Mathematical Analysis I A
Faculdade de Ciências e Tecnologia
Departamento de Matemática
Teacher in charge
Bento José Carrilho Miguens Louro
The student is supposed to obtain the rigorous knowledge of the notion of limit (of sequences and functions) and the main associated results. It is also an objective the differential calculus on one variable, including the analytical study of real functions of one real variable.
The student must be familiar with mathematics taught in the final year of high school.
1. Real numbers. Topological notions in R.
2. Mathematical induction.
3. Sequences of real numbers. Limits. Infinite limits. Limits at infinite. Monotone sequences. Convergent sequences. Subsequences. Upper limit and lower limit. Cauchy sequence. Completeness of R.
4. Single real variable functions: limits and continuity. Properties of continuous functions; Bolzano’s theorem. Uniform continuity. Lispschitz continuous functions. Cantor’s theorem.
5. Differential calculus. Derivatives. Fundamental theorems: Rolle, Darboux, Lagrange and Cauchy. Indeterminate forms. Cauchy and L’Hospital rules. Taylor’s formula. MacLaurin’s formula. Extrema, concavity and inflection points.
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. Ellis, R.; Gullick, D. - Calculus with Analytic Geometry, 5ª edição, Saunders College Publishing, 1994.
5. Figueira, M. - Fundamentos de Análise Infinitesimal, Textos de Matemática, vol. 5, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, 1996.
6. Hunt, R. - Calculus, 2ª edição, Harper Collins, 1994.
7. Larson, R.; Hostetler, R.; Edwards, B. - Calculus with Analytic Geometry, 5ª edição, Heath, 1994.
8. Larson, R.; Hostetler, R.; Edwards, B. - Calculo, Vol. 1, 8ª edição, McGraw-Hill, 2006.
9. Santos Guerreiro, J. - Curso de Análise Matemática, Livraria Escolar Editora, 1989.
10. Sarrico, C. - Análise Matemática, Leituras e Exercícios, Gradiva, 1997.
11. Spivak, M. - Calculus, World Student Series Edition, 1967.
12. Stewart, J. - Calculus, 3ª edição, Brooks/Cole Publishing Company, 1995.
13. Swokowski, E. W. - Cálculo com Geometria Analítica, 2ª edição, Makron Books, McGraw-Hill, 1994.
14. Taylor, A.; Mann, R. - Advanced Calculus, 2ª edição, Xerox College Publishing, 1972.
Classes consist on an oral explanation which is illustrated by examples and the resolution of some exercises. Most results are proven. Students have access to copies of the theory and proposed exercises. Some of the exercises are solved in class, the remaining are left to the students as part of their learning process.
Students must attend, at least, two thirds of the classes and must deliver two thirds of the homework. There are two 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.