Public University of Navarre



Academic year: 2015/2016 | Previous academic years:  2014/2015  |  2013/2014  |  2012/2013  |  2011/2012 
International Double Bachelor's degree in Economics, Management and Business Administration at the Universidad Pública de Navarra
Course code: 176104 Subject title: MATHEMATICS
Credits: 6 Type of subject: Basic Year: 1 Period: 1º S
Department: Mathematics
Lecturers:
INDURAIN ERASO, ESTEBAN   [Mentoring ]

Partes de este texto:

 

Module/Subject matter

Module: Quantitative methods.

Subject matter: Mathematics.

Up

Descriptors

Basic tools of Differential Calculus, Integral Calculus and Linear Algebra.

Up

General proficiencies

  • CG03: Oral and written communication in their mother tongue.
  • CG05: Computer skills relevant to the field of study.
  • CG07: The capacity to solve problems.
  • CG09: The capacity to work as part of a team.
  • CG16: Working in high-pressure environments.
  • CG17: A capacity for self-reliant learning.
  • CG01: A capacity for analysis and synthesis.
  • CG02: A capacity for organisation and planning.
  • CG03: Oral and written communication in their mother tongue.
  • CG04: Oral and written communication in a foreign language.
  • CG05: Computer skills relevant to the field of study.
  • CG06: The ability to search for and analyse information from different sources.
  • CG07: The capacity to solve problems.
  • CG08: The capacity to make decisions.
  • CG09: The capacity to work as part of a team.
  • CG14: Critical and self-critical skills.
  • CG16: The capacity to work in high-pressure environments.
  • CG17: A capacity for self-reliant learning.

Up

Specific proficiencies

  • CE03: To discern relevant information and data in microeconomic and macroeconomic data which a non-professional would be unable to recognise.
  • CE04: To perform economic analysis applying professional criteria, preferably criteria based on the use of technical instruments.
  • The students should get used to working at a proficiency level with the most fundamental techniques of Calculus, namely: real numbers, basic binary operations, maps and real-valued functions, single equations, systems of equations, and so on.
  • The students should improve their ability to state, pose questions related to, mathematically interpret and finally solve, problems arising in Economics, as well as to criticize, compare, discuss, evaluate and draw conclusions from the results obtained.
  • The students should be familiar with the scope and different possible uses of the contents of the subject matter, not only as a tool for the better understanding of several other complementary subjects, but also (and not less important), as relevant devices to be used for Decision-Making in typical situations coming from Economics, Management and/or Business Administration studies.
  • CE02: To identify sources of economic information relevant to their particular enterprise and their contents.
  • CE03: To discern information and data relevant to their particular enterprise which a non-professional would be unable to recognise.
  • CE04: To analyse business management problems, applying professional criteria based on the use of technical instruments.
  • The students should get used to working at a proficiency level with the most fundamental techniques of Calculus, namely: real numbers, basic binary operations, maps and real-valued functions, single equations, systems of equations, and so on.
  • The students should improve their ability to state, pose questions related to, mathematically interpret and finally solve, problems arising in Economics, as well as to criticize, compare, discuss, evaluate and draw conclusions from the results obtained.
  • The students should be familiar with the scope and different possible uses of the contents of the subject matter, not only as a tool for the better understanding of several other complementary subjects, but also (and not less important), as relevant devices to be used for Decision-Making in typical situations coming from Economics, Management and/or Business Administration studies.

Up

Learning outcomes

Learning Outcomes Contents Training activities Evaluation
R15 - R17 Basic elements of linear algebra and differential and integral calculus Theoretical sessions. Practical sessions. Preparation of assignments, individually or in groups. Individualized tutoring or in small groups. Personal study. Tests of control. Working in groups. Final exam.
R16 - R18 Mathematical optimization Theoretical sessions. Practical sessions. Preparation of assignments, individually or in groups. Individualized tutoring or in small groups. Personal study. Tests of control. Working in groups. Final exam.

