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Code: | LEM21112 | Sigla: | MA |

Áreas Científicas | |
---|---|

Classificação | Área Científica |

OFICIAL | Matemática |

Ativa? | Yes |

Página Web: | https://moodle.ips.pt/2122/course/view.php?id=1855 |

Unidade Responsável: | Departamento de Matemática |

Curso/CE Responsável: |

Sigla | Nº de Estudantes | Plano de Estudos | Anos Curriculares | Créditos UCN | Créditos ECTS | Horas de Contacto | Horas Totais |
---|---|---|---|---|---|---|---|

EM | 112 | Plano de Estudos | 2 | - | 6 | 60 | 162 |

Docente | Responsabilidade |
---|---|

Ana Isabel Celestino de Matos |

Theorethical and Practical : | 4,00 |

Type | Docente | Turmas | Horas |
---|---|---|---|

Theorethical and Practical | Totais | 3 | 12,00 |

Paula Cristina Martins dos Reis | 4,00 | ||

Ana Isabel Celestino de Matos | 4,00 | ||

Carla Cristina Morbey Rodrigues | 4,00 |

The objective of Matemática Aplicada is to learn key areas of mathematics that are necessary for understanding, modeling, predicting, identify and solve problems of general engineering and mechanical engineering in particular, providing powerful tools for a less elementary approach on the curricular units in engineering courses.

In each topic, students must acquire the following skills.

1. Multiple integrals

Identify and geometrically represent the integration region.

Recognize the most effective integration order.

When necessary, make the most appropriate change of variable.

Calculate areas, volumes, masses, centers of mass and moments of inertia using multiple integrals.

**2. Ordinary Differential Equations (ODE)**

Recognize and solve the following five types of 1st order ODE: separable variable ODE; homogeneous ODE; linear ODE; Bernoulli EDO and exact ODE.

Identify the characteristic polynomial of an nth order linear ODE with constant coefficients and determine and classify the roots of this polynomial.

Determine the general solution of an nth order homogeneous linear ODE with constant coefficients.

Determine the general solution of an nth order linear ODE with constant coefficients, not homogeneous, in certain specific cases.

**3. Laplace transforms**

Calculate the Laplace transform of a function using the definition.

Determine, using the properties, direct and inverse Laplace transforms. Solve linear equations with constant coefficients with Laplace transforms.

**4. Series**

4.1. Numerical Series

Identify geometric, arithmetic and telescopic series, study their nature and calculate the sum when convergent.

Identify a series of non-negative terms and apply the most appropriate of the following tests to determine their nature: Integral Test; 1st and 2nd Comparison Tests; Reason Test; nth root Test.

Identify alternating series and study their nature (simple/absolute convergence or divergence).

**4.2. Power Series**

Determine the convergence range of a power series.

Represent functions in power series of (x-a), when necessary using the Derivation and Primitive Power Series Theorems.

**5. Fourier series**

Determine the Fourier coefficients and the Fourier series of a periodic function.

Geometrically interpret periodic functions, even/odd periodic functions and perform even/odd periodic extensions.

1. Multiple integrals

Recognize the most effective integration order.

When necessary, make the most appropriate change of variable.

Calculate areas, volumes, masses, centers of mass and moments of inertia using multiple integrals.

Recognize and solve the following five types of 1st order ODE: separable variable ODE; homogeneous ODE; linear ODE; Bernoulli EDO and exact ODE.

Identify the characteristic polynomial of an nth order linear ODE with constant coefficients and determine and classify the roots of this polynomial.

Determine the general solution of an nth order homogeneous linear ODE with constant coefficients.

Determine the general solution of an nth order linear ODE with constant coefficients, not homogeneous, in certain specific cases.

Determine, using the properties, direct and inverse Laplace transforms. Solve linear equations with constant coefficients with Laplace transforms.

4.1. Numerical Series

Identify geometric, arithmetic and telescopic series, study their nature and calculate the sum when convergent.

Identify a series of non-negative terms and apply the most appropriate of the following tests to determine their nature: Integral Test; 1st and 2nd Comparison Tests; Reason Test; nth root Test.

