Study programme competencies |
Code
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Study programme competences / results
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A3 |
Evaluate and foreseeing, from relevant data, the development of a company. |
A4 |
Elaborate advisory reports on specific situations of companies and markets |
A6 |
Identify the relevant sources of economic information and to interpret the content. |
A8 |
Derive, based on from basic information, relevant data unrecognizable by non-professionals. |
A9 |
Use frequently the information and communication technology (ICT) throughout their professional activity. |
A10 |
Read and communicate in a professional environment at a basic level in more than one language, particularly in English |
A11 |
To analyze the problems of the firm based on management technical tools and professional criteria |
A12 |
Communicate fluently in their environment and work by teams |
B1 |
CB1-The students must demonstrate knowledge and understanding in a field of study that part of the basis of general secondary education, although it is supported by advanced textbooks, and also includes some aspects that imply knowledge of the forefront of their field of study |
B2 |
CB2 - The students can apply their knowledge to their work or vocation in a professional way and have competences typically demostrated by means of the elaboration and defense of arguments and solving problems within their area of work |
B3 |
CB3- The students have the ability to gather and interpret relevant data (usually within their field of study) to issue evaluations that include reflection on relevant social, scientific or ethical |
B4 |
CB4-Communicate information, ideas, problems and solutions to an audience both skilled and unskilled |
B5 |
CB5-Develop skills needed to undertake further studies learning with a high degree of autonomy |
B10 |
CG5-Respect the fundamental and equal rights for men and women, promoting respect of human rights and the principles of equal opportunities, non-discrimination and universal accessibility for people with disabilities. |
C1 |
Express correctly, both orally and in writing, in the official languages of the autonomous region |
C4 |
To be trained for the exercise of citizenship open, educated, critical, committed, democratic, capable of analyzing reality and diagnose problems, formulate and implement knowledge-based solutions oriented to the common good |
C5 |
Understand the importance of entrepreneurial culture and know the means and resources available to entrepreneurs |
C6 |
Assess critically the knowledge, technology and information available to solve the problems and take valuable decisions |
C7 |
Assume as professionals and citizens the importance of learning throughout life. |
C8 |
Assess the importance of research, innovation and technological development in the economic and cultural progress of society. |
Learning aims |
Learning outcomes |
Study programme competences / results |
Understand the basic concepts of the euclidean space IRn. |
A8 A11
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Determine if a set is open, closed, bounded, compact and convex. |
A8 A11
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Understand the concept of function of several variables. |
A8 A11
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Draw the level set of a function of two variables. |
A8 A11
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Understand the concept of continuous function. |
A8 A11
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Determine if a function is continuous or not. |
A8 A11
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Recognize a linear function. |
A8 A11
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Recognize a quadratic form. |
A8 A11
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Classify a quadratic form by examining the signs of the principal minors and by eigenvalues. |
A8 A11
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Classify a constrained quadratic form. |
A8 A11
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Calculate and interpret partial derivatives and elasticities. |
A4 A8 A11
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B1 B2 B5 B10
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C1 C7
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Find the Taylor polynomial of a function. |
A8 A11
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Calculate the partial derivatives of a compounded function. |
A8 A11
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Use the existence theorem to analyze if a equation defines an implicit real function. |
A8 A11
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Find the partial derivatives and elasticities of an implicit function, and interpret them. |
A8 A11
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Analyze the concavity/convexity of a function. |
A8 A11
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Formulate mathematical programming problems. |
A3 A4 A6 A8 A9 A10 A11
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B1 B2 B3 B4 B5 B10
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C1 C4 C5 C6 C7 C8
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Distinguish between local and global optima. |
A8 A11
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Graphically solving an optimization problem |
A8 A11
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B3
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Analyze the existence of global optima using the Weierstrass theorem. |
A8 A11
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Find the critical points of a function of several variables. |
A8 A11
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Classify the critical points using the second-order conditions. |
A8 A11
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Determine the local or global character of the optima of an unconstrained problem. |
A8 A11
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Formulate economic problems as mathematical programs with equality constraints. |
A8 A11
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Find the critical points of a mathematical program with equality constraints. |
A8 A11
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Classify the critical points and interpret the Lagrange multipliers. |
A8 A11
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Determine the local or global character of the optima of an equality-constrained problem. |
A8 A11
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Know the structure and basic properties of a linear program. |
A8 A11
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Formulate simple economic problems as linear programs. |
A3 A4 A8 A11 A12
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B1 B2 B3 B4 B5 B10
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C1 C4 C6 C7 C8
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Solve linear programs by the simplex algorithm. |
A3 A4 A6 A8 A9 A11
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B1 B2 B3 B4 B5 B10
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C1 C4 C5 C6 C7 C8
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Contents |
Topic |
Sub-topic |
1. The euclidean space IRn. |
The vector space IRn.
