Pierluigi Zezza, Metodi matematici per le scienze economiche e aziendali, 2009, Carocci Editore, Roma, pg. 260, Euro 23.50. ISBN 9788843048182
Texts for exercises:
Eserciziario done by the teachers of the course (available on elearning)
Giuseppe Zwirner, Istituzioni di matematiche per gli studenti delle facoltà di chimica, agraria, scienze naturali, economia-commercio e statistica, Parte prima, ed. Cedam-Padova, 1975
Suggested readings for the prerequisites:
Giuseppe Anichini, Antonio Carbone, Paolo Chiarelli, Giuseppe Conti, Precorso di matematica, 2010, Pearson Education, pag. 272, Euro 17,00 - ISBN 9788871925899
Roberto D'Ercole - Precorso di Matematica per Economia e Scienze, 2011, Pearson Education, pag.264 Euro 17,00 - ISBN 9788871926308
Paolo Boieri, Giuseppe Chiti, Precorso di matematica, 1994, Zanichelli, Euro 22,30 - ISBN 9788808091864
Pierluigi Zezza, Metodi matematici per le scienze economiche e aziendali, 2009, Carocci Editore, Roma, pg. 260, Euro 23.50. ISBN 9788843048182
Texts for exercises:
Eserciziario done by the teachers of the course (available on elearning)
Giuseppe Zwirner, Istituzioni di matematiche per gli studenti delle facoltà di chimica, agraria, scienze naturali, economia-commercio e statistica, Parte prima, ed. Cedam-Padova, 1975
Suggested readings for the prerequisites:
Giuseppe Anichini, Antonio Carbone, Paolo Chiarelli, Giuseppe Conti, Precorso di matematica, 2010, Pearson Education, pag. 272, Euro 17,00 - ISBN 9788871925899
Roberto D'Ercole - Precorso di Matematica per Economia e Scienze, 2011, Pearson Education, pag.264 Euro 17,00 - ISBN 9788871926308
Paolo Boieri, Giuseppe Chiti, Precorso di matematica, 1994, Zanichelli, Euro 22,30 - ISBN 9788808091864
Learning Objectives - Last names D-L
1. mathematical tools for the analysis of real functions of one real variable;
2. applications to simple economic problems.
Learning Objectives - Last names M-P
The students will learn the basic elements of the scientific method and of calculus: limits, continuity, and derivatives.
The aim of this course is to provide the necessary tools for developing and studying mathmatical methods using functions of a single variable.
The students will be able to develop a complicated logical-mathematical argument through elementary steps.
Students will be able to appreciate the importance of assumptions in theorems.
Students will be able to define and apply the basic ideas of calculus as limits and derivatives.
Students will be able to solve equations approximately, using appropriate algorithms.
Students will be ablve to descrive how the problems of calcolusm arise from economic, or social, or political, or environments phenomema.
Students will be able to model these problems in mathematical terms.
Students will be able to solve these problems and obtain results which will improve their understanding of reality.
Prerequisites - Last names D-L
1. Sets of numbers; Arithmetic.
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.
2. Basic algebra, equations, inequalities.
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. Linear systems with two equations
and two unknowns. Equations containing ratios of polynomials. Equations
with radicals. Inequalities. Solving inequalities of degree one and degree two,
inequalities, with ratios of polynomials, with radicals.
3. Sets and logic.
Elementary set terminology: being an element of a set, inclusion, intersection, union, complementary set, empty set. Logical implications: suffcient condition, necessary condition.
4. Analytic geometry.
Cartesian coordinates in the plane. Pythagorean theorem, Distance between
two points. Equation of the line. Parallel lines and perpendicular lines. Equation of the parabola.
Prerequisites - Last names Q-Z
1. Sets of numbers; Arithmetic.
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.
2. Basic algebra, equations, inequalities.
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. Linear systems with two equations
and two unknowns. Equations containing ratios of polynomials. Equations
with radicals. Inequalities. Solving inequalities of degree one and degree two,
inequalities, with ratios of polynomials, with radicals.
3. Sets and logic.
Elementary set terminology: being an element of a set, inclusion, intersection, union, complementary set, empty set. Logical implications: suffcient condition, necessary condition.
