Mathematics I is part of the module of basic subjects in the first year of this Degree. Mathematics II complements it. It provides the mathematical tools for the analysis of functions of one variable and the basics of Linear Algebra. Mathematics II completes it addressing the analysis of functions of several variables.

This is the first course in mathematics in the degree. It is covered in the first half of the first year. Its content is basic, aimed at reviewing and reinforcing previous knowledge acquired in the secondary school. The first part is devoted to the analysis of functions of a single variable (limits, continuity, derivatives, graphs, integrals). The last part is devoted to Algebra Lineal.

Given its basic and application-oriented character, it is used both in courses that require basic mathematics and those that require mathematics with a certain level of complexity (i.e. Statistics, Econometrics, Microeconomics). This course will teach students how to make some basic calculus, as graphical representation of functions, derive and integrate functions and solve linear systems of equations.

Given the importance of logical thinking and mathematical knowledge in the professional field, mathematics will improve the professional profile of graduate students.

This course assumes a minimal background of elementary calculus, equation-solving and manipulation of mathematical expressions, which is covered in secondary school. Namely, that covered in first and second year of secondary school in "humanidades y ciencias sociales".

The objective of this subject is to achieve the student to learn and understand the basic concepts of Differential and Integral Calculus of functions of one variable and Linear Algebra which are necessary for this Degree. It must provide the student with this background with a suitable balance between conceptual and practical content. Learning the sheer mechanics is not enough: a rational use of the instruments requires the understanding of the conceptual scaffolding that supports them. Therefore the objective is the student to reach a sufficient level of familiarity and understanding of the syllabus. As to the skills, beyond the most specific ones related to the content of the syllabus (obvious and superfluous to enumerate, e. g., understanding of the meaning of the notion of derivative and knowing how to derive, etc.), it can be underlined as the transversal skills to which this subject should particularly help to develop the rigorous logic reasoning and the precise formulation of propositions.

Specific skills:

- Familiarity with notation and mathematical language,

- Skillful manipulation of algebraic expressions.

- Knowledge of the basic properties of real-valued functions.

- Capacity to understand economic notions and problems stated in mathematical language. Initiation to their formulation.

- Solving and interpreting optimization problems.

General skills:

- Capacity to organizing and planning one's own study.

- Capacity for analytic and critical thought.

- Capacity to communicate by oral and written means in precise terms.

- Capacity to work both in an autonomous way and within a team.

- Develop learning skills in order to improve autonomy, be it for addressing further studies or self-education in a constantly changing world.

1. Preliminaries

1.1 Numbers: N, Z, Q, R

1.2 The straight line R: Operations with real numbers, integer and fractional powers, absolute value

1.3 Order in R, intervals, bounded above/below, infimum (supremum), maximum (minimum)

1.4 The plane R2, straight lines and other important curves



2. Real-valued functions of a single variable

2.1 Notion of real-valued function of a single variable

2.2 Graphical representation of a real-valued function of a single variable

2.3 Basic functions: linear, quadratic, polynomials, rational, trigonometric

2.4 Continuity: meaning and definition. Bolzano's Theorem on intermediate values



3. Differential calculus

3.1 Derivative. Interpretation of the sign and magnitude of the derivative

3.2 Linear approximation

3.3 Calculation of derivatives

3.4 Derivative of a compound function: Chain rule

3.5 Derivative of a function implicitly defined by an equation

3.6 Mean-value theorem of the derivative (Lagrange's theorem)

3.7 Second-order derivative. Interpretation: concavity and convexity

3.8 Second-order approximation

3.9 Local and global maxima and minima

- First-order condition for interior points

- Second-order condition for interior points

3.10 Study and representation of functions



4. Integral calculus

4.1 Calculation of the total accumulated/consumed from the accumulation/consummation rate: "indefinite" integral or anti-derivative and definite integral

4.2 Calculus of primitives: immediate primitives, integration by parts, integration by substitution or change of variable

