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  4. Mathematical Analysis I

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

Code

100008

Credits

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.