE OF THE WRITTEN TEST IS EQUAL TO THE SUM OF POINTS ASSIGNED TO THE INDIVIDUAL QUESTIONS CARRIED OUT BY THE STUDENT. DURING THE WRITTEN TEST IT IS NOT ALLOWED TO CONSULT TEXTS, USE PCS AND MOBILE PHONES; THE USE OF THE CALCULATOR IS PERMITTED ONLY IN CERTAIN CASES.
THE ORAL EXAM, LASTING ABOUT 30 MINUTES, CONSISTS OF A DISCUSSION ON DEMONSTRATION OF THEOREMS PROPOSED DURING THE LESSONS AND ON THE THEORETICAL AND METHODOLOGICAL CONTENT INDICATED IN THE PROGRAM, IN ORDER TO ASK FOREST NOT ONLY THE LEVEL OF KNOWLEDGE AND SKILLS OF UNDERSTANDING STUDENT, BUT ALSO THE ABILITY TO EXPOSE THE TOPICS WITH THE APPROPRIATE TERMINOLOGY.
NO INTERCURSE TESTS ARE PROVIDED. THE ORAL EXAM WILL BE USED TO ASSESS THE STUDENT'S ABILITY TO EXPOSE THE MATHEMATICAL CONCEPTS AND THEOREMS PROVEN DURING THE LESSONS IN A CLEAR AND RIGOROUS WAY.

	Texts
	• P. MARCELLINI - C. SBORDONE, “ANALISI MATEMATICA UNO “, LIGUORI EDITORE • P. MARCELLINI - C. SBORDONE, “ELEMENTI DI ANALISI MATEMATICA UNO “, LIGUORI EDITORE • D'APICE-MANZO, "VERSO L'ESAME DI MATEMATICA 1" CUES EDITORE • NOTES DISTRIBUTED DURING THE LESSONS IN-DEPTH TEXT BOOKS • E. GIUSTI “ANALISI MATEMATICA I“, BOLLATI BORINGHIERI