Up

Methodology

The methodology used throughout this subject matter aims to give a positive answer to the following challenges (among others):

  • Lead the student to be responsible of his/her own process of learning and acquiring knowledge.
  • Establish a preference where the practical items are more important than the theoretical ones. To do so, the lecturer will furnish the basically tools to solve problems, but not disregarding the most important theoretical facts, that will allow the students to know why things are as they are.
  • Enable and recommend the systematic use of computer devices and the access to the Internet.

To do so, the lecturer will develop the scheduled activities in the following way:

  1. Theoretical classes (that will include exercises) will be taught in the lecture room by the lecturer in charge of this subject matter. Along these lectures, the lecturer will introduce and develop the most important facts, enlighting them by means of the inclusion of a wide list of examples and practical cases to be analyzed. Therefore, the applications are understood as a base and ground on which the theory leans, and not only as a mere list of exercises. The exposition made by the lecturer will show the main lines and aspects of the subject matter. The complementary use of the bibliography, access to the Internet, or use of the tutorials and consulting hours to complete, improve and reinforce the items shown in the lecture room, is a task to be done by the students.
  2. Practical and problem-solving sessions, in which the lecturer will state and help the students to solve (in the lecture room) typical problems related to the subject matter. To do so, the lecturer will select, when available, ``case studies", that is: key problems that perhaps are not trivial or straightforward, and have some special difficulty. These problems should show in a clear way how the most important theoretical facts interact to get a solution, output or result. These kinds of lectures will require an active (and compulsory) participation of the students that should have tried to solve (in advance) the problems stated. Having this in mind, it is very important here that the students work in teams in which they will comment, discuss, analyze, get solutions and draw conclusions from the problems they solve.
  3. The personal work of the student is crucial here, since the subject is exigent and forces the student to update, at the same speed in which it is taught, the amount of matter shown in the lecture room. To cope with this, the student must develop techniques of self-training. To control all this process, the interaction student-lecturer through tutorials and consulting hours is compulsory.
  4. The student must get used to working in teams as a complement to his/her individual process of learning. Having this in mind, the lecture should also provide the students with sessions of "group-tutorials", in which a team of students will expose the achievements obtained (working in group) on schemes and models to be analyzed and discussed.
  5. Systematic  "control tests" should be scheduled by the lecturer to be done in the lecture room by the students in a period of, say, once a forthnight, to have a clear idea about how the students are understanding and using the main concepts and ideas. This is a way to control the "continuity” in the way in which the students interact with the subject matter. Notice that to assimilate the main ideas of Mathematics, a process of continuous learning is required.

Up

Relationship between formative activities and proficiencies/learning outcomes

Training activity Methodology Proficiencies developed
Theoretical sessions Lecture focused on explaining concepts illustrated with examples CG03, CE03, CE04,CG01, CE02, CE03, CE04
Practical sessions Realization of classroom exercises in small groups (4 people). Problems will be resolved preferably in an economic environment using, when deemed advisable, computing tools. Practices with the field of microeconomics. Oral presentation of the results. CG03, CG05, CG07, CG09, CG16, CE03, CE04, CG01, CG07, CG08, CG14, CE02, CE03, CE04
Preparation of assignments, individually or in groups. Exercises and problems with non-presential character work. Sometimes individual and sometimes work in group. CG03, CG05, CG07, CG09, CG16, CE03, CE04, CG01, CG02, CG03, CG04, CG05, CG06, CG07, CG08, CG09, CG14, CG17, CE02,CE03, CE04
Individualized tutoring or in small groups Working sessions customized teacher-student or between the teacher and a reduced group of students. CG07, CG09,CG01,CG03
Personal study and examination.   CG03, CG07, CG16, CG17, CE03, CE04, CG01, CG02, CG03, CG05, CG06, CG07, CG16, CG17, CE02, CE03, CE04

Up

Languages

English.

Up

Evaluation

  • Continuous assessment: active participation in the course, tests of control and working in groups, individually or in small groups. 40% of the final score.
  • Regular assessment (individual): in which the students should solve problems or case studies. 60% of the final score and it is compulsory to get at least 4 points (up to 10) in order to pass this subject matter. This part is recoverable, on a date fixed by the academic calendar for the evaluation of recovery.

