Carmine MONETTA | MATHEMATICS I
Carmine MONETTA MATHEMATICS I
cod. 0612200001
MATHEMATICS I
0612200001 | |
DEPARTMENT OF INDUSTRIAL ENGINEERING | |
EQF6 | |
CHEMICAL ENGINEERING | |
2023/2024 |
OBBLIGATORIO | |
YEAR OF COURSE 1 | |
YEAR OF DIDACTIC SYSTEM 2016 | |
AUTUMN SEMESTER |
SSD | CFU | HOURS | ACTIVITY | |
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MAT/05 | 9 | 90 | LESSONS |
Objectives | |
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At the end of the course the student will acquire the basic elements of mathematical analysis and linear algebra, the demonstration techniques and related results, as well as the ability to use appropriate calculation tools. Knowledge and understanding Understanding of the terminology used in mathematical analysis; knowledge of demonstration methodologies; knowledge of the fundamental concepts of mathematical analysis. Notions of: vector algebra; numerical sets; real functions; equations and inequalities; numerical sequences; limits of a function; continuous functions; derivative of a function; fundamental theorems of differential calculus; study of the graph of a function; matrices and linear systems; vector spaces; linear transformations and diagonalization; analytic geometry. Applied knowledge and understanding - engineering analysis Knowing how to apply the theorems and rules studied to solving problems. Knowing how to identify the most appropriate methods to efficiently solve a mathematical problem. Knowing how to express propositions in mathematical language. Knowing how to perform calculations with limits, derivatives, simple calculations with vectors and matrices. Be able to develop the study of the graph of a function. Applied knowledge and understanding - engineering design Knowing how to structure a proof with mathematical rigor. Knowing how to build methods and procedures for solving problems. Making judgements – engineering practice Knowing how to apply the acquired knowledge to contexts different from those presented during the course. Transversal skills - communication skills Knowing how to work in a team. Knowing how to orally expose the topics covered by the course. Transversal skills - ability to learn Knowing how to deepen the topics covered using teaching materials other than those proposed during the course. |
Prerequisites | |
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FOR THE SUCCESSFUL ACHIEVEMENT OF THE GOALS OF THE COURSE, STUDENTS ARE REQUIRED TO HAVE THE FOLLOWING PREREQUISITES: -KNOWLEDGE OF ALGEBRA, WITH PARTICULAR REFERENCE TO: ALGEBRAIC, LOGARITHMIC AND EXPONENTIAL EQUATIONS AND INEQUALITIES. -KNOWLEDGE RELATED TO TRIGONOMETRY, WITH PARTICULAR REFERENCE TO THE BASIC TRIGONOMETRIC FUNCTIONS. |
Contents | |
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NUMERICAL SETS (THEORY HOURS AND EXERCISES HOURS 4/4): INTRODUCTION TO SET THEORY. OPERATIONS ON SUBSETS. 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. REAL FUNCTIONS (THEORY HOURS AND EXERCISES HOURS 6/4): 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. NUMERICAL SEQUENCES (THEORY HOURS AND EXERCISES HOURS 4/2): DEFINITIONS. BOUNDED, CONVERGENT, DIVERGENT AND OSCILLATING SEQUENCES. MONOTONE SEQUENCES. NEPERO’S NUMBER. CAUCHY'S CRITERION FOR CONVERGENCE. LIMITS OF A FUNCTION (THEORY HOURS AND EXERCISES HOURS 6/4): DEFINITION. RIGHT AND LEFT-HAND LIMITS. UNIQUENESS AND COMPARISON THEOREMS. OPERATIONAL IDENTITIES AND INDETERMINATE FORMS. NOTABLE SPECIAL LIMITS. CONTINUOUS FUNCTIONS (THEORY HOURS AND EXERCISES HOURS 6/4): DEFINITION. CONTINUITY AND DISCONTINUITY. WEIERSTRASS, ZEROS, BOLZANO THEOREMS. UNIFORM