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 consists of problem solving by the students under the guidance of the lecturer. A strong interaction between notions and their practical application is emphasised. 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. Detected non-continuous real functions of two or three variables at a given point;        

5. Compute the directions of greatest growth of a real function of two or three variables;

6. Solve a constrained extrema problem.     

Work Placement(s)

No

Syllabus

I. Sequences and functions of a real variable

I.1 Sequences

I.2 Trigonometric and hyperbolic functions and their inverses

I.3 Limits, continuity and differentiability of functions of a real variable

 

II. Integration

II.1 Primitives

II.2 Riemann integral and applications

II.3 Improper integrals

 

III. Real functions of two or three variables

III.1 Limits and continuity

III.2 Partial derivatives, directional derivatives and chain rule

III.3 Tangent plane

III.6 Maxima and minima. Lagrange Multipliers.

Head Lecturer(s)

Joana Margarida Mavigné de Andrade Alves de Sousa Nunes da Costa

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: Princípos de Análise Matemática Aplicada, McGraw-Hill, Lisboa (1994)

[3] Earl W. Swokowski, Cálculo com geometria analítica Vol I e Vol II, Makron Books (1995)

[4] Ana d'Azevedo Breda, Joana Nunes da Costa: Cálculo com funções de várias variáveis. McGraw-Hill, Lisboa (1996).