Mathematical Analysis I


This course aims at the development of analytical reasoning and calculus, essential for the remaining courses in the undergraduate program. The main learning fields are Differential Calculus and Integral Calculus for real functions with one variable.

General characterization





Responsible teacher

Patrícia Santos Ribeiro


Weekly - Available soon

Total - Available soon

Teaching language

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


There are no attendance requirements.


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.