Mathematical Analysis I

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

Continuous assessment
2 or more midterm exams: 100.0%

Final assessment
Exam: 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.