CONTINUITY. DERIVATIVE OF A FUNCTION (THEORY HOURS AND EXERCISES HOURS 4/4): 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. FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS: ROLLE, CAUCHY, LAGRANGE THEOREMS AND COROLLARIES. DE L'HOSPITAL THEOREM. MAXIMA AND MINIMA. TAYLOR AND MAC-LAURIN FORMULAS. GRAPH OF A FUNCTION (THEORY HOURS AND EXERCISES HOURS 4/4): ASYMPTOTES OF A GRAPH. LOCAL MAXIMA AND MINIMA. CONCAVE AND CONVEX FUNCTIONS AT A POINT, INFLECTION POINTS. DRAWING GRAPH. MATRICES (THEORY HOURS AND EXERCISES HOURS 2/2): DEFINITIONS AND PROPERTIES. DETERMINANT AND ITS COMPUTATION: LAPLACE’S FORMULA. RANK OF A MATRIX. INVERSE OF A MATRIX. LINEAR SYSTEMS OF EQUATIONS (THEORY HOURS AND EXERCISES HOURS 2/2): DEFINITIONS, MATRIX EQUATION, CONSISTENCY AND INCONSISTENCY, NUMBER OF SOLUTIONS. ROUCHÉ-CAPELLI THEOREM. CRAMER THEOREM AND RULE. GAUSSIAN ELIMINATION ALGORITHM. DISCUSSION OF LINEAR SYSTEMS WITH A PARAMETER. VECTOR SPACES (THEORY HOURS AND EXERCISES HOURS 2/2): THE VECTOR SPACE STRUCTURE. LINEAR DEPENDENCE AND INDEPENDENCE OF VECTORS. GENERATORS. BASES. DIMENSION OF A VECTOR SPACE AND RELATED RESULTS. VECTOR SUBSPACES. DIAGONALIZATION (THEORY HOURS AND EXERCISES HOURS 2/2): EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND EQUATION. ALGEBRAIC AND GEOMETRIC MULTEPLICITIES. DEFINITION OF DIAGONALIZATION AND ITS CHARACTERIZATIONS (FOR MATRICES). SUFFICIENT CONDITION FOR THE DIAGONALIZATION. ANALYTICAL GEOMETRY (THEORY HOURS AND EXERCISES HOURS 4/4): PLANE AND SPACE CARTESIAN COORDINATE SYSTEM. EQUATION OF A LINE (CARTESIAN, PARAMETRIC) AND OF A PLANE. PARALLELISM OF LINES AND PLANES. PROPER AND IMPROPER BUNDLE OF STRAIGHT LINES. NUMERICAL SERIES (THEORY HOURS AND EXERCISES HOURS 4/2): INTRODUCTION TO MAIN NUMERICAL SERIES. |
Teaching Methods | |
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ATTENDANCE AT THE LECTURES IS STRONGLY RECOMMENDED. |
Verification of learning | |
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THERE WILL BE WRITTEN EXERCISES AND AN ORAL INTERVIEW. WRITTEN EXERCISES: IT CONSISTS IN SOLVING TYPICAL PROBLEMS. SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED. ORAL INTERVIEW: IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING. FINAL EVALUATION: THE FINAL MARK, EXPRESSED ON A SCALE FROM 18 TO 30 (POSSIBLY WITH LAUDEM), DEPENDS ON THE MARK OF THE WRITTEN EXAM, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW. CRITERION CORRESPONDING TO THE ACHIEVEMENT OF A MINIMUM THRESHOLD TO PASS THE EXAM: THE KNOWLEDGE BASIS IS ACCEPTABLE IN RELATION TO SOME OF THE PROGRAM CONTENT. THE ABILITY TO SOLVE PROBLEMS EXTENDS TO SIMPLE "STANDARD" PROBLEMS, FOLLOWING ROUTINE PROCEDURES. TRANSFERABLE SKILLS ARE RUDIMENTAL. CRITERION CORRESPONDING TO THE ACHIEVEMENT OF THE EXCELLENCE: THE KNOWLEDGE BASE COVERS THE ENTIRE PROGRAM. THE CONCEPTUAL UNDERSTANDING IS FULL AND DEEP. TROUBLESHOOTING PROCEDURES ARE ADEQUATE TO THE NATURE OF THE PROBLEM. PERFORMANCE IN TRANSFERABLE SKILLS IS GENERALLY VERY GOOD. |
Texts | |
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THEORY - P. MARCELLINI - C. SBORDONE, "ANALISI MATEMATICA UNO", LIGUORI EDITORE (1996). - C. DELIZIA, P. LONGOBARDI, M. MAJ, C. NICOTERA, "MATEMATICA DISCRETA", MCGRAW-HILL (2009). EXERCISES - P. MARCELLINI - C. SBORDONE, "ESERCITAZIONI DI MATEMATICA I - VOL. 1 E 2", LIGUORI EDITORE (2016). - C. DELIZIA, P. LONGOBARDI, M. MAJ, C. NICOTERA, "MATEMATICA DISCRETA", MCGRAW-HILL (2009). |
More Information | |
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THE COURSE LANGUAGE IS ITALIAN. CONTACT THE LECTURER AT MTOTA@UNISA.IT |
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