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# Mathematical Analysis I E

7910

### Department

Departamento de Matemática

7.0

### Teacher in charge

Ana Cristina Malheiro Casimiro, Ana Maria de Sousa Alves de Sá, Bento José Carrilho Miguens Louro, José Maria Nunes de Almeida Gonçalves Gomes

17

78

Português

### Objectives

Learn the basic topics of Mathematical Analysis. It is intended that the students acquire elementary techniques of calculus for the Physics, Chemistry and Engineering. Moreover, they should develop solid methods of logical reasoning.

### Prerequisites

The student must be familiar with mathematics taught in high school.

### Subject matter

1. Topological notions in IR.  Mathematical induction. Sequences of real numbers.

2. Single real variable functions: limits and continuity. Properties of continuous functions; Bolzano’s theorem.

4. Primitives. Primitivation by parts and by change of variables. Computation of primitives of rational, irrational and transcendent functions.

5. Integral calculus. Riemann integral. Fundamental theorem of calculus; Mean Value Theorem; Barrow’s formula. Computation of areas of plane figures. Improper Integrals.

3.Differential calculus. 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.

### Bibliography

Main book : Análise Matemática 1, Bento Louro e Ana Sá.

Other:

1. Cálculo com geometria analítica, Earl W. Swokowski, MacGraw-Hill,1983.
2. SALAS, HILLE - Calculus, one and several variables, John Wiley Sons, Inc, 1995.
3. APOSTOL, T. - Calculus, Blaisdell, 1967.
4. CAMPOS FERREIRA, J. - Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982.
5. STEWART, J. - Calculus, 3ª edição, Brooks/Cole Publishing Company, 1995.
6.  Anton, Bivens, Davis - Calculus, 8th Edition, Wiley, 2005 .

### Teaching method

Lectures consists of an oral explanation of the theory, illustrated by examples and applications.  Problem solving sessions are devoted to exercises and problems that student are ought to solve, under teacher''''''''s supervision. Autonomous and regular study by the student is expected outside the classroom. Homework exercices may be requested by the problem-solving sessions tacher''''''''s.

Questions or difficulties experienced by the student can be discussed  during classes, in  a predetermined period devoted to tutorials or in a particular schedule settled with the teacher.