Course guide

# Mathematics

Academic year 2021 / 2022
Last updated on: 22/06/2021
Approval date: 30/06/2021

## Grado (bachelor's degree)

Bachelor's Degree in Economics

## Branch

Social and Legal Sciences

Formación Básica

Matemáticas

1

1

6

Core course

## Teaching staff

### Theory

• Juan Calvo Yagüe. Groups: Y
• Julia García Cabello. Groups: A y B
• Philippe Bechouche . Groups: C y D

### Practice

• Alejandro Ramos Lora. Groups: 3, 4 y 8
• Juan Calvo Yagüe. Groups: 9
• Julia García Cabello. Groups: 1 y 2
• Manuel Garzón Martínez. Groups: 10
• Philippe Bechouche . Groups: 5, 6 y 7

## Timetable for tutorials

### Juan Calvo Yagüe

juancalvo@ugr.es
• Thursday de 10:00 a 13:00
• Wednesday de 10:00 a 13:00

### Julia García Cabello

cabello@ugr.es
• Monday de 8:00 a 8:30
• Tuesday de 8:00 a 8:30
• Wednesday de 8:00 a 13:00

### Philippe Bechouche

phbe@ugr.es
Second semester
• Monday de 9:00 a 15:00 (Online o Fccee. Desp B02)
First semester
• Monday de 10:00 a 15:00 (Tuturia Online)
• Monday de 15:00 a 15:30 (Fccee. Desp B02)
• Tuesday de 15:00 a 15:30 (Fccee. Desp B02)

### Alejandro Ramos Lora

ramoslora@ugr.es
• de 11:00 a 13:00 (Desp 60. Fac Ciencias)

### Manuel Garzón Martínez

manuelgarzon@ugr.es
• Monday de 16:00 a 18:00

## Prerequisites and recommendations

Pre-university level of mathematics

## Brief description of course content (According to the programme’s verification 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

## Skills

### General Skills

• CG02  - Habilidad de comprensión cognitiva
• CG03  - Capacidad de análisis y síntesis
• CG04  - Capacidad de organización y planificación
• CG08  - Ability to make decisions.
• CG09  - Capacidad para la toma de decisiones
• CG16  - Ethical commitment.
• CG17  - Capacidad de aprendizaje y trabajo autónomo
• CG24  - Capacidad de aplicar los conocimientos en la práctica

### Subject-specific Skills

• CE11  - Conocer y aplicar los conceptos básicos de Matemáticas
• CE12  - Utilizar las herramientas adecuadas del Algebra lineal y del Cálculo Diferencial en el análisis económico
• CE13  - Aprender métodos de integración y su aplicación al campo económico-empresarial
• CE14  - Conocer las series numéricas y aprender a calcular el valor de la suma en las series geométricas

## 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.

## Planned learning activities

### Theory Syllabus

1. Basic notions on single-variable functions
1. Intervals. Domain and range of a function.
2. Elementary functions. Properties.
3. Functions in Economics: supply, demand, incomes, costs, benefits, utility.
4. Limit of a function. Continuity.
5. Bolzano’s theorem.  Applications.

1. Differential and Integral Calculus of single-variable functions
1. Derivatives: geometric interpretation and applications.
2.  Antiderivatives (primitive functions).
3.  Definite integrals. Barrow’s rule.
4. Optimization of single-variable functions

2. Optimization of single-variable functions
1. Increase and decrease intervals. Concave and convex functions.
2. Local and global extrema. Weierstrass theorem.

1. Basic notions on matrices
1.  General knowledge about matrices: notation, operations and properties.
2.  Computing determinants.
3. Computing the inverse of a matrix.

1. System of linear equations
1. Row reduction. Rank of a matrix.
2. Gaussian elimination.
3. Rouché- Fröbenius theorem (Rouché–Capelli theorem). Cramer’s rule.
4.  Homogeneous systems.

1. Matrix diagonalisation
1. Computing eigenvalues and eigenvectors
2. Equivalent matrices. Diagonalisation: the diagonal and the invertible matrices.
3. Economic applications.

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

### Practical Syllabus

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.

Computer-based practices:

• Practice 1. Representation of single-variable functions. Derivatives and antiderivatives.
• Practice 2. Operating with matrices. Solving systems of linear equations. Matrix diagonalization.

• 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)

• 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 learning resources/tools

Department of Applied Mathematics: http://www.ugr.es/~mateapli

## 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 examination diet

1. In the continuous assessment, that 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.
2. 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.
3. 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 examination diet

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-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”).

### Single final assessment (evaluación única final)

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”).

