Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Contents

This subject matter introduces the fundamentals of Calculus and Linear Algebra, and has the following objectives:

• Leading the previous mathematical knowledge and background of the students to a more logical way of reasoning based on structures.
• Knowing and using the main techniques, notations, results and devices related to Differential ad Integral Calculus.
• Applying the new mathematical tools to the analysis and discussion of problems and case-studies, emphasizing the ones coming from contexts related to Economics, Business Administration and/or Decision-Making.

Up

Descriptors

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

Up

General proficiencies

1. The students should develop skills to achieve a fluent communication with their environment, as well as an ability enough to share their experiences and improvements with other students, working in teams.

2. The students should be able to read, write and communicate in more than one language.
In this very case, since the subject matter will be taught in English, it is indeed obvious that the students should be used to working in English. Needless to say that they should have a good level of knowledge of the English language (B2-C1 recommended, if not compulsory).

3. The students should get accustomed to using daily the information technologies and communication tools, having in mind the importance that those technologies will represent along all their future professional work.

4. The students should get used to identifying the main sources of information related to Economics, and, obviously, they should also be able to extract the economic meaning and contents that come from those sources.

5. The students should be able to obtain relevant information from economic databases.
They should do this in a professional way and at an outstanding level, not available to (or not reachable by) non-professional people.

6. The students should tackle the problems through professional criteria leaning on the use of technical skills and tools.

Up

Specific proficiencies

1. 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.

2. 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.

3. 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.

4. The students should reinforce their self-criticism and use it to get a feedback on their own process of learning.

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

Evaluation

• Periodically-scheduled control tests consisting of the resolution of problems and a multi-response test, and practical works to be analyzed in a team, and discussed in the lecture room as well as in group-tutorials: 40% of the final score.
• Final exam (written) in which the students should solve problems or case studies. Some kind of hint previously authorized by the lecturer (e.g.: textbook or excerpt, calculators, computer devices, etc.), could eventually be allowed: 60% of the final score.

From these activities, the only one that could be recovered in a second call, so improving the final mark, is the part corresponding to the final exam. That is, there will be two calls to do the final exam, and the second one can be interpreted as a new opportunity to improve the performance of such a part in the final score. His weight in the final score is again a 60%, keeping the remaining 40% of the score as the corresponding to the marks got along the semester through the other items to be computed, namely the ones aforementioned periodically-scheduled control tests consisting of the resolution of problems and a multi-response test plus  the practical works to be analyzed in a team, and discussed in the lecture room as well as in group-tutorials.

Up

Agenda

PART 1. MATRIX CALCULUS (LINEAR ALGEBRA)

1. Matrix operations.

2. Inverse matrices.

3. Determinants

4. Systems of linear equations.

PART 2. DIFFERENTIAL CALCULUS

5. Real functions, limits and continuity.

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

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

8. Homogeneous functions.

PART 3. OPTIMIZATION

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

10. Optimization (functions of several real variables).

PART 4. INTEGRAL CALCULUS

11. Primitive integral of a real function.

12. Definite integral.

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:

• 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.
• 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

Languages

Spanish.

Up