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Elements of Mathtematics I

Code:

MI05

Sigla:

EM I

Áreas Científicas
Classificação	Área Científica
CNAEF	Mathematics

Ocorrência: 2022/2023 - 1S

Ativa?	Yes
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	26	Plano de Estudos	1	-	6	60	162

Língua de trabalho

Portuguese

Objetivos

The general objective of this course unit is to provide students with the basic mathematical knowledge required in the professional training of a top professional technician.

Resultados de aprendizagem e competências

By the end of term time, students should be able to:

1. Identify the properties of a real function.


2. Characterise inverse trigonometric functions.


3. Interpret the notion of limit of a function and calculate the limit of a function.


4. Analise the function continuity and apply the theorems of Bolzano and Weierstrass.


5. Interpret the concept of derivative of a function and calculate the derivative of a function at a point by definition.


6. Analise the differentiability of a function in an open interval and apply the derivative rules to calculate the derivative function.


7. Apply the theorems of Rolle, Lagrange and Cauchy.


8. Apply the Taylor's theorem to a k-times differentiable function.

Modo de trabalho

Presencial

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

Previous mathematical knowledge acquired through to secondary school, in particular fractional number and polynomials operations; equation and inequality solving; elementary properties of a real function.

Programa

1. Real Functions of Real Variable
1.1. Introduction to mathematical language and logical operations.
1.2. Generalities about real functions of real variable.
1.3. Study of inverse trigonometric functions.
1.4. Notion of limit; lateral boundaries; properties and operations.
1.5. Continuous functions, properties and continuity extension.
1.6. Fundamental theorems of continuous functions.

2. Differential Calculus in R
2.1. Notion of derivative of a function: definition and interpretations in geometric and physical terms; equations of the lines tangent and normal to the graph of a function at a point.
2.2. Lateral derivatives; differentiability and their properties; derivation rules; derived from the compound function and the inverse function; derived from inverse trigonometric functions; notion of differential.
2.3. Fundamental theorems of differentiable functions.
2.4. Derivatives of higher order; Taylor and Maclaurin formulas (Lagrange remnants). Application to the study of monotony, extremes and concavities.

Bibliografia Obrigatória

Campos Ferreira, J.; Introdução à Análise Matemática - 12ª edição, Fundação Calouste Gulbenkian, 2018. ISBN: 978-972-31-1388-4

Bibliografia Complementar

Larson, R., Hostetler, R. P., Edwards, B. H.; Cálculo – Vol. I – 8ª edição, McGraw Hill, 2006
Thomas, G.; Cálculo, Vol. 1 - 11ª Edição, Pearson, 2009

Métodos de ensino e atividades de aprendizagem

Student-centered approach is the pedagogical methology applied. During classes, the fundamental concepts on the different subjects of the course unit are firstly presented, illustrated by some application examples. Afterwards, students will carry out exercises to consolidate knowledge on the covered topics through collaborative working group, under the guidance of the teacher.

Tipo de avaliação

Distributed evaluation with final exam

Componentes de Avaliação

Designation	Peso (%)
Exame	0,00
Teste	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

The approval in this UC (curricular unit) can be obtained through two assessment processes: Continuous Evaluation or Exam Evaluation.

CONTINUOUS EVALUATION

The Continuous Assessment assumes:

º A1: Conducting individual exercises in synchronous class;

º A2: Two Tests (with ratings rounded to the nearest tenth)

The Continuous Assessment Score (NAC): Mean A1*0.15+Average A2*0.85

Access to continuous assessment requires attendance of at least 75% of classes. The conditions for passing the continuous assessment are as follows:

If the NAC (rounded to units) is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to the aforementioned average, provided that in any of the tests the grade was greater than or equal to 0 values.
If the NAC (rounded to units) is greater than or equal to 18, the student must take an oral test, obtaining as a final grade the average of the classification obtained and of the referred oral test. If the student does not attend the oral test, the final classification will be 17 values.
Recovery of one of the tests

The conditions for its realization will be as follows:

in order to meet the passing conditions (NAC is greater than or equal to 10 and classification in both Tests greater than or equal to 7.0 values), a student who has a classification greater than or equal to 7.0 in one of the Tests has the option of performing the recovery of one and only one of the Tests, on the same day and time of the Normal Season Exam;
a student who has scored less than 7.0 in one of the Tests, has not been able to take it or has given up, will only be able to make the recovery of that Test;
approved students cannot recover a Test in order to improve their grade.

EVALUATION BY EXAM
Students who have not obtained approval for Continuous Assessment may take an exam, being approved as long as they obtain a grade of 10 or higher.

NOTE: In any of the evaluation processes, whenever the final classification is greater than or equal to 18 values, the student must carried out an oral test, obtaining as a final grade the average of the classifications of the written test and of the said oral test . If the student does not attend the oral test, the final classification will be 17 values.

Fórmula de cálculo da classificação final

Average A1*0,15+ Average Média A2*0,85
or
Exam evaluation

Avaliação especial (TE, DA, ...)

Working students, high-level athletes, association leaders and students under the Religious Freedom Law must address, until the second academic week of the semester, to the head of the Curricular Unit to present their pertinent specificities, in accordance with the terms of the respective diplomas under penalty of failure to enforce them for lack of objective conditions.

Melhoria de classificação

In this course unit passed students of this academic year may only apply to their improvement classification in the supplementary period exam.