Course guide of Mathematics (2261111)

Pre-university level of mathematics

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

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.

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.

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.

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

• 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  

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”).

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”).

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”).

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