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Teaching Language

Course Content

Suggested readings

Learning Objectives

Prerequisites

Teaching Methods

Further information

Type of Assessment

Course program

Course year

First year - First Semester

Belonging Department

Economics and Management (DISEI)

Course Type

Single education field course

Scientific Area

SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES

Credits

9

Teaching Hours

72

Teaching Term

13/09/2017 ⇒ 07/12/2017

Attendance required

No

Type of Evaluation

Final Grade

Course Content

show

Course program

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Lectureship

- Last names A-C FIORINO LUCIA
- Last names A-C VILLANACCI ANTONIO
- Last names A-Z COLUCCI DOMENICO
- Last names A-Z VALORI VINCENZO
- Last names D-L MANCINI CECILIA
- Last names M-P GORI MICHELE
- Last names Q-Z MENICUCCI DOMENICO

italian

Italian

Italian

Italian.

Italian.

Sets. Real numbers. Real functions of one real variable. Limits for functions. Continuous functions. Differential calculus.

Functions of one variable and their properties. Derivatives and their use. Limits and continuity. Single variable optimization. Introduction to functions of several variables. Partial derivatives.

Sets. Real numbers. Real functions of one real variable. Limits for functions. Continuous functions. Differential calculus.

Sets. Real numbers. Real functions of one real variable. Limits for functions. Continuous functions. Differential calculus.

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

K. Sydsaeter, P. Hammond, A. Strom.

Metodi Matematici per l'Analisi Economica e Finanziaria.

Pearson, 2015.

Metodi Matematici per l'Analisi Economica e Finanziaria.

Pearson, 2015.

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

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

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

The goal of this course is to provide mathematical tools which allow to build and understand simple economic models which rely on real functions of one real variable.

KNOWLEDGE: Elements of mathematical analysis. Differential calculus for functions of a single variable. Introduction to multi-variable calculus.

COMPETENCE: The course aims at providing students with tools required for building and studying mathematical models that use real functions of one or more variables, typically found in economic and business applications. Students will learn the use of calculus for the study of functions and of simple optimization problems.

COMPETENCE: The course aims at providing students with tools required for building and studying mathematical models that use real functions of one or more variables, typically found in economic and business applications. Students will learn the use of calculus for the study of functions and of simple optimization problems.

The goal of this course is to provide mathematical tools which allow to build and understand simple economic models which rely on real functions of one real variable.

The goal of this course is to provide mathematical tools which allow to build and understand simple economic models which rely on real functions of one real variable.

The natural numbers. The integers. The rationals. An intuitive idea of real numbers. Arithmetic operations and their properties. Percentages. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Inequalities and manipulation of inequalities. Absolute value. Powers, roots and their properties. Simplifying expressions.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

The natural numbers. The integers. The rationals. An intuitive idea of real numbers. Arithmetic operations and their properties. Percentages. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Inequalities and manipulation of inequalities. Absolute value. Powers, roots and their properties. Simplifying expressions.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals. Summations and Newton's Binomial formula.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

Most of these topics are covered in Ch. 0 (paragraphs 0.1 to 0.14 included), in Ch. 1 (paragraphs 1.4 to 1.7 included) and in Ch. 2 (Par. 2.5) of the textbook.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals. Summations and Newton's Binomial formula.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

Most of these topics are covered in Ch. 0 (paragraphs 0.1 to 0.14 included), in Ch. 1 (paragraphs 1.4 to 1.7 included) and in Ch. 2 (Par. 2.5) of the textbook.

The natural numbers. The integers. The rationals. An intuitive idea of real numbers. Arithmetic operations and their properties. Percentages. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Inequalities and manipulation of inequalities. Absolute value. Powers, roots and their properties. Simplifying expressions.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

The natural numbers. The integers. The rationals. An intuitive idea of real numbers. Arithmetic operations and their properties. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Inequalities and manipulation of inequalities. Absolute value. Powers, roots and their properties. Simplifying expressions.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

The natural numbers. The integers. The rationals. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Percentages. An intuitive idea of real numbers. Absolute value. Powers, roots and their properties.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.

Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.

Class lectures. The course lenght is 12 weeks with three classes per week.

Class lectures, seminars, online guided self assessment, weekly homework assignments.

The course length is 12 weeks with three classes a week.

The course length is 12 weeks with three classes a week.

Class lectures. The course lenght is 12 weeks with three classes per week.

Class lectures. The course lenght is 12 weeks with three classes per week.

Class lectures. The course lenght is 12 weeks with two classes per week and a practice class per week.

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

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

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

Exam via web . Further information on the exam rules are available in the Regolamento on the Moodle page of the course.

There is a written exam including ten multiple choice questions and 2/3 problems.

