Mathematical Analysis I C
Code
10341
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Ana Cristina Malheiro Casimiro, Manuel Valdemar Cabral Vieira, Paula Alexandra da Costa Amaral
Weekly hours
6
Total hours
122
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. Continuity and reciprocal bijections.
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 Notes
Ana Alves de Sá and Bento Louro, Análise Matemática I, FCT-UNL, 2010
Recommended Bibliography
- Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons Inc., 1999
- Jaime Campos Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982
- Rod Haggarty, Fundamentals of Mathematical Analysis, Prentice Hall, 1993
- Carlos Sarrico, Análise Matemática, Leituras e Exercícios, Gradiva, 1997
- João Paulo Santos, Cálculo numa variável real, Coleção Ensino da Ciência e da Tecnologia, IST Press, 2012
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.