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 diagonalization.
• 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 optimization 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
   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.
2. Differential Calculus of single-variable functions.
   1. Derivatives: geometric interpretation and applications.
   2. Derivatives of elementary functions.
3. Optimization of single-variable functions
   1. Increase and decrease intervals. Concave and convex functions.
   2. Local and global extrema. Weierstrass theorem.
4. Integral Calculus of single-variable functions.
   1. Antiderivatives (primitive functions).
   2. Definite integrals. Barrow’s rule.
5. Basic notions on matrices
   1. General knowledge about matrices: notation, operations and properties.
   2. Computing determinants.
   3. Computing the inverse of a matrix.
6. System of linear equations
   1. Row reduction. Rank of a matrix.
   2. Gaussian elimination.
   3. Rouché- Fröbenius theorem. Cramer’s rule.
   4. Homogeneous systems.
7. Matrix diagonalization
   1. Computing eigenvalues and eigenvectors
   2. Equivalent matrices. Diagonalization: the diagonal and the invertible matrices.
   3. Economic applications.
8. 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.

Seminars / Workshops
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.

• Carvajal, A., Hammond, P.J., Strøm, A. and Sydsaeter, K. Essential mathematics for economic analysis. Harlow UK : Pearson Global Editions, 2021. (online)
• Haeussler, E.F., Paul, R.S. and Wood R.J., Introductory mathematical analysis: For Business, Economics, and The life and Social sciences. Harlow UK: Pearson Global Editions, 2021.
• Larson, R B., R P. Hostetler y B. H. Edwards. Cálculo y geometría analítica. Vol. I (9 Ed.) Mc Graw-Hill, Madrid, 2011.
• Sydsaeter, K. Further mathematics for economic analysis. Harlow : Prentice Hall, 2008.
• Sydsaeter, K., Hammond, P.J. and Stom, A. Essential mathematics for economic analysis. Harlow : Pearson Education Limited, 2016.

• Alegre P. y otros. Matemáticas Empresariales. Ed. AC.
• Balbás A. y otros. Análisis Matemático para la Economía (I y II). Ed. AC.
• Haeussler J.R y Paul R.S. Matemáticas para Administración, Economía, Ciencias Sociales y
  de la Vida. Ed. Prentice Hall, 1997.
• Lax, P.D. and Terrell, M.S.Calculus With Applications. New York, NY : Springer New York : Imprint: Springer, 2014. (online)
• Merino, L. M. y E. Santos. Algebra Lineal con métodos elementales. Thomson, 2006.
• Sydsaeter, K., Hammond, P.J., Matemáticas para el Análisis Económico. Prentice Hall, 2008.
• Stewart J. Cálculo Diferencial e integral. México : International Thomson Editores, 2001.
• Zill, D. y Wright, W. Cálculo de una variable. Mc Graw Hill, 2011.

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  

According to University of Granada Assessment and Grading Regulations (see http://secretariageneral.ugr.es/bougr/pages/bougr71/ncg712/!), continuous and single final
assessment are proposed for this subject.

Continuous assessment will be the default choice, unless another option be formally requested to the Head of the Department (University of Granada Assessment and Grading Regulations). Continuous assessment is divided into two blocks. The score of each block is obtained by gathering the score of a partial test and other activities such as exercises, online tests, seminars/workshops, blackboard exhibitions, homeworks, etc. The breakdown of the grades is the following:

• Block I, related with lessons 1 to 4, will score 5 points maximum.
• Block II, related with lesson 5 to 8, will score 5 points maximum.

The final grade will be:

• The sum of both block grades if this is greater than or equal to 5 points.
• If the sum is less than 5 points, students could make an exam of the blocks where the obtained mark is less than 50% of the maximum score (2 points in Block I, and 3 points in Block II).The final exam will consist of a global test comprising both blocks mentioned before with the same score (that is, maximum score 5 points each).
• If a student takes the part corresponding to one block in the final test, the student drops the previous score in this block. The grade obtained in each block in the final test will substitute the one obtained during the semester. The final grade will be the sum of both block marks.

A single final test on the theoretical and practical contents of the course with a maximum score of 10 points.

The single final assessment will comprise a single test with a maximum score of 10 points. Every detail on the single final assessment regulations by UGR can be found at the following URL:

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

Date and place of the exam will be set by the Faculty (as well as the final exam in the continuous assessment).

All aspect related with both continuous and final assesment will be guided by current assessment
regulations by UGR (http://secretariageneral.ugr.es/bougr/pages/bougr112/_doc/examenes%21)

Information of interest to students with disabilities and/or Specific Educational Support Needs (NEAE): Service and support management (https://ve.ugr.es/servicios/atencionsocial/estudiantes-con-discapacidad).

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).