In order to pass the exam, the student needs to pass a written exam and then an oral exam.

Further information on the exam rules are available in the Regolamento on the Moodle page of the course.

Further information on the exam rules are available in the Regolamento on the Moodle page of the course.

In order to pass the exam, the student needs to pass a written exam and then an oral exam.

Further information on the exam rules are available in the Regolamento on the Moodle page of the course.

In order to pass the exam, the student needs to pass a written exam and then an oral exam.

Further information on the exam rules are available in the Regolamento on the Moodle page of the course.

Further information on the exam rules are available in the Regolamento on the Moodle page of the course.

Set theory.

Real numbers. Algebraic and order properties. Geometric representation of real numbers. The real line. Topology of the real line. Neighborhood of a point. Interior points and accumulation points. Bounded sets. Supremum and infimum of a set.

Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the composite function. Monotone functions. Strict monotonicity and invertibility. Supremum and Infimum of a function. Maximum/minimum of a function and maximizers/minimizers. The exponential function. The logarithmic function. Power functions, absolute value, sine, cosine, tangent and piecewise defined functions.

Limit of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Limits of rational functions, of logarithmic functions, and of the exponential functions. Indeterminate forms. Special limits.

Continuous functions. Definition and examples. Continuity of elementary functions. Operations with functions and continuity. The intermediate value theorem. Local maxima and minima and local minimizer/maximizer.

Weierstrass theorem.

Definition of derivative. Tangent line. Rules of differentiation for sum, product, quotient and composition. Fermat Theorem, Rolle and Lagrange Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima.

Real numbers. Algebraic and order properties. Geometric representation of real numbers. The real line. Topology of the real line. Neighborhood of a point. Interior points and accumulation points. Bounded sets. Supremum and infimum of a set.

Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the composite function. Monotone functions. Strict monotonicity and invertibility. Supremum and Infimum of a function. Maximum/minimum of a function and maximizers/minimizers. The exponential function. The logarithmic function. Power functions, absolute value, sine, cosine, tangent and piecewise defined functions.

Limit of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Limits of rational functions, of logarithmic functions, and of the exponential functions. Indeterminate forms. Special limits.

Continuous functions. Definition and examples. Continuity of elementary functions. Operations with functions and continuity. The intermediate value theorem. Local maxima and minima and local minimizer/maximizer.

Weierstrass theorem.

Definition of derivative. Tangent line. Rules of differentiation for sum, product, quotient and composition. Fermat Theorem, Rolle and Lagrange Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima.

Sets theory.

Real numbers. Algebraic and order properties. Geometric representation of real numbers, irrationality of the square root of 2. Intervals, upepr/lower bound of a set, maximum and minimum of a set, sets bounded from above/below, infimum/supremum of a set, completeness of the set of the real numbers.

Neighborhood of a point. Interior points and accumulation points of a set, open/closed sets.

Real functions of one real variable. Domain and graph of a function. Image and inverse image. Injective and inverse functions. Sum, product, quotient and composition of functions. Restriction of a function. Monotone functions. Strict monotonicity and invertibility. Bounded functions, Supremum and infimum of a function on a set. Maximum/minimum of a function om a set. Basic functions: linear functions, quadratic functions, exponential functions, logarithmic functions, power functions, absolute value, sine, cosine, tangent and piecewise defined functions.

Limits of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient of functions. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Vertical and horizontal asymptote. Limits of the basic functions. Indeterminate forms. Special limits.

Continuous functions. Definition and examples. Continuity of some basic functions. Operations with functions and continuity. The intermediate value theorem. Weierstrass theorem.

Differential calculus.

Definition of first derivative of a function. Tangent line. Relation between continuity and differentiability of a function. Rules of differentiation for sum, product, quotient and composition. Local maxima and minima and local minimizer/maximizer. Stationary points. Relation among stationary and local maximizers/minimizers.

Rolle and Lagrange Theorems. Sign of the first derivative and monotonicity. De l'Hospital Theorems.

Higher order derivatives. Second order derivative. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima. Analysis of a function. Taylor theorem for the Taylor formula up to the first order.

Real numbers. Algebraic and order properties. Geometric representation of real numbers, irrationality of the square root of 2. Intervals, upepr/lower bound of a set, maximum and minimum of a set, sets bounded from above/below, infimum/supremum of a set, completeness of the set of the real numbers.

Neighborhood of a point. Interior points and accumulation points of a set, open/closed sets.

Real functions of one real variable. Domain and graph of a function. Image and inverse image. Injective and inverse functions. Sum, product, quotient and composition of functions. Restriction of a function. Monotone functions. Strict monotonicity and invertibility. Bounded functions, Supremum and infimum of a function on a set. Maximum/minimum of a function om a set. Basic functions: linear functions, quadratic functions, exponential functions, logarithmic functions, power functions, absolute value, sine, cosine, tangent and piecewise defined functions.

