**Academic Year/course: 2017/18**

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**417 - Degree in Economics**

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**27402 - Mathematics I**

__Syllabus Information__

__Syllabus Information__

**Academic Year:**

**Subject:**

**Faculty / School:**

**Degree:**

**ECTS:**

**Year:**

**Semester:**

**Subject Type:**

**Module:**

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**5.1. Methodological overview**

The objective of this subject is that the students should develop the analytical skills, rigour and intuition needed for using mathematical concepts and results and that they should be able to apply these abilities to the analysis of problems of an economic nature. Therefore, the teaching should aim to provide students with a solid mathematical knowledge and to train them in a way of reasoning that will allow them thereafter to successfully solve a wide variety of questions in an economic scenario.

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**5.2. Learning tasks**

The program offered to the students to help them achieve the learning results includes the following activities**:**

**1. ****Theoretical lessons** which will be based on lectures to present the concepts and results corresponding to the contents. At the same time, some exercises will be solved with the participation of the students to help them comprehend the theoretical concepts presented. These classes are face-to-face and will be given to the full group.

Time allotted: 1,2 ECTS credits (30 hours).

**2. ****Practical lessons**, in which the students will apply the theoretical results in order to solve, with the teacher’s help, more complete exercises, and problems of an economic nature. Problem sheets will be available for the students and the teacher will announce in advance the problems that will be solved in each practical lesson so that the students can prepare them beforehand. These classes are face-to-face and will be given separately to each subgroup.

Time allotted: 1.2 ECTS credits (30 hours each subgroup).

**3. ****Seminars** (practical classes P6), which may consist of a number of different activities designed to support the learning process, including: follow-up of some simple projects that had been assigned to small teams of students and the presentation of these projects; answering questions that students may have regarding some of the contents taught; solving problems of an economic nature by using some of the mathematical tools taught during the classes, etc. These seminars may also be devoted to the teaching of more advanced topics, intended for the students interested in learning some further mathematical tools that would allow them to deal with more general problems. In this way, the students are shown that both Mathematics and Economics are vibrant sciences with many facets to be studied.

Time allotted: Pending of the agreement of the Department Committee

*4. **Out of class work**: 3.6 ECTS credits*

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**5.3. Syllabus**

**TABLE OF CONTENTS**

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**Preliminaries: **Determinants.Applications: calculation of the rank of a matrix, calculation of the inverse matrix and Cramer’s Rule

**Chapter 1. Diagonalization of square matrices **

1.1. : Spanning sets. Basis.

1.2. Eigenvalues and eigenvectors of a square matrix: definition and calculation.

1.3. Diagonalization of a square matrix.

1.4. Application to the calculation of matrix powers.

**Chapter 2. Real quadratic forms**

2.1. Quadratic forms: definition. Matrix expression and polynomial expression.

2.2. Diagonal expression of a quadratic form.

2.3. Classification of a quadratic form according to its sign.

2.4. Constrained quadratic forms.

**Chapter 3. Functions from **** to **

3.1. Preliminaries: topological concepts.

3.2. Functions: domain, range and graph. Level sets of scalar functions.

3.3. Continuity of a function.

3.4. Differentiation of a function. Partial derivatives. Gradient vector. Jacobian matrix.

3.5. Differentiability. Directional derivative of differentiable functions.

3.6. Differentiation of composed functions: Chain’s Rule. Tree diagrams.

3.7. Higher order derivatives. Schwarz’s Theorem. Hessian matrix. Taylor’s Theorem.

3.8. Implicit function Theorem. Differentiation of implicit functions.

3.9. Homogeneous functions. Euler’s Theorem.

3.10. Basic integration methods of function of one variable. Barrow’s Rule.

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**5.4. Course planning and calendar**

The calendar will be explained to the students in the presentation of the subject.

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**5.5. Bibliography and recommended resources**

The recommended readings are usually available in the University Library's data base (go to 'bibliografia recomendada' in 'biblioteca.unizar.es').