# Course guide of Mathematics (2351111)

## Grado (bachelor's degree)

## Branch

## Module

## Subject

## Year of study

## Semester

## ECTS Credits

## Course type

## Teaching staff

### Theory

- María Victoria Fernández Muñoz. Grupos: A y B
- Julia García Cabello. Grupo: Z
- Joaquín Francisco Sánchez Lara. Grupos: C y D
- Elena Torres Lozano. Grupo: D

### Practice

- María Victoria Fernández Muñoz Grupos: 1, 2, 3 y 4
- Carlos Borja Pulido García Grupo: 7
- Joaquín Francisco Sánchez Lara Grupos: 5 y 6
- Elena Torres Lozano Grupo: 8

## Timetable for tutorials

### María Victoria Fernández Muñoz

Email- First semester
- Monday de 12:30 a 13:30
- Tuesday de 10:30 a 13:30
- Wednesday de 08:30 a 09:30
- Thursday de 12:30 a 13:30
- Second semester
- Wednesday
- 10:30 a 12:30
- 15:30 a 16:30

- Thursday de 15:30 a 16:30
- Friday de 08:30 a 10:30

### Julia García Cabello

Email- First semester
- Monday
- 08:00 a 08:30 (Petición de Tutoría Presencial por Email)
- 12:30 a 14:30 (Petición de Tutoría Presencial por Email)

- Tuesday
- 08:00 a 08:30 (Petición de Tutoría Presencial por Email)
- 12:30 a 14:30 (Petición de Tutoría Presencial por Email)

- Wednesday de 08:00 a 08:30 (Petición de Tutoría Presencial por Email)
- Thursday de 08:00 a 08:30 (Petición de Tutoría Presencial por Email)
- Second semester
- Monday de 08:00 a 13:00 (Petición de Tutoría Presencial por Email)
- Tuesday de 08:00 a 08:30 (Petición de Tutoría Presencial por Email)
- Wednesday de 08:00 a 08:30 (Petición de Tutoría Presencial por Email)

### Joaquín Francisco Sánchez Lara

Email- First semester
- Tuesday de 11:30 a 14:00 (Desp B02 - Fac Empresariales)
- Thursday de 09:30 a 13:00 (Desp B02 - Fac Empresariales)
- Second semester
- Tuesday de 10:00 a 13:00 (Desp B02 - Fac Empresariales)
- Wednesday de 10:00 a 13:00 (Desp B02 - Fac Empresariales)

### Elena Torres Lozano

Email- Monday de 12:30 a 13:30
- Wednesday de 09:30 a 10:30

### Carlos Borja Pulido García

Email- Thursday de 11:00 a 13:00

## Prerequisites of recommendations

Pre-university level of mathematics

## Brief description of content (According to official validation report)

The contents developed in the program are mathematical calculation and Linear Algebra:

• Basic concepts of real functions of one variable.

• Differential and integral calculus of real functions of one variable.

• Optimisation of functions of one variable.

• Basics of vectors and matrices.

• Solving systems of linear equations.

• Matrix diagonalisation.

• Numerical sequences and series

## General and specific competences

### General competences

- CG01. Ability to learn and work autonomously.
- CG02. Ability to analyse and search for information from a variety of sources applicable to the field of study.
- CG04. Ability to work in a team.
- CG06. Ability to analyse and summarise.
- CG08. Problem-solving skills in economic and business contexts.

### Specific competences

- CE09. Know and apply theoretical concepts and instrumental techniques and tools for solving economic problems in real-life scenarios.
- CE13. Know the basic mathematical and statistical techniques applied to the economic-business field, and quantitatively analyze the economic-business reality and Interrelate the knowledge acquired in various subjects of the degree in the field of mathematics, statistics and economic theory
- CE14. Know and apply the basic concepts of Mathematics

### Transversal competences

- CT02. Assess, on the basis of the relevant information records, the situation and foreseeable evolution of a company, issue reports on specific company and market situations, and make decisions on the basis of the resulting information.
- CT03. Be able to plan and control the overall management or the various divisions of a company.

