27408 - Mathematics II
Basic Education
5.2. Learning tasks
The program offered to the students to help them achieve the learning results includes the following activities:
1. Theoretical lessons which will be based on lectures to present the concepts and results corresponding to the contents. At the same time, some exercises will be solved with the participation of the students to help them comprehend the theoretical concepts presented. These classes are face-to-face and will be given to the full group.
Time allotted: 1,2 ECTS credits (30 hours).
2. Practical lessons, in which the students will apply the theoretical results in order to solve, with the teacher’s help, more complete exercises, and problems of an economic nature. Problem sheets will be available for the students and the teacher will announce in advance the problems that will be solved in each practical lesson so that the students can prepare them beforehand. These classes are face-to-face and will be given separately to each subgroup.
Time allotted: 1.2 ECTS credits (30 hours each subgroup).
3. Seminars (practical classes P6), which may consist of a number of different activities designed to support the learning process, including: follow-up of some simple projects that had been assigned to small teams of students and the presentation of these projects; answering questions that students may have regarding some of the contents taught; solving problems of an economic nature by using some of the mathematical tools taught during the classes, etc. These seminars may also be devoted to the teaching of more advanced topics, intended for the students interested in learning some further mathematical tools that would allow them to deal with more general problems. In this way, the students are shown that both Mathematics and Economics are vibrant sciences with many facets to be studied.
Time allotted: Pending of the agreement of the Department Committee
4. Out of class work: 3.6 ECTS credits
5.3. Syllabus
Chapter 1: Mathematical programs
1.1. General formulation of a mathematical program. Classification.
1.2. Definitions and properties. Weierstrass’ Theorem.
1.3. Graphical solving.
1.4. Introduction to convexity:
1.4.1. Convex sets. Definition and properties.
1.4.2. Convex and concave functions. Definitions and properties.
1.4.3. Convex programs.
Chapter 2: Programming without constraints
2.1. Problem’s formulation.
2.2. Local optima:
2.2.1. First order conditions for the existence of a local optimum.
2.2.2. Second order conditions for the existence of a local optimum.
2.3. Global optima: convex programs.
Chapter 3: Programming with equality constraints
3.1. Problem’s formulation.
3.2. Local optima:
3.2.1. First order conditions for the existence of a local optimum.
3.2.2. Second order conditions for the existence of a local optimum.
3.3. Global optima: convex programs and Weierstrass’ Theorem.
3.4. Economic interpretation of the Lagrange’s multipliers.
Chapter 4: Linear programming
4.1. Formulation of a problem of linear programming.
4.2. Solutions of a linear program. Basic feasible solutions.
4.3. Characterization of the optimal basic feasible solutions. Simplex’ Algorithm.
4.4. Introduction to the sensitivity analysis.
4.5. Introduction to the dual program.
Chapter 5: Introduction to ordinary differential equations
5.1. Introduction to the dynamical analysis.
5.2. Concept of differential equation, solution and types of solution.
5.3. First order ordinary differential equations:
5.3.1. Separable equations.
5.3.2. Linear first order equations.
5.4. Linear differential equations of order n with constant coefficients.
5.5. Qualitative analysis: equilibrium points and stability.