Mathematical Analysis I

Instance: 2008/2009 - 1S

Programmes

Teaching - Hours Recitations: 5 Type Teacher Classes Hours Recitations Totais 6 30,00 Fernanda Campos de Sousa   10,00 Isabel Maria Silva   10,00 Victor Gonçalves de Sousa   10,00

Docência - Responsabilidades Teacher Responsabilidade Fernanda Maria Campos de Sousa Regente

This is a Draft, waits for Course Director validation.

Teaching Language

Portuguese

Objectives, Skills and Learning Outcomes

Consolidation and spread, in some extent, the knowledge in Mathematics the students have gathered in secondary school. Development of scientific and mathematical way of thinking. Understanding, manipulation and application of the concepts of one variable integration and series. Indication of a basic set of mathematical knowledge required by other subjects for civil engineering. Development of the ability to apply new mathematical concepts.

Program

Mathematical methodology, foundations of logic, real numbers axiomatic, some topologic concepts. Bolzano-Weierstrass theorem. Real functions of real variables: continuity and limit. Intermediate values and Weierstrass theorems. Differential calculus for one variable functions: definitions and geometric interpretation. Rolle, Lagrange and Cauchy theorems. Derivatives of inverse and composed functions. Practical rules and applications of derivation. Integral calculus of real functions of real variables: Riemann integral, integral operations, undefined integrals. Fundamental theorem of calculus, Barrow´s formula and mean-value theorem. Integration of rational functions. Integration by parts and integration by substitution. Improper integrals. Calculation of areas in the plan. Numerical sequences and series: Cauchy sequence, convergence and sum of a serie, absolute and conditional convergence. Series of nonnegative terms: comparison, d'Alembert, Cauchy, integral  criteria. Alternating series: Leibniz test. Sequences and series of functions: pointwise and uniform convergences. Power series. Polynomial approximation, Taylor polynomials and formula.

Main Bibliography

Complementary Bibliography

Teaching Procedures

Concepts and important results are presented in theoretical classes, with geometric interpretation, when possible, and enlightening examples. Some constructive demonstrations are presented. There will be a strong call for concepts understanding and calculation ability. The students will be alerted to the available computational tools, their performance and limitations. In practical classes, the students are guided to the resolution of selected problems.

Evaluation Type

Distributed evaluation without final exam

Registered evaluation and occupation components

Admission to Exams

Limits for absences to classes are determined by Art. 4.1 ( 25 % of the expected practical classes).

Final mark

Two write tests, M1 and M2.
The Final Classification (CF) is
CF=0.4M1+0.6M2.
(M1 and M2 -20 values)

Special Assignments

"Mini-test", only for the students with a final mark of 9. This test will be an additional chance for the students that are close to reach the minimum needed on this subject.

Special evaluation (TE, DA, ...)

SPECIAL RULES FOR MOBILITY STUDENTS: Proficiency in Portugues; Evaluation by exam and/or coursework(s) defined in accordance with student profile.

Improvement of Final/Distributed Classification

According art. 10.2. from General Rules for Evaluation, improvement of classification will be submitted to the previous formula

Comments

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Working time estimated out of classes: 4 hours