## Objectives (Expressed as expected learning outcomes)

1. Acquisition of the basic techniques of mathematics.

2. Gain the ability to lay out economic and business problems with mathematical language.

3. Relate the knowledge acquired with the typical concepts of other subjects of the degree (Statistics, Economic Theory, Accounting ...).

4. Solve problems in the economic and business environment using the most appropriate mathematical techniques.

5. Analyse the economic and business reality quantitatively.

6. Calculate the value of the sums in geometric series.

7. Adequately interpret graphs of functions of one variable.

8. Calculate derivatives and primitives of elementary functions.

9. Solve optimisation of functions of one variable.

10. Solve symbolically abstract matrix equations.

11. Calculate the determinants of low dimensional square matrices.

12. Calculate the inverse matrices of regular low dimensional matrices.

13. Calculate and interpret the eigenvalues and eigenvectors of square matrices.

14. Apply abstract knowledge to problems formulated with economic terminology.

## Detailed syllabus

### Theory

1. Basic notions on single-variable functions

- Intervals. Domain and range of a function.
- Elementary functions. Properties.
- Functions in Economics: supply, demand, incomes, costs, benefits, utility.
- Limit of a function. Continuity.
- Bolzano’s theorem. Applications.

2. Differential and Integral Calculus of single-variable functions

- Derivatives: geometric interpretation and applications.
- Antiderivatives (primitive functions).
- Definite integrals. Barrow’s rule.

3. Optimization of single-variable functions

- Increase and decrease intervals. Concave and convex functions.
- Local and global extrema. Weierstrass theorem.

4. Basic notions on matrices

- General knowledge about matrices: notation, operations and properties.
- Computing determinants.
- Computing the inverse of a matrix.

5. System of linear equations

- Row reduction. Rank of a matrix.
- Gaussian elimination.
- Rouché- Fröbenius theorem (Rouché–Capelli theorem). Cramer’s rule.
- Homogeneous systems.

6. Matrix diagonalisation

- Computing eigenvalues and eigenvectors
- Equivalent matrices. Diagonalisation: the diagonal and the invertible matrices.
- Economic applications.

7. Sequences and series of real numbers

- Sequences of real numbers, operators on sequences, arithmetic and geometric sequences.
- Series of real numbers. Convergence and series convergence tests.
- Sum of a geometric series.

### Practice

Seminars / Workshops (This is non-scoring activity)

At least a one seminar will be performed, whose contents will be selected amongst the following ones:

Seminar 1: Demand and supply equations. Surplus and shortages.

Seminar 2: Taylor series approximation.

Seminar 3: Optimization of basic functions in Economics and Business.

## Bibliography

### Basic reading list

*García Cabello J., Matemáticas imprescindibles en la Administración de empresas:**Ejemplos prácticos y aplicaciones. Ed. Fleming, (2016).**Haeussler J.R y Paul R.S. Matemáticas para Administración, Economía, Ciencias Sociales y de la Vida. Ed. Prentice Hall.**Larson, R B., R P. Hostetler y B. H. Edwards. Cálculo y geometría analítica. Vol. I (9 Ed.) Mc- Graw-Hill, Madrid, (2011).**Merino, L. M. y E. Santos. Algebra Lineal con métodos elementales. Ed. Thomson, (2006).**Stewart J. Cálculo Diferencial e integral. Ed. Thomson.**Sydsaeter, K., Hammond, P.J., Matemáticas para el Análisis Económico. Ed. Prentice Hall.**Zill, D. y Wright, W. Cálculo de una variable. Mc Graw Hill, (2011)*

### Complementary reading

*Alegre P. y otros. Matemáticas Empresariales. Ed. AC.**Balbás A. y otros. Análisis Matemático para la Economía (I y II). Ed. AC.**Caballero R. y otros. Matemáticas Aplicadas a la Economía y la Empresa. Ed. Pirámide.*

## Recommended links

## Teaching methods

- MD01. Docencia presencial en el aula
- MD02. Estudio individualizado del alumno, búsqueda, consulta y tratamiento de información, resolución de problemas y casos prácticos, y realización de trabajos y exposiciones.
- MD03. Tutorías individuales y/o colectivas y evaluación

