Identifying Data  2019/20  
Subject (*)  Mathematics II  Code  611G02010  
Study programme 


Descriptors  Cycle  Period  Year  Type  Credits  
Graduate  2nd fourmonth period 
First  Basic training  6  
Language 


Teaching method  Facetoface  
Prerequisites  
Department  Economía 

Coordinador 



Lecturers 



Web  http://moodle.udc.es  
General description 
O obxectivo deste curso é presentar aos alumnos os conceptos básicos do cálculo diferencial en varias variables e a programación matemática, que serán necesarios para a aprendizaxe doutras disciplinas do grao e para a súa carreira futura. O estudante deberá entender os conceptos básicos presentados e os resultados que os relacionan, e aplicar ese coñecemento de forma adecuada e rigorosa para resolver problemas prácticos. Farase unha énfase especial na aplicación dos contidos do curso a problemas de natureza económica e á interpretación dos resultados obtidos. Tamén se pretende axudar os alumnos a desenvolver habilidades xenéricas, como a capacidade de análise e síntese, a capacidade de razoamento lóxico, a capacidade de resolución de problemas, o pensamento crítico, a aprendizaxe independente, ou a capacidade de recuperar e utilizar información de varias fontes. 

Contingency plan 

Learning aims 
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 secondorder 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 equalityconstrained 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 
Contents 
Topic  Subtopic 
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.  Firstorder necessary conditions. Secondorder conditions. The convex case. Sensitivity analysis. 
7. Equalityconstrained optimization  Formulation. Firstorder necessary conditions: the Lagrange theorem. Secondorder conditions. The convex case. Sensitivity analysis. 
8. Linear programming.  Formulation of linear programs. Basic feasible solutions. Fundamental theorems. The simplex algorithm. 
Planning 
Methodologies / tests  Competencies  Ordinary class hours  Student’s personal work hours  Total hours 
Introductory activities  A6 A9 A12 C1  1  0  1 
Multiplechoice 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 
Methodologies  Description 
Introductory activities  It will be the presentation of the course (one hour). 
Multiplechoice questions  There will be two multiplechoice 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 inclass 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. 
Personalized attention 


Assessment 
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 
Multiplechoice questions  A10 B2 B3 B4  There will be two multiplechoice 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 partialtime enrollment must fill the same requirements for assessment that students on fulltime enrollment. Continuous assessment will consist of two inclass multiplechoice quizzes (10% each) and two inclass exams (15% each). Nonattendance 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 nonjustified absence. The qualification of NOTTAKEN 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 nongraphic and nonprogrammable. Exams written in pencil will not be admitted. Virtual Platform: It will be used the Moodle virtual platform (http://moodle.udc.es). 
Sources of information 
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, McGrawHill I. Pérez Grasa, G. Jarne y E. Minguillón (1997). Matemáticas para la economía: álgebra lineal y cálculo diferencial . Madrid, McGrawHill I. Pérez Grasa, G. Jarne y E. Minguillón (2001). Matemáticas para la economía: programación matemática y sistemas dinámicos . Madrid, McGrawHill 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, McGrawHill R. M. Barbolla, E. Cerdá y P. Sanz (2001). Optimización. Cuestiones, ejercicios y aplicaciones a la economía . Madrid, Prentice Hall 

Recommendations 
Subjects that it is recommended to have taken before  

Subjects that are recommended to be taken simultaneously 
Subjects that continue the syllabus 
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). 