Applied Mathematics II

Differential calculus in IR.

Expository method combined wikh solving exercises.

The course program consists of three chapters.

The first chapter involves studying matrices, including matrix calculus and solving systems of linear equations. It is intended with this chapter that students know several techniques for solving systems of linear equations that allow them to have another approach to solving problems in their area of study. It also seeks to prepare students for the application of these concepts in subsequent disciplines of their course.

The second chapter addresses the analysis of real functions of two real variables, involving definitions and properties, study of limits and continuity, calculation of derivatives, differentials and extremes subject or not to equality restrictions. It is intended that students are able to study real functions of two real variables, in particular functions involved in problems in the area of ​​Business Management. In this sense, the optimization of functions is of great importance in this chapter.

The last chapter focuses on the study of some numerical methods for determining zeros and extremes of functions, using, in particular, the bisection method and Newton’s method.

I.                    Matrix Algebra

Matrices
1.1.  Definitions. General concepts
1.2.  Operations and their properties
1.3.  Matrix transposition
1.4.  Reduction. Characteristic. Gaussian elimination.
1.5.  Inverse of a matrix. Gauss-Jordan method.
 
Linear equations AX=B
2.1.  Definitions. General concepts
2.2.  Solving systems with Gauss elimination method
2.3. Systems discussion
2.4.  Solving systems using the inverse of A
 
Determinants
3.1.  Definitions
3.2.  Properties. Reduction method.
3.3.  Laplace Theorem
3.4.  Applying determinants to
3.4.1.    solve systems with Cramer’s rule
3.4.2.     determinate inverse matrices
 
4. Examples
 
II.                  Differential Calculus of real functions of two real variable
 

1. Definitions
2. Limits and continuity
3. Partial derivatives of first and second order
4. Differentials
5. The Chain rule
6. Maxima and Minima
6.1.  Critical Bound
6.2.  Bounded region. Lagrange multipliers
7. Examples

III. Numerical methods for real functions of real variable
 

1. Finding roots of a function using the bisection method and the Newton’s method
2. Finding maxima and minima of a function using the bisection method and the Newton’s method
3. Examples

NAO

Main Bibliography:

Will be available on Nonio:

• the theoretical notes
• the practical sheets and respective resolutions
• the form
• the tests carried out in previous academic years.

As a complementary bibliography, we recommend:

1- A. P. Santana e J. F. Queiró, Introdução à Álgebra Linear, Gradiva, 2010

2 – M. Carvalho, Cálculo Matricial, ISCAC, Coimbra, 2005

3 – Larson, Holstetler and Edwards, Cálculo. Vol I e II, São Paulo. Ed McGraw Hill, 2006

4- A.d’A. Breda e J.N. da Costa, Cálculo com Funções de Várias Variáveis, McGraw-Hill, 1996

5 – L.J. Goldstein, D.C. Lay e D.I. Schneider, Matemática Aplicada: Economia, Administração e Contabilidade, Bookman, Porto Alegre, 8ª Edição, 2000

6 – C. Pires, Cálculo para Economistas, McGraw-Hill, 2001

7- E. W. Swokowski, Cálculo com Geometria Analítica, Volume 1 e 2, Makron Books, Ltda., São Paulo, 2ª Edição, 1994