## Assessment methods (Instruments, criteria and percentages)

### Ordinary assessment session

- In the continuous assessment, the attendance to the corresponding assessment activities is obligatory. Lack of attendance to the assessment activities on the dates and the places specified for it will be understood as a waiver of the right of performance of these activities.
- In order to pass the course under this option, a final mark equal o bigger than 5 is required. Otherwise, the course is considered to be failed. Dates and places for assessment activities will be made public sufficiently in advance.
- In the continuous assessment option, the total score is the sum of all scores corresponding to assessment activities. These are the following:

- Two test-type controls throughout the semester corresponding to units 1,2,3 and 4, 5, 6 respectively. They are non-eliminatory while containing both theory and practical questions (including those of seminars and computer practicals) related to the matter of study. Each of these scored a maximum of 3 points.These shall be carried out either face-to-face or throughout the Prado platform, depending on the current sanitary situation.
- Two additional exams on computer practicals which score 0.5 points each.These shall be carried out either face-to-face or throughout the Prado platform, depending on the current sanitary situation. A final written exam which scores a maximum of 3 points (30% weight versus 70% weight corresponding to all other assessment activities). Date and place for the final written exam will be made public by the Faculty of Economic and Business Sciences. These shall be carried out either face-to-face or throughout the Prado platform, depending on the current sanitary situation.

All these assessment activities could be complemented with personal interviews lectures students, if required by lectures.

The explanations given by students in such interviews therein will be binding when scoring these activities. These shall be carried out either face-to-face or throughout Google Meet sessions, depending on the current sanitary situation.

Students with no attendance to either the 2 partial tests or a partial and the final exam (that is, 2 of the 3 written exams) will have the final mark “Not Having Been Submitted” (“No Presentado”).

### Extraordinary assessment session

It will consist of a single written exam which will be graded on a 0-10 scale (scoring a maximum of 10 points). In order to pass the course under this option, a final mark equal o bigger than 5 is required. Otherwise, the course is considered to be failed.

Date and place for the final written exam will be made public by the Faculty of Economic and Business Sciences. These shall be carried out throughout the Prado platform in case that face-toface assessments are not allowed for the Sanitary Authorities’ s dictations.

Students with no attendance to such final written exam (that scores a maximum of 10 points) will have the final mark “Not Having Been Submitted” (“No Presentado”).

### Single final assessment

According to the Rules for Assessment and grading of the students of the University of Granada (latest changes approved by the Governing Board of 26th October 2016,

http://secretariageneral.ugr.es/bougr/pages/bougr112/_doc/examenes%21) the assessment of students’ academic performance will reflect public, objective and impartial criteria, and will preferably be continuous.

Nevertheless, the students may apply for a single final assessment (article 8 of the current Rules for Assessment, which provides for the taking of a single final assessment). Students may apply for either in the first two weeks of teaching of the subject or two weeks following change of matriculation. Application is to be made through the electronic system (https://sede.ugr.es/sede/catalogo-de-procedimientos/solicitud-evaluacion-unica-final.html), citing and accrediting the reasons for not being able to undergo the system of continuous assessment (reasons of employment, health, disability or any other correctly justified cause), with the understanding that this assessment is undertaken in a single academic act in order to accredit that the student has acquired in full the competencies described.

On one hand, lack of application for a single final assessment will be understood as a waiver of the right of such assessment. On the other hand, those students who have been granted a single final assessment are not eligible to apply for a continuous assessment.

Single Final Assessment shall consist of a single written exam which will be graded on a 0-10 scale (scoring a maximum of 10 points). In order to pass the course under this option, a final mark equal o bigger than 5 is required. Otherwise, the course is considered to be failed. Date and place for the final written exam will be made public by the Faculty of Economic and Business Sciences. These shall be carried out throughout the Prado platform in case that face-to-face assessments are not allowed for the Sanitary Authorities’ s dictations.

Students with no attendance to such final written exam (that scores a maximum of 10 points) will have the final mark “Not Having Been Submitted” (“No Presentado”).

## Additional information

- PRADO online platform
- Student Guides, where schedules, methodologies, timetables are fully described at https://fccee.ugr.es/pages/docencia/guias_titulaciones