Mathematics I
Áreas Científicas 
Classificação 
Área Científica 
OFICIAL 
Matemática 
Ocorrência: 2021/2022  1S
Ciclos de Estudo/Cursos
Sigla 
Nº de Estudantes 
Plano de Estudos 
Anos Curriculares 
Créditos UCN 
Créditos ECTS 
Horas de Contacto 
Horas Totais 
LTE 
63 
Plano de Estudos 
1 
 
6 
75 
162 
Docência  Responsabilidades
Língua de trabalho
Portuguese
Obs.: Português
Objetivos
The aim of the Mathematics I curricular unit is to provide students the basic mathematical knowledge referred to in the syllabus of the curricular unit and necessary for their training as Higher Technicians or Engineers.
Resultados de aprendizagem e competências
In each topic, students must acquire the following learning outcomes:

Identify the properties of real variable real functions, interpret the limit concept and compute function limits.

Analyze the continuity of a function and apply the Bolzano and Weierstrass theorems.

Characterize the inverse of a trigonometric function.

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

Analyze the differentiability of a function on an open interval and apply the rules of derivation, Rolle, Lagrange and Cauchy theorems.

Apply the Taylor formula of a function at a point.

Recognize and determine antiderivatives.

Integration by parts, by substitution and integration of rational functions.

Understand the Riemann integral and apply the fundamental theorem of integral calculus.

Compute definite integrals using Barrow's formula.

Calculate areas and volumns using definite integrals.

Calculate improper integrals with unbounded endpoints and with unbounded functions.
Modo de trabalho
Presencial
Prérequisitos (conhecimentos prévios) e corequisitos (conhecimentos simultâneos)
Mathematics of Secondary School.
Programa
 Real functions of one real variable
 Basic notions.
 Limits; onesides limits; properties.
 Continuous functions, properties, continuous extension.
 IntermediateValue, ExtremeValue and inverse function theorem.
 Inverse trigonometric functions.
 Differential calculus in ℝ
 Derivative and its interpretations; tangent and normal lines to a graph.
 Differentiable function, properties; differentiation rules; chain rule and inverse function theorem; derivatives of inverse trigonometric functions;
 Fundamental theorems: Rolle’s, MeanValue and Cauchy’s MeanValue theorem; Hospital’s rule.
 Higherorder derivatives; Taylor’s and Maclaurin’s formulas (Lagrange’s remainder). Monotony, local extrema, concavities.
 Integral calculus in ℝ
 Antiderivatives; properties. Methods of integration: by parts, by substitution, of rational functions.
 Riemann integral; properties. Indefinite integral; properties.
 Fundamental Theorem of Calculus, Barrow’s formula.
 Integration by parts and by substitution.
 Areas and volumns of solids of revolution.
 Improper integrals.
Bibliografia Obrigatória
DMAT; Apontamentos editados pelo Departamento de Matemática (Available in Moodle)
Bibliografia Complementar
Stewart, J.; Calculus, Cengage Learning, 2014
Apostol, T.; Cálculo, Vol. I  2ª edição, Reverté, 1994
Demidovitch, B.; Problemas e Exercícios de Análise Matemática, Escolar Editora, 2010
Métodos de ensino e atividades de aprendizagem
The Curricular Unit (CU) Mathematics I has a teaching load of 5 hours per week distributed over 3 hours of theoreticalpractical class (TP) and 