The first and second opportunities will be graded in the same way. Students with partial-time enrollment must fill the same requirements for assessment that students on full-time enrollment.

Continuous assessment will consist of two in-class multiple-choice quizzes (10% each) and two in-class exams (15% each). Non-attendance to more than four class sessions (lecture, practice or seminar) will lead to not computing the continuous assessment qualification, which represents a 50% of the final mark. To qualify an absence as justified or not we will follow the provisions of Article 12, points 1 and 5, of the Normas de avaliación, revisión e reclamación das cualificacións dos estudos de grao e mestrado universitarios. In case of disrespectful behavior with peers or teacher, or using electronic devices (tablet, computer, telephone, ...) or other material unrelated to the class activities, you will be required to leave the classroom, and it will be counted as a non-justified absence.

The qualification of NOT-TAKEN will also be awarded to the student who has only participated in assessment activities that have a weighting below 20% of the final grade, regardless of the qualification obtained.

The final grade for students applying to the call of December will be the qualification of the final exam valued on 10 points.

Conditions for carrying out exams: During the examination, you cannot have access to any device that allows communication with the outside and/or storage of information. Entry to the examination room with these devices may be denied. The student may use a scientific calculator non-graphic and non-programmable. Exams written in pencil will not be admitted.

Virtual Platform: It will be used the Moodle virtual platform (http://moodle.udc.es).

Learning outcomes	Study programme competences
Identify the notable sets of a subset of IRn.	A8 A11
Understand the basic concepts of the euclidean space IRn.	A8 A11
Determine if a set is open, closed, bounded, compact and convex.	A8 A11
Understand the concept of function of several variables.	A8 A11
Draw the level set of a function of two variables.	A8 A11
Understand the concept of continuous function.	A8 A11
Determine if a function is continuous or not.	A8 A11
Recognize a linear function.	A8 A11
Recognize a quadratic form.	A8 A11
Classify a quadratic form by examining the signs of the principal minors.	A8 A11
Classify a constrained quadratic form.	A8 A11
Calculate and interpret partial derivatives and elasticities.	A4 A8 A11	B1 B2 B5 B10	C1 C7
Find the Taylor polynomial of a function.	A8 A11
Calculate the partial derivatives of a compounded function.	A8 A11
Use the existence theorem to analyze if a equation defines an implicit real function.	A8 A11
Find the partial derivatives and elasticities of an implicit function, and interpret them.	A8 A11
Analyze the concavity/convexity of a function.	A8 A11
Formulate mathematical programming problems.	A3 A4 A6 A8 A9 A10 A11	B1 B2 B3 B4 B5 B10	C1 C4 C5 C6 C7 C8
Distinguish between local and global optima.	A8 A11
Graphically solving an optimization problem	A8 A11	B3
Analyze the existence of global optima using the Weierstrass theorem.	A8 A11
Find the critical points of a function of several variables.	A8 A11
Classify the critical points using the second-order conditions.	A8 A11
Determine the local or global character of the optima of an unconstrained problem.	A8 A11
Formulate economic problems as mathematical programs with equality constraints.	A8 A11
Find the critical points of a mathematical program with equality constraints.	A8 A11
Classify the critical points and interpret the Lagrange multipliers.	A8 A11
Determine the local or global character of the optima of an equality-constrained problem.	A8 A11
Know the structure and basic properties of a linear program.	A8 A11
Formulate simple economic problems as linear programs.	A3 A4 A8 A11 A12	B1 B2 B3 B4 B5 B10	C1 C4 C6 C7 C8
Solve linear programs by the simplex algorithm.	A3 A4 A6 A8 A9 A11	B1 B2 B3 B4 B5 B10	C1 C4 C5 C6 C7 C8

Topic	Sub-topic
1. The euclidean space IRn.	The vector space IRn. Inner product. Norm. Distance. Interior, closure, isolated, limit and boundary points. Open and closed sets. Compact sets.
2. Functions of several variables.	Basic concepts. Graphical representation of real functions. Level sets. Limit of a function at a point. Continuity. Linear functions. Quadratic forms. Classification. Constrained quadratic forms.
3. Derivatives of functions of several variables.	Partial derivatives. Partial derivatives of higher order. Class one function Chain's Rule. Taylor's theorem. Implicit function theorem.
4. Convexity of sets and functions.	Convex sets. Properties. Convex functions. Properties. Characterization of twice continuously differentiable convex functions.
5. Introduction to mathematical programming.	Formulation of a mathematical program. Local and global optima. Graphic solving. Basic Theorems in optimization.
6. Unconstrained optimization.	First-order necessary conditions. Second-order conditions. The convex case. Sensitivity analysis.
7. Equality-constrained optimization	Formulation. First-order necessary conditions: the Lagrange theorem. Second-order conditions. The convex case. Sensitivity analysis.
8. Linear programming.	Formulation of linear programs. Basic feasible solutions. Fundamental theorems. The simplex algorithm.

