# Mathematical Analysis I

## Objectives

The main purpose of Analysis 1 is to provide a clear understanding and a skillful use of concepts of one variable real
analysis, in view of a solid analytical background required by Information Management subsequent disciplines.

## General characterization

100008

5.0

##### Responsible teacher

José Maria Nunes de Almeida Gonçalves Gomes

##### Hours

Weekly - Available soon

Total - Available soon

##### Teaching language

Portuguese. If there are Erasmus students, classes will be taught in English

### Prerequisites

There are no attendance requirements.

### Bibliography

Sydsæter, K, Hammond, P., Essential Mathematics for Economic Analysis, 2nd ed., Prentice Hall, 2006. Campos Ferreira, J., Introdução à Análise Matemática, 8ª ed., Fundação Calouste Gulbenkian, 2005. Azenha, A., Jerónimo, M.A., Elementos de Cálculo Diferencial e Integral em IR e IRn, McGraw-Hill, 1995.

### Teaching method

Lectures and practical classes for solving exercises.

### Evaluation method

Continuous Evaluation System (1st season)

• Final grade is calculate by the following formula: intermediate tests (T1, T2, T3) during semester (minimum grade in each test: 7,5 points). Final grade: 30%T1+40%T2+30%T3
Final Exam (only 2nd season)
• Exam (100%) (minimum grade: 9,5 points)

## Subject matter

1 The IR set
Basic concepts.
Topological notions.
2 Real functions of one real variable
Generalities about real functions of one real variable.
Notion of limit; lateral limts, properties and operations.
Continuous functions: definition and properties of continuous functions.
Theorems of Bolzano and Weierstrass .
3 Differential Calculus on IR
Derivative of a function: definition of the tangent line equation.
One-sided derivatives; differentiability; relationship between differentiability and continuity of a function; derivation rules; derivative of the composite function.
Fundamental theorems: theorems of Rolle, Lagrange and Cauchy; Cauchy rule; indeterminate forms.
Derivatives from the higher order; formula of Taylor and MacLaurin.
Extremes of functions; concavity and inflection points; asymptotes; sketch graph of a function.
4 Integral Calculus in IR
Antiderivative: definition and General methods to compute antiderivatives.
Integral Calculus: Riemann integral; Fundamental theorems of integral calculus; calculating areas of plane figures.

## Programs

Programs where the course is taught: