MATHEMATICS
Instructional goals
The course aims to provide first-year economics students with the main
mathematical tools that are used to analyze and solve a wide range of
problems in applied in economics, business, finance and insurance. This
will be their first step towards achieving a broad training in quantitative
disciplines, as well as a good methodological training for the analysis and
critical interpretation of economic and business dynamics. This will allow
them to acquire the useful tools for the formulation, implementation and the support to the decisions in the professional context.
Intended learning outcomes
Knowledge and understanding:
By the end of the course, students will have acquired a wide range of
mathematical tools and will be able to understand and solve numerous
theoretical and applied problems in economics, business and finance.
Ability to apply knowledge and understanding:
By the end of the course, students will be able to apply quantitative tools
to business decisions, to the analysis of economic phenomena and to the
resolution of economic and business problems in the work contexts in
which they will operate.
Autonomy of judgment:
By the end of the course, students will be able to develop a critical
capacity in identifying the most suitable solution to the proposed
problem, even in highly innovative contexts.
Communication skills:
By the end of the course, students will be able to develop communication
skills through written papers and oral interviews and to highlight the
relationships between the theoretical concepts and the most important
economic and managerial applications.
Learning ability:
By the end of the course, students will be able to stay up to date and
improve their skills in examining the quantitative aspects of economics
and management.
Course Contents
The mathematical analysis for functions of a single variable allows the
students to introduce the fundamental mathematical tools (limits,
derivatives, integrals) for the definition and study of the mathematical
models in economics.
The functions of several variables make it possible to develop models
that are closer to economic reality. To this aim, the tools already
introduced for the functions of single variable are suitably adapted to the
new context.
The study of vectors, matrices and linear systems then allows the design
of models capable of including large amounts of data.
The explanation is accompanied by economic examples to illustrate the
importance of the acquired tools in the context of the study program.
Reference Books
Textbook:
(AG): Angelo Guerraggio, Matematica (quarta edizione), Pearson Italia.
Additional material on the website. Learn.luiss.it
Teaching Methods
Teaching will include the following activities:
- In presence teaching
- In presence exercise sessions
- material for in-depth study available on the learn.luiss.it page of the
course.
Assessment Method
The exam includes an intermediate written test, a final written test and
an oral test.
Thesis assignment criteria
Interview with the teacher after the exam.
Week 1
Introduction to the course. Natural numbers, integers, rational and real numbers. Set theory. Vectors: definitions, operations, geometric representation, vector spaces, linear combination. Linearly dependent and independent vectors, rank. (Ch.15) Exercises on the topics of the week and description of economic applications.
Week 2
Matrices: definitions and characteristics, the determinant, Sarrus' rule, algebraic complement and general rule. Minor, rank, and characteristic. Operations between matrices and inverse matrix. (Ch. 15 and Ch. 16). Exercises on the topics of the week and description of economic applications.
Week 3
Linear equations. Systems of linear equations, Cramer's theorem, Rouché-Capelli theorem. Parametric systems. (Ch.16) Exercises on the topics of the week and description of economic applications.
Week 4
The notion of a function (Ch.2) Elementary functions and their initial properties: linear, quadratic, hyperbolic, power, exponential, and logarithmic functions (Ch.3). Exercises on the topics of the week and description of economic applications.
Week 5
Limit of a function: An informal definition. A unified (formal) definition, from the general definition to specific cases. Right-hand and left-hand limits.
Week 6
Continuous functions and the calculation of limits (Ch.7) Derivative of a function: Definition, geometric and economic interpretations. Calculation of the derivative. (Beginning of Ch.8). Exercises on the topics of the week and description of economic applications.
Week 7
Differentiability and continuity, higher derivatives (End of Ch.8). Rolle's Theorem, Lagrange's Mean Value Theorem, and Taylor's Theorem. De l'Hopital's Rule and indeterminate forms (Beginning of Ch.9). Exercises on the topics of the week and description of economic applications.
Week 8
Finding the maximum and minimum points of a function. Optimization problems. Convexity and concavity of a function, inflection points in function analysis (End of Ch.9). Integral of a function: immediate and nearly immediate cases (Beginning of Ch.10). Exercises on the topics of the week and description of economic applications.
Week 9
Integration by parts and substitution (End of Ch.10) The definite integral: Construction, definition, main properties, existence, and calculation (Ch.11). Exercises on the topics of the week and description of economic applications.
Week 10
Functions of two variables: Introduction, continuity, partial derivatives (Beginning of Ch.14). Exercises on the topics of the week and description of economic applications.
Week 11
Unconstrained and constrained optimization problems (End of Ch.14). Exercises on the topics of the week and description of economic and sustainability applications.
Week 12
In the final week of the course, we will focus on the economic applications of the mathematical theories studied. We will analyze how derivatives and integrals can solve real-world problems in economics and management. This summary aims to consolidate the understanding and practical application of these mathematical tools.