Mathematical Analysis I

Year
1
Academic year
2023-2024
Code
01017810
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 will assume 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 will be 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. Analyse the continuity and differentiability of real functions of real variable;

2. Draw curves in polar and parametric coordinates;

3. Compute derivatives and primitives of elementary functions;

4. Solve problems involving linear approximations, differential and implicit differentiation;

5. Use the Fundamental Theorem of Calculus to compute areas, lengths and volumes of solids of revolution;

6. Solve first order differential equations by separable of variables and solve linear differential equations of first order;

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

Work Placement(s)

No

Syllabus

I. Functions and curves 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 Parameterized 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 Separable differential equations

III.2 First order linear differential equations. 

Head Lecturer(s)

Adérito Luís Martins Araújo

Assessment Methods

Continuous assessment
2 or more midterm exams: 100.0%

Final assessment
Exam: 100.0%

Bibliography

[1] Adérito Araújo, Análise Matemática I, Notas de Curso, Coimbra, 2018. (Disponível online)
[2] James Stewart, Cálculo, volumes I e II, tradução da 8ª edição norte-americana, Cengage Learning, 2017.
[3] Edwin "Jed" Herman e Gilbert Strang (entre outros), Calculus, volumes 1, 2 e 3, OpenStax, 2018. (Disponíveis online em: https://openstax.org/details/books/calculus-volume-1; https://openstax.org/details/books/calculus-volume-2; https://openstax.org/details/books/calculus-volume-3)
[4] Gilbert Strang, Calculus, Wellesley-Cambridge Press, 1991. (Disponível online em: https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/)
[5] Elon Lages Lima, Curso de Análise, volume 1, 11a edição, Projecto Euclides, IMPA, 2004.
[6] Jaime Carvalho e Silva: Princípios de Análise Matemática Aplicada, McGraw-Hill, Lisboa (1994)