Up

Contents

Differential calculus: a study of the functions of one and several variables, properties, operations, graphical representations, continuity, differentiability, asymptotic behavior. Homogeneous functions. Optimization, extreme of functions of one and several variables, unconstrained optimization, optimization with equality constraints. Integral calculus: indefinite and definite integral. Calculation of primitives. Applications to the calculation of areas and other applications. Matrices, determinants, operations and applications: Matrix calculus. Discussion and solving of systems of linear equations. Applications to Economics and Business Models.

Up

Agenda

1. MATRIX CALCULUS (LINEAR ALGEBRA)

1.1. Matrices

1.2. Transpose of a matrix

1.3. Addition of matrices.

1.4. Product of a matrix by a scalar number.

1.5. Product of matrices.

1.6. Inverse matrices.

1.7. Elementary transformations of a matrix.

1.8. Successive powers of a matrix.

1.9. Determinants

1.10. Determinants and inverse matrices.

1.11. Rank of a matrix.

1.12. Systems of linear equations.

2. DIFFERENTIAL CALCULUS

2.1. The real line and the n-dimensional space.

2.2. Real functions.

2.3. Limits.

2.4. Continuity.

2.5. Derivatives (functions of a single real variable).

2.6. Partial derivatives (functions of several real variables).

2.7. Optimization (functions of a single real variable).

2.8. Concave and convex functions.

2.9. Optimization without constraints (functions of two real variables).

2.10. Optimization with constraints.

3. INTEGRAL CALCULUS

3.1. Primitive integral of a real function.

3.2. Definite integral.

4: APPENDICES

4.1. Homogeneous functions.

4.2 Optimization theory (functions of several real variables).

Up

Bibliography

Access the bibliography that your professor has requested from the Library.


By means of the tool “MiAulario”, the students will have access to schemes, lists of exercises, links to complementary material, etc., related to the subject matter.

We do not recommend any particular textbook on this subject. There are many possible books on Differential Calculus, Integral Calculus, Linear Algebra and related items, even from a point of view of Economics or reportedly addressed to students of Economics and/or Business Administration, that can be found in any "average-size" universitary library.

Perhaps the reference "Mathematical models in the Social, Management ald Life Sciences" , by D.N. Burghes and A. D. Wood (Ellis Horwood. Chichester, UK. 1984) is an excellent reference to find "case-studies" related to the main concepts to be developed throughout the semester in this subject matter.

For the sake of completeness, we include below a list (not exhaustive, and by no means the only possible one) of texts in Spanish that can be used in the preparation of some lectures.

LIST (texts in Spanish) FOLLOWS:

  • Arya, Jagdish C.; Lardner, Robin W. (2005) "Matemáticas Aplicadas a la Administración y la Economía" (5ª edición) Pearson Educación.
  • Baum, A. (1992) “Cálculo aplicado” Limusa.
  • Caballero, R. y otros (1993) “Matemáticas aplicadas a la economía y a la empresa (380 ejercicios resueltos y comentados)” Pirámide.
  • Calvo y otros (2003) “Problemas resueltos de matemáticas aplicadas a la Economía y la Empresa” Paraninfo.
  • Canceló, J.R. y otros (1987) “Problemas de álgebra lineal para economistas” Tebar Flores.
  • García Güemes, A. (1992) “Matemáticas aplicadas a la Empresa” A.C.
  • Hoffman, L.D. y Bradley, G.L. (1998) “Cálculo aplicado a Administración, Economía y Ciencias Sociales” (6ª edición) McGraw-Hill.
  • Martínez Estudillo, F. J. (2005) “Introducción a las Matemáticas para la Economía” Desclée de Brouwer.
  • Muñoz Alamillos, A. y otros (2003) “Problemas de matemáticas para economía, administración y dirección de empresas” Ediciones Académicas.
  • Sammamed y otros “Matemáticas I. Economía y Empresa. Problemas” Centro de Estudios Ramón Areces.
  • Vázquez Cueto, J.M. (2002) “Matemáticas Empresariales: Ejercicios planteados y resueltos” CEURA.

Up