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Properties of numbers. Approximate calculations, error propagation, rounding, estimates and orders of magnitude. Equations and inequalities. Systems of linear equations. Systems of linear inequalities with an unknown. Absolute value. Systems of linear equations: generalities about determinants. Cramer's Rule. Graphic resolution and approximation of the zeros of a polynomial function. Cartesian coordinates in the plane. Equations of lines; conditions of parallelism and perpendicularity between two lines; distance between two points; angle between two lines. Parabolas with their axis parallel to the y axis. Monometric and non-monometric systems. Direct and inverse proportionality. Circumference; particular equations of ellipse, hyperbola and parabola. Powers and logarithms in the real field. The number "e"; natural logarithm. Logarithmic and semi-logarithmic scales. Arithmetic and geometrical sequences. Trigonometry: some identities and trigonometric equations. Real functions of a real variable (polynomial functions, power functions, exponential and logarithmic functions, trigonometric functions, and radicals, fract functions, functions of functions, etc.); values and image of a function. Operations on functions. Particular cases of the development of Fourier. Functions with absolute value. Finite and infinite limit of a function at one point or at infinity. Continuity of a function. Increasing and decreasing functions; maximum and minimum of a function; asymptotes; inverse function; functional composition. Derivatives of the functions of a variable: definition of incremental ratio and of derivative and their geometric meaning. Derivatives of elementary functions. Differential; Taylor formula of 1st degree (linearization of a function). Derivation rules: derivative of the sum, of the product and the quotient of two functions, of the function composed of functions. Derivatives of higher order. Study of a function with qualitative methods and derivatives. Hint to the partial derivatives for the functions of two variables and to the total differential. Rule of De l'Hospital. Problem of areas: approximation and definite Integral: definition and properties. Fundamental theorem of integral calculus; primitives of a function and indefinite integrals. Integration by decomposition and by substitution. Application to the calculation of areas. Mean value theorem and its geometric meaning. Line and surface integrals (background). Linear differential equations of the first order. Equations with separable variables. Examples of homogeneous second order differential equations. Initial conditions. Cauchy problem. Statistics: histograms and other forms of representation; averages, dispersion and quadratic deviation. Normal distribution. Regression line in two character distributions. Probability calculation elements: definitions, properties, combinatorial calculation elements, conditional probability and Bayes theorem. Exercises: Mathematization problems. Reading and interpretation of graphs and tables. Numerical exercises on the subjects carried out.

Basic mathematics, learnt in any 5-year upper secondary school course.

Villani, Gentili, Matematica, McGraw-Hill. Abate, Matematica e statistica. Le basi per le scienze della vita. McGraw-Hill.

The teaching methodology adopted for the Mathematics course will be characterized by dialogued frontal lessons, that is interactive lessons, during which students can clarify doubts, apply their knowledge and evaluate the skills achieved. The student will find on the e-learning platform the slides and the teaching materials (exam program, suggested texts) useful for the preparation of the exam. The slides are a guide and sometimes an integration, but cannot replace the suggested texts and the lectures given by the teacher.

Attendance is strongly recommended.

The assessment of the Mathematics course is carried out through an exam consisting of a written test, followed by an oral exam. The written test will last about 120-150 minutes. The aim of the exam is to certify the skills in the application of the acquired knowledge and in the use of mathematical tools to solve problems. A basic knownledge of the treated topics and an analogous ability to solve mathematical problems is required to pass the exam with minimum grade 18/30. To achieve a score of 30/30 cum laude, the student must prove to have acquired an excellent knowledge of all the topics covered during the course and the ability to choose suitable mathematical tools to effectively solve the assigned problems, by presenting clear, correct and complete arguments and using appropriate mathematical language.

Lectures and exercises in the classroom.

Numerical calculation calls and data representation Units of measurement and conversion factors. Representation and properties of numbers. Approximate calculations and error propagation. Significant figures and rounding. Estimates and orders of magnitude. Percentages. Representation of data. Sequences. Arithmetic and geometric sequences. Recursively defined sequences. The number e (Euler’s constant). Linear algebra. Matrices: sum, product, determinants, and inverse. Systems of linear equations: elimination method and Gaussian algorithm. Rank of a matrix and solvability conditions of linear systems. Analytic geometry. Cartesian coordinates in the plane. Monometric and dimetric systems. Lines and segments. Conic sections: parabolas, circles, ellipses, and hyperbolas. Functions of real variables. Definition and classification of real functions. Injective, surjective, and bijective functions. Operations on functions: arithmetic operations, composition, inversion, and piecewise definition. Polynomial, rational, power, exponential, logarithmic, trigonometric, and absolute value functions. Limits of functions, continuity, and qualitative study of functional behavior. Differential calculus. Definition and computation of derivatives of elementary functions. Geometric meaning of the derivative. Differentiation rules for sums, products, quotients, and compositions. Higher-order derivatives. Analysis of monotonicity, concavity, extrema, and inflection points. Taylor series expansion. Integral calculus. Indefinite integrals and antiderivatives; properties of the indefinite integral. Computation of antiderivatives of simple functions. Definite integrals and the Fundamental Theorem of Calculus. Improper integrals. Differential equations. Initial conditions and Cauchy’s theorem. Separable differential equations. First-order linear differential equations. Bernoulli’s equation. Mathematical models for selected natural phenomena. Probability and combinatorics. Definition and properties of probability. Elements of combinatorial analysis. Conditional probability and Bayes’ theorem. Computational tools and algorithmic thinking (Developed in parallel with the previous content). Basic concepts for the construction of simple algorithms. Use of spreadsheets and software tools for programming, visualization, and symbolic manipulation.

Basic mathematics common to all five-year upper secondary school.

V. Villani, G. Gentili, Matematica, Mc Graw Hill Tasks from previous examinations

In presence, suggested

The assessment of the Mathematics course is carried out through an exam consisting of a written test, followed by an oral exam. Five exam sessions will be set during the academic year. The written test, lasting 1,5 hour, consists of 4 closed questions and 1 open question. The oral, to be made in the same session of the written one, consists of some questions related to the written test. The objective of the exam is to certify the skills in the application of the acquired knowledge and in the use of mathematical tools to solve problems. The open-ended questions and those proposed during the oral examination have, moreover, the objective of certifying the ability to build suitable arguments to support the adopted solution approach and the use of a correct specific language. A further element that can be considered for assessment purposes will be the result of a formative written test, proposed in the middle of the course.

Lectures in which fundamental concepts, definitions, theorems and methods of calculation are presented, with particular attention to the logical and formal aspects of the discipline. The exercises are dedicated to solving problems and application examples, in order to consolidate the knowledge acquired and develop the ability to apply it to concrete contexts. Activities on the use of computer tools (such as symbolic and graphic calculation software, spreadsheets or programming environments) to facilitate an experimental and interactive approach to the study of content.