Teacher
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RARITA' LUIGI
(syllabus)
EDUCATIONAL AIMS The course presents the basic elements of Mathematical Analysis. The educational objectives of the course include the acquisition of results and demonstration techniques, as well as the ability to use the related calculation tools. Knowledge and understanding of the following aspects is expected: 1) First approaches to real numbers and functions. 2) Equations and inequalities. 3) Complex numbers. 4) Matrices. 5) Linear systems. 6) Domains and limits of real functions of one real variable. 7) Calculation of limits and continuous functions. 8) Derivatives of real functions of one real variable. 9) Study of function. 10) Indefinite and definite integrals, 11) Numerical series. 12) Functions of two variables. Knowledge and understanding of applicative type are expected on the following aspects: 1) Applications of the theorems and rules studied for solving real problems in the world of transport. 2) Knowing how to identify the best solution strategies for equations in the complex field. 3) Ability to perform calculations with limits, derivatives, integrals and series. 4) Be able to study a real function of a real variable and draw a qualitative graph. 5) Formalization of procedures for solving optimization problems in the case of two-variable functions.
DELIVERY TEACHING 96 ONLINE VIDEO LESSONS (12 TEACHING UNITS - TWO HOURS DURATION FOR EACH CFU).
INTERACTIVE TEACHING N. 2 INTERACTIVE LESSONS FOR CFU. N. 5 THEMATIC DISCUSSIONS ON THE TEACHING FORUM (TOPIC) AND N. 2 POST FOR CFU AS DEFINED BY THE GUIDELINES ON THE EDUCATION OF THE PQA. N. 2 E-TIVITIES EACH 5 CFU. N. 2 TESTS FOR EACH CFU WITH 8 MULTIPLE CHOICE QUESTIONS.
COURSE PROGRAM Module 1. First approaches to numbers and real functions: Axioms of real numbers, and related properties. Set theory: generality and representation; intersection, union, set of parts, difference; Cartesian product. Natural, integer, rational, real numbers. Upper and lower extremes, maximum and minimum of a numerical set. Natural, integer and rational numbers. Functions: domain, invertibility, increasing and decreasing functions. Elementary functions.
Module 2. Equations and inequalities: First and second degree equations. First and second degree inequalities. Inequalities of higher than second degree, fractional inequalities and systems of inequalities. Irrational, exponential and logarithmic inequalities.
Module 3. Complex numbers: Complex numbers in Cartesian form, trigonometric form, exponential form and polar form. Rationalization of complex numbers in Cartesian form. Properties of the product and of the ratio between complex numbers in trigonometric, exponential and polar form. Powers and roots of complex numbers. Solution of equations in the complex domain.
Module 4. Matrices: Operations between matrices: addition, subtraction, multiplication between matrices and non-commutativity properties; multiplication between a scalar and a matrix. Determinants of square matrices: generalities; calculation rules; properties of the determinants. Rank of matrices: generality; calculation in the case of square and non-square matrices; relationship between rank of matrices and linear independence/dependence of row/column vectors.
Module 5. Linear systems: Matrix formulation of a linear system. Characterization of the solutions of a linear system: uniqueness of solution, infinite solutions, incompatibility; geometric meaning of a linear system and relation with its solutions. Solving rules: inverse matrix method; Cramer's method. Homogeneous linear systems. Formalization of the solutions of a linear system via procedures that foresee elementary operations between rows and columns of complete and incomplete matrices.
Module 6. Domains and limits of real functions of one real variable: Real functions of a real variable: techniques for the computation of the domain. Intuitive approach to the definition of limit: geometric meaning of a finite limit when the independent variable tends to a finite/infinite value; geometric meaning of an infinite limit when the independent variable tends to a finite/infinite value. Analytical formalization of finite/infinite limit when the independent variable tends to a finite/infinite value. Limits of compound functions. Uniqueness of the limit theorem. Comparison theorem. Indeterminate forms. Infinite and infinitesimal. Asymptotes of a function.
Module 7. Calculation of limits and continuous functions: Calculation procedures for limits in an indeterminate form. Notable limits and their applications. Continuous functions in a point and in an interval. Points of discontinuity of a function. Weierstrass theorem, sign permanence theorem, zeros theorem, intermediate value theorem.
Module 8. Derivatives of real functions of a real variable: Limit of the incremental ratio and definition of derivative. Geometric meaning of the derivative. Derivability and continuity. Derivatives of elementary functions. Derivatives of compound functions. Relative maxima and minima for real functions of a real variable. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Concave, convex functions, and inflection points. De L’Hospital's theorem.
Module 9. Study of function: Procedures for the study of a function and for the sketch of its qualitative graph. Insights about fractional rational functions.
