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Mathematical Analysis 2016/2017

  • 6 ECTS
  • Taught in Portuguese
  • Both continuous and final Assessment

Objectives

With this Curricular Unit the student must able to:
-describe and apply methods and techniques necessary for modeling continuous phenomenon, necessary for other Curricular Units in the course;
-solve, analytically and numerically, problems by applying concepts and techniques from differential and integral calculus, function approximation by function series;
-analyze, criticize and explain the obtained results;
-apply mathematical knowledge for problem solving;
- abstract and know how to express the logical reasoning needed to solve problems.

Recommended Prerequisites

Knowledge acquired in Mathematics of Elementary and Secondary Education.

Teaching Metodology

The lecture, demonstrative and interrogative methods will be used for the introduction of the concepts, definitions and properties, always accompanied by examples and
problems to be resolved in discussion with the students. Wherever possible the teacher will appeal to the geometric interpretation of the problems and some demonstrations
using software. Support texts of the all the topics will also be provided.
Exercises and problems will be proposed to the students for being solved individually or in small groups. Later the solutions and also the resolution strategies used will be
discussed with the entire class.

Body of Work

1. Differential and Integral Calculus in R
Functions review
Antiderivatives
Techniques of integrations
Definite integrals and applications
2. Infinite series
Numerical series
Function series
3. Elements of numerical analisys
Theory of errors
Numerical differentiation and integration

Recommended Bibliography

- Dowling, E. (2009) Mathematical methods for business and economics, McGraw-Hill
- Simões, V. (2009). Análise Matemática 1: Resumo da Matéria + Problemas Resolvidos. Orion. ISBN: 9789728620141.
- Simões, V. (2011). Análise Matemática 2: Resumo da Matéria + Problemas Resolvidos (vol. 1). Orion. ISBN: 9789728620172.
- Sá, A. A., Louro, B. (2009). Sucessões e Séries: Teoria e Prática. Escolar Editora, ISBN: 978-972-592-238-5.
- Anton, H. “Calculus”, 10th Revised Edition, John Wiley & Sons, ISBN: 9781118721414, ISBN-10: 1118721411.
- Scilab: Free Open Source Software for Numerical Computation (2013) http://www.scilab.org/.
- WolframAlpha: Computational Knowledge Engine(2013) http://www.wolframalpha.com/

Complementary Bibliography

- Avelino, C.P., Machado, L.M. F. (2010). “Primitivas - Teoria e Exercícios Resolvidos”, Publindústria, ISBN: 9789728953591.
- Hilmonas, A., Howard, A. (2005). “Cálculo: Conceitos e Aplicações” , Livros Téc. e Cient. Editora, ISBN: 9788521614166.
- Khan Academy (2013) https://www.khanacademy.org/

Weekly Planning

WEEK 1 - Information regarding the Curricular Unity (CU): teacher, syllabus, learning outcomes, bibliography, evaluation method, office hours, moodle page.
Application of a diagnostic test.
Review of polynomial, exponential and logarithmic functions.
WEEK 2 - Review of derivation rules.
WEEK 3 – Antiderivatives of real variable functions : definition, properties, immediate primitives.
WEEK 4 - Techniques of integrations: integration by parts, integration of rational fractions.
WEEK 5 – Integration by substitution.
WEEK 6 – Definite Integrals: Properties, Fundamental Formula of Integral Calculus.
WEEK 7 – Computation of areas.
WEEK 8 - Mini-test 1 (MT1). Numerical Series: Brief review of sequences. Definitions and properties.
WEEK 9 – Geometric and Mengoli series.
WEEK 10 – Series of non-negative terms: Convergence criteria.
WEEK 11 - Alternating series. Geometrical series
WEEK 12 – Riemman series. Power series.
WEEK 13 – Taylor and MacLaurin Formulas.
WEEK 14 – Numerical Analysis elements. Theory of errors .
WEEK 15 - Numerical differentiation and integration.

Demonstration of the syllabus coherence with the curricular unit's objectives

Topics 1 and 2 of this Curricular Unity contribute directly to the first objetive, allowing the student to acquire knowledge, methods and techniques necessary for modeling
continuous phenomena that will be necessary in other Curricular Units of the course but also in situations that could arise in their professional future. This topics will also contribute to the second objetive which is also reinforced by topic 3 that will contribute, although in a basic form, for the student to be able to solve problems numerically.
Finally, topic 4 will contribute directly to the last three objetives, since it will allow the student to apply in an ative way mathematical knowledge, criticizing and discussing
the obtained results and the strategies for resolution that were used.

Demonstration of the teaching methodologies coherence with the curricular unit's objectives

Combining lecturing, demonstration and questioning will not only allow the transmission of new knowledge and skills, but also enables student participation in the learning process, encouraging group dynamics and individual work. Problem solving, individually or in group, will provide that, in an ative way, the student develops oral, writing and criticism skills as well as the ability to adapt to new situations.

relevant generic skillimproved?assessed?
Adapting to new situationsYesYes
Commitment to effectivenessYes 
Commitment to qualityYes 
Ethical and responsible behaviourYes 
Event organization, planning and managementYes 
Foreign language proficiency  
IT and technology proficiencyYesYes
Problem Analysis and AssessmentYes 
Problem-solvingYesYes
Self-assessmentYes 
TeamworkYesYes
Written and verbal communications skillsYesYes
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