Identify alternating series and study their nature (simple/absolute convergence or divergence).

Determine the convergence range of a power series.

Represent functions in power series of (x-a), when necessary using the Derivation and Primitive Power Series Theorems.

Geometrically interpret periodic functions, even/odd periodic functions and perform even/odd periodic extensions.

Double and triple integrals: definition, properties and applications. Change of variables in double and triple integrals.

Laplace transform and inverse Laplace transform: definitions and properties. Solving linear equations with constant coefficients with Laplace transforms.

Convergent series, properties and convergence tests. Absolute convergence.

Domain of convergence, differentiation and integration of power series. Taylor series. Representation of functions in power series.

Fourier series of a periodic function. Computation of the Fourier coefficients. Fourier series of some usual functions. Properties of Fourier series. Convergence of Fourier series.

Gabriel E. Pires; Cálculo diferencial e integral em IR^n, 2ª edição, IST press, 2014

E. Kreyszig; Advanced Engineering Mathematics, John Wiley & Sons, 2011 (Advised for students who do not read Portuguese)

H. Anton; Calculus: A New Horizon, Vol. 1, 6th edition, John Wiley & Sons, 1999 (Advised for students who do not read Portuguese)

M. A. Ferreira; Integrais Múltiplos Equações Diferenciais - Exercícios, 1ª edição, Edições Sílabo, 1995

M. O. Baptista, M. A. Silva; Equações Diferenciais e Séries - Exercícios, 1ª edição, Edições Sílabo, 1994

B. Demidovich; Problemas e Exercícios de Análise Matemática, McGraw-Hill, 1993

N. Piskounov; Cálculo diferencial e integral, Vol. I e Vol. 2, Edições Lopes da Silva, 1977

T. M. Apostol; Cálculo Vol. I e Vol. II, Reverté, 1993

J. Campos Ferreira; Introdução à Análise Matemática, 4ª edição, Fundação Calouste Gulbenkian, 1991

G. B. Thomas, M. D. Weir, J Hass; Cálculo, Vol. II, 11ª edição, Pearson Education, 2009

João P. Santos; Cálculo numa variável real, IST press, 2016

Pedro M. Girão; Introdução à análise complexa, séries de Fourier e equações diferenciais, IST press, 2014

Gabriel E. Pires; Cálculo diferencial e integral em IR^n, 2ª edição, IST press, 2014

Vasco Simões; Análise Matemática 2, Edições Orion, 2011

Matemática Aplicada (Applied Mathematics) has a teaching load of 4h/week of theoretical-practical classes, in which the fundamental concepts are presented, proved some results and solved exercises that illustrate each topic.

In these classes students should acquire an overview of the themes and their interconnections, learn the correct and objective formulation of mathematical definitions, the precise enunciation of propositions and practice the deductive reasoning, as well as learn some applications to engineering of the various notions presented.

The consolidation of knowledge by students will be based on reading the materials provided and autonomous resolution of exercises, using the study material recommended, the bibliography and the support of the teachers in their office hours.

All the information and specific materials of Matemática Aplicada will be available on its page in the Moodle platform.

**Observation:**

- daytime classes will be on-site;

- the class in after-work hours will be remotely, on the Microsoft Teams platform, in the Applied Mathematics team 2021/2022, in the specific channel of the class.

In these classes students should acquire an overview of the themes and their interconnections, learn the correct and objective formulation of mathematical definitions, the precise enunciation of propositions and practice the deductive reasoning, as well as learn some applications to engineering of the various notions presented.

The consolidation of knowledge by students will be based on reading the materials provided and autonomous resolution of exercises, using the study material recommended, the bibliography and the support of the teachers in their office hours.

All the information and specific materials of Matemática Aplicada will be available on its page in the Moodle platform.

- daytime classes will be on-site;

- the class in after-work hours will be remotely, on the Microsoft Teams platform, in the Applied Mathematics team 2021/2022, in the specific channel of the class.

Designation | Peso (%) |
---|---|

Exame | 100,00 |

Total: |
100,00 |

Designation | Tempo (Horas) |
---|---|

Estudo autónomo | 102,00 |

Frequência das aulas | 60,00 |

Total: |
162,00 |

A student can obtain approval by continuous assessment and, should he fail, by exam assessment.

