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METHODS AND TECHNIQUES OF MATHEMATICS

Rosanna MANZO METHODS AND TECHNIQUES OF MATHEMATICS

1212500003
DEPARTMENT OF MANAGEMENT & INNOVATION SYSTEMS
EQF6
DIPLOMATIC, INTERNATIONAL AND GLOBAL SECURITY STUDIES
2024/2025

OBBLIGATORIO
YEAR OF COURSE 1
YEAR OF DIDACTIC SYSTEM 2019
AUTUMN SEMESTER
CFUHOURSACTIVITY
963LESSONS
CIRO D'APICE T Curriculum
ROSANNA MANZO Curriculum
ExamDate
D'APICE11/12/2024 - 15:00
D'APICE11/12/2024 - 15:00
Objectives
THE COURSE PRESENTS THE BASIC ELEMENTS OF MATHEMATICAL ANALYSIS AND LINEAR ALGEBRA.
THE EDUCATIONAL OBJECTIVES CONSIST IN THE ACQUISITION OF BASIC MATHEMATICAL TECHNIQUES AND METHODS, NECESSARY FOR A REPRESENTATION AND ANALYSIS OF THE MAIN ASPECTS OF REALITY AND IN THE ABILITY OF SOLVING MATHEMATICAL AND APPLIED PROBLEMS.

KNOWLEDGE AND UNDERSTANDING
THE STUDENT WILL ACQUIRE KNOWLEDGE ABOUT:
•NUMERIC SETS,
•REAL FUNCTIONS OF A REAL VARIABLE,
•LIMITS,
•CONTINUOUS FUNCTIONS,
•DERIVATIVES,
•INTEGRALS,
•MATRICES AND LINEAR SYSTEMS,
•REAL FUNCTIONS OF TWO REAL VARIABLES,
•ORDINARY DIFFERENTIAL EQUATIONS.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING
THE STUDENT WILL BE ABLE TO:
•CALCULATE SIMPLE FUNCTION LIMITS,
•CALCULATE THE DERIVATIVE OF A FUNCTION,
•DRAW THE QUALITATIVE GRAPH OF A FUNCTION,
•INTERPRET THE GRAPH OF A FUNCTION,
•CALCULATE SIMPLE INTEGRALS,
•PERFORM OPERATIONS WITH MATRICES,
•SOLVE LINEAR SYSTEMS,
•DETERMINE MAXIMA AND MINIMA OF FUNCTIONS OF TWO VARIABLES,
•SOLVE SIMPLE ORDINARY DIFFERENTIAL EQUATIONS,
•APPLY THE LEARNED TECHNIQUES AND METHODS FOR THE RESOLUTION OF REAL PROBLEMS.

MAKING JUDGEMENTS
THE STUDENT WILL BE ABLE TO IDENTIFY THE MOST APPROPRIATE METHODS TO EFFICIENTLY SOLVE MATHEMATICAL PROBLEMS, OF BOTH THEORETICAL AND APPLICATIVE TYPE.

COMMUNICATION SKILLS
THE STUDENT WILL BE ABLE TO DESCRIBE THE TREND OF ECONOMIC, GEOGRAPHICAL, SOCIAL AND POLITICAL PHENOMENA BY THE LANGUAGE OF MATHEMATICS.

LEARNING SKILLS
THE STUDENT WILL BE ABLE TO APPLY THE ACQUIRED KNOWLEDGE WITHIN SOCIAL AND POLITICAL CONTEXTS.
Prerequisites
FOR THE SUCCESSFUL ACHIEVEMENT OF THE PREDEFINED OBJECTIVES AND, IN PARTICULAR, FOR AN ADEQUATE UNDERSTANDING OF THE CONTENTS PROVIDED BY THE TEACHING, KNOWLEDGE RELATED TO EQUATIONS AND INEQUALITIES ARE PARTICULARLY USEFUL AND, THEREFORE, REQUIRED TO THE STUDENT.

