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Mathematical Applications B

Code:

MI21

Sigla:

AMB

Áreas Científicas
Classificação	Área Científica
OFICIAL	Matemática

Ocorrência: 2023/2024 - 2S

Ativa?	Yes
Página Web:	https://moodle.ips.pt/2324/course/view.php?id=1635
Unidade Responsável:	Departamento de Matemática
Curso/CE Responsável:	Professional Technical Higher Education Course in Industrial Maintenance

Ciclos de Estudo/Cursos

Sigla	Nº de Estudantes	Plano de Estudos	Anos Curriculares	Créditos UCN	Créditos ECTS	Horas de Contacto	Horas Totais
TSPMI	33	Plano de Estudos	1	-	6	60	162

Língua de trabalho

Portuguese

Objetivos

The objective of this course is to provide students with basic knowledge of complex numbers and some notions of Linear Algebra (namely matrices, determinants, vector calculus and analytic geometry in the plane).

It is intended that students understand the notions and elementary properties of these topics, so that they know how to use these mathematical tools, necessary in their training as senior technicians.

 

Resultados de aprendizagem e competências

In each topic students should acquire the following skills:

1. Identify the algebraic, trigonometric, and exponential representations of a complex number and perform operations with complex numbers.
2. Perform algebraic operations with matrices.
3. Use elementary operations to transform a matrix into the echelon form and solve systems of linear equations using the Gaussian elimination method.
4. Calculate the determinant of a matrix and apply its properties to calculate the inverse of a matrix and solve systems of linear equations.
5. Perform operations with vectors and apply their properties.
6. Calculate inner product, cross product and scalar triple product of vectors.
7. Recognize lines and conic sections and identify their geometric elements.

Modo de trabalho

Presencial

Pré-requisitos (conhecimentos prévios) e co-requisitos (conhecimentos simultâneos)

Knowledge acquired in the Curricular Unit of Fundamentals of Calculus and in Mathematics of the previous academic path.

Programa

1. Complex Numbers
 1.1. Introduction to complex numbers: algebraic form, conjugate and modulus.
 1.2. Operations with complex numbers.
 1.3. Trigonometric form and exponential form of complex numbers.
  1.4. Nth roots of a complex number.
2. Matrix Calculus
   2.1. Matrices basic notions and algebraic operations.
   2.2. Classifying and solving systems of linear equations.
3. Determinants
   3.1. Definition of determinant, its properties and computing methods
   3.2. Applications of determinants: computing the inverse matrix using the adjoint matrix; Cramer's rule.
4. Vector Calculus
   4.1. Vector notion and operations.
   4.2 Inner product of vectors, norm and unit vector of a vector and their properties.
   4.3. Angle between two vectors, orthogonal projection; orthogonal and orthonormal sets of vectors.
   4.4. Cross product and scalar triple product of vectors: definitions, properties and applications.
5. Analytic Geometry in the Plane
   5.1. Slope and equations of a line in the plane, parallel lines and perpendicular lines.
   5.2. Distance between lines; distance from a point to a line.
   5.3. Conic sections: circumference, ellipse, hyperbola and parabola.

Bibliografia Obrigatória

Apontamentos e exercícios disponíveis na página da UC no Moodle; elaborados por docentes do Departamento de Matemática

Bibliografia Complementar

Maria Adelaide Carreira e Maria Susana Metello de Nápoles; Variável Complexa – Teoria Elementar e Exercícios Resolvidos, McGraw Hill, 1997
Geraldo Ávila; Variáveis Complexas e Aplicações, Livros Técnicos e Científicos Editora, 1990
John H. Mathews and Russel W. Howell; Complex Analysis for Mathematics and Engineering, Jones and Bartlett Publishers, 1997
Earl W. Swokowski and Jeffery A. Cole; PRECALCULUS – Functions and graphs, Thomson – Brooks/Cole, 2005
Howard Anton and Chris Rorres; Elementary Linear Algebra, John Wiley & Sons, 2000
Alfredo Steinbruch e Paulo Winterle; Álgebra Linear, McGraw Hill, 1987
António Monteiro; Álgebra Linear e Geometria Analítica, McGraw Hill, 2001
Luz, C., Matos, A. e Nunes S.; Álgebra Linear (Volume I), ESTSetúbal/IPS, 2002. ISBN: 972-8431-16-9
Maria Augusta Ferreira Neves, Maria Teresa Coutinho Vieira, Alfredo Gomes Alves; Livro de Texto - Matemática 12º ano, Porto Editora, 1991
Maria Augusta Ferreira Neves, Maria Teresa Coutinho Vieira, Alfredo Gomes Alves; Exercícios de Matemática - Matemática 12º ano, Porto Editora, 1991