The student can also choose to do only the exam assessment.

All assessments will be in person, except if the Government, the IPS President or the Director of ESTSetúbal decide otherwise (due to the pandemic).

In such case, the assessments will be remote and the evaluation rules and procedures will be updated and published on the Moodle page.**Continuous assessment**The continuous assessment is based on two tests.Let NT1 and NT2 be the grades of the tests (rounded to tenths).

The final grade, CF, will be calculated by the formula CF=0.5xNT1+0.5xNT2.

Assuming that the tests are carried out in person, the conditions of approval are the the following:

- if CF is greater than or equal to 10 and less than 17, the student is approved with a final grade equal to CF, provided that both test grades are greater than or equal to 7.0;

- if CF is greater than or equal to 17, the student must do an oral test.

The final grade will be the average of these two grades. If the student does not attend the oral test, the final grade will be 16 values.

A student may recover a test even if he has not taken it or has dropped out.

It will not be possible to recover a test with a view to improving the grade.

To recover the test, the student will have to register, according to the terms and deadlines that will be indicated in due time.

Assuming that the exam is carried out in person, let E be the grade obtained in the exam (rounded to the units):

- if E is greater than or equal to 10 and less than 17, the student is approved with final grade E;

- if E is greater than or equal to 17, the student must do an oral test. The final grade will be the average of these two grades. If the student does not attend the oral test, the final grade will be 16 values.

Assuming that the tests are carried out in person:

- if CF is greater than or equal to 10 and less than 17, the student is approved with a final grade equal to CF, provided that both test grades are greater than or equal to 7.0;

- if CF is greater than or equal to 17, the student must do an oral test. The final grade will be the average of these two grades.

If the student does not attend the oral test, the final grade will be 16 values.

Students covered by Article 271 of the IPS Student Performance Assessment Guidelines contact the teacher responsible, by email to ana.matos@estsetubal.ips.pt, until the second week of the semester, to present their relevant specificities.

According to Article 11 of the IPS Student Performance Assessment Guidelines.

Enrolment for tests/exams is required up to one week before the date of the test/exam, on the Moodle page.

It is mandatory to present an official identification document on tests and exams.

In tests or exams the student can only leave the room one hour after the start and itimplies the final delivery of the test/exam.

The only consult forms allowed on tests and exams are the ones provided by the teachers (copies are available in the Moodle page).

The use of calculators in tests or exams is not allowed.

The handling or displaying of mobile phones or any other means of remote communication during testes ou exames is not allowed.

The tests and exames are individual, so any type of communication with third parties is strictly prohibited, with the exception of the teacher who will be monitoring the tests.

**2.** The rules for remote assessments, if necessary, will be published in due time on the Moodle page.

**3.** In case of fraud suspicion or other circumstance that leads to the need to confirm the knowledge evidenced in the resolution of a test or exam, the student may be called to a session, in person, which will focus only on that knowledge. If the student does not attend this session, without due justification, the test or exam will be considered invalid.

**4.** According to the IPS rules, whenever there is a situation of proven fraud in assessments, carried out in person or remote, it will be canceled and the student will be subject to the application of the IPS Disciplinary Regulation for Students.

**5.** The office hours of the teachers will take place on the TEAMS platform and its schedule will be available on the Moodle page. The student may request face-to-face support, which is subject to prior appointment by email and the teacher confirmation of its viability.

**6.** Communication with students is carried out exclusively through the institutional e-mail. It is up to the student to periodically consult his e-mail account in the IPS domain and use it to communicate with the teachers and IPS services.

The only consult forms allowed on tests and exams are the ones provided by the teachers (copies are available in the Moodle page).

The use of calculators in tests or exams is not allowed.

The handling or displaying of mobile phones or any other means of remote communication during testes ou exames is not allowed.

The tests and exames are individual, so any type of communication with third parties is strictly prohibited, with the exception of the teacher who will be monitoring the tests.

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Página gerada em: 2024-11-14 às 21:27:05

Página gerada em: 2024-11-14 às 21:27:05