MANDATORY PREPARATORY TEACHINGS
NONE.
Contents
NUMERICAL SETS.
(LECTURE/EXERCISE/LABORATORY HOURS 2/0/0)
INTRODUCTION. OPERATIONS ON SUBSETS OF A SET. INTRODUCTION TO REAL NUMBERS. EXTREMES OF A NUMERICAL SET. INTERVALS OF R. NEIGHBORHOODS, POINTS OF ACCUMULATION. CLOSED SETS AND OPEN SETS.

REAL FUNCTIONS OF A REAL VARIABLE.
(LECTURE/EXERCISE/LABORATORY HOURS 4/2/0)
DEFINITION. DOMAIN, CODOMAIN AND GRAPH OF A FUNCTION. EXTREMES OF A REAL FUNCTION. MONOTONE FUNCTIONS. COMPOUND FUNCTIONS. INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER AND N-TH ROOT FUNCTIONS, EXPONENTIAL FUNCTION, LOGARITHMIC FUNCTION, POWER FUNCTION.

BASIC NOTIONS ON EQUATIONS AND INEQUALITIES.
(LECTURE/EXERCISE/LABORATORY HOURS 1/4/0)
EQUATIONS OF THE FIRST ORDER. QUADRATIC EQUATIONS. IRRATIONAL EQUATIONS. EXPONENTIAL AND LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. LINEAR INEQUALITIES. SECOND ORDER INEQUALITIES. FRACTIONAL INEQUALITIES. IRRATIONAL INEQUALITIES. EXPONENTIAL AND LOGARITHMIC INEQUALITIES. SYSTEMS OF INEQUALITIES.

NUMERICAL SEQUENCES (BASIC ELEMENTS).
(LECTURE/EXERCISE/LABORATORY HOURS 1/0/0)
BOUNDED, CONVERGENT, OSCILLATING AND DIVERGENT SUCCESSIONS. MONOTONE SEQUENCES.

LIMITS OF A FUNCTION.
(LECTURE/EXERCISE/LABORATORY HOURS 2/4/0)
DEFINITION. RIGHT LIMIT AND LEFT LIMIT. UNIQUENESS THEOREM OF THE LIMIT. THEOREM OF PERMANENCE OF SIGN. COMPARISON THEOREMS. OPERATIONS AND INDETERMINATE FORMS. KNOWN LIMITS.

CONTINUOUS FUNCTIONS.
(LECTURE/EXERCISE/LABORATORY HOURS 2/2/0)
DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM. INTERMEDIATE VALUE THEOREM. ZEROS THEOREM.

DERIVATIVE OF A FUNCTION.
(LECTURE/EXERCISE/LABORATORY HOURS 2/4/0)
DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRICAL MEANING, THE TANGENT LINE TO THE GRAPH OF A FUNCTION. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY FUNCTIONS. DERIVATIVES OF COMPOUND FUNCTIONS. HIGHER ORDER DERIVATIVES.

FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS.
(LECTURE/EXERCISE/LABORATORY HOURS 4/2/0)
ROLLE THEOREM. CAUCHY THEOREM. LAGRANGE THEOREM AND COROLLARIES. DE L’HOSPITAL THEOREM. CONDITIONS FOR RELATIVE MAXIMA AND MINIMA.

STUDY OF THE GRAPH OF A FUNCTION.
(LECTURE/EXERCISE/LABORATORY HOURS 1/5/0)
ASYMPTOTES OF A GRAPH. SEARCH OF RELATIVE MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTIONS POINTS. GRAPH OF A FUNCTION BY ITS CHARACTERISTIC ELEMENTS.

INTEGRATION OF FUNCTIONS OF ONE VARIABLE.
(LECTURE/EXERCISE/LABORATORY HOURS 2/4/0)
DEFINITE INTEGRAL AND GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS. DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS.

ELEMENTS OF LINEAR ALGEBRA.
(LECTURE/EXERCISE/LABORATORY HOURS 2/4/0)
MATRICES AND DETERMINANTS. RESOLUTION OF LINEAR SYSTEMS: REDUCTION TO ROW ECHELON FORM.

FUNCTIONS OF SEVERAL VARIABLES.
(LECTURE/EXERCISE/LABORATORY HOURS 2/4/0)
DEFINITIONS. LIMIT AND CONTINUITY. PARTIAL DERIVATIVES. SCHWARZ’S THEOREM. GRADIENT AND DIFFERENTIABILITY. DIRECTIONAL DERIVATIVES. RELATIVE MINIMA AND MAXIMA.

ORDINARY DIFFERENTIAL EQUATIONS.
(LECTURE/EXERCISE/LABORATORY HOURS 1/2/0)
DEFINITIONS. PARTICULAR AND GENERAL INTEGRAL. EXAMPLES. THE CAUCHY PROBLEM. FIRST-ORDER DIFFERENTIAL EQUATIONS. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. RESOLUTION METHODS.

TOTAL LECTURE/EXERCISE/LABORATORY HOURS 26/37/0
Teaching Methods
THE TEACHING INCLUDES THEORETICAL LECTURES IN THE CLASSROOM FOR A TOTAL OF 26 HOURS AND CLASSROOM EXERCISES FOR A TOTAL OF 37 HOURS.
ATTENDANCE OF CLASSROOM LECTURES AND EXERCISES, ALTHOUGH NOT MANDATORY, IS STRONGLY RECOMMENDED FOR THE FULL ACHIEVEMENT OF THE LEARNING OBJECTIVES.
Verification of learning
THE FINAL EXAM AIMS TO EVALUATE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE THEORETICAL LECTURES AND THE CLASSROOM EXERCISE SESSIONS; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; THE SKILL OF PROVING THEOREMS; THE SKILL OF SOLVING PROBLEMS; THE ABILITY TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHODS IN PROBLEMS SOLVING; THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO THE SOLUTION OF DIFFERENT EXERCISES COMPARED TO THOSE PRESENTED DURING THE EXERCISE SESSIONS.
THE EXAM NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES CONSISTS OF A WRITTEN TEST, PRELIMINARY FOR THE ORAL EXAMINATION, AND AN ORAL EXAMINATION.
THE WRITTEN TEST CONSISTS OF THE RESOLUTION OF EXERCISES BASED ON WHAT HAS BEEN PROPOSED IN THE FRAMEWORK OF THE THEORETICAL LECTURES AND EXERCISE SESSIONS. THE WRITTEN TEST, WHICH THE STUDENT WILL HAVE TO FACE IN TOTAL AUTONOMY, WILL LAST 2 AND A HALF HOURS.
THE RESULT OF THE WRITTEN TEST IS PASSED OR FAILED.
STUDENTS WHO PASS THE WRITTEN TEST WILL HAVE TO TAKE AND PASS THE ORAL TEST.
THE ORAL TEST IS MAINLY DEVOTED TO ASSERT THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING AND COVERS DEFINITIONS, THEOREMS PROOFS, AND EXERCISES SOLVING.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE ORAL TEST.
LAUDE WILL BE ATTRIBUTED TO STUDENTS WHO SHOW EXCELLENT KNOWLEDGE OF THE CONTENT OF THE COURSE, AND EXCELLENT EXPOSURE SKILLS COMBINED WITH THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE FOR THE RESOLUTION OF PROBLEMS NOT ADDRESSED DURING THE COURSE.
Texts
C. D’APICE, R. MANZO: “VERSO L'ESAME DI MATEMATICA 1, RACCOLTA DI ESERCIZI CON SVOLGIMENTO”, MAGGIOLI EDITORE, APOGEO EDUCATION, 2015.
C. D’APICE, T. DURANTE, R. MANZO: “VERSO L'ESAME DI MATEMATICA 2, RACCOLTA DI ESERCIZI CON SVOLGIMENTO”, MAGGIOLI EDITORE, APOGEO EDUCATION, 2015.
SUPPLEMENTARY TEACHING MATERIALS WILL BE AVAILABLE IN THE TEACHING SECTION WITHIN THE UNIVERSITY'S E-LEARNING AREA (HTTP://ELEARNING.UNISA.IT), ACCESSIBLE TO STUDENTS IN THE COURSE VIA THE UNIQUE UNIVERSITY CREDENTIALS.
More Information
TEACHING IS PROVIDED IN ITALIAN.
Lessons Timetable

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