Overall objectives:

1 - To equip the students with some theoretical and practical knowledge of Mathematics which is necessary for general and academic purposes.
2 - To consolidate and to develop the knowledge that was previously acquired by the student.
3 - Ability to work with matrices; knowing how to apply elementary operations to a matrix; ability to solve systems of linear equations using matrix calculus; knowing how to compute eigenvalues and eigenvectors of a matrix and to know some of its applications.
4 - Ability to analytically manipulate algebraic and transcendental functions; knowing the concepts of limits and continuity; manipulate in a reasonable manner differential calculus, both in R and in Rn; ability to generalise, to some extent, some of the concepts previously learned and to use them in other contexts of higher level courses.

Syllabus:

1 - 1. Matrix calculus: Matrices; Basic operations with matrices; Inverse of a matrix; Solving linear systems using the inverse matrix and by the Gauss Elimination Method; Determinants; Cramer's Rule; Eigenvalues and eigenvectors; Power of a matrix Examples of computer-assisted matrix calculations
2 - Real valued functions of real variable: Basic concepts; Algebraic operations with functions; Composite function; Limits; Continuity; Differential calculus; Differentiation techniques; Derivative of the composite function; Derivative of the inverse function; Higher order derivatives; Rolle, Lagrange and Cauchy Theorems; Graphical representation of functions; Evaluation of indeterminate forms;
3 - Real valued functions of vector variable: Graphical representation and contour lines; Limits and continuity; Partial derivatives; Gradient and Hessian Matrix; Maxima and Minima; the method of Lagrange multipliers; Examples of computer-assisted study of real valued functions of vector variable

Literature/Sources:

Piskounov, N. , 1990 , Cálculo Diferencial e Integral , Editora Lopes da Silva
Apostol Tom M. , 1983 , Cálculo , Editora Reverté Ltda
Agudo, F. Dias , 1992 , Introdução à Algebra Linear E Geometria Analítica , Escolar Editora
Ferreira, J. Campos , 2005 , Introdução à Análise Matemática , Fundação Calouste Gulbenkian
Lang, Serge , 1986 , Introduction to Linear Algebra , Springer Verlag
Monteiro, António , 2001 , Álgebra linear e geometria analítica , McGraw-Hill
Courant, R. John , 1989 , Introduction to Calculus and Analysis I , Springer

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Use of the whiteboard for presentation, explanation of subject material and problem solving. Use of the computer and projector to help visualize and better understand the concepts. Usage of adequate software (e.g. Geogebra) or graphing calculator to better visualize some concepts and / or to help in exploring problems with more complexity. Evaluation: two tests (one at the middle and one at the end of the semester), each one with a weight of 50% on the final grade. In the appeal season, students who have failed to pass the course during the normal season can choose between retaking one or the two tests, each corresponding to 50% of the final grade.