Mathematics for Business

Objectives

One first goal The general objective of the subject is the knowledge of optimization techniques for functions of a real variable. Finding the optimum of a function is vital in the economic and business context: maximizing profits and minimizing costs are two examples. A second goal is to learn the R programming language that provides a wide variety of statistical, numerical and graphical techniques. A programming language is made up of mathematical and logical expressions, formalisms and rules. The R language is free software, the most used for data analysis, and its use and learning will continue in subsequent courses.

The course is divided into two parts. In the first we will learn the general theory for the optimization of functions of real variables, with or without restrictions, making special mention of the theory of convexity and the Kuhn-Tucker conditions. The objective is also to introduce linear programming, through which innumerable real problems of transportation, resource allocation, production scheduling can be formulated... In this direction, this part of the subject covers the mathematical tools that serve as a basis for more specific areas such as Operations Management, Strategic Management or Operations Research. In the second part of the course, the objective is to know the R language and know how to perform a basic descriptive analysis of a data set. This knowledge will be fundamental in the statistics and quantitative methods subjects of the following courses.

The instrumental nature of the subject taught in this subject involves a mostly practical approach. However, in order to be able to solve mathematical problems and exercises, it is essential to know and understand a series of theoretical concepts and to know how to apply them correctly in their resolution. It is a requirement of the subject to have taken the subject "Mathematics", being absolutely necessary to derive functions with ease and solve systems of linear equations using the Gauss pivot method.

Learning outcomes

1. Understands the relationships between magnitudes and expresses them in terms of functions

2. Derivate functions of one and several variables with fluency

3. Apply differential calculus appropriately

4. Pose and solve optimization problems

5. Develops correctly in the general use of ICT and especially in the technological environments specific to the professional field

6. Apply their knowledge, understanding and problem-solving skills in complex or professional and specialized work areas that require the use of creative and innovative ideas.

7. Collects and interprets data and information on which to base their conclusions, including reflections on social, scientific or ethical issues.

Competencies

Specific skills

• Interpret mathematical and statistical data related to the company, to aid in business decision making, using digital technology tools as needed.

Basic skills

• Students can apply their knowledge to their work or vocation in a professional manner and have competencies typically demonstrated through drafting and defending arguments and solving problems in their field of study.
• Students have the ability to gather and interpret relevant data (usually within their field of study) in order to make judgments that include reflection on relevant social, scientific and ethical issues.

Core skills

• Be a critical thinker before knowledge in all its dimensions. Show intellectual, cultural and scientific curiosity and a commitment to professional rigor and quality.
• Display professional skills in complex multidisciplinary contexts, working in networked teams, whether face-to-face or online, through use of information and communication technology.

Content

1. Optimization of functions of one variable:
   1. Maxima and minima of continuous functions on an interval. Weierstrass theorem.
   2. Topological concept of domain boundary.
   3. Maxima and minima of differentiable functions.
   4. Criteria for classifying extrema for differentiable functions.
   5. Absolute maximums and minimums. General criterion for determining maximums and minimums.
   6. Linear approximation of functions.
2. Optimization of functions of several free variables:
   1. Topology in the plane and the extreme value theorem for continuous functions.
   2. Maxima and minima of differentiable functions.
   3. General criterion for determining maximums and minimums.
   4. Hessian criterion for the classification of extremes of differentiable functions.
   5. Convex sets. Convex and concave functions.
   6. Hessian criterion for determining concave and convex functions.
   7. Taylor's approximation.
3. Nonlinear programming:
   1. Lagrange multipliers for functions of two variables and an equality constraint.
   2. Economic interpretation of the Lagrange multiplier.
   3. General case of n variables and equality restrictions.
   4. Mathematical programming of two variables and a constraint with inequality.
   5. General case and Kuhn-Tucker conditions.
4. Linear programming:
   1. Algebra review: Gauss and Kramer.
   2. Graphical resolution of the case of two variables.
   3. The general standard program.
   4. The simplex algorithm.
5. The R programming language:
   1. Introduction.
   2. Objects: vectors, matrices, data sets, lists, functions...
   3. Basic mathematical operations.
   4. Matrices and systems of linear equations.
   5. Numerical optimization.

Evaluation

The course evaluation consists of different actions:

• Participation: 5% (Non-refundable)
• Monitoring and carrying out exercises: 15% (Non-refundable).
• Carrying out practical questionnaires on R: 10% (Non-recoverable).
• Tests:
  • R control: 30% (Non-recoverable)
  • Final synthesis test: 40% (Recoverable*, Minimum grade of 4).

*Only the final synthesis test can be retaken.

To obtain the final grade for the course, you must obtain a minimum grade of 4 in the final synthesis test.

Methodology

We will combine theoretical classes with practical classes where students will carry out exercises and present the solutions together to be able to discuss them. We will do practical classes to get started in the use of the R programming language via the RStudio software.

All classes are face-to-face.

Bibliography

Key references

• Larson, R., ; Edwards, B.H. (2010). Cálculo I (9 ed.). McGraw-Hill.
• Larson, R., Hostetler, R.P., Edwards, B.H. (2006). Cáculo II (8 ed.). McGraw-Hill.
• Sydsaeter, K., Hammond, P. (2011). Matemáticas para el Análisis Económico (2 ed.). Prentice Hall.

Further reading

Teachers will provide complementary bibliography and compulsory reading throughout the course via the Virtual Campus.