Subject: Calculus I

Scientific Area:

Mathematics

96 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - Provide the students with basic theoretical and practical knowledge to be used as the fundamental mathematical tools more advanced courses. In particular, it seeks to convey the fundamentals of the theory of mathematical analysis to a variable with the corresponding applications to specific problems suited to the various areas of knowledge

# Syllabus:

1 - Matrices: Basics of matrices. Matrices operations. Classification Matrices.
2 - Determinants of square matrices: Calculation of determinants. Rule Sarrus. Laplace theorem and its applications. Applications of determinants to the linear systems: systems of linear equations. Gauss. Cramer's rule. Calculation of the inverse of an invertible matrix.
3 - Eigenvalues ​​and eigenvectors: Basic concepts of eigenvalues ​​and eigenvectors. Own subspace associated with an eigenvalue. Algebraic and geometric multiplicity. Characteristic polynomial. Cayley-Hamilton theorem. Diagonalization.
4 - Real functions of real variable: Domain contradomínimo. Function: continuous, composite, monotonous, inverse, exponential. Logarithmic. Theorems of Bolzano and Weierstrass. Limits.
5 - Diferenciabilidade.Teoremas fundamental calculus: Rolle, Lagrange and Darboux. Rule of Cauchy. Calculation of indeterminacies. Polynomial approximation of functions. Taylor's formula. Application of these concepts to the study of an entire function.
6 - Integration of functions of one variable. Indefinite integral. Immediate integration by parts and by substitution. Definite integral and its application. Fundamental theorem of integral calculus. Derivative of an integral
7 - Sequences of real numbers: limited sequences, convergent sequences, funfamentais theorems, infinite limits and infinity. Mathematical Induction.

# Literature/Sources:

Anton Howard , 1992 , Calculus , John Wiley & Sons
Luís T. Magalhães , 1998 , Álgebra Linear como Introdução a Matemática Aplicada , Texto Editora
T. Apostol , 1998 , Cálculo , Ed. Reverté, Lda.
James Stewart , 2005 , Cálculo (2005) , Thomson
R.Courant, F. John , 1989 , Introduction to Calculus and Analysis , Springer-Verlag

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
The theoretical classes were essentially expositive, always using practical examples. The practical classes are based on the resolution of exercises and their discussion. The evaluation is made 3 tests (frequencies) to be solved individually. The frequencies have weights 30% (6 values), 35% (7 values) and 35% (7 values), each of these frequencies have a minimum grade of 3 values.