Luigi RARITA' | MATHEMATICS FOR ECONOMICS

cod. 0212700117

MATHEMATICS FOR ECONOMICS

	0212700117
	DIPARTIMENTO DI SCIENZE AZIENDALI - MANAGEMENT & INNOVATION SYSTEMS
	EQF6
	BUSINESS MANAGEMENT
	2020/2021

CURRICULUM ACCOUNTING, FINANCE AND AUDITING

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2014
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	SECS-S/06	10	60	LESSONS	BASIC COMPULSORY SUBJECTS

PROFESSORS

	CIRO D'APICE T Curriculum
	GERARDO DURAZZO Curriculum

	Objectives
	STUDENTS WILL HAVE AT THEIR DISPOSAL MATHEMATICAL TOOLS THAT ARE FUNDAMENTAL FOR A SUITABLE QUANTITATIVE APPROACH TO THE ISSUES OF A BUSINESS AND FINANCIAL NATURE THAT WILL BE ADDRESSED DURING THE DEGREE COURSE. STUDENTS WILL BE ABLE TO APPLY THE QUANTITATIVE LEARNED TOOLS TO SOLVE SOME CLASSIC PROBLEMS IN ECONOMICS AND FINANCIAL CHOICES. KNOWLEDGE AND UNDERSTANDING THE TEACHING AIMS AT ACQUIRING THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS AND LINEAR ALGEBRA: NUMERICAL SETS, REAL FUNCTIONS, BASIC NOTIONS OF EQUATIONS AND INEQUALITIES, LIMITS, CONTINUOUS FUNCTIONS, DERIVATIVES, STUDY OF A FUNCTION,. THE SPECIFIC LEARNING OUTCOMES ACTUALLY CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS IN THE ABILITY TO SOLVE EXERCISES AND TO DEAL, IN A CONSTRUCTIVE MANNER, WITH TEXTBOOKS FOR A SUFFICIENTLY INDEPENDENT APPROACH TO THE PROBLEM SOLVING. THE STUDENTS WILL MASTER FUNDAMENTAL MATHEMATICAL INSTRUMENTS THAT WILL ALLOW THEM TO HAVE A QUANTITATIVE APPROACH TO THE MANAGERIAL AND FINANCIAL TOPICS THAT WILL BE ENCOUNTERED DURING THE DEGREE COURSE. APPLYING KNOWLEDGE AND UNDERSTANDING STUDENTS WILL BE ABLE TO APPLY THE QUANTITATIVE INSTRUMENTS TO THE RESOLUTION OF CLASSICAL ECONOMIC PROBLEMS AND OF FINANCIAL DECISIONS. IN PARTICULAR THEY WILL BE ABLE TO APPLY THEOREMS AND RULES IN PROBLEMS SOLVING, TO COMPUTE LIMITS, DERIVATIVES, INTEGRALS, TO PERFORMING THE GRAPHICAL STUDY OF A FUNCTION, TO COMPUTE CALCULUS WITH MATRICES AND TO SOLVE LINEAR SYSTEMS, TO COMPUTE DERIVATIVES AND MINIMA AND MAXIMA FOR MULTIVARIATE FUNCTIONS. MAKING JUDGEMENTS BEING ABLE TO IDENTIFY THE BEST METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM. COMMUNICATION SKILLS BEING ABLE TO EXPLAIN VERBALLY A TOPIC OF THE COURSE. LEARNING SKILLS BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.

STUDENTS WILL HAVE AT THEIR DISPOSAL MATHEMATICAL TOOLS THAT ARE FUNDAMENTAL FOR A SUITABLE QUANTITATIVE APPROACH TO THE ISSUES OF A BUSINESS AND FINANCIAL NATURE THAT WILL BE ADDRESSED DURING THE DEGREE COURSE. STUDENTS WILL BE ABLE TO APPLY THE QUANTITATIVE LEARNED TOOLS TO SOLVE SOME CLASSIC PROBLEMS IN ECONOMICS AND FINANCIAL CHOICES.

KNOWLEDGE AND UNDERSTANDING
THE TEACHING AIMS AT ACQUIRING THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS AND LINEAR ALGEBRA: NUMERICAL SETS, REAL FUNCTIONS, BASIC NOTIONS OF EQUATIONS AND INEQUALITIES, LIMITS, CONTINUOUS FUNCTIONS, DERIVATIVES, STUDY OF A FUNCTION,.
THE SPECIFIC LEARNING OUTCOMES ACTUALLY CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS IN THE ABILITY TO SOLVE EXERCISES AND TO DEAL, IN A CONSTRUCTIVE MANNER, WITH TEXTBOOKS FOR A SUFFICIENTLY INDEPENDENT APPROACH TO THE PROBLEM SOLVING.
THE STUDENTS WILL MASTER FUNDAMENTAL MATHEMATICAL INSTRUMENTS THAT WILL ALLOW THEM TO HAVE A QUANTITATIVE APPROACH TO THE MANAGERIAL AND FINANCIAL TOPICS THAT WILL BE ENCOUNTERED DURING THE DEGREE COURSE.

APPLYING KNOWLEDGE AND UNDERSTANDING
STUDENTS WILL BE ABLE TO APPLY THE QUANTITATIVE INSTRUMENTS TO THE RESOLUTION OF CLASSICAL ECONOMIC PROBLEMS AND OF FINANCIAL DECISIONS.
IN PARTICULAR THEY WILL BE ABLE TO APPLY THEOREMS AND RULES IN PROBLEMS SOLVING, TO COMPUTE LIMITS, DERIVATIVES, INTEGRALS, TO PERFORMING THE GRAPHICAL STUDY OF A FUNCTION, TO COMPUTE CALCULUS WITH MATRICES AND TO SOLVE LINEAR SYSTEMS, TO COMPUTE DERIVATIVES AND MINIMA AND MAXIMA FOR MULTIVARIATE FUNCTIONS.

MAKING JUDGEMENTS
BEING ABLE TO IDENTIFY THE BEST METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM.

COMMUNICATION SKILLS
BEING ABLE TO EXPLAIN VERBALLY A TOPIC OF THE COURSE.

LEARNING SKILLS
BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.

	Prerequisites
	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. (1,2) 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. (3/4) TOTAL HOURS: (28/32)

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.
(1,2)
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.
(3/4)

TOTAL HOURS: (28/32)

	Teaching Methods
	THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 20 HOURS AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 28 HOURS. ATTENDANCE TO THE COURSE IS MANDATORY. IN ORDER TO PARTICIPATE TO THE FINAL ASSESSMENT AND TO GAIN THE CREDITS CORRESPONDING TO THE COURSE, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE HOURS OF ASSISTED TEACHING ACTIVITIES.

	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 EXERCISES; THE ABILITY TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHODS IN EXERCISES 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.

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 EXERCISES; THE ABILITY TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHODS IN EXERCISES 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.

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	TEACHING PROVIDED IN ITALIAN.

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Luigi RARITA' | MATHEMATICS FOR ECONOMICS