Limits of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient of functions. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Vertical and horizontal asymptote. Limits of the basic functions. Indeterminate forms. Special limits.

Continuous functions. Definition and examples. Continuity of some basic functions. Operations with functions and continuity. The intermediate value theorem. Weierstrass theorem.

Differential calculus.

Definition of first derivative of a function. Tangent line. Relation between continuity and differentiability of a function. Rules of differentiation for sum, product, quotient and composition. Local maxima and minima and local minimizer/maximizer. Stationary points. Relation among stationary and local maximizers/minimizers.

Rolle and Lagrange Theorems. Sign of the first derivative and monotonicity. De l'Hospital Theorems.

Higher order derivatives. Second order derivative. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima. Analysis of a function. Taylor theorem for the Taylor formula up to the first order.

Set theory.

Real numbers. Algebraic and order properties. Geometric representation of real numbers. The real line. Topology of the real line. Neighborhood of a point. Interior points and accumulation points. Bounded sets. Supremum and infimum of a set.

Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the composite function. Monotone functions. Strict monotonicity and invertibility. Supremum and Infimum of a function. Maximum/minimum of a function and maximizers/minimizers. The exponential function. The logarithmic function. Power functions, absolute value, sine, cosine, tangent and piecewise defined functions.

Limit of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Limits of rational functions, of logarithmic functions, and of the exponential functions. Indeterminate forms. Special limits.

Continuous functions. Definition and examples. Continuity of elementary functions. Operations with functions and continuity. The intermediate value theorem. Local maxima and minima and local minimizer/maximizer.

Weierstrass theorem.

Definition of derivative. Tangent line. Rules of differentiation for sum, product, quotient and composition. Fermat Theorem, Rolle and Lagrange Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima.

Real numbers. Algebraic and order properties. Geometric representation of real numbers. The real line. Topology of the real line. Neighborhood of a point. Interior points and accumulation points. Bounded sets. Supremum and infimum of a set.

Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the composite function. Monotone functions. Strict monotonicity and invertibility. Supremum and Infimum of a function. Maximum/minimum of a function and maximizers/minimizers. The exponential function. The logarithmic function. Power functions, absolute value, sine, cosine, tangent and piecewise defined functions.

Limit of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Limits of rational functions, of logarithmic functions, and of the exponential functions. Indeterminate forms. Special limits.

Continuous functions. Definition and examples. Continuity of elementary functions. Operations with functions and continuity. The intermediate value theorem. Local maxima and minima and local minimizer/maximizer.

Weierstrass theorem.

Definition of derivative. Tangent line. Rules of differentiation for sum, product, quotient and composition. Fermat Theorem, Rolle and Lagrange Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima.

Set theory.

Real numbers. Algebraic and order properties. Geometric representation of real numbers. The real line. Topology of the real line. Neighborhood of a point. Interior points and accumulation points. Bounded sets. Supremum and infimum of a set.

Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the composite function. Monotone functions. Strict monotonicity and invertibility. Supremum and Infimum of a function. Maximum/minimum of a function and maximizers/minimizers. The exponential function. The logarithmic function. Power functions, absolute value, sine, cosine, tangent and piecewise defined functions.

Limit of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Limits of rational functions, of logarithmic functions, and of the exponential functions. Indeterminate forms. Special limits.

Continuous functions. Definition and examples. Continuity of elementary functions. Operations with functions and continuity. The intermediate value theorem. Local maxima and minima and local minimizer/maximizer.

Weierstrass theorem.

Definition of derivative. Tangent line. Rules of differentiation for sum, product, quotient and composition. Fermat Theorem, Rolle and Lagrange Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima.

Real numbers. Algebraic and order properties. Geometric representation of real numbers. The real line. Topology of the real line. Neighborhood of a point. Interior points and accumulation points. Bounded sets. Supremum and infimum of a set.

Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the composite function. Monotone functions. Strict monotonicity and invertibility. Supremum and Infimum of a function. Maximum/minimum of a function and maximizers/minimizers. The exponential function. The logarithmic function. Power functions, absolute value, sine, cosine, tangent and piecewise defined functions.

Limit of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Limits of rational functions, of logarithmic functions, and of the exponential functions. Indeterminate forms. Special limits.

Continuous functions. Definition and examples. Continuity of elementary functions. Operations with functions and continuity. The intermediate value theorem. Local maxima and minima and local minimizer/maximizer.

Weierstrass theorem.

Definition of derivative. Tangent line. Rules of differentiation for sum, product, quotient and composition. Fermat Theorem, Rolle and Lagrange Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima.