Syllabus
EN
IT
Lorenzo Peccati, Sandro Salsa, Annamaria Squellati. Mathematics for economic business.
Written test with exercises and theoretical questions
Learning Objectives
LEARNING OUTCOMES:
Knowledge and understanding of the basic principles useful for a deep comprehension of the main concepts of mathematical analysis, particularly related to sequences, series and functions of one variable. The objective is to apply basic notions on mathematical analysis to economic modelling with a rigorous approach.
KNOWLEDGE AND UNDERSTANDING:
The course develops basic topics of mathematical analysis. It provides basic knowledge of economic modeling and financial mathematics.
APPLYING KNOWLEDGE AND UNDERSTANDING:
The structure of the course consists of a theoretical part (definitions and theorems with proofs), the discussion of useful examples for economic theory, supplemented by fundamental practical exercises in which concrete methods are developed for the solution of basic problems in mathematical analysis and linear algebra. The understanding of proofs allows students to have the right approach for facing different types of problems of moderate difficulty, in the field of economic modeling and in the understanding, based on quantitative methods, of social phenomena.
MAKING JUDGEMENTS:
The course includes proofs of theorems and analytic properties. This aspect allows students to construct and develop logical arguments with a clear identification of assumptions and conclusions, to recognize correct proofs, and to identify incorrect or incomplete reasoning. Various examples of applications to economic models are also studied, which enable students to propose and analyze mathematical models useful for social sciences.
COMMUNICATION SKILLS:
To succeed the written exam the student must develop the necessary scientific rigor required to describe the analytical solution of a problem. The course therefore provides some tools needed to communicate, rigorously, scientific results in the social sciences.
LEARNING SKILLS:
The course provides basic tools for the development of further studies, both in Mathematics and in Economics. The theoretical study provides the ability to deal independently with new problems of medium difficulty.
Knowledge and understanding of the basic principles useful for a deep comprehension of the main concepts of mathematical analysis, particularly related to sequences, series and functions of one variable. The objective is to apply basic notions on mathematical analysis to economic modelling with a rigorous approach.
KNOWLEDGE AND UNDERSTANDING:
The course develops basic topics of mathematical analysis. It provides basic knowledge of economic modeling and financial mathematics.
APPLYING KNOWLEDGE AND UNDERSTANDING:
The structure of the course consists of a theoretical part (definitions and theorems with proofs), the discussion of useful examples for economic theory, supplemented by fundamental practical exercises in which concrete methods are developed for the solution of basic problems in mathematical analysis and linear algebra. The understanding of proofs allows students to have the right approach for facing different types of problems of moderate difficulty, in the field of economic modeling and in the understanding, based on quantitative methods, of social phenomena.
MAKING JUDGEMENTS:
The course includes proofs of theorems and analytic properties. This aspect allows students to construct and develop logical arguments with a clear identification of assumptions and conclusions, to recognize correct proofs, and to identify incorrect or incomplete reasoning. Various examples of applications to economic models are also studied, which enable students to propose and analyze mathematical models useful for social sciences.
COMMUNICATION SKILLS:
To succeed the written exam the student must develop the necessary scientific rigor required to describe the analytical solution of a problem. The course therefore provides some tools needed to communicate, rigorously, scientific results in the social sciences.
LEARNING SKILLS:
The course provides basic tools for the development of further studies, both in Mathematics and in Economics. The theoretical study provides the ability to deal independently with new problems of medium difficulty.
Prerequisites
Being able to solve first and second order equations, inequalities, fractional inequalities, exponential, logarithmic and trigonometric inequalities.
Basic knowledge of properties of exponential, logarithmic and trigonometric functions.
Basic knowledge of properties of exponential, logarithmic and trigonometric functions.
Program
Basic properties of functions.
Set theory. Set Operations: union, intersection, difference, complement. Intuitive construction of number sets.
Real valued functions.
Linear, quadratic and polynomial functions. Exponential, Logarithmic and Trigonometric functions. Domain, Range of a function. Increasing, decreasing and monotonic functions.
Injective, surjective functions. The inverse function.
Sequences.
Definition of limit of a sequence, uniqueness of the limit of a sequence (with proof), monotonic sequences. Existence of the limit of a monotonic and bounded sequence. Comparison Theorem (with proof). Notable limits.
Series.
Definition and examples. Conditions for convergence of geometric series (with proof).
Limit of real functions of one real variable.
Definition. Uniqueness of the limit. Comparison Theorem.
Continuity: definition and main theorems. Characterization of the discontinuity points of a function: jumps and removable and essential discontinuities.
Derivative of a real functions of one real variable.
Necessary condition for differentiability (with proof).
Maxima and minima of a function: necessary and sufficient conditions.
Tangent line and 1st order Taylor approximation.
Convex and concave functions. Second-order conditions for maxima and minima.
Set theory. Set Operations: union, intersection, difference, complement. Intuitive construction of number sets.
Real valued functions.
Linear, quadratic and polynomial functions. Exponential, Logarithmic and Trigonometric functions. Domain, Range of a function. Increasing, decreasing and monotonic functions.
Injective, surjective functions. The inverse function.
Sequences.
Definition of limit of a sequence, uniqueness of the limit of a sequence (with proof), monotonic sequences. Existence of the limit of a monotonic and bounded sequence. Comparison Theorem (with proof). Notable limits.
Series.
Definition and examples. Conditions for convergence of geometric series (with proof).
Limit of real functions of one real variable.
Definition. Uniqueness of the limit. Comparison Theorem.
Continuity: definition and main theorems. Characterization of the discontinuity points of a function: jumps and removable and essential discontinuities.
Derivative of a real functions of one real variable.
Necessary condition for differentiability (with proof).
Maxima and minima of a function: necessary and sufficient conditions.
Tangent line and 1st order Taylor approximation.
Convex and concave functions. Second-order conditions for maxima and minima.
Books
Lorenzo Peccati, Sandro Salsa, Annamaria Squellati. Mathematics for economic business.
Bibliography
Knut Sydsæter, Peter Hammond, Arne StrØm, Essential Mathematics for Economic Analysis
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Teaching methods
Theoretical lectures and exercise classes
Exam Rules
Written test with exercises and theoretical questions
Attendance Rules
Attendance is not compulsory but highly recommended