Within the curriculum of the Degree in Business Administration, two mathematics subjects are included: Mathematics I and Mathematics II; both in the first year. The knowledge about univariate calculus acquired in Mathematics I will be basic for the subject Mathematics II, in which the multivariate calculus is studied; these will be fundamental for the analysis and understanding of economic models. The optimization of functions of one and n-variables, with or without constraints, is a very important tool for making decisions in a wide range of economical situations; therefore, we will study some mathematical procedures in order to solve optimization problems. However, we must not forget that the objective of this subject is to provide the students with the quantitative instruments to be able to raise and analyze rigorously economic problems. For that reason, we suggest exercises related to specific economic contexts, from which the students must be able to identify the main components to formulate the corresponding mathematical model to solve it. In addition, we will incorporate the use of the computer to solve problems.

SPECIFIC COMPETENCES

* An ability to manage basic concepts and techniques of multivariate calculus.

* An ability to justify the procedures and the formulation of logical arguments properly using deductive reasoning.

* An ability to formalise quantifiable phenomena related to economic and business science through mathematical models.



TRANSVERSAL COMPETENCES

* Develop learning skills to acquire a high degree of autonomy in order to undertake studies later, as well as to improve their own self-training in a context of continuous changes and innovations.

* Know how to search, identify, analyse and synthesize information from various sources, with critical capacity to assess the situation and foreseeable evolution of a company, make reasoned judgments and take decisions.

* Capacity for written and oral communication.

* Ability to work with responsibility and respect, initiative and leadership.



KEYWORDS

Calculus for functions of n-variables; Unconstrained and constrained optimization of functions of n-variables; Linear programming; Multiple integral

Unit 1.- Functions of n-variables

1.1 Notions of topology.

1.2 Functions of multiple variables

1.3 Graphic representation. level curves

1.4 Limit of a function.

1.5 Operations with limits.

1.6 Continuity of functions of multi-variable functions.

1.7 Important continuous functions.





Unit 2.- The Derivative for functions of n-variables

2.1 Partial differentiation. Geometrical interpretation

2.2 Gradient vector.

2.3 Second order partial derivatives. Hessian matrix.

2.4 Linear approximation. Plane tangent to a surface

2.5 Taylor series expansion.

2.6 Derivative of a compound function. Chain rule.

2.7 Implicit functions. Derivative.

2.8 Homogeneous function. Properties.



Unit 3.- Optimization of functions of one variable

3.1 Previous definitions. Types of optima

3.2 Necessary conditions for unconstrained local optimum.

3.3 Second order conditions.

3.4 Optimization over an interval

3.5 Concave and convex functions. Points of inflection.



Unit 4.- Optimization of functions of n-variables

4.1 General approach to an optimization problem.

4.2 Types of optima. Theorem of extreme values.

4.3 Unconditional optimization. Necessary condition of local optimum.

4.4 Sufficient condition of local optimum.

4.5 Concave and convex functions. Sufficient condition of global optimum.

4.6 An application: the method of least squares.

Appendix: Quadratic forms.





Unit 5.- Constrained optimization

5.1 General approach and graphic resolution.

5.2 Direct method of resolution by elimination of variables.

5.3 Method of Lagrange multipliers: first-order condition.

5.4 Sufficient condition of local optimum.

5.5 Economic interpretation of the Lagrange multiplier.



Unit 6.- Linear programming

6.1 Geometric representation of small-dimensional problems.

6.2 Introduction to the theory of duality.

6.3 Theorems of duality.

6.4 Economic interpretation of dual variables.

6.5 Solving linear programs using the SOLVER complement of the Excel spreadsheet.



Unit 7.- Multiple integral

7.1 General approach.

7.2 Double integral over a rectangular region.

7.3 Applications of double integrals

Lectures (75%); practical classes (25%).

Practical classes are individual resolution workshops.





- In the event that the health situation, due to the Covid-19, does not allow face-to-face teaching, the teaching would be taught at a distance, using the tools that the University makes available to us.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 75
  • Team projects (problem solving, project design)) (%): 25

GUIDANCE ON CONTINUOUS EVALUATION

Final written test: up to 7.5 points of the mark.

Individual evaluation of practical classes (problem solving, project design, Individual work): up to 2.5 points of the mark.





WAIVER

Students may waive continuous assessment up to 4 weeks before the end of classes

This waiver must be submitted in writing to the course teaching staff.

The students that waive continuous evaluation will get their total mark by means of the final written test.



In the event that the health situation does not allow the final test to be carried out in person, the procedure specified in the eGela student guide for such situations would be activated.

The same criteria as in the ordinary evaluation.

However, those students who have been subject to continuous evaluation may waive it and choose to get their total mark by means of a final written test.

Both in the reprography service of the Faculty of Economics and Business (Gipuzkoa section) and in the virtual platform of the UPV/EHU (eGela), students will have at their disposal the material to be able to follow the lectures, as well as as the collection of exercises proposed for each topic.

Basic bibliography

- Barrios, Javier; Carrillo, Marianela;Gil, María Candelaria; González, Conceptción; Pertano, Celina (2022): Análisis de funciones en economía y empresa. Un enfoque interdisciplinar. 2ª edición. Ediciones Díaz de Santos.

- Sydsaeter, K. Hammond, P. y Carvajal, A. (2012): Matemáticas para el Análisis Económico. Editorial Pearson. Madrid 2012 (2ª edición).

- Sydsaeter, Knut; Hammond, Peter; Strom, Arne; Carvajal, Andrés (2021): Essential Mathematics for Economic Analysis. Pearson Education Limited, 6ª edición. Harlow, Reino Unido.

In-depth bibliography

- Barbolla, R., Cerdá, E. y Sanz, P. (2006): Optimización (Cuestiones, ejercicios y aplicaciones a la economía). Editorial Prentice Hall. Madrid.
- Caballero, R. et al (2000): Matemáticas aplicadas a la Economía y a la Empresa. 380 ejercicios resueltos y comentados. Editorial Pirámide. Madrid.
- Calvo, M. et al. (2003): Problemas resueltos de Matemáticas Aplicadas a la Economía y a la Empresa. Editorial AC. Madrid.
- Hoffman, L. y Bradley, G. (2004): Cálculo aplicado para Administración, Economía y Ciencias Sociales. Editorial McGraw-Hill. Bogotá (8ª edición).
- Larson, R. y Edwards, B.H. (2010): Cálculo 1 (de una variable). Editorial McGraw Hill. México.
- Larson, R. y Edwards, B.H. (2010): Cálculo 2 (de varias variables). Editorial McGraw Hill. México.
- Stewart, J. (2006): Cálculo (conceptos y contextos). Editorial Thomson. México D.F. (5ª edición).

Web addresses

https://www.wolframalpha.com
http://reshmat.ru/linear_programming_online.html