# Mathematical Analysis I

## Objectives

In this curricular unit it is intended that students develop their logical reasoning and calculation skills, essential for the learning of other curricular units of their cycle of studies. The main goals are the learning and consolidation of fundamental knowledge of Differential and Integral Calculus for real functions of a real variable.

## 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

It is recommended the frequency of Mathematic A in secondary education.

### Bibliography

Sydsæter, K, Hammond, P., Essential Mathematics for Economic Analysis, 2nd ed., Prentice Hall, 2006.;

Sarrico, Carlos, *Análise Matemática, Leituras e exercícios*, Gradiva.;

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 Assessment ****( 1st period)**

The continuous assessment consists of conducting, during the academic semester, 2 tests** T _{1,} T_{2}** and a final exam

**E**.

__The tests have no minimum grade.__

__The exam has minimum grade of 8.5__.

The final grade is calculated as follows: 20%**T _{1}**+20%

**T**+60%

_{2}If, exceptionally, it is not possible to carry out an assessment in person, there will be the possibility of do the same online followed by an oral exam, but the situation will be analyzed case by case.

**Exam Assessment ****(only 2nd period)**

**Final exam** (**100%**) (face-to-face and with minimum score of 9.5).

## Subject matter

1. The IR set

1.1 Basic concepts.

1.2 Topological notions.

2. Real functions of one real variable

2.1 Generalities about real functions of one real variable.

2.2 Notion of limit; lateral limts, properties and operations.

2.3 Continuous functions: definition and properties of continuous functions.

2.4Theorems of Bolzano and Weierstrass .

3. Differential Calculus on IR

3.1 Derivative of a function: definition of the tangent line equation.

3.2 One-sided derivatives; differentiability; relationship between differentiability and continuity of a function; derivation rules; derivative of the composite function.

3.3 Fundamental theorems: theorems of Rolle, Lagrange and Cauchy; Cauchy rule; indeterminate forms.

3.4 Derivatives from the higher order; formula of Taylor and MacLaurin.

3.5 Extremes of functions; concavity and inflection points; asymptotes; sketch graph of a function.

4. Integral Calculus in IR

4.1 Antiderivative: definition;

4.2 General methods to compute antiderivatives.

4.3 Integral Calculus: Riemann integral;

4.4 Fundamental theorems of integral calculus;

4.5 Calculation of areas of plane figures.

## Programs

Programs where the course is taught: