Identifying Data 2019/20
Subject (*) Mathematics II Code 611G02010
Study programme
Grao en Administración e Dirección de Empresas
Descriptors Cycle Period Year Type Credits
Graduate 2nd four-month period
First Basic training 6
Language
Spanish
Galician
Teaching method Face-to-face
Prerequisites
Department Economía
Coordinador
Seijas Macias, Jose Antonio
E-mail
antonio.smacias@udc.es
Lecturers
Blanco Louro, Amalia
Lema Fernández, Carmen Socorro
Rodríguez González, David
Seijas Macias, Jose Antonio
E-mail
amalia.blanco.louro@udc.es
carmen.lemaf@udc.es
david.rodriguez.gonzalez@udc.es
antonio.smacias@udc.es
Web http://moodle.udc.es
General description O obxectivo deste curso é presentar aos alumnos os conceptos básicos do cálculo diferencial en varias variables e a programación matemática, que serán necesarios para a aprendizaxe doutras disciplinas do grao e para a súa carreira futura. O estudante deberá entender os conceptos básicos presentados e os resultados que os relacionan, e aplicar ese coñecemento de forma adecuada e rigorosa para resolver problemas prácticos. Farase unha énfase especial na aplicación dos contidos do curso a problemas de natureza económica e á interpretación dos resultados obtidos.
Tamén se pretende axudar os alumnos a desenvolver habilidades xenéricas, como a capacidade de análise e síntese, a capacidade de razoamento lóxico, a capacidade de resolución de problemas, o pensamento crítico, a aprendizaxe independente, ou a capacidade de recuperar e utilizar información de varias fontes.
Contingency plan

Study programme competencies
Code Study programme competences
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
Identify the notable sets of a subset of IRn. A8
A11
Understand the basic concepts of the euclidean space IRn. A8
A11
Determine if a set is open, closed, bounded, compact and convex. A8
A11
Understand the concept of function of several variables. A8
A11
Draw the level set of a function of two variables. A8
A11
Understand the concept of continuous function. A8
A11
Determine if a function is continuous or not. A8
A11
Recognize a linear function. A8
A11
Recognize a quadratic form. A8
A11
Classify a quadratic form by examining the signs of the principal minors. A8
A11
Classify a constrained quadratic form. A8
A11
Calculate and interpret partial derivatives and elasticities. A4
A8
A11
B1
B2
B5
B10
C1
C7
Find the Taylor polynomial of a function. A8
A11
Calculate the partial derivatives of a compounded function. A8
A11
Use the existence theorem to analyze if a equation defines an implicit real function. A8
A11
Find the partial derivatives and elasticities of an implicit function, and interpret them. A8
A11
Analyze the concavity/convexity of a function. A8
A11
Formulate mathematical programming problems. A3
A4
A6
A8
A9
A10
A11
B1
B2
B3
B4
B5
B10
C1
C4
C5
C6
C7
C8
Distinguish between local and global optima. A8
A11
Graphically solving an optimization problem A8
A11
B3
Analyze the existence of global optima using the Weierstrass theorem. A8
A11
Find the critical points of a function of several variables. A8
A11
Classify the critical points using the second-order conditions. A8
A11
Determine the local or global character of the optima of an unconstrained problem. A8
A11
Formulate economic problems as mathematical programs with equality constraints. A8
A11
Find the critical points of a mathematical program with equality constraints. A8
A11
Classify the critical points and interpret the Lagrange multipliers. A8
A11
Determine the local or global character of the optima of an equality-constrained problem. A8
A11
Know the structure and basic properties of a linear program. A8
A11
Formulate simple economic problems as linear programs. A3
A4
A8
A11
A12
B1
B2
B3
B4
B5
B10
C1
C4
C6
C7
C8
Solve linear programs by the simplex algorithm. A3
A4
A6
A8
A9
A11
B1
B2
B3
B4
B5
B10
C1
C4
C5
C6
C7
C8

Contents
Topic Sub-topic
1. The euclidean space IRn. The vector space IRn.
Inner product. Norm. Distance.
Interior, closure, isolated, limit and boundary points.
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 Ordinary class hours 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
 
Personalized attention 1 0 1
 
(*)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 two 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 The group will be divided into two subgroups for the seminars.
Practical test: There will two in-class practical exams.
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
Problem solving
Seminar
Description
The student will be able to contact the teacher by the following means:
- Moodle (using the forums or direct messages).
- Email.
- Personal tutoring in the office (at the official dates or at other dates upon request).
- Seminars in small groups (group tutorials).

Assessment
Methodologies Competencies Description Qualification
Practical test: A8 A11 B1 B2 B3 B4 B5 C1 There will be two presential exams. Each of them will represent a 15% of the final grade (1.5 points each). 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. 30
Mixed objective/subjective test A10 B2 B3 B4 The final (presential) exam will represent a 50% of the final mark (5 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. 50
Multiple-choice questions A10 B2 B3 B4 There will be two multiple-choice presential exams. Each of them will represent a 10% of the final grade (1 point each). 20
 
Assessment comments
The first and second opportunities will be graded in the same way. Students with partial-time enrollment must fill the same requirements for assessment that students on full-time enrollment.

Continuous assessment will consist of two in-class multiple-choice quizzes (10% each) and two in-class exams (15% each). Non-attendance to more than four class sessions (lecture, practice or seminar) will lead to not computing the continuous assessment qualification, which represents a 50% of the final mark. To qualify an absence as justified or not we will follow the provisions of Article 12, points 1 and 5, of the  Normas de avaliación, revisión e reclamación das cualificacións dos estudos de grao e mestrado universitarios. In case of disrespectful behavior with peers or teacher, or using electronic devices (tablet, computer, telephone, ...) or other material unrelated to the class activities, you will be required to leave the classroom, and it will be counted as a non-justified absence.

The qualification of NOT-TAKEN will also be awarded to the student who has only participated in assessment activities that have a weighting below 20% of the final grade, regardless of the qualification obtained.

The final grade for students applying to the call of December will be the qualification of the final exam valued on 10 points.

Conditions for carrying out exams: During the examination, you cannot have access to any device that allows communication with the outside and/or storage of information. Entry to the examination room with these devices may be denied. The student may use a scientific calculator non-graphic and non-programmable. Exams written in pencil will not be admitted.

Virtual Platform: It will be used the Moodle virtual platform (http://moodle.udc.es).

Sources of information
Basic K. Sydsæter, P. J. Hammond y P. Carvajal (2012). Matemáticas para el análisis económico . Madrid, Pearson

Complementary 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


Recommendations
Subjects that it is recommended to have taken before
Mathematics I/611G02009

Subjects that are recommended to be taken simultaneously

Subjects that continue the syllabus

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).



(*)The teaching guide is the document in which the URV publishes the information about all its courses. It is a public document and cannot be modified. Only in exceptional cases can it be revised by the competent agent or duly revised so that it is in line with current legislation.