Inner product. Norm. Distance.
Open and closed sets.
Compact sets. |
2. Functions of several variables. |
Basic concepts.
Graphical representation of real functions. Level sets.
Limit of a function at a point.
Continuity.
Linear functions.
Quadratic forms. Classification. Constrained quadratic forms. |
3. Derivatives of functions of several variables. |
Partial derivatives.
Partial derivatives of higher order. Class one function
Chain's Rule.
Taylor's theorem.
Implicit function theorem. |
4. Convexity of sets and functions. |
Convex sets. Properties.
Convex functions. Properties.
Characterization of twice continuously differentiable convex functions. |
5. Introduction to mathematical programming. |
Formulation of a mathematical program.
Local and global optima.
Graphic solving.
Basic Theorems in optimization. |
6. Unconstrained optimization. |
First-order necessary conditions.
Second-order conditions.
The convex case.
Sensitivity analysis. |
7. Equality-constrained optimization |
Formulation.
First-order necessary conditions: the Lagrange theorem.
Second-order conditions.
The convex case.
Sensitivity analysis. |
8. Linear programming. |
Formulation of linear programs.
Basic feasible solutions.
Fundamental theorems.
The simplex algorithm. |
Planning |
Methodologies / tests |
Competencies / Results |
Teaching hours (in-person & virtual) |
Student’s personal work hours |
Total hours |
Introductory activities |
A6 A9 A12 C1 |
1 |
0 |
1 |
Multiple-choice questions |
A10 B2 B3 B4 |
2 |
7 |
9 |
Mixed objective/subjective test |
A10 B2 B3 B4 |
3 |
15 |
18 |
Guest lecture / keynote speech |
A3 A4 A8 A9 A11 A12 B1 B5 C6 C7 |
15 |
15 |
30 |
Seminar |
B10 C4 C5 C8 |
2 |
4 |
6 |
Practical test: |
A8 A11 B1 B2 B3 B4 B5 C1 |
2 |
8 |
10 |
Problem solving |
A6 B1 |
25 |
50 |
75 |
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Personalized attention |
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1 |
0 |
1 |
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(*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students. |
Methodologies |
Methodologies |
Description |
Introductory activities |
It will be the presentation of the course (one hour). |
Multiple-choice questions |
There will be several multiple-choice exams. These exams will have questions with several given answers --only one will be correct-- related to theoretical and practical concepts covered in the course. |
Mixed objective/subjective test |
At the end of the course, there will be a mixed (theoretical/practical) exam. This exam will take place at the official date determined by the Faculty. |
Guest lecture / keynote speech |
There will be 15 hours of keynote speech, that will be focused on the exposition of the theoretical contents. |
Seminar |
They will be several seminars with personalized attention of character essentially practical. These seminars will preferably be face-to-face. Sufficiently in advance, the dates, times and classrooms will be published for ecah group. |
Practical test: |
There will be several practical tests along the term. These tests will consist of one or several questions to which will have to answer by writing and justifying properly the answers. |
Problem solving |
There will be 25 hours of problem solving classes, which will be focused on the formulation and solving of problems related to the practical contents of the subject. |
Personalized attention |
Methodologies
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Problem solving |
Seminar |
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Description |
The students will have of the following roads of communication:
- Asynchronous Communication:
-Platform Moodle (by means of the use of the forums or direct messages).
-Email of the teachers. For asynchronous queries.
- Synchronous communication:
-Personal Tutoring using the periods of time fixed by the teachers of the subject.
-Seminars (Group tutoring).
Also it will be possible tutoring in other dates and different hours to the established, previous application by part of the students. |
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Assessment |
Methodologies
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Competencies / Results |
Description
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Qualification
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Practical test: |
A8 A11 B1 B2 B3 B4 B5 C1 |
There will be two presential exams, their weighting in the final evaluation is 20% (2 points). In this exams, the reasoning capacity of the students will be especially valued. |
20 |
Mixed objective/subjective test |
A10 B2 B3 B4 |
The final (presential) exam will represent a 60% of the final mark (6 points). It will be valued a good understanding of the concepts, the use of appropriate reasoning, the proper use of mathematical language, and the skills in formulating and solving problems. |
60 |
Multiple-choice questions |
A10 B2 B3 B4 |
Throughout the course there will be two multiple choice tests (multiple choice), their weighting in the final evaluation is 20% (2 points). |
20 |
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Assessment comments |
A) EVALUATION REGULATIONS 1. Conditions for making of the examinations and test, and identification of students During the realisation of the examinations will not be able to have access to any device that allow the communication with the outside and/or the storage of information. It will be able to deny the entrance to the classroom with this type of devices. They will not admit the examinations written with pencil. The students will have to identify by means of DNI or equivalent for making the tests of evaluation. 2. Use of calculator The calculator that can be used must not have ANY of the following characteristics: Possibility of transmitting data, being programmable, graphic display, solving equations, operations with matrix, calculation of determinants, derivatives and/or integrals, storage of alphanumeric data. When it has any of these characteristics, it will be withdrawn. If during the development of the exam a calulator is used that is not allowed, the same measures will be adopted as when the students are copying. B) QUALIFICATION TYPES 1. Not taken qualification It will award the qualification of NOT TAKEN to the student that only participate in activities of evaluation that have a weight less than 20% of the final qualification, with independence of the qualification obtained. 2. Students part time (or with dispenses of attendence): it will be evaluated according to the same norms that the rest of students C) EVALUATION OPPORTUNITIES 1. First opportunity Continuous evaluation Continuous evaluation will consist of two multiple choice tests (multiple choice questions) and two presential exams (practical test) in the classroom. Their weighting in the final evaluation is 40% (4 points) Final exam Mixed objective/subjective test. The final exam will represent a 60% of the final mark (6 points) In addition, students can obtain up to one point for active participation in classes, seminars and personal tutoring, which will be added to the mark obtained in the continuous evaluation and in the final exam. 2. Second opportunity In the second opportunity there will be a mixed objective/subjective test and the qualification will be the highest of the following two options: - Sum of the marks obtained in the continuous evaluation of the first opportunity (maximum four points of the multiple choice questions and practical test carried out) and in the mixed objective/subjective test of the second opportunity (maximum six points) - Qualification obtained in the mixed objective/subjective test of the second opportunity valued on ten points.
3. Opportunity in Advance: The final qualification of the student that request this opportunity will be the obtained in the face-to-face examination valued on ten points.
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Sources of information |
Basic
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K. Sydsæter, P. J. Hammond y P. Carvajal (2012). Matemáticas para el análisis económico . Madrid, Pearson |
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Complementary
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R. Caballero, S. Calderón, T. P. Galache, A. C. González, Mª. L. Rey y F. Ruiz (2000). Matemáticas aplicadas a la economía y la empresa. 434 ejercicios resueltos y comentados . Madrid, Pirámide
E. Minguillón, I. Pérez Grasa y G. Jarne (2004). Matemáticas para la economía. Libro de ejercicios. Álgebra lineal y cálculo diferencial. Madrid, McGraw-Hill
I. Pérez Grasa, G. Jarne y E. Minguillón (1997). Matemáticas para la economía: álgebra lineal y cálculo diferencial . Madrid, McGraw-Hill
I. Pérez Grasa, G. Jarne y E. Minguillón (2001). Matemáticas para la economía: programación matemática y sistemas dinámicos . Madrid, McGraw-Hill
M. Hoy, J. Livernois, C. McKenna, R. Rees y T. Stengos (2001). Mathematics for economics. Cambridge, MA, The MIT Press
A. C. Chiang y K. Wainwright (2006). Métodos fundamentales de economía matemática . Madrid, McGraw-Hill
R. M. Barbolla, E. Cerdá y P. Sanz (2001). Optimización. Cuestiones, ejercicios y aplicaciones a la economía . Madrid, Prentice Hall |
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Recommendations |
Subjects that it is recommended to have taken before |
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Subjects that are recommended to be taken simultaneously |
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Subjects that continue the syllabus |
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Other comments |
It is advisable to have passed Mathematics I. Students must be familiar with the concepts and fundamental results of linear algebra (matrices, determinants and systems of linear equations), and differential calculus in one variable (limit, continuity, derivative, elasticity, optima, convexity). |
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