4. Analytic geometry.
Cartesian coordinates in the plane. Pythagorean theorem, Distance between
two points. Equation of the line. Parallel lines and perpendicular lines. Equation of the parabola.
Teaching Methods - Last names D-L
Class lectures. The course lenght is 12 weeks with three classes per week.
Further information - Last names D-L
Tutor junior.
Also for the academic year 2014-2015, it will be possible to attend review/exercise classes with some senior students.
Students can attend classes in any of the four offered courses, but they are invited to attend the course corresponding to the first letter of their last name.
Students have to take the exam with the professors teaching the course corresponding to the first letter of their last name.
Type of Assessment - Last names D-L
There will be a written exam. Reaching a passing grade on the written exam gives access to an oral exam. If this last is passed then the exam is succesfully concluded. For further information on the exam rules, please, contact prof.
villanacci at antonio.villanacci@unifi.it or at 055 - 43 74 691.
Type of Assessment - Last names Q-Z
There will be a written exam. A passing grade on the exam is a passing grade
of the course. For further information on the exam rules, please, contact prof.
villanacci at antonio.villanacci@unifi.it or at 055 - 43 74 691.
Course program - Last names D-L
1. The role of mathematics in applications.
2. Introduction to mathematical reasoning
3. 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, from
above/from below. Supremum and inÖmum of a set.
4. Functions. Domain, image and graph of a function. Injective and bijective
functions. Inverse functions. Composition of functions. Domain of the
composite function. The exponential function. The logarithmic function.
The basic trigonometric functions.
5. Monotone functions. Strict monotonicity and invertibility. Functions
bounded above/below. Supremum and InÖmum of a function; maximum
and minimum of a function.
6. Sequences. Limit of a sequence. Uniqueness of the limit.
7. Limit of a function. Uniqueness of the limit. Sign invariance theorem.
Comparison or squeeze theorem. Left and right limits. Operations with
limits: multiplication by a number, sum, product, quotient. Limit of
composite functions (change of variables). Limits at infinity, infinite limits.
Infinities and Infinitesimals. Limits of rational functions, of logarithmic
functions, and of the exponential functions. Indeterminate forms.
8. Continuous functions. DeÖnitions and examples. Continuity of elementary
functions. Operations with functions and continuity. The intermediate
value theorem. Bisection method. Global and local maxima and minima.
Weierstrass theorem.
9. Definition of derivative. Tangent line. Rules of di§erentiation for sum,
product, quotient and composition. Fermat Theorem, Rolle and Lagrande
Theorem, de líHospital Theorem. Sign of derivative and monotonicity.
Convex functions. Sign of second derivative and convexity. Sign of the
second derivative as a su¢ cient conditions for local maxima and minima.
Taylor polynomial
Course program - Last names Q-Z
1. The role of mathematics in applications.
2. Introduction to mathematical reasoning
3. 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, from
above/from below. Supremum and inÖmum of a set.
4. Functions. Domain, image and graph of a function. Injective and bijective
functions. Inverse functions. Composition of functions. Domain of the
composite function. The exponential function. The logarithmic function.
The basic trigonometric functions.
5. Monotone functions. Strict monotonicity and invertibility. Functions
bounded above/below. Supremum and InÖmum of a function; maximum
and minimum of a function.
6. Sequences. Limit of a sequence. Uniqueness of the limit.
7. Limit of a function. Uniqueness of the limit. Sign invariance theorem.
Comparison or squeeze theorem. Left and right limits. Operations with
limits: multiplication by a number, sum, product, quotient. Limit of
composite functions (change of variables). Limits at infinity, infinite limits.
Infinities and Infinitesimals. Limits of rational functions, of logarithmic
functions, and of the exponential functions. Indeterminate forms.
8. Continuous functions. DeÖnitions and examples. Continuity of elementary
functions. Operations with functions and continuity. The intermediate
value theorem. Bisection method. Global and local maxima and minima.
Weierstrass theorem.
9. Definition of derivative. Tangent line. Rules of di§erentiation for sum,
product, quotient and composition. Fermat Theorem, Rolle and Lagrande
Theorem, de líHospital Theorem. Sign of derivative and monotonicity.
Convex functions. Sign of second derivative and convexity. Sign of the
second derivative as a su¢ cient conditions for local maxima and minima.
Taylor polynomial