# Diplomatic, International and Global Security Studies | METHODS AND TECHNIQUES OF MATHEMATICS

## Diplomatic, International and Global Security Studies METHODS AND TECHNIQUES OF MATHEMATICS

 1212500003 DEPARTMENT OF MANAGEMENT & INNOVATION SYSTEMS EQF6 DIPLOMATIC, INTERNATIONAL AND GLOBAL SECURITY STUDIES 2022/2023

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2019 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 9 63 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS
 CIRO D'APICE T
ExamDate
D'APICE15/02/2023 - 15:00
D'APICE15/02/2023 - 15:00
D'APICE13/04/2023 - 15:00
Objectives
THE COURSE PROPOSES TO GIVE TO STUDENTS THE FUNDAMENTAL MATHEMATICAL TECHNIQUES AND METHODS FOR DESCRIBING AND ANALYSING THE MAIN ASPECTS OF THE WORLD AROUND US.
STUDENTS MUST BE ABLE TO LEARN A RIGOROUS AND ANALYTIC METHOD OF REASONING AND TACKLING PROBLEMS. IN PARTICULAR THEY MUST BE ABLE TO SKETCH AND INTERPRET GRAPHS OF REAL FUNCTIONS OF ONE REAL VARIABLE, TO COMPUTE MAXIMA AND MINIMA FOR MULTIVARIATE FUNCTIONS, TO CALCULATE DEFINITE AND INDEFINITE INTEGRALS, TO SOLVE EQUATION SYSTEMS AND SIMPLE MATHEMATICAL PROBLEMS MODELLED BY ORDINARY DIFFERENTIAL EQUATIONS
THE TEACHING AIMS AT ACQUIRING THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS: NUMERICAL SETS, REAL FUNCTIONS, BASIC NOTIONS OF EQUATIONS AND INEQUALITIES, LIMITS, CONTINUOUS FUNCTIONS, DERIVATIVES, STUDY OF A FUNCTION, INTEGRATION OF A SINGLE VARIABLE FUNCTION, MATRICES AND LINEAR SYSTEMS, MULTIVARIABLE FUNCTIONS, ORDINARY DIFFERENTIAL EQUATIONS.
THE SPECIFIC LEARNING OUTCOMES ACTUALLY CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS IN THE ABILITY TO SOLVE MATHEMATICAL AND APPLIED PROBLEMS.
STUDENTS WILL BE ABLE TO APPLY THE QUANTITATIVE LEARNED TOOLS TO SOLVE PROBLEMS.
IN PARTICULAR, THEY WILL BE ABLE TO COMPUTE LIMITS, DERIVATIVES, INTEGRALS, TO STUDY THE GRAPH OF A FUNCTION, TO COMPUTE CALCULUS WITH MATRICES AND TO SOLVE LINEAR SYSTEMS, TO CALCULATE DERIVATIVES AND MINIMA AND MAXIMA FOR MULTIVARIATE FUNCTIONS, TO SOLVE ORDINARY DIFFERENTIAL EQUATIONS.
Prerequisites
FOR A SUCCESSFUL ACHIEVEMENT OF THE PROPOSED GOALS AND, IN PARTICULAR, FOR A PROPER UNDERSTANDING OF THE CONTENTS SCHEDULED FOR THE TEACHING, STUDENTS ARE REQUIRED TO MASTER THE FOLLOWING PARTICULARLY USEFUL KNOWLEDGE ABOUT EQUATION AND INEQUALITIES.
MANDATORY PREPARATORY TEACHINGS
NONE.
Contents
NUMERICAL SETS.
INTRODUCTION TO SET THEORY. 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.
(2,0)
REAL FUNCTIONS.
DEFINITION. DOMAIN, CODOMAIN AND GRAPH OF A FUNCTION. EXTREMES OF A REAL FUNCTION. MONOTONE FUNCTIONS. COMPOSITE FUNCTIONS. INVERTIBLE FUNCTIONS. ELEMENTARY FUNCTIONS: N-TH POWER AND N-TH ROOT FUNCTIONS, EXPONENTIAL, LOGARITHMIC FUNCTION, POWER FUNCTION.
(4,2)
BASIC NOTIONS OF EQUATIONS AND INEQUALITIES.
EQUATIONS OF FIRST ORDER. QUADRATIC EQUATIONS. IRRATIONAL EQUATIONS. EXPONENTIAL AND LOGARITHMIC EQUATIONS. SYSTEMS OF EQUATIONS. LINEAR INEQUALITIES. INEQUALITIES OF THE SECOND ORDER. FACTIONAL INEQUALITIES. IRRATIONAL INEQUALITIES. EXPONENTIAL AND LOGARITHMIC INEQUALITIES. SYSTEMS OF INEQUALITIES.
(1,4)
LIMITS OF A FUNCTION.
DEFINITION. RIGHT AND LEFT HAND-SIDE LIMITS. UNIQUENESS THEOREM. COMPARISON THEOREMS. OPERATIONS AND INDETERMINATE FORMS. KNOWN LIMITS.
(2,4)
NUMERICAL SEQUENCES (BASIS ELEMENTS)
DEFINITION. NUMERICAL SEQUENCES. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES.
(1,0)
CONTINUOUS FUNCTIONS.
DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS THEOREM. ZEROS THEOREM.
(2,2)
DERIVATIVE OF A FUNCTION
DEFINITION. LEFT AND RIGHT DERIVATIVES. GEOMETRIC MEANING, THE TANGENT LINE TO THE GRAPH OF A FUNCTION. DIFFERENTIABILITY AND CONTINUITY. DERIVATION RULES. DERIVATIVES OF ELEMENTARY FUNCTIONS. DERIVATIVES OF COMPOSITE FUNCTION. HIGHER ORDER DERIVATIVES.
(2,4)
FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS.
ROLLE'S THEOREM. CAUCHY'S THEOREM. LAGRANGE'S THEOREM AND COROLLARIES. THEOREM OF DE L'HOSPITAL. CONDITIONS FOR MAXIMA AND MINIMA.
(4,2)
GRAPH OF A FUNCTION
ASYMPTOTES OF A GRAPH. SEARCH OF MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. GRAPH OF A FUNCTION BY ITS CHARACTERISTIC ELEMENTS.
(1,5)
INTEGRATION OF ONE VARIABLE FUNCTIONS
DEFINITION OF INDEFINITE INTEGRAL. BASIC INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND GEOMETRICAL MEANING. FUNDAMENTAL THEOREM OF CALCULUS.
(2,4)
MATRICES AND LINEAR SYSTEMS
MATRICES AND DETERMINANTS. LINEAR SYSTEMS: ROUCHE-CAPELLI THEOREM, CRAMER THEOREM.
(2,4)
MULTIVARIABLE FUNCTIONS.
DEFINITIONS. LIMITS AND CONTINUITY. WEIERSTRASS THEOREM. PARTIAL DIFFERENTIATION. SCHWARZ THEOREM. GRADIENT AND DIFFERENTIABILITY. DIRECTIONAL DERIVATIVES. LOCAL MINIMA AND MAXIMA.
(2/4)
ORDINARY DIFFERENTIAL EQUATIONS.
DEFINITIONS. PARTICULAR AND GENERAL INTEGRAL. EXAMPLES. CAUCHY PROBLEM. EXISTENCE AND UNIQUENESS THEOREM. FIRST ORDER DIFFERENTIAL EQUATIONS. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. SOLUTIONS METHODS.
(1/2)

TOTAL HOURS: (26/37)
Teaching Methods
THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 26 HOURS AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 37 HOURS.
THE FREQUENCY OF CLASSROOM LECTURES AND EXERCISES, WHILE NOT REQUIRED, IS STRONGLY RECOMMENDED IN ORDER TO OBTAIN FULL ACHIEVEMENT OF THE LEARNING OBJECTIVES..
Verification of learning
WITH REGARD TO THE LEARNING OUTCOMES OF THE TEACHING, 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 DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.
THE EXAM NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES CONSISTS IN A WRITTEN TEST, PRELIMINARY WITH RESPECT TO THE ORAL EXAMINATION, AND IN AN ORAL INTERVIEW.
THE WRITTEN TEST CONSISTS IN SOLVING PROBLEMS IMPLEMENTED ON THE BASIS OF WHAT HAS BEEN PROPOSED IN THE FRAMEWORK OF THE THEORETICAL LECTURES AND EXERCISE SESSIONS. SUCH A WRITTEN TEST, THAT THE STUDENT WILL HAVE TO FACE IN TOTAL AUTONOMY, WILL LAST 2 AND HALF HOURS
THERE WILL BE TWO MID-TERM TESTS CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE, WHICH IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENT ON THESE TOPICS AT THE FINAL WRITTEN TEST.
IN THE CASE OF PRODUCTION OF A SUFFICIENT PROOF, IT WILL BE EVALUATED THROUGH QUALITATIVE SCALES (RANGES OF MARKS).
THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND WILL COVER DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE RANGE OF MARKS OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.
Texts
WRITTEN NOTES GIVEN BY THE TEACHER.
C. D’APICE, T. DURANTE, R. MANZO, VERSO L’ESAME DI MATEMATICA I, MAGGIOLI, 2015.
C. D’APICE, R. MANZO, VERSO L’ESAME DI MATEMATICA II MAGGIOLI, 2015.