Mathematical Analysis I

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 will be 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 that successfully completes this course will be able to:
i) compute the limit of a sequence or function beyond the scope of those studied in High School;
ii) integrate elementary functions, powers of trigonometric and hyperbolic functions and rational functions;
iii) use the Fundamental Theorem of Calculus to compute areas of plane figures, volumes of solids and lengths of curves;
iv) solve a differential equation with separable variables;
v)  solve a linear differential equation of order one;
vi) solve linear differential equations of order larger than one.

Work Placement(s)

No

Syllabus

I.  Real sequences and real functions of a single variable
I.1 Elementary topology of the reals. The supremum axiom
I.2 Sequences of real numbers: limits and basic properties
I.3 Trigonometric and hyperbolic functions and their inverses
I.4 Limits, continuity and differentiability of real functions of a single variable
I.5 Rolle’s Theorem, Lagrange Mean Value Theorem and its corollaries. Cauchy’s rule

II.   Integration
II.1 Introduction to integration. Elementary integration
II.2 Integration by parts and integration of powers of trigonometric and hyperbolic functions
II.3 Integration of rational functions
II.4 Integration by substitution
II.5 The Riemann integral of a function and the Fundamental Theorem of Calculus
II.6 Volume of solids of revolution and lengths of curves
II.7 Numerical integration
II.8 Improper integrals

III.   Ordinary differential equations
III.1 Equations with separable variables
III.2 Linear equations of order one
III.3 Linear equations of order larger than one

Head Lecturer(s)

Maria João Rodrigues Ferreira

Assessment Methods

Final Assessment
Exam: 100.0%

Assessment
Frequency: 100.0%

Bibliography

[1] Stewart, J., Cálculo, Volumes I e II, 5ª edição, Pioneira, S. Paulo. (2006)
[2] Zill, D. G., Equações Diferenciais com aplicações em modelagem, Thomson, S. Paulo. (2003)
[3] J. Campos Ferreira, Introdução à Análise Matemática, 11ª edição, Fundação Calouste Gulbenkian, Lisboa, 7a. Edição, (2014).
[4] J. Carvalho e Silva, Princípios de Análise Matemática Aplicada,  McGraw-Hill, (2005).
[5] Carlos Sarrico, Análise Matemática, Leitura e exercícios, 6ª edição, Colecção Trajectos Ciência n. 4, Gradiva, (2005).