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

Rules for Assessment and grading of the students of the University of Granada (article 8)

## SCENARIO A (CONTACT AND ONLINE TEACHING AND LEARNING)

### Timetable

TIMETABLE

According to Official Academic Organization Plan, Department of Applied Mathematics http://www.ugr.es/~mateapli

Prof. Dr.Julia García Cabello:

L: 8:00 a 8:30

M: 8:00 a 8:30

Mx: 8:00 a 13:00

### Tools for tutorial support (Include virtual tools for tutorial support)

TOOLS FOR TUTORIALS

(Indicate which digital tools will be used for tutorials)

Available at:   http://mateapli.ugr.es

Tutorial will be mainly conducted  by either Google Meet or the UGR email .

• Individual tutorials shall be appointed upon  students’s  request
• Group tutorials could be proposed by  the lectures when necessary

### Adjustment measures for assessment methods (Instruments, criteria and percentages)

• According to the UGR Normative for Adapting the Teaching to the COVID situation held in the academic course 2020-2021 and approved by the Governing Board of UGR on 25-06-20, in scenario A teaching  is considered as a multimodal (hybrid) system combining as many  face-to-face activities  as possible with virtual/online activities (synchronous sessions) together with formative actions  in order to boost the autonomous students learning.
• The ratio  between these two types of learning  (face-to-face and virtual one) shall be fixed by the Faculty of Business and Economics depending on the Sanitary Authorities’ s dictations.  Virtual sessions should be focus on theoretical issues while the face-to-face activities shall  be devoted to both resolving problems and the corresponding assessment sessions.
• Computer practicals shall be carried out in a virtual mode in order to prevent infection by personal contact.
• Virtual sessions shall be performed through the UGR authorized online platforms (Google Meet for instance) with synchronous activities as a priority. However, if Sanitary Authorities should impose an asynchronous scenario, all sessions should be recorded/streamed and stored in  Google drive to be shared with all students. In such case, complementary activities (tutorials, seminars…) shall be carried  out in order to  strengthen training of students. Homework submission shall be performed throughout the UGR authorized online platforms: Prado, Google Meet, Google Drive or by means of the UGR email.
• Additionally, special attention shloud be paid to delivering teaching material to students throughout the online platforms Prado and/or  Google Drive.

### Ordinary examination diet

1. In the continuous assessment, that 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.
2. 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.
3. 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  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 examination diet

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-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”).

### Single final assessment (evaluación única final)

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”).

## SCENARIO B (SUSPENSION OF CONTACT ACTIVITIES)

### Timetable

TIMETABLE

According to Official Academic Organization Plan, Department of Applied Mathematics http://www.ugr.es/~mateapli

Prof. Dr.Julia García Cabello:

L: 8:00 a 8:30

M: 8:00 a 8:30

Mx: 8:00 a 13:00

### Tools for tutorial support (Include virtual tools for tutorial support)

• TOOLS FOR TUTORIALS

(Indicate which digital tools will be used for tutorials)

Available at:   http://mateapli.ugr.es

Tutorial will be mainly conducted  by either Google Meet or the UGR email .

• Individual tutorials shall be appointed upon  students’s  request
• Group tutorials could be proposed by  the lectures when necessary

### Adjustment measures for assessment methods (Instruments, criteria and percentages)

• According to the UGR Normative for Adapting the Teaching to the COVID situation held in the academic course 2020-2021 and approved by the Governing Board of UGR on 25-06-20, in scenario B teaching  is considered as virtual in all its activities with a major priority for synchronous sessions.
• In case of an asynchronous scenario should be imposed, virtual sessions shall be performed through the UGR authorized online platforms (Google Meet). In an asynchronous scenario, all sessions should be recorded/streamed and stored in  Google drive to be shared with all students. In such case, complementary activities (tutorials, seminars…) shall be carried  out in order to  strengthen training of students.
• Homework submission shall be performed throughout the UGR authorized online platforms: Prado, Google Meet, Google Drive or by means of the UGR email.
• Computer practicals shall be carried out in a virtual mode in order to prevent infection by personal contact.
• Additionally, special attention should be paid to delivering teaching material to students throughout the online platforms Prado and/or  Google Drive.

### Ordinary examination diet

1. In the continuous assessment, that 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.
2. 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.
3. 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 throughout the Prado platform.
• Two additional exams on computer  practicals which score 0.5 points each.These shall be carried out throughout the Prado platform.
• 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 throughout the Prado platform.

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 Google Meet sessions.

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 examination diet

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.

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 (evaluación única final)

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”).