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Applied Mathematics
| Code: | LEM21112 | Sigla: | MA |
| Áreas Científicas | |
|---|---|
| Classificação | Área Científica |
| OFICIAL | Matemática |
Ocorrência: 2023/2024 - 1S
| Ativa? | Yes |
| Página Web: | https://moodle.ips.pt/2324/course/view.php?id=1796 |
| Unidade Responsável: | Departamento de Matemática |
| Curso/CE Responsável: |
Ciclos de Estudo/Cursos
| Sigla | Nº de Estudantes | Plano de Estudos | Anos Curriculares | Créditos UCN | Créditos ECTS | Horas de Contacto | Horas Totais |
|---|---|---|---|---|---|---|---|
| EM | 170 | Plano de Estudos | 2 | - | 6 | 60 | 162 |
Docência - Responsabilidades
| Docente | Responsabilidade |
|---|---|
| Carla Cristina Morbey Rodrigues |
Docência - Horas
| Theorethical and Practical : | 4,00 |
| Type | Docente | Turmas | Horas |
|---|---|---|---|
| Theorethical and Practical | Totais | 3 | 12,00 |
| Ana Isabel Celestino de Matos | 8,00 | ||
| Carla Cristina Morbey Rodrigues | 4,00 |
Língua de trabalho
PortugueseObjetivos
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.
Resultados de aprendizagem e competências
In each topic, students must acquire the following skills.
1. Multiple integrals
Identify and geometricallyrepresent 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
Identify and geometricallyrepresent 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.
Modo de trabalho
PresencialPré-requisitos (conhecimentos prévios) e co-requisitos (conhecimentos simultâneos)
Knowledge acquired in the curricular units (UC) Mathematics I and Mathematics II of the current curricular plan. (Elements of Mathematics I and Elements of Mathematics II of the CTeSP are equivalent to Mathematics I) or in the UC Mathematical Analysis I, Analysis Mathematics II and Linear Algebra and Analytical Geometry from the previous curricular plan.Programa
1. Multiple integrals
Double and triple integrals: definition, properties and applications. Change of variables in double and triple integrals.
2. Ordinary Differential Equations (ODE)
Definitions and examples. First order ODE: solving methods and techniques for separable variables equations; homogeneous equations; linear equations; Bernoulli equations and exact equations. Nth order linear equations with constant coefficients: properties and solving methods.
3. Laplace transforms
Laplace transform and inverse Laplace transform: definitions and properties. Solving linear equations with constant coefficients with Laplace transforms.
4. Series
4.1.Numerical series
Convergent series, properties and convergence tests. Absolute convergence.
4.2. Power series
Domain of convergence, differentiation and integration of power series. Taylor series. Representation of functions in power series.
5. Fourier 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.
Double and triple integrals: definition, properties and applications. Change of variables in double and triple integrals.
2. Ordinary Differential Equations (ODE)
Definitions and examples. First order ODE: solving methods and techniques for separable variables equations; homogeneous equations; linear equations; Bernoulli equations and exact equations. Nth order linear equations with constant coefficients: properties and solving methods.
3. Laplace transforms
Laplace transform and inverse Laplace transform: definitions and properties. Solving linear equations with constant coefficients with Laplace transforms.
4. Series
4.1.Numerical series
Convergent series, properties and convergence tests. Absolute convergence.
4.2. Power series
Domain of convergence, differentiation and integration of power series. Taylor series. Representation of functions in power series.
5. Fourier 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.
Bibliografia Obrigatória
Apontamentos e exercícios elaborados por docentes do DMAT; disponíveis na página Moodle da UCJames Stewart; Calculus, Brooks/Cole Publishing Company, 1999. ISBN: 0-534-35949-3 (Advised for students who do not read Portuguese)
Gabriel E. Pires; Cálculo diferencial e integral em IR^n, 2ª edição, IST press, 2014
Bibliografia Complementar
E. Kreyszig; Advanced Engineering Mathematics, John Wiley & Sons, 2011 (Advised for students who do not read Portuguese)Azenha & M. A. Jerónimo; Elementos de cálculo diferencial e integral em IR e IRn, McGraw Hill, 2006
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
Observações Bibliográficas
All books in english listed in the Complementary Bibliography are available at the ESTSetúbal Library.Métodos de ensino e atividades de aprendizagem
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.
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.
Tipo de avaliação
Evaluation with final examComponentes de Avaliação
| Designation | Peso (%) |
|---|---|
| Exame | 100,00 |
| Total: | 100,00 |
Componentes de Ocupação
| Designation | Tempo (Horas) |
|---|---|
| Estudo autónomo | 102,00 |
| Frequência das aulas | 60,00 |
| Total: | 162,00 |
Obtenção de frequência
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.
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 (rounded to units).
The conditions of approval are the the following:
- if CF is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to CF, provided that both test grades are greater than or equal to 7.5;
- if CF is greater than or equal to 18, 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 17 values.
Retrieving a test
- In order to meet the approval conditions (final average greater than or equal to 10 and classification in both tests greater than or equal to 7.5 points), a student may recover one and only one of the tests, on the same day and time of the first date of exam.
- 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.
Exam Assessment
Students who choose not to carry out the continuous assessment or fail do obtain approval on it may attend the regular exams.
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 18, the student is approved with final grade E;
- if E is greater than or equal to 18, 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 17 values.
Comment
The tests have a duration of two hours and the exams of two and a half hours.
Fórmula de cálculo da classificação final
Continuous assessment
Let NT1 and NT2 be the grades of the tests (rounded to tenths) and CF=0.5xNT1+0.5xNT2 (rounded to units).
Let NT1 and NT2 be the grades of the tests (rounded to tenths) and CF=0.5xNT1+0.5xNT2 (rounded to units).
Assuming that the tests are carried out in person:
Exam Assessment
Let E be the grade obtained in the exam (rounded to the units):
- if CF is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to CF, provided that both test grades are greater than or equal to 7.5;
- if CF is greater than or equal to 18, 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 17 values.
Exam Assessment
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 18, the student is approved with final grade E;
- if E is greater than or equal to 18, 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 17 values.
Avaliação especial (TE, DA, ...)
Students covered by article 256º of the Regulation of Academic Activities and Guidelines for Evaluation and School Performance of IPS Students must, until the second week of the beginning of the semester, contact the person responsible for the curricular unit, through the email carla.rodrigues@estsetubal.ips.pt, to present their pertinent specificities, under the terms provided for in the respective diplomas, under penalty of them not being able to be executed due to lack of objective conditions.
Melhoria de classificação
According to Article 11º of the IPS Guidelines for the Evaluation of School Performance of IPS Students, the improvement of classification may occur in the academic year of registration, in the appeal season, or in the academic year following the one of approval, in any of the exam evaluation seasons, with the exception of the special season.
Observações
Assessment rules (tests and exams)
Enrolment for tests/exams is required up to one week before the date of the test/exam, on the Moodle page.
Note:
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.
According to the IPS rules, whenever there is a situation of proven fraud in assessments, it will be canceled and the student will be subject to the application of the IPS Disciplinary Regulation for Students.
Student office hours:
The office hours of the teachers will be available on the Moodle page.
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.
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 register for the tests up to one week before the date of its realization, which consists of:
- completion of the specific survey on the UC page on Moodle;
- acquisition and delivery in the Reprography of a notebook of checkered sheets of proof (with 1 cover and 5 continuation sheets), with the headers to be filled, being the name and number of student recorded on the proper sheet corresponding to your course.
- 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.
Note:
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.
According to the IPS rules, whenever there is a situation of proven fraud in assessments, it will be canceled and the student will be subject to the application of the IPS Disciplinary Regulation for Students.
Student office hours:
The office hours of the teachers will be available on the Moodle page.
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.
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