Matemáticas

General information

Type:	BAS	Curs:	1	Period:	S semester
ECTS Credits:	6 ECTS

Teaching Staff:

Group	Teacher	Department	Language
Sec: A	Núria Agell Jané	Operaciones, Innovación y Data Sciences	CAT
Sec: A	Xari Rovira Llobera	Operaciones, Innovación y Data Sciences	CAT

Group	Teacher	Department	Language
Sec: B	Xari Rovira Llobera	Operaciones, Innovación y Data Sciences	ESP
Sec: B	Francesc Prats Duaygues	Esade	ESP

Group	Teacher	Department	Language
Sec: C	Xari Rovira Llobera	Operaciones, Innovación y Data Sciences	CAT

Prerequisites

Basic knowledge of the functions of a variable.

Workload distribution

Lectures: 46 hours
Participatory sessions: 20 hours
Independent study: 108 hours
Tutorials / feedback: 6 hours

COURSE CONTRIBUTION TO PROGRAM

Mathematics is an essential tool for quantitative information analysis, the creation and interpretation of models to explain the economic and financial reality of the business environment and for the development of structured reasoning processes.

Course Learning Objectives

After completing this course, students should be able to:

- Recognise and use mathematical language fluently in specific situations.
- Understand, relate and use the concepts and theoretical models of basic matrix algebra and one and two-variable calculus to apply them over the course of their academic and professional career.
- Use mathematical reasoning and demonstrate basic propositions.
- Demostrate rigorous dedductive proceses.

Competences

1. Knowledge acquisition, comprehension and structuring
5. Ability for continuous learning/Ability to learn continually
2. Application of knowledge to achieve results
4. Conveying information and/or knowledge

1. Knowledge acquisition, comprehension and structuring

5. Ability for continuous learning/Ability to learn continually

2. Application of knowledge to achieve results

4. Conveying information and/or knowledge

Relation between Activities and Competences

	1	5	2	4
Completion of exercises, class participation and tutorials
Weekly tests
Mid-term exam
Final exam
Lectures

CONTENT

1. Matrices and simultaneous equations Aims of this block: Upon completing this section, students should be able to: -Use the matrix calculus language. -Work fluently with symbolic matrices. -Know how to resolve problems and develop simple equations involving matrices and linear equations. Content: -Definitions and types of matrices. Vectors and Euclidean space (Rn) -Operations with matrices. -Determinants. -Linear combination and independence. -Matrix rank. -Linear systems. Activities: -5 hours of lectures 3 hours of participatory classes
2. Matrix diagonalisation Aims of this blcok: Upon completing this section, students should be able to: -Master matrix calculus language and use symbolic notations. -Calculate square matrix eigenvalues and eigenvectors. -Understand square matrix and symmetrical matrix diagonalisation processes. Content: -Square matrix eigenvalues and eigenvectors. -Square matrix diagonalisation and that of symmetrical matrices. Activities: -5.5 hours of lectures 1.5 hours of participatory classes
3. Quadratic forms Aims of this block: After completing this block, students should be able to -Master the language of matrix calculus and the use of symbolic notation - Recognise the concepts and applications of quadratic forms -Classify quadratic forms -Master the concepts of scalar product, norm and distance in Rn (Euclidean space) Content: -Definition and classification of quadratic forms -Sign of a quadratic form -Scalar product, norm and distance Activities: -5 hours of lectures -3 hours of participatory classes
4. Real numbers and elemental functions Aims of the section: Upon completing this section, students should be able to: -Use one-variable calculus language. -Use different number sets, functional elements and related elements. Content: -Real numbers. -Operations. -Elemental functions: Expression, domain, range and graphs. -Inverse function. Activities: 3.5 hours of lectures 1.5 hours of participatory classes
5. Study of one-variable functions Aims of this block: Upon completing this section, students should be able to: -Use one-variable calculus language. -Understand and use the concepts of limits, continuity and function derivatives. -Interpret simple-function graphs. -Develop simple reasoning using these concepts in concrete applications. Content: -Functional limit. -Indeterminate forms. -Continuity and types of function discontinuity. -Derivative of a function. -Derived function. -Calculus rules. -Derivable function properties. -Derivative applications to interpret graph functions. Activities: 7 hours of lectures 3 hours of participatory classes
6. Study of two-variable functions Aims of this block: Upon completing this section, students should be able to: -Use two-variable calculus language. -Understand and use concepts related to two-variable functions. -Represent and interpret level curves and use them in specific applications. Content: -Basic geometric concepts. -Level curves: Representing two-variable functions. -Differentiable functions: Partial and directional derivatives. -Critical points. Necessary condition of extremality. -Conditions for second-order determination of extremes. -Constrained optimisation of functions. Resolving graphs. Activities: 10.5 hours of lectures 4.5 hours of participatory classes
7. Simple and double integrals Aims of this block: Upon completing this section, students should be able to: -Use integral calculus language. -Understand and use the integral concept and resolve simple integrals. -Represent areas on a plane and resolve some double integrals. Content: -The defined integral of a real function. -Indefinite integral: Antiderivatives. -Double integral concept. Properties. -Double integral calculus. -Fubini's Theorem. -Simple and double improper integrals. Activities: 9 hours of lectures 3 hours of participatory classes

1. Matrices and simultaneous equations

Aims of this block:
Upon completing this section, students should be able to:
-Use the matrix calculus language.
-Work fluently with symbolic matrices.
-Know how to resolve problems and develop simple equations involving matrices and linear equations.

