Subject: Mathematical Analysis I

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

Overall objectives:

1 - The acquisition of the foundations of the theory of Mathematical Analysis in IR.
2 - To obtain mastery in reasoning processes and to develop concepts formalisation skill. Developing abstraction skills.
3 - Developing abstraction skills. Knowledge of proofing methods and being able to select the most adequate one for solving specific problems.

Syllabus:

1 - Real numbers: field, order, axiom of supremum, topological notions.
2 - Natural numbers. Induction method. Sequences: limits, subsequences, Squeeze Theorem for Sequences, Cauchy sequences. Numerical series: convergence; geometric series; convergence criterias, absolute convergent series.
3 - Real funtions in one variable. The exponencial, logarithmic and trigonometic functions, the inverse trigonometric functions, hiperbolic functions. Limits and continuity. Uniform continuity. Differentiability - fundamental theorems. Cauchy rule and indeterminate expressions.
4 - Taylor's formula and applications. Power series and interval of absolute convergence. Sequences and series of functions: Pointwise and Uniform convergence; fundamental results.

Literature/Sources:

Apostol Tom M. , 1983 , Cálculo , Reverté Ltda
J. Campos Ferreira , 2005 , Introdução à Análise Matemática , Fundação Calouste Gulbenkian, 8ª edição
Courant R. e John F. , 1989 , Introduction to Calculus and Analysis , Springer-Verlag
Spivak, M. , 2006 , Calculus , Cambridge Univers . Press

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
In the lectures the students will become familiarized with the results and their proofs. For the tutorial classes it will be created and provided worksheets with practical and theoretical problems. The evaluation of the curricular unit will focus on two written tests, each with a weight of 45% of the final grade, and an oral presentation, worth 10% of the final grade. Each test shall have a duration of 2h30m e the oral presentation will be about solving a theoretical exercise or a theorem, assigned randomly to each student, from a list made know by the teacher. During the supplementary Exam period students who have not successfully completed the curricular unit can choose between retaking one of the tests or doing an exam which includes all the contents of the curricular unit.