B026182 - Laboratory on Mathematics for Economic and Statistical Applications

Italian

Real numbers. Real functions of real variables. Limits of functions. Continuous functions. Differential calculus. Introduction to functions of several variables.

Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati Boringhieri.

On the part "Introduction to functions of several variables", instructors will provide reading material.

The goal of this course is to provide mathematical tools which allow to build and understand simple economic models. At the end of the course the student will have to know the mathematical concepts and the theorems presented. With regard to these concepts and theorems, the student will also need to be able to understand and appropriately use the formalism and the syntax, solve exercises and problems and perform simple deductive reasoning.

Natural numbers, whole numbers and rational numbers. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Percentages. Real numbers (intuitive idea). Absolute value. Powers and roots. Polynomials. Sum and product of polynomials. Square and cube of a binomial. Notable products. Factorization of simple polynomials. Rational expressions. Sum and product of rational expressions. Identity. Equations and solutions of an equation. Inequalities and solutions of an inequality. First and second degree equations and inequalities. Equations and inequalities of a higher degree. Equations and inequalities with rational expressions. Irrational equations and inequalities. Equation systems and inequalities. Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems of two equations in two unknowns. Parallelism and perpendicularity of two lines. Equation of the parabola. Circumference equation. Set theory: inclusion, intersection, union, complements and empty set.

Lectures and tutorials both in classroom and through recordings.

The course has an internet page on the platform Moodle, which provides further information on the course.

The learning assessment takes place through a written test and a possible oral test. The written test and the oral test are aimed to verify:
-the knowledge acquired regarding the mathematical concepts and theorems presented during the course,
-the understanding and use of the formalism and the syntax related to the concepts studied,
-the ability to apply the acquired knowledge to solve exercises and problems,
-the ability to perform simple deductive reasoning.

IMPORTANT INFORMATION. Due to the emergency related to the virus SARS-CoV-2, the learning assessment might be modified. In such a case students will be promptly informed.

Further details on the learning assessment are available on the moodle page of the course.

Students can avoid reading proofs of theorems with *.

Real numbers. Operations and orders. Geometric representation of real numbers. Theorem* on the irrationality of the square root of 2. Intervals, upper bound and lower bound of a set, maximum and minimum of a set, sets which are bounded above or bounded below, upper and lower bound, completeness property. Neighborhood of a point, interior points, accumulation points, open sets, closed sets.

Real functions. The concept of function. Real functions of real variable. Domain and graph of a function. Inverse image and image. Injective functions and inverse functions. Sum, product, quotient and composition of functions. Restrictions of functions. Monotonic functions, strict monotonicity and invertibility. Bounded above and below functions, upper bound and lower bound of a function on a set, maximum points and minimum points of a function on a set, maximum value and minimum value of a function on a set. Elementary functions: linear functions, quadratic functions, exponential function, logarithm function, power functions, absolute value function. Piecewise defined functions.

Limit of a function. Limit of a function at a point. Theorem of uniqueness of the limit. Theorem of the permanence of the sign. Right limit and left limit. Theorem* on the limit of the sum of functions. Theorem* on the limit of the product of functions. Theorem* on the limit of the quotient of functions. Theorem* on the limit of the composition of functions (change of variables). Infinite limits and limits to infinity. Horizontal and vertical asymptotes. Limits of elementary functions. Indefinite forms and special limits.

Continuous functions. Definition of continuity of a function. Continuity of elementary functions. Theorem* on the continuity of the sum of functions. Theorem* on the continuity of the product of functions. Theorem* on the continuity of the quotient of functions. Theorem* on the continuity of the composition of functions. Theorem* of the zeros. Intermediate value theorem for continuous functions. Theorem* of Weierstrass.

Differential calculus. Definition of differentiability of a function. Derivative of a function. Tangent line to the graph of a function. Theorem on the relationship between differentiability and continuity. Theorem* on the derivative of the sum of functions. Theorem* on the derivative of the product of functions. Theorem* on the derivative of the quotient of functions. Theorem* on the derivative of the composition of functions. Local maximum points and local minimum points of a function. Stationary points. Relationship between local maximum/minimum points and stationary points (Fermat's theorem). Rolle's theorem. Lagrange's theorem. Theorem on the relationship between the sign of the first derivative and the monotonicity of a function. Theorems* of de l'Hôpital.

Derivatives of higher order. Second order derivative. Concave and convex functions. Theorem* on the relationship between convexity and concavity of a function and the sign of the second derivative. Theorem* on the sign of the second derivative as a sufficient condition for local maxima and minima. Study of the graph of a function.

Functions of two variables. Introduction to the functions of two variables. Domain of a function. Graphical representation, level curves. Neighborhood of a point, interior points, accumulation points. Limit of a function at a point. Definition of continuity of a function. Continuity of elementary functions. Partial derivatives. Partial derivatives and monotonicty. Tangent plane to the graph of a function. Theorem* of the total differential.