Universidade Nova de Lisboa

Universidade Nova de Lisboa

Faculdade de Ciências e Tecnologia

Mathematical Analysis I

Code

11504

Academic unit

Faculdade de Ciências e Tecnologia

Department

Departamento de Matemática

Credits

6.0

Teacher in charge

Ana Maria de Sousa Alves de Sá

Weekly hours

6

Total hours

75

Teaching language

Português

Objectives

Domain of the basic techniques required for the Mathematical Analysis of real functions of real variable.

The students should acquire not only calculus capabilities fundamental to the acquisition of some of the knowledge lectured in Physics, Chemistry and other Engineering subjects,  but also to develop methods of solid logic reasoning and analysis.

Being a first course in Mathematical Analysis, it introduces some of the concepts which will be deeply analyzed and generalized in subsequent courses.

Prerequisites

The student must master the mathematical knowledge lectured until the end of Portuguese High School.

Subject matter

1. Topology - Mathematical Induction - Sequences

Basic topology of the real numbers. Order relation.

Mathematical induction.

Generalities about sequences. Convergence of a sequence and properties for calculus of limits. Subsequences. Bolzano-Weierstrass theorem. 

2. Limits and Continuity

Generalities about real functions of real variable. Convergence according to Cauchy and Heine. Calculus properties.

Continuity of a function at a given point. Properties of continuous functions. Bolzano theorem. Weierstrass theorem.

3. Differenciability

Generalities. Fundamental theorems: Rolle, Lagrange and Cauchy. Calculus techniques for limits. Taylor formula and applications.

4. Indefinite Integration

Introduction. Indefinite integration by parts. Indefinite integration by substitution.  Indefinite integration of rational functions.

5. Riemann Integration

Introduction. Fundamental theorems. Definite integration by parts and by substitution. Some applications.

Improper integration.

Bibliography

Adopted text

  1. Ana Alves de Sá e Bento Louro, Cálculo Diferencial e Integral em ℝ

Recommended Bibliography

  1. Alves de Sá, A. e Louro, B. - Cálculo Diferencial e Integral em ℝ, Exercícios Resolvidos, Vol. 1, 2, 3
  2. Anton, H. - Cálculo, um novo horizonte, 6ª ed., Bookman, 1999
  3. Campos Ferreira, J. - Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982
  4. Carlos Sarrico, Análise Matemática, Leituras e Exercícios, Gradiva, 1997
  5. Larson, R.; Hostetler, R.; Edwards, B. - Calculus with Analytic Geometry, 5ª ed., Heath, 1994
  6. Figueira, M. - Fundamentos de Análise Infinitesimal, Textos de Matemática, vol. 5, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, 1996

Teaching method

Theoretical classes consist in a theoretical exposition illustrated by application examples.

Practical classes consist in the resolution of  application exercises for the methods and results presented in the theoretical classes.

Any questions or doubts will be addressed during the classes, during the weekly sessions specially programmed to attend students or in individual sessions previously scheduled between professors and students.

Courses