Mathematical Analysis I

Recommended Prerequisites

High School.

Teaching Methods

The classes are of theoretical and theoretical-practical type. The teaching methods in the theoretical components will be predominantly expository. In the practical components problems will be solved under the guidance of the teacher.

The strong interaction between concepts and their practical application must be discussed, giving as much as possible a central role in the visualization and analysis of particular situations before making a progressive abstraction of the concepts being introduced.

Some tutorial support will be available to help the students solving the proposed tasks.

Learning Outcomes

Give the students the basic knowledge of differential and integral calculus for real functions of a real variable. Know how to solve differential equations and apply this knowledge in modeling and solving problems. The relevance of the results presented in the context of applications should be emphasized. It is also intended that students acquire a knowledge of the concepts in order to assess the scope and limitations of the materials studied and their applications.

The course aims at developing the following skills: analysis and synthesis, organization and planning, oral and written communication, problem-solving skills and computational ability. On the personal level it also allows to develop self-learning skills and independent thinking.

Work Placement(s)

No

Syllabus

I- Revisions. 

I.1 Numerical sequences: limits and basic. 

I.2 Real functions of one real variable: limits, continuity and differentiation

I.3 Inverse trigonometric functions

I.4 Hyperbolic functions and inverse hyperbolic functions. 

II- Integration of Real functions of one real variable. 

II.1 Anti-derivatives

II.2 Defined Integral. Applications: areas, volumes and curve length. 

II.3 Numerical integration. 

II.4 Improper integrals

III- Integration of complex functions of real variable. 

III.1 Complex numbers (revision) 

III.2 Introduction to complex functions of real variable

III.3 Integration of complex functions of real variable

IV- Approximation of function by series

IV.1 Numerical serials

IV.2 Function sequences

IV.3 Function series

IV.4 Taylor Series

IV.5 Fourier series: trigonometric version and complex version

Head Lecturer(s)

Jorge Manuel Sentieiro Neves

Assessment Methods

Final assessment
Exam: 100.0%

Continuous assessment
Resolution Problems: 50.0%
Mini Tests: 50.0%

Bibliography

• Júlio S. Neves, Apontamentos Teóricos de Análise Matemática, 2011.
• Carlos Sarrico, Análise Matemática, Leitura e exercícios,  Colecção: Trajectos Ciência n. 4, Gradiva, 1997.
• J. Carvalho e Silva, Princípios de Análise Matemática Aplicada,  McGraw-Hill, 1994.
• J. Campos Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, Lisboa, 7a. Edição, 1999.
• Earl W. Swokowski, Cálculo com Geometria Analítica (Volumes I e II), McGraw-Hill, São Paulo, 1983.