27308 -- Mathematics II

Curso:

1

Semestre:

2

Créditos:

6.0

Universidad de Zaragoza

Learning outcomes that define this course

1

- To have gained good skills in using mathematical language, both in comprehension and writing.
- To be able to identify the fundamental elements of an optimisation problem: variables, objective function and constraints.
- To be able to formulate static optimisation problems: unconstrained, and with equality and/or inequality constraints.
- To know how to solve an optimisation problem by the graphical method, when that is possible.
- To be able to evaluate whether or not a mathematical programme meets the conditions that allow it to be solved by the techniques learnt.
- To be able to distinguish between critical points and extrema (optima).
- To be able to discriminate between local and global optima.
- To be able to distinguish between necessary conditions and sufficient conditions for local optimality.
- To be able to calculate the critical points by solving the system of equations obtained by applying the first-order conditions for local optimality, both for unconstrained cases and for problems with equality constraints.
- To know how to classify the obtained critical points by using the second-order conditions, both for unconstrained optimisation programmes and for problems with equality constraints.
- To be able to apply the conditions which guarantee that an optimum is global.
- To be able to interpret economically the Lagrange multipliers obtained in an optimisation problem with equality constraints.
- To be able to evaluate whether a mathematical programme is linear. If it is, they must know how to solve it by the graphical method (when that is possible) and by the simplex algorithm.
- When varying a parameter of a linear optimisation programme, the students must be able to analyse how the solution changes, without solving the new problem.
- To be able to use some computer programmes to find the solution to an optimisation problem and to be able to interpret the results obtained.
- To be able to identify a dynamic process in an economic scenario and be able to represent this process (when possible) by an ordinary differential equation.
- To understand the concept of the solution of an ordinary differential equation and to be able to distinguish between general solution and particular solution.
- To be able to discriminate between a first-order differential equation and a linear differential equation of order n.
- To be able to identify whether a first-order differential equation is with separable variables, homogeneous, exact, or of linear type, and to know how to solve the equation by the appropriate method.
- For a linear differential equation with constant coefficients, they must be able to write the complementary (homogeneous) equation and obtain its general solution.
- To be able to find a particular solution of a linear differential equation with constant coefficients.
- To have the know-how to calculate the general solution of a linear differential equation with constant coefficients.
- To be able to work out the solution of a linear differential equation of order n with constant coefficients, given n initial conditions.

Introduction

Mathematics II is a basic-training subject with a value of 6 ECTS credits and it is taught during the second semester of the first year. It is based on and complements Mathematics I, a subject of the first semester of the first year.

The subject Mathematics II consists of two different parts: Mathematical Programming and Dynamical Analysis, which, respectively, apply to two different points of view of economic reality. After learning the first part, the students will be able to formulate and solve a wide variety of classical optimisation problems: both linear and non-linear, whether unconstrained or with equality and/or inequality constraints. In the case of optimisation programmes where both the objective function and the constraints are linear, the solving technique used is the simplex algorithm. This topic may be used to connect the traditionally-taught solving methods with the use of computer software, which simplifies the calculations and introduces students to professional practice.

The second part, Dynamical Analysis, is concerned with solving differential equations and with the analysis of the solutions. Its inclusion in the syllabus is necessary because many of the processes that Economic Analysis deals with are non-static. Some examples of these dynamic processes are: optimal economic growth, optimal management of renewable and non-renewable resources, optimal long-term investment, etc.