Teacher
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PIERRI ANNA
(syllabus)
1. First approaches to real numbers and functions: Axioms of real numbers, and related properties. Set theory: generalities and representation; intersection, union, set of parts, difference; Cartesian product. Natural, integer, rational, real numbers. Upper and lower extremes, maximum and minimum of a number set. Natural, integer and rational numbers. Functions: domain, invertibility, increasing and decreasing. Elementary functions. 2. Equations and inequalities: Equations of first and second degree. Disequations of first and second degree. Disequations of degree greater than two, fraternal inequalities and systems of inequalities. Irrational, exponential and logarithmic inequalities. 3. Complex numbers: Complex numbers in Cartesian form, trigonometric form, exponential form and polar form. Rationalization of complex numbers in Cartesian form. Properties of product and ratio between complex numbers in trigonometric, exponential and polar form. Powers and roots of complex numbers. Solving equations in the complex field. 4. Matrices: Operations between matrices: addition, subtraction, multiplication between matrices and properties of noncommutativity; multiplication between a scalar and a matrix. Determinants of square matrices: generalities; calculation rules; properties of determinants. Rank of matrices: generalities; computation in the case of square and nonsquare matrices; relationship between rank of matrices and linear independence/dependence of row/column vectors. 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 to its solutions. Solving rules: inverse matrix method; Cramer's method. Homogeneous linear systems. Formalization of solutions of a linear system by procedures involving elementary operations between rows and columns of complete and incomplete matrices. 6. Domains and limits of real functions of a real variable: Real functions of real variable: techniques for calculating the domain. Intuitive approach to limit definition: 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 theorem of the limit. Comparison theorem. Indeterminate forms. Infinities and infinitesimals. Asymptotes of a function. 7. Calculation of limits and continuous functions: Calculation procedures for limits occurring in indeterminate form. Notable limits and their applications. Continuous functions at a point and in an interval. Points of discontinuity of a function. Weierstrass theorem, permanence of sign theorem, zero theorem, intermediate value theorem. 8. Derivatives of real functions of a real variable: Incremental ratio limit 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 real variable. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Concave, convex functions and points of inflection. De L'Hospital's theorem. 9. Function study: Procedures for studying a function and plotting its qualitative graph. Insights into rational fraternal functions.
Methods of in-progress profit verifications The degree of student learning is constantly monitored through verification tools and methodologies. In particular, in order to make the verification and certification of training outcomes feasible, the lecturer and tutor will take into account the following aspects: 1. automatic tracking of training activities by the system - reporting; 2. didactic and technical monitoring (at the level of quantity and quality of interactions, adherence to didactic deadlines, delivery of scheduled papers, etc.); 3. formative verifications in itinere, including for self-assessment (e.g. multiple choice tests, true/false, sequence of questions with different difficulty, simulations, concept maps, papers, group projects, etc.); 4. final proficiency exam, in which not only the performance in the exam is taken into account and valued, but also the work done online (activities carried out at a distance, quantity and quality of online interactions, etc.). Evaluation, within this framework, takes into account several aspects: a. the outcome of a number of intermediate tests, if any (in terms of online tests, development of papers, 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 test.
(reference books)
- Paolo Marcellini, Carlo Sbordone, Elements of Mathematical Analysis 1, Simplified version for new undergraduate courses, Liguori Editore, 2016. - Ciro D'Apice, Rosanna Manzo, Toward the Mathematical Analysis 1 exam, Maggioli Editore, 2015. Supplementary teaching materials will be available in the teaching section within the university platform
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