Métodos de ensino e atividades de aprendizagem

Mathematical Applications B has 4 hours/week of theoretical-practical classes.
During the classes, the fundamental concepts of the different subjects are presented and some exercises are solved. During the class, under the guidance of the teacher, students will solve some exercises.
The consolidation of knowledge by students will be based on reading the materials provided, the autonomous resolution of exercises using the study material provided and recommended and the support of the teacher in his office hours.

Tipo de avaliação

Evaluation with final exam

Componentes 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

Approval can be obtained either by continuous assessment or by exam assessment.

Continuous Assessment
The continuous assessment is based on two tests and two mini-tests. For first time first year students, access to continuous assessment requires attendance of at least 75% of the classes up to each test.

Let T1 and T2 be the grades of the tests and MT1 and MT2 the grades of the mini-tests (all on a scale of 0 to 20 values, rounded to tenths), the final grade will be calculated from the weighted average:

CF=((T1+T2)/2)×0.8+((MT1+MT2)/2)×0.2

with the following conditions:

• If CF (rounded to the nearest unit) 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 8.0 values.
• If CF (rounded to the nearest unit) is greater than or equal to 18, the student must take an oral exam. 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.

Recovery of tests
To meet the approval conditions (final average greater than or equal to 10 and both tests grades greater than or equal to 8.0 values), 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 retrieve a test even if he has not taken it or has dropped out. Recovering a test to improve a grade will not be allowed.
The recovery of tests is subject to the confirmation, according with the time between the dates of the 2nd test and of the first date of exam.

Exam Assessment
Students who choose not to carry out the continuous assessment or fail to obtain approval on it may attend the regular exams.
Let E be the grade obtained in the exam (rounded to the nearest unit):

• 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 mini-tests last twenty-five minutes, the tests one hour and a half and the exams two hours and a half.

Fórmula de cálculo da classificação final

Let T1 and T2 the be the grades of the tests and MT1 and MT2 the grades of the mini-tests (all on a scale of 0 to 20 values, rounded to tenths), and CF=((T1+T2)/2)×0.8+((MT1+MT2)/2)×0.2 (rounded to the nearest unit):

• 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 8 values.
• If CF is greater than or equal to 18, the student must take an oral exam. 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 nearest unit):

• 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 under Article 256 of the IPS Student Performance Assessment Guidelines students must contact the teacher responsible by e-mail (ana.matos@estsetubal.ips.pt), until the second week of the semester, to present their relevant specificities.

Melhoria de classificação

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

Observações

1. Assessment rules (mini-tests, tests and exams)

• Enrolment for any assessment test and exam is mandatory up to one week before the date of the assessment, on the Moodle page.  


• In any assessment is mandatory to present an official identification document.


• In any assessment, only the consultation form provided by the teacher for that assessment will be allowed; a copy of it will be available on the Moodle page.


• The use of calculators is not allowed in tests and exams.


• The handling or displaying of mobile phones or any other mean of remote communication during tests or exams is not allowed (mobile phones must be turned off).


• In tests and exams the student can only leave the room after one hour and it implies the final delivery of the test or exam.

2. In circumstances that lead to the need to confirm the knowledge evidenced in the resolution of a test or exam, the student may be called to a face-to-face session, which will focus only on this knowledge. If the student does not attend this session, without due justification, the test will be considered invalid.
3. In the first week of classes, students are registered on the Mathematical Applications B page on the IPS Moodle platform where all the support materials and information are available.
4. The schedule of the teacher’s office hours will be available on the Moodle page.
5. All communication with the students is carried out exclusively through the institutional e-mail. It is up to the student to periodically consult the Moodle page and his e-mail account in the IPS domain and use this e-mail to communicate with the teachers and IPS services.