# Course guide of Mathematics (2261111)

## Grado (bachelor's degree)

## Branch

## Module

## Subject

## Year of study

## Semester

## ECTS Credits

## Course type

## Teaching staff

### Theory

### Practice

## Timetable for tutorials

### 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)

## 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

- CG02. Cognitive comprehension skills.
- CG03. Ability to analyse and summarise.
- CG04. Ability to organise and plan.
- CG08. Problem-solving skills.
- CG09. Ability to make decisions.
- CG16. Ability to engage in critical and self-critical reasoning.
- CG17. Ability to learn and work autonomously.
- CG24. Ability to apply knowledge to practice.

### Specific competences

- CE11. Know and apply the basic concepts of Mathematics.
- CE12. Use the appropriate tools of Linear Algebra and Differential Calculus in economic analysis.
- CE13. Learn integration methods and their application to the economic and business field.
- CE14. Know about numerical series and learn how to calculate the sum in geometrical series.

## 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.
- Optimization of single-variable functions

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.

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.

## 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

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

## Teaching methods

- MD01. Face-to-face teaching in the classroom
- MD02. Individual work by the student; retrieval, consultation and processing of information; problem solving and practical case studies; and completion of assignments and presentations
- MD03. Individual and/or group tutoring and evaluation

## Assessment methods (Instruments, criteria and percentages)

### Ordinary assessment session

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 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

https://fccee.ugr.es/pages/docencia/guias_titulaciones

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

http://secretariageneral.ugr.es/bougr/pages/bougr112/_doc/examenes%21

Información de interés para estudiantado con discapacidad y/o Necesidades Específicas de Apoyo Educativo (NEAE): Gestión de servicios y apoyos (https://ve.ugr.es/servicios/atencion-social/estudiantes-con-discapacidad).