Methodologies / tests	Competencies	Ordinary class hours	Student’s personal work hours	Total hours
Introductory activities	A6 A9 A12 C1	1	0	1
Multiple-choice questions	A10 B2 B3 B4	2	7	9
Mixed objective/subjective test	A10 B2 B3 B4	3	15	18
Guest lecture / keynote speech	A3 A4 A8 A9 A11 A12 B1 B5 C6 C7	15	15	30
Seminar	B10 C4 C5 C8	2	4	6
Practical test:	A8 A11 B1 B2 B3 B4 B5 C1	2	8	10
Problem solving	A6 B1	25	50	75

Personalized attention		1	0	1

(*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students.

Methodologies	Description
Introductory activities	It will be the presentation of the course (one hour).
Multiple-choice questions	There will be two multiple-choice exams. These exams will have questions with several given answers --only one will be correct-- related to theoretical and practical concepts covered in the course.
Mixed objective/subjective test	At the end of the course, there will be a mixed (theoretical/practical) exam. This exam will take place at the official date determined by the Faculty.
Guest lecture / keynote speech	There will be 15 hours of keynote speech, that will be focused on the exposition of the theoretical contents.
Seminar	The group will be divided into two subgroups for the seminars.
Practical test:	There will two in-class practical exams.
Problem solving	There will be 25 hours of problem solving classes, which will be focused on the formulation and solving of problems related to the practical contents of the subject.

Methodologies	Competencies	Description	Qualification
Practical test:	A8 A11 B1 B2 B3 B4 B5 C1	There will be two presential exams. Each of them will represent a 15% of the final grade (1.5 points each). It will be valued a good understanding of the concepts, the use of appropriate reasoning, the proper use of mathematical language, and the skills in formulating and solving problems.	30
Mixed objective/subjective test	A10 B2 B3 B4	The final (presential) exam will represent a 50% of the final mark (5 points). It will be valued a good understanding of the concepts, the use of appropriate reasoning, the proper use of mathematical language, and the skills in formulating and solving problems.	50
Multiple-choice questions	A10 B2 B3 B4	There will be two multiple-choice presential exams. Each of them will represent a 10% of the final grade (1 point each).	20

Assessment comments
The first and second opportunities will be graded in the same way. Students with partial-time enrollment must fill the same requirements for assessment that students on full-time enrollment. Continuous assessment will consist of two in-class multiple-choice quizzes (10% each) and two in-class exams (15% each). Non-attendance to more than four class sessions (lecture, practice or seminar) will lead to not computing the continuous assessment qualification, which represents a 50% of the final mark. To qualify an absence as justified or not we will follow the provisions of Article 12, points 1 and 5, of the Normas de avaliación, revisión e reclamación das cualificacións dos estudos de grao e mestrado universitarios. In case of disrespectful behavior with peers or teacher, or using electronic devices (tablet, computer, telephone, ...) or other material unrelated to the class activities, you will be required to leave the classroom, and it will be counted as a non-justified absence. The qualification of NOT-TAKEN will also be awarded to the student who has only participated in assessment activities that have a weighting below 20% of the final grade, regardless of the qualification obtained. The final grade for students applying to the call of December will be the qualification of the final exam valued on 10 points. Conditions for carrying out exams: During the examination, you cannot have access to any device that allows communication with the outside and/or storage of information. Entry to the examination room with these devices may be denied. The student may use a scientific calculator non-graphic and non-programmable. Exams written in pencil will not be admitted. Virtual Platform: It will be used the Moodle virtual platform (http://moodle.udc.es).

Basic	K. Sydsæter, P. J. Hammond y P. Carvajal (2012). Matemáticas para el análisis económico . Madrid, Pearson

Complementary	R. Caballero, S. Calderón, T. P. Galache, A. C. González, Mª. L. Rey y F. Ruiz (2000). Matemáticas aplicadas a la economía y la empresa. 434 ejercicios resueltos y comentados . Madrid, Pirámide E. Minguillón, I. Pérez Grasa y G. Jarne (2004). Matemáticas para la economía. Libro de ejercicios. Álgebra lineal y cálculo diferencial. Madrid, McGraw-Hill I. Pérez Grasa, G. Jarne y E. Minguillón (1997). Matemáticas para la economía: álgebra lineal y cálculo diferencial . Madrid, McGraw-Hill I. Pérez Grasa, G. Jarne y E. Minguillón (2001). Matemáticas para la economía: programación matemática y sistemas dinámicos . Madrid, McGraw-Hill M. Hoy, J. Livernois, C. McKenna, R. Rees y T. Stengos (2001). Mathematics for economics. Cambridge, MA, The MIT Press A. C. Chiang y K. Wainwright (2006). Métodos fundamentales de economía matemática . Madrid, McGraw-Hill R. M. Barbolla, E. Cerdá y P. Sanz (2001). Optimización. Cuestiones, ejercicios y aplicaciones a la economía . Madrid, Prentice Hall

Other comments
It is advisable to have passed Mathematics I. Students must be familiar with the concepts and fundamental results of linear algebra (matrices, determinants and systems of linear equations), and differential calculus in one variable (limit, continuity, derivative, elasticity, optima, convexity).