Teacher
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PIETRANERA Ileana
(syllabus)
Mathematics and statistical analysis
Sets and their operations, various notations. Numeric sets: from N to R, need for an expansion. Set of rational numbers, set of reals and field structure. Absolute value of a number and triangular inequality. Interval definition: closed, open; definition of around. Definition of function and related classification; even and odd functions: examples, recognition and graphic consequences. Elementary functions. Summation symbol and its use to indicate polynomials of degree n. Function classification. Domain and sign of rational and irrational, transcendent algebraic functions.
Definition of finite limit for x tending towards a finite value: limit verification. Infinite limit for x tending towards a finite value: check; vertical asymptote. Finite and infinite limits for x which tends to infinity: horizontal asymptote; oblique asymptote of fractional rational algebraic functions. Continuous function in one point and in an interval. Classification of discontinuity: the function part integer, the functions whose graph admits vertical asymptotes; eliminable discontinuity: functions defined in branches. Elementary functions as continuous functions. Uniqueness of the limit theorem. Limits of continuous functions Comparison theorem or "of the carabinieri". Theorems on continuous functions: Weierstrass', intermediate values, existence of zeros. Definition of the number of Napier. Notable limits of transcendent functions and their consequences: relative examples. Definition of derivative and its geometric meaning. Application to physics and other applied sciences. Functions that cannot be differentiated in a point, classification of the points of non-derivability: angular points, cusps and inflections with a vertical tangent; examples of all kinds. Derivatives of elementary functions with proof. Derivative of a sum and a product with proof and application examples. Relations between differentiability and continuity. Definition of increasing and decreasing function. Derivative of the reciprocal function of a differentiable function (with proof) and of a quotient: application examples. Compound functions: definitions, examples, derivative of a compound function. Complete study of a function. Definition of relative max and min and Fermat's theorem. Search for relative max and min (with the method of studying the sign of the first derivative) and absolutes. Rolle, Cauchy and Lagrange theorems: examples and geometric meanings. De l'Hopital's theorem. Comparison of infinitesimals and comparison of infinites. Calculation of successive derivatives. Convexity and concavity of a curve, determination of inflections with oblique tangents: determination of inflectional tangents. Development of a Taylor / Mc Laurin series function: application to some notable 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. Null, identical matrix, transposed matrix of a date. Determinant of a matrix and its calculation in the 2x2 and 3x3 cases (Sarrus rule). General rule for calculating determinants of nxn matrices: Laplace's 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 near immediate integrals. Definition of integral defined by means of Riemann sums. Integral mean theorem. Integral function and fundamental theorem of integral calculus, Torricelli Barrow's theorem. Applications to the calculation of areas and applications to physics. Integration of fractional rational algebraic functions with 2nd degree denominator with positive delta, zero. Integrals attributable to the arctangent Indefinite integrals solvable by substitution. Integration rule by parts. Relative exercises: integral of (senx) ^ 2; integrals that are solved by iterating the procedure. Generalized and improper integrals, numerical approximation of the integrals defined with the rectangle method.
Introduction of the functions of two variables: domain and graph in three-dimensional space. Level curves. Continuity of functions of two variables, meaning and calculation of first and second partial derivatives, Hessian matrix; 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
Statistic analysis Descriptive statistics: tables and graphs, histograms and pie charts. Position indices: mean, mode and median percentiles and quartiles, weighted average. Dispersion indices: variance and standard deviation. Evaluation of uncertainties in measurements: absolute, relative and percentage errors. Bimodal statistical tables: joint distributions; dependence and independence. Least squares method, Covariance, Linear correlation coefficient. Combinatorial calculus. Simple and repetitive provisions; simple and repetitive permutations; simple combinations and with repetitions; binomial coefficient in its various forms, power of a binomial with Newton's formula. Classical definition of probability: relative examples. Total and compound probability theorems. The concept of random variable and discrete and continuous distributions. Discrete distribution: binomial distribution, definition and related examples. 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 normal distribution: relative examples
(reference books)
TESTO BASE P.Marcellini, C.Sbordone
Elementi di calcolo – Versione semplificata per I nuovi corsi di laurea
Ed. Liguori
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