4.3 Definite integral. Barrow's rule

4.4 Improper integrals: unbounded function or/and unbounded interval



Second part: Algebra



5. The spaces R2, R3 and Rn

5.1 Points and vectors. Operations: sum and product by a scalar; scalar product

5.2 Straight lines and planes

5.3 Linear relationships: linear combination, linear dependence

6. Systems of linear equations and matrices

6.1 Systems of linear equations

6.2 Matrices

6.3 The simplest systems of linear equations: systems in row echelon

6.4 Gaussian elimination

6.4 Homogeneous systems



7. Linear systems, vectors and matrices

7.1 Rank of a matrix and rank of a vector's system

7.2 Corollaries of the results relative to the solution of a linear system:

Linear dependence/independence

Base



8. Determinants

8.1 Determinant and rank. Rank of a matrix and rank of a vector's system

8.2 Calculation of the inverse matrix

The course is based on three types of classes: theoretical, practical and seminars, where the use of mathematical symbolic language, rigorous reasoning and both autonomous and group work, will be fostered.

In theoretical classes the main features of each topic will be presented and illustrated by examples, and students will be oriented to the study with the support of material available in the "virtual room" and basic references. Material required for each class should be available in advance. Explanations will be complemented with the students' participation by solving and discussing exercises. Autonomous work and capacity to argue rigorously using mathematical language will be fostered.

In practical classes the theoretical knowledge acquired in theoretical classes will be applied in different situations. Ideally, these classes should aim at developing their skill to formulate precisely, solve and present mathematically tractable problems. Some of them will be proposed days in advance of addressing their solution in class so as to give the students the opportunity to handle them by themselves.

As a part of the continuous evaluation system, there will be one test towards the half of the semester, which will allow students to check their level and guide them towards where concentrate their work in order to improve their results.

• Sistema de Evaluación Continua
• Sistema de Evaluación Final
• Herramientas y porcentajes de calificación:
  • Prueba escrita a desarrollar (%): 50
  • test (%): 50

The objective of the evaluation is to assess the level of understanding and learning achieved by students.

Students enrolled in the course are entitled to a final exam which will be the same for all students of all groups (and grades).

In addition there will be a first partial test, towards the middle of the semester, corresponding to the first part of the course. Students who had obtained a score equal to or greater than 6 points out of 10 in this first partial test may choose one of the following two options:

a) write a second partial test corresponding to the second part of the course, which will coincide in date and time with the final examination of the course. In such case the final grade shall be the average of the marks obtained in the first and second partial tests;

(b) take part in the final exam.

The final marks of the students writing the final exam will be the one obtained in this examination. Those students who do not write either the final examination or the second partial test, provided that they are entitled to it, will be marked as not submitted to the course.

Both partial tests and the final exam will consist of questions which allow the assessment of the degree of assimilation of the content of the course, that is, knowledge of the techniques and understanding of the basic concepts supporting them. To achieve this goal, these tests will be composed of practical exercises and some questions of theoretical nature, such as definitions of basic concepts and the statement of important results.

If the sanitary conditions allow it, the exams will be in-person. However, given the exceptional and unpredictable circumstances under the "new normal", if it is not possible to conduct an in-person exam, it will be done on-line through the eGela platform. In this case, teachers may request an oral explanation of the tests carried out, through eGela, BlackBoard Collaborate or similar platforms.

A single final examination will be held on the date specified in the Centre's official examination calendar. In this second final exam the student will write a test that assesses the full content of the course.

If the sanitary conditions allow it, the exam will be in-person. However, given the exceptional and unpredictable circumstances under the "new normal", if it is not possible to conduct an in-person exam, it will be done on-line through the eGela platform. In this case, teachers may request an oral explanation of the tests carried out, through eGela, BlackBoard Collaborate or similar platforms.

Available files in the "virtual room".

Bibliografía básica

Knuth Sydsaeter and Peter Hammond: Essential Mathematics for Economic Analysis, Prentice Hall.

Bibliografía de profundización

A. Chiang: Fundamental Methods of Mathematical Economics. McGraw-Hill.

Direcciones web

https://www.wolframalpha.com/