Mathematical Analysis I A
Code
10969
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
9.0
Teacher in charge
Ana Margarida Fernandes Ribeiro
Weekly hours
6
Total hours
80
Teaching language
Português
Objectives
The goals of the course include:
- a basic understanding of the special language, notation, and point of view of calculus
- the ability to solve basic computational problems involving derivatives and integrals
- a basic understanding of the fundamental theorems of calculus
Prerequisites
The student must be familiar with mathematics taught in the final year of high school.
Subject matter
1. Real numbers. Topological notions in IR. Mathematical induction.
2. Sequences of real numbers. Limits. Infinite limits. Limits at infinite. Monotone sequences. Convergent sequences. Subsequences. Upper limit and lower limit. Cauchy sequence. Completeness of IR.
3. Single real variable functions: limits and continuity. Properties of continuous functions; Bolzano’s theorem. Uniform continuity. Lispschitz continuous functions. Cantor’s theorem.
4. Differential calculus. Derivatives, physical and geometric interpretations and properties. Fundamental theorems: Rolle, Darboux, Lagrange and Cauchy. Cauchy rule. Taylor’s formula and applications. Extrema, concavity and inflection points.
Bibliography
1. Alves de Sá, A.; Louro, B. - Sucessões e Séries - Teoria e Prática, Livraria Escolar Editora, 2008.
2. Apostol, T. - Calculus, Blaisdell, 1967.
3. Campos Ferreira, J. - Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982.
4. Elon Lages Lima - Curso de Análise - Projeto Euclides, Rio de Janeiro, 1989.
5. Figueira, M. - Fundamentos de Análise Infinitesimal, Textos de Matemática, vol. 5, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, 1996.
6. Lang, S. - Undergraduate Analysis, Springer, 1983.
7. Rudin, W. - Principles of Mathematical Analysis, McGraw-Hill, 1976
Teaching method
Classes consist on theoretical lectures, illustrated by examples and applications, and on problem solving. Most results are proven. Students have access to a well adapt bibliography and to the proposed exercises. Some of the exercises are solved in class, the remaining are left to the students as part of their learning process.
Evaluation method
The absence unjustified to the classes must be at most of 3 classes. There are three mid-term tests that can substitute the final exam in case of approval. Otherwise the student must pass the final exam. More detailed rules are available in the portuguese version.