Mathematical Analysis I
1
2023-2024
01017810
Mathematics
Portuguese
Face-to-face
SEMESTRIAL
6.0
Compulsory
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)
NoSyllabus
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)