Teacher
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PIETRANERA Ileana
(syllabus)
Sets and their operations, various notations. Numerical sets: from N to R, need for enlargement. Definition of function and its classification; even and odd functions: examples, recognition and graphic consequences. Summation symbol and its use to indicate polynomials of degree n. Set of rational numbers, set of reals. The absolute value of a number and triangular inequality. Definition of interval: closed, open; definition of around.
Classification of functions. Domain and sign of rational and irrational, transcendent algebraic functions. Definition of finite limit for x tending to a finite value: limit verification. Infinite limit for x tending to a finite value: verification; vertical asymptote. Finite and infinite limits for x tending to infinity: horizontal asymptote; oblique asymptote of fractal rational algebraic functions.
A function that continues at a point and in a range. Classification of discontinuities: the function part integer, the functions whose graph admit vertical asymptotes; eliminable discontinuity: functions defined by branches. Elementary functions as continuous functions. Limit uniqueness theorem. Continuous function limits Comparison theorem or "of the carabinieri". Theorems on continuous functions: Weierstrass, intermediate values, existence of zeros. Definition of the Nepero number. Notable limitations of transcendent functions and their consequences: relative examples.
Definition of derivative and its geometric meaning. Non-derivable functions at a point, classification of points of non-derivability: angular points, cusps and vertical tangent flexes; examples of all kinds. Derivatives of elementary functions with proof. Derivative of a sum and a product with demonstration and application examples. Relationships between derivability and continuity. Definition of ascending and decreasing function. Derivative of the reciprocal function of a derivable function (with proof) and of a quotient: examples of application. Compound functions: definitions, examples, derivative of a compound function. Complete study of a function. Definition of max and relative min and Fermat's theorem. Search for relative max and min (with the method of studying the sign of the first derivative) and absolute max and min. De l'Hospital's theorem( applications only ). Comparison of infinitesimals and comparison of infinites. Calculation of subsequent derivatives. Convexity and concavity of a curve, determination of oblique tangent flexes: determination of inflectional tangents. Development of a Taylor/Mc Laurin series function: application to some remarkable functions. Invertible functions: definition and examples. Derivative of the inverse function of a date: case of arcsinx, arcosx and arctanx.
Introduction to matrices, matrix writing, sum of matrices and product rows by columns. A null, identical, transposed matrix of a date. Determinant of a matrix and its calculation in cases 2x2 and 3x3 (Sarrus rule). Rank of a matrix: definition and determination. Definition and determination of the inverse of a given matrix. Linear systems of n equations in m unknowns: Rouchè–Capelli theorem and Cramer's theorem (case n x n). Indefinite integrals: definition of primitive functions, immediate and almost immediate integrals. Definition of integral defined by Riemann sums. Integral mean theorem. Integral function and fundamental theorem of integral calculus, Torricelli Barrow's theorem. Applications to the calculation of areas. Integration of fracted rational algebraic functions with 2nd degree denominator with positive, zero delta. Integrals attributable to the archtangent Indefinite integrals solvable by substitution. Integration rule by parts. Relative exercises: integral of (senx)^2; integrals that are resolved by iterating the procedure. Generalized and improper integrals, numerical approximation of integrals defined by the method of rectangles. Introduction of the functions of two variables: domain and graph in three-dimensional space. Contour lines. Continuity of functions of two variables, meaning and calculation of first and second partial derivatives, Hessian matrix and determinant; search for free relative maximums and minimums. Differential equations: order of an equation, general integral, particular integral: Cauchy problem. Differential equations with separable variables: relative examples Linear first-order differential equations: relative examples Population dynamics. The concept of mathematical model. From Malthus' model to Verhulst's model: the logistic function. Study of both models as solutions of differential equations (with proof and relative problems). Statistical analysis Descriptive statistics: tables and graphs, histograms and pie charts. Position indices: average, fashion and median percentiles and quartiles, weighted average. Dispersion indices: variance and standard deviation. Assessment of uncertainties in measurements: absolute, relative and percentage errors. Least squares method, Covariance, Linear correlation coefficient. Combinatorial calculus. Simple and repetitive arrangements; simple and repeating permutations; simple combinations and with repetitions; binomial coefficient in its different forms, power of a binomial with Newton's formula. Classical definition of probability: relative examples. Frequentist and subjective definition of probability. Total probability theorem. The concept of random variable and discrete and continuous distributions. Expected value and variance of a random variable distribution Discrete distributions: binomial distribution, definition and relative examples; Continuous distributions: normal distribution: density function, Gauss curve. Introduction of the distribution function and use of tables for the calculation of the probability of random events with standardized normal distribution: adequate examples of each topic covered have been carried out.
(reference books)
TESTO BASE P.Marcellini, C.Sbordone
Elementi di calcolo – Versione semplificata per I nuovi corsi di laurea
Ed. Liguori
Per gli esercizi viene lasciato libero lo studente di utilizzare qualunque eserciziario di analisi e calcolo delle probabilità e statistica sia in forma cartacea che on line.
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