Module 10. Indefinite and definite integrals: Primitive and indefinite integral. Immediate integrals of elementary and compound functions. Integration rules and methods. Integrals by parts and by substitution. Definite integral and its geometric meaning. Mean theorem. Integral function and fundamental theorem of integral calculus. Fundamental formula of integral calculus.
Module 11. Numerical series: Connection between numerical sequences and numerical series. Convergent, divergent and indeterminate series. Geometric, harmonic and telescopic series. Convergence criteria for series with positive terms: comparison, ratio and root criterions. Leibniz criterion for the convergence of alternating series.
Module 12. Functions of two variables: Domains and topology for functions of two variables. Limits, continuity and derivability. Differentiability. Relative maxima and minima for functions of two variables: generality; first order necessary condition. Second order sufficient condition.
METHOD OF VERIFYING PROFIT IN INTINERE The degree of student learning is constantly monitored through verification tools and methodologies. In particular, in order to make the verification and certification of the training results feasible, the teacher and the tutor will consider the following aspects: 1) automatic tracking of training activities by the system - reporting; 2) didactic and technical monitoring (in terms of quantity and quality of interactions, compliance with didactic deadlines, delivery of the planned papers, etc.); 3) ongoing training checks, also for self-assessment (eg multiple choice tests, true/false, sequence of questions with varying difficulty, simulations, concept maps, papers, group projects, etc.); 4) final exam, during which it is considered and valued not only the performance during the exam, but also the work done online (remote activities, quantity and quality of online interactions, etc.) . In this framework, the evaluation takes into account several aspects: a) the result of a certain number of intermediate tests, if foreseen (in terms of online tests, development of works, etc.); b) the quality and quantity of participation in online activities (frequency and quality of interventions that can be monitored through the platform); c) the results of the final exam. It is specified that the final test is only of "written" type. The oral test is optional and is useful only and exclusively to refine the evaluation of the written test. The topics of the oral exam, if carried out, are agreed between the teacher and the student. All the collected data will be, in general, evaluated by the teacher to give the student an evaluation that is objective and consistent with the university's aims, taking into account both the summative aspects and the formative aspects.
ASSESSMENT METHOD AND AIMS OF THE FINAL EXAM The access to the exam is subordined to the recognition of attendance, which will be recognized, at the booking time for the exam, by an appropriate certificate, that will certify the development of the didactic verification activities in progress and at the level of the work made in the various exercises. The exam will consist of a written test, with an optional oral one. The final evaluation will be expressed out of 30, with possible honors depending on the degree of maturity reached by the student. The exam aims at the didactic aims described as follows. In particular, the student will have to: a) be able to determine the domain of a real function of a real variable; b) know how to find the solutions of an equation in the complex domain; c) be able to recognize the best resolution strategy for limits of real functions of a real variable; d) know how to study the qualitative graph of a real function of a real variable, identifying any asymptotes, any relative and absolute maximum/minimum, and any inflection points; e) be able to recognize a possible methodology to solve an indefinite or definite integral; f) know how to study the features of a numerical series; g) be able to identify possible relative maxima/minima for functions of two variables. The written test foresees a score for all the previous aims, from point a) to point g). Each goal has a variable score. Honors will be awarded in those particular cases in which particular maturity is shown in solving the exam questions.
KNOWLEDGE AND UNDERSTANDING IN TERMS OF EXPECTED OUTCOMES (DUBLIN DESCRIPTOR NO. 1) Knowledge of the theory of real functions of a real variable will allow the student to acquire awareness of the possible analytical modeling of phenomena that describe the real world of transport. Knowledge of integral theory will allow to properly estimate the volume of traffic flows in transport networks. The knowledge of numerical series theory will allow to define convergence techniques for ad hoc numerical schemes for the simulation of transport networks. Knowledge of the theory of functions of two variables will allow the student to define suitable optimization techniques for traffic performance in transport networks.
SKILLS TO APPLY KNOWLEDGE AND UNDERSTANDING IN TERMS OF EXPECTED OUTCOMES (DUBLIN DESCRIPTOR NO. 2) The student will be able to use the knowledge related to the theory of functions, integration and numerical series to solve analytical/numerical problems commonly encountered in normal transport networks. The student will be able to use knowledge on functions of several variables to formulate traffic optimization problems in transport networks.
(reference books)
• Paolo Marcellini, Carlo Sbordone, Elementi di Analisi Matematica 1, Versione semplificata per i nuovi corsi di laurea, Liguori Editore, 2016. • Ciro D’Apice, Rosanna Manzo, Verso l’esame di Matematica 1, Maggioli Editore, 2015. • Ciro D’Apice, Tiziana Durante, Rosanna Manzo, Verso l’esame di Matematica 2, Maggioli Editore, 2015. Additional teaching material will be available in the section dedicated to teaching within the university platform.
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