Content:
-Definitions and types of matrices. Vectors and Euclidean space (Rn)
-Operations with matrices.
-Determinants.
-Linear combination and independence.
-Matrix rank.
-Linear systems.

Activities:
-5 hours of lectures
3 hours of participatory classes

2. Matrix diagonalisation

Aims of this blcok:
Upon completing this section, students should be able to:
-Master matrix calculus language and use symbolic notations.
-Calculate square matrix eigenvalues and eigenvectors.
-Understand square matrix and symmetrical matrix diagonalisation processes.

Content:
-Square matrix eigenvalues and eigenvectors.
-Square matrix diagonalisation and that of symmetrical matrices.

Activities:
-5.5 hours of lectures
1.5 hours of participatory classes

3. Quadratic forms

Aims of this block:
After completing this block, students should be able to
-Master the language of matrix calculus and the use of symbolic notation
- Recognise the concepts and applications of quadratic forms
-Classify quadratic forms
-Master the concepts of scalar product, norm and distance in Rn (Euclidean space)
Content:
-Definition and classification of quadratic forms
-Sign of a quadratic form
-Scalar product, norm and distance

Activities:
-5 hours of lectures
-3 hours of participatory classes

4. Real numbers and elemental functions

Aims of the section:
Upon completing this section, students should be able to:
-Use one-variable calculus language.
-Use different number sets, functional elements and related elements.

Content:
-Real numbers.
-Operations.
-Elemental functions: Expression, domain, range and graphs.
-Inverse function.

Activities:
3.5 hours of lectures
1.5 hours of participatory classes

5. Study of one-variable functions

Aims of this block:
Upon completing this section, students should be able to:
-Use one-variable calculus language.
-Understand and use the concepts of limits, continuity and function derivatives.
-Interpret simple-function graphs.
-Develop simple reasoning using these concepts in concrete applications.

Content:
-Functional limit.
-Indeterminate forms.
-Continuity and types of function discontinuity.
-Derivative of a function.
-Derived function.
-Calculus rules.
-Derivable function properties.
-Derivative applications to interpret graph functions.

Activities:
7 hours of lectures
3 hours of participatory classes

6. Study of two-variable functions

Aims of this block:
Upon completing this section, students should be able to:
-Use two-variable calculus language.
-Understand and use concepts related to two-variable functions.
-Represent and interpret level curves and use them in specific applications.

Content:
-Basic geometric concepts.
-Level curves: Representing two-variable functions.
-Differentiable functions: Partial and directional derivatives.
-Critical points. Necessary condition of extremality.
-Conditions for second-order determination of extremes.
-Constrained optimisation of functions. Resolving graphs.

Activities:
10.5 hours of lectures
4.5 hours of participatory classes

7. Simple and double integrals

Aims of this block:
Upon completing this section, students should be able to:
-Use integral calculus language.
-Understand and use the integral concept and resolve simple integrals.
-Represent areas on a plane and resolve some double integrals.

Content:
-The defined integral of a real function.
-Indefinite integral: Antiderivatives.
-Double integral concept. Properties.
-Double integral calculus.
-Fubini's Theorem.
-Simple and double improper integrals.

Activities:
9 hours of lectures
3 hours of participatory classes

Relation between Activities and Contents

	1	2	3	4	5	6	7
Completion of exercises, class participation and tutorials
Weekly tests
Mid-term exam
Final exam
Lectures

Methodology

Lectures and participatory classes:

Lectures:Faculty will combine theoretical explanations and exercises. 15 minutes of each of the Thursday classes are given over to a weekly test. Students will be able to consult the theoretical notesavailable on the course website when completing these tests.

Participatory classes: in the first class, work groups will be created to discuss the exercises.
The participatory class consists of 2 parts:
- Discussion of the previously assigned exercises which the students will have completed prior to the participatory class. In this first part, faculty will address any queries arising about the exercises in the lists corresponding to each block.
- Working in groups of 3 (or a maximum of 4) students, students will complete exercises set by faculty during the class. Groups must hand in these exercises at the end of the class.

Compulsory tutorials: After each participatory class on Thursday, a group of students will have to attend a compulsory tutorial to assess their competences.

Voluntary tutorials, which will he given in person or on-line, are available to those students that require help with completing the exercises and the use of the different tools.

ASSESSMENT

ASSESSMENT BREAKDOWN

Description	%
Completion of exercises, class participation and tutorials	10
Weekly tests	30
Mid-term exam	20
Final exam	40

Assessment criteria

The correct use of concepts and theoretical models, and mastery of mathematical notation achieved through the completion of exercises that must be undertaken both through independent, out-of-class study as well as through group-work activities during the participatory classes.
The ability to carry out reasoning and simple demonstrations will be assessed through discussion and problem-solving activities in the classroom during the participatory classes and through weekly tests (ongoing assessment) along with the mid-term and final exams.
Also, through question and answer sessions during the required tutoriels, faculty will assess if students understand what's demanded of them in the question at hand and if they can clearly and succinctly explain the steps taken to resolve the problem and, if applicable, distinguist the different ways of addressing the problem.

The students' final marks will be based on the following activities and their respective percentages:
10% - Exercises, class participation and tutorials
30% - Weekly tests. This component will consist of the average mark obtained on all the weekly tests after discarding the 4 worst marks
20% - Mid-term exam on the first 3 blocks dedicated to algebra. Students who receive a mark under 5 (out of 10) on this exam will have to retake this material as part of the final exam in January.
40% - Final exam in January on the 4 blocks dedicated to calculas.
There is no minimum mark for any of these 4 components.

For students who must retake the course or those who can only sit the exam, 100% of the mark for the course will be the mark they obtain on the final exam in January, completing both the calculus and algebra exams. If students fail to take either of these, they will be considered to have not taken the exam as a whole.

If the mark on the final exam is below a 5, the mark they obtain in the re-sit will represent 100% of the mark for the class.

For students to be able to asess their own learning and progress, they will receive feedback from faculty throughout the course. This feedback will be given in different ways depending on the evaluation activity (corrected weekly tests, feedback during participatory classy and tutorials, voluntary tutorials, etc.).

The degree to which students acquire the competencies foreseen for this course will be assessed based on the problem-solving activities carried out during participatory classes and discussions during the required tutorials. This assessment may be reflected as a mark from 0 to 10 and, in addition, it will be reflected by means of indicators used in the students' different courses, serving to track each student's progress throughout the programme.

Bibliography

Short bibliography:

Blanco, S., et al. Matemáticas empresariales I. Enfoque teórico-práctico. Vol 1: "Álgebra lineal". Vol 2: "Cálculo diferencial". Editorial AC-Thomson, 2003.

Cámara, A., et al. Problemas resueltos de matemáticas para economía y empresa. Editorial AC-Thomson, 2003.

Jarne, G, et al. Matemáticas para la economía. Álgebra lineal y cálculo diferencial. McGraw-Hill, 2001.

Larson, R., et al. Cálculo, vol. 1 & 2. McGraw Hill, 2006 (8th ed.).

Chiang, Alpha C. Métodos fundamentales de economía matemática. McGraw Hill, DL, 1994 (3rd ed.).

Camacho, E. (ed.). Fundamentos de cálculo para economía y empresa. Delta Publicaciones, 2006.

Getino, J. et al. Fundamentos de matemáticas. Introducción al álgebra lineal y al cálculo en una y varias variables. Delta Publicaciones, 2007.

Sydsaeter, K et al. Matemáticas para el análisis económico. Ed. Pearson, (2nd ed.).

Timetable and sections

Group	Teacher	Department
Sec: A	Núria Agell Jané	Operaciones, Innovación y Data Sciences
Sec: A	Xari Rovira Llobera	Operaciones, Innovación y Data Sciences

Timetable Sec: A

From 2014/9/15 to 2014/12/18:
Each Monday from 14:30 to 16:00.
Each Thursday from 10:00 to 11:30.
Each Thursday from 8:00 to 10:00.
Each Thursday from 11:30 to 12:00.
Each Monday from 16:00 to 17:00.

From 2014/10/1 to 2014/12/17:
Each Wednesday from 15:00 to 16:00.

Saturday 2014/10/25 from 9:00 to 12:00.

Group	Teacher	Department
Sec: B	Xari Rovira Llobera	Operaciones, Innovación y Data Sciences
Sec: B	Francesc Prats Duaygues	Esade

Timetable Sec: B

From 2014/9/15 to 2014/12/18:
Each Monday from 13:45 to 14:15.
Each Thursday from 17:00 to 18:30.
Each Monday from 12:15 to 13:45.
Each Thursday from 18:30 to 19:00.
Each Thursday from 15:00 to 17:00.
Each Thursday from 14:00 to 15:00.

From 2014/10/2 to 2014/12/18:
Each Thursday from 12:00 to 13:00.

Saturday 2014/10/25 from 9:00 to 12:00.

Group	Teacher	Department
Sec: C	Xari Rovira Llobera	Operaciones, Innovación y Data Sciences

Timetable Sec: C

From 2014/9/15 to 2014/12/18:
Each Thursday from 17:00 to 18:30.
Each Monday from 12:15 to 13:45.
Each Thursday from 18:30 to 19:00.
Each Thursday from 15:00 to 17:00.
Each Thursday from 14:00 to 15:00.

From 2014/10/2 to 2014/12/18:
Each Thursday from 12:00 to 13:00.

Saturday 2014/10/25 from 9:00 to 12:00.

Matemáticas (BBA50000)