# Mathematical Analysis I

**Year**

1

**Academic year**

2022-2023

**Code**

01019327

**Subject Area**

Mathematics

**Language of Instruction**

Portuguese

**Mode of Delivery**

Face-to-face

**Duration**

SEMESTRIAL

**ECTS Credits**

6.0

**Type**

Compulsory

**Level**

1st Cycle Studies

## Recommended Prerequisites

Mathematics A from the Portuguese High School Curriculum.

## Teaching Methods

The teaching in this course assumes two formats: theoretical and example classes. During a theoretical class teaching is mostly expository. During an example class teaching will consist of problem solving by the students under the guidance of the lecturer. A strong interaction between notions and their practical application is emphasized. In this task, the visualization and the analysis of concrete examples takes on a central role and prepares the way for the abstract definitions. Tutorial support is available to students to help them on the tasks assigned by the lecturers.

## Learning Outcomes

The student who successfully completes this course will be able to:

1. Compute limits of sequences and functions beyond those studied in High School;

2. Compute derivatives and primitives of elementary functions;

3. Use the Fundamental Theorem of Calculus to compute areas and lengths;

4. Solve a separable differential equation;

5. Solve a first order linear differential equation;

6. Solve higher order linear differential equations;

7. Solve problems involving applications of differential equations in mathematical modeling.

## Work Placement(s)

No## Syllabus

I. Functions of a real variable

I.1 Trigonometric and hyperbolic functions and their inverses

I.2 Limits, continuity and differentiability of functions of a real variable

I.3 Parametrized curves and polar coordinates

II. Integration

II.1 Primitives

II.2 Riemann integral and applications

II.3 Improper integrals

III. Ordinary Differential Equations

III.1 First order linear differential equations: the separable case and the linear case

III.2 Higher order linear differential equations: annihilator, reduction of order, variation of parameters methods.

## Head Lecturer(s)

Jaime Maria Monteiro de Carvalho e Silva

## Assessment Methods

Final assessment

*Exam: 100.0%*

Continuous assessment

*2 or more midterm exams: 100.0%*

## Bibliography

[1] James Stewart: Cálculo, Volumes I e II, Cengage Learning (tradução da 8ª edição norte-americana) 2017

[2] Jaime Carvalho e Silva: Princípios de Análise Matemática Aplicada, McGraw-Hill, Lisboa (1994).

[3] Dennis G. Zill: Equações Diferenciais com aplicações em modelagem. Cengage Learning (tradução da 10ª edição norte-americana), 2016.