The course Mathematics II is part of the basic modules in the first year of this Bachelor's degree. It is the logical complement to the subject Mathematics I, generalizing and developing for functions in several variables, the analysis carried out in the course Mathematics I for functions of one variable.

In this course, we will study some specific methods and techniques, inspired by typical processes in quantitative economic and business decision making, with the emphasis put on Optimization.



After a short general introduction containing some preliminaries with respect to topology, convex sets, concave and convex functions, we mainly focus on three major subjects: analysis, optimization and integral calculus. For each subject we rely on the basic knowledge, understanding and skills previously acquired in other courses in the Bachelor's degree, after which we deepen and expand our study of these themes, and apply them to (social) economics.



The "Analysis" section deals with functions in two or several variables. We look at topics such as function evaluation and derivatives relating to the study of extrema, a rather important subject within a faculty of applied economic sciences.



In the section on integral calculus, we focus on the calculation of double definite integrals. Both two last major sections combined provide the students the way how specific types of (business-) economic problems can be solved by switching to mathematical models.



The third part studies maxima and minima of functions, with or without constraints, with equality constraints and with inequalities. The general results can then be used to generate the basic theorems of linear programming as a special case. The main results are presented on comparative statics. The practical side of the course treats both mathematical and economic applications.



Without losing the basic and applied nature of the course Mathematics I, this course develops and deepens in the linear and non-linear programming tools. This makes it a good support for other courses in the curriculum requiring a sound mathematical basis as, for example, the introduction to the statistical methods.

Taking into account the importance that logical thinking, abstraction and sound mathematical knowledge have in a good part of the professional scope for this Bachelor's degree, this course, together with Mathematics I, will allow students to improve their final portfolio.



At the start of this course the student should have acquired the following proficiencies:

Knowledge corresponding to the final attainment level of secondary school (direction humanities and social sciencies);

Broad basic knowledge of arithmetic, algebra, as well as limits, derivatives, function evaluation and integrals in the case of functions in one variable;

The knowledge and competencies acquired in the course "Mathematics I" of the first Bachelor-year.

The course aims to provide the students with a thorough understanding of the basis of both Differential and Integral Calculus of functions in several variables, as well as Optimization, which are essential to the several Bachelor's degrees offered at our Faculty. The students will be presented the knowledge in that light of an appropriate balance between the conceptual and practical contents. In other words, learning the bare mechanics will not be enough: a rational use of these tools above mentioned requires an understanding of the conceptual framework that supports them.

The goal is therefore that the students grasp an adequate level of familiarity and comprehension of the syllabus.



As for the aptitudes, beyond the more specific in relation to the course syllabus (such as understanding the meaning of the notions of partial derivative as well as Integral besides knowing how to perform their calculation, etcetera), one may emphasize complementary generic skills.

Among them, a meticulous logical reasoning and precise formulation of propositions should not be left unmentioned.





Specific competences of the course:



- To understand and optimally handle the notation and mathematical language.

- To use algebraic expressions correctly.

- To extend the notions acquired in the context of functions of one variable to functions in several variables.

- To get to know the conceptual basis of both Differential and Integral Calculus of real functions in several variables.

- To understand concepts and problems of an economic sphere expressed in mathematical language.

- To pose, solve and analyze optimization problems where functions in several variables play a role.

- To pose, solve and analyze practical exercises involving Differential and Integral Calculus of functions in several variables.





General competences:



- Efficiency in the use basic tools needed in the analysis of economic problems.

- Ability to abstract and analytical thinking.

- Capacity for critical reflection.

- Ability to organize and plan the study itself.

- Ability to communicate in writing using precise language.

- Ability to work independently and in teams.

- Ability to work in teams, with responsibility and respect, initiative and leadership.

- Development of learning skills to acquire a high degree of autonomy, both in order to undertake further studies as of his own self-education, in a frame subject to continuous changes and innovations.

1 Differential Calculus

1.1 Introduction.

1.2 Notion of a set in R and Rn. Representation of sets.

1.3 Real functions in several variables.

1.4 Limits and Continuity (in connection to functions of several variables).

1.5 Differential Calculus (in connection to functions of several variables).

1.6 Functions of a special nature.



2 Optimization in several variables

2.1 Fundamental concepts and general results.

2.2 Optimization without restrictions.

2.3 Optimization with equality restrictions.

2.4 Optimization limited to certain sets.

2.5 Introduction to Linear Programming.



3 Integral Calculus

3.1 Concept of a double integral over bounded areas.

3.2 Properties of double integrals.

3.3 Calculation of double integrals over bounded areas.Theorem of Fubini.

The methodology is based on three types of teacher-student contact. The use of mathematical and symbolic language will always be encouraged as well as rigorous and formal reasoning.

Lectures (class contact teaching)

Each topic introduced is inspired by (economic, social) examples. This motivating introduction is followed by a thorough discussion of the context, to enable students to become familiar with both the meaning and the mechanics of the mathematical topic. Examples in the text are completely worked out and the steps in the calculations are explained with the outmost mathematical rigor.

Practice sessions

Most topics make their first appearance in a lecture, while a few will appear in these practice sessions. Calculus can not be learned by watching someone do it. Because of this, during these sessions, students will be asked to work on skill-based exercises as well as word problems. Word problems are an integral part of teaching calculus in a life science context.



During these classes special emphasis will be made on the ability of students to model, solve and analyze complex problems in a logical and critical manner.



Seminars

A distinctive feature of these features is the diverse applications that it highlights: geometric extrema problems, linear programming, etc... These sessions will have a self-contained treatment of the content, thus permitting a discussion of the different approach to solutions. Full solutions to the exercises are provided and detailed suggestions for further reading are given.

• Sistema de Evaluación Continua
• Sistema de Evaluación Final
• Herramientas y porcentajes de calificación:
  • Prueba escrita a desarrollar (%): 50
  • 2 pruebas parciales (%): 50

The objective is to assess the level of understanding and learning achieved by students. This will be implemented by the organization of two tests and a final exam. The two tests, the first of which will take place when the first chapter has been seen and the second one after chapter 2, will enable students to check their understanding of the content of the course and guide their work to improve, if possible, their results. The final grade will be based on the best of the following two options: (a) the points which result from applying a 50% weight to the final examination and the other 50% to the continuous assessment tests, and (b) the points which result from applying 100% weight to the final exam. In the case of students whose choice of only final evaluation has been granted, their final exam will provide 100% of their grades.

If a student does not write the final exam, his/her qualification will be "No presentado".

The final exam will be the same for all the students of all the groups in the first Bachelor-year.

These exams will consist of questions meant to evaluate the degree of assimilation of the theoretical content of the syllabus, that is, the knowledge of the techniques studied in the course and the understanding of the basic notions that support them. Therefore, exercises solved with the help of purely mechanical algorithms will not be found in these tests, whereas theoretical questions such as definitions of basic concepts or the statement of important results/propositions will form the core of them.

In the second chance the students will be able to get 100% of the qualification.

In the event that, due to the Covid-19, circumstances prevent the face-to-face realisation of any

or all of the tests, the evaluation will be exactly the same, except that these tests will be carried out electronically through the eGela platform.

In this second final exam the student will write a test that assesses the full content of the course.

If this exam cannot be done in the faculty, it will be done electronically through the eGela

platform.

Documents, with illustrative examples of the theory, exercises, written by the teacher, as well as links to content that extend the classroom work and which are available on the virtual platform eGela, are reading.

Bibliografía básica

* Essential Mathematics for Economic Analysis, Knuth Sydsaeter and Peter Hammond, Prentice Hall, (1996).

* Calculus (fourth edition), Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards, Heath and Company (1990).

Bibliografía de profundización

* Mathematic Applications to Business, Economy and Social Sciences, Ronald J. Harshbarger, James J. Reynolds, Mc Graw Hill, (2004)

Direcciones web

www.geogebra.org