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Elements of Mathtematics I

Code: PARC02     Sigla: EMI

Áreas Científicas
Classificação Área Científica
CNAEF Mathematics

Ocorrência: 2021/2022 - 1S

Ativa? Yes
Página Web:
Unidade Responsável: Departamento de Matemática
Curso/CE Responsável: Professional Technical Higher Education Courses in Networks and Information Systems

Ciclos de Estudo/Cursos

Sigla Nº de Estudantes Plano de Estudos Anos Curriculares Créditos UCN Créditos ECTS Horas de Contacto Horas Totais
TSPPAR 44 Plano de Estudos_2015_16 1 - 6 60 162

Docência - Responsabilidades

Docente Responsabilidade
César Rodrigo Fernandez

Docência - Horas

Theorethical and Practical : 4,00
Type Docente Turmas Horas
Theorethical and Practical Totais 1 4,00
César Rodrigo Fernandez 4,00

Língua de trabalho



Providing students with the basic mathematical knowledge on real functions of a single real variable required in the professional training of a top professional technician.

Resultados de aprendizagem e competências

1. Identification of properties of functions on a real variable
2. Characterization of the inverse of a trigonometric function.
3. Interpretation of the concept of limit value and computation of such limits on functions.
4. Analysis of the continuity of a function and application of Bolzano's and Weierstrass' theorems.
5. Interpretation of the concept of derivative and computation of the derivative of a function on any point, by definition.
6. Analysis of the differentiability of a function on an open interval and application of derivation rules.
7. Application of Rolle's, Lagrange's and Cauchy's Theorems.
8. Application of Taylor's formula of a function on a point.

Modo de trabalho


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

Mathematical knowledge corresponding to average secondary education, namely rational number an polynomial arithmetic, solving equations and inequalities, and elementary notions of real functions of a single real variable.


1. Real functions of a real variable
1.1. Introduction to mathematical language and logical operations.
1.2. Generalities about real functions of a real variable.
1.3. Study of inverse trigonometric functions.
1.4. Notion of limit; lateral limits; properties and operations.
1.5. Continuous functions, properties and extension by continuity.
1.6. Fundamental theorems of continuity.
2. Differential calculus
2.1. Derivative of a function: concept, geometrical perspective and physical perspective; tangent and normal straight lines at a specific point.
2.2. Lateral derivatives; differentiability and its properties; derivative rules; derivatives of composite and inverse functions; inverse trigonometric functions derivatives; differential.
2.3. Fundamental theorems of differentiation.
2.4. Derivatives of higher order; Taylor and Maclaurin formulas (with Lagrange error bound). Study of monotony, extrema and concavity of a function.

Bibliografia Obrigatória

Departamento de Matemáica ESTS; Textos de Apoio editados pelo Departamento de Matemática (

Bibliografia Complementar

Ron Larson, Robert P. Hostetler, Bruce H. Edwards; Cálculo Vol.1, McGraw-Hill. ISBN: 85-86804-56-8
Azenha A.; Jerónimo M.A.; Cálculo diferencial e integral em IR e IRn, McGraw-Hill. ISBN: 972-8298-03-X
Tom M. Apostol; Calculus - Vol 1, Wiley International
Análise Real; F.R. Dias Agudo, Escolar Editora,. ISBN: 972-9241-53-8
J.P. Santos; Cálculo numa variável real, IST Press. ISBN: 978-989-8481-18-4
Demidovitch B.; Problemas e exercícios de Análise Matemática, Escolar Editora
F. R. Dias Agudo; Análise real, Escolar Editora
Campos Ferreira J.; Introdução à Análise Matemática, Fundação Calouste Gulbenkian

Métodos de ensino e atividades de aprendizagem

Classes are theoretical-practical where the fundamental concepts of the different subjects of the program are presented and some exercises that illustrate the topics are solved. Then students will carry out, under the guidance of the teacher, a set of exercises that will allow them to obtain an understanding of the topics covered.

Central notions are introduced with a rigorous mathematical language, and its properties are analysed, identifying the most relevant results regarding them, with the help of examples and counter-examples. Solving of exercises, first by direct application of the studied properties, and later by extension of these ideas to a broader areas, will simplify illustrate and solve any doubts among the students about the meaning of these objects and its properties.

Proposal of practical exercises for the students to solve outside of the classroom and the presentation of these resolutions within the classroom allows to fix the adquired knowledge.

Tools for the lesson: Whiteboard, Computer with projector, visual teaching environments (Moodle/Teams), Lesson Notes and Exercise collection.

Tipo de avaliação

Distributed evaluation with final exam

Componentes de Avaliação

Designation Peso (%)
Teste 100,00
Total: 100,00

Componentes de Ocupação

Designation Tempo (Horas)
Estudo autónomo 102,00
Total: 102,00

Obtenção de frequência

Students have two systems for evaluation: continuous evaluation or evaluation by exam.


Continuous evaluation assumes a compulsory presence in at least 75% ot presential lessons and consists on the execution of 4 Mini-tests (open answer questions)

Taking as MT1, MT2, MT3, MT4 the Mini-test results (evaluated from 0 to 5 points, rounded to the decimal point), the final score FS (rounded to units) is computed as follows:

FS = MT1 + MT2 + MT3 + MT4.

Conditions to obtain positive evaluation are:

If FS is greater or equal 10 and lower than 18, the student has positive evaluation with FS as final score, provided that MT1+MT2 and MT3+MT4 are both greater or equal to 3.5 points.
In the case that the student doesn't have positive evaluation due to conditions indicated on (1.), he/she may choose to take an alternative exam to improve the lowest of the results MT1+MT2 or MT3+MT4, provided that one of these sums is equal or greater than 3.5 points.


Students that don't reach a positive result in the continuous evaluation may take an exam, with results in the inteber scale between 0 and 20 points, and where the positive evaluation is for those students that get 10 or more points.

In any of these evaluation systems, if the final score is greater or equal 18, the student will be subject to an additional oral exam. The final score in this case will be the mean of the written and oral evaluations. If the student does not attend the oral exam, its final score will be 17 points.

Fórmula de cálculo da classificação final

FS = MT1 + MT2 + MT3 + MT4.

FS - Final score (integer values range 0-20)

MT1 - Mini-test 1 score (values range 0-5 rounded to decimal point)

MT2 - Mini-test 2 score (values range 0-5 rounded to decimal poi3t)

MT3 - Mini-test 3 score (values range 0-5 rounded to decimal point)

MT4 - Mini-test 4 score (values range 0-5 rounded to decimal point)

Avaliação especial (TE, DA, ...)

Working Students, competition sportlers, student associations leaders and students demanding application of Religious Freedom Laws should contact the Curricular Unit Responsible teacher on this subject, no later than two weeks after the start of the teaching activities, indicating the corresponding specific circumstances, as declared by the corresponding norms. The timely application is needed so that the corresponding measures can be applied, with the needed objective conditions.


Each mini-test has a duration of 60 minutes; The alternative exam has 90 minutes, and the exam is 2 hours and 30 minutes.
In all evaluation processes the presentation of an ID with photo is compulsory.
In all written exams student may consult an specific formula sheet provided by the teachers.
The use or merely reaching for any electronic devices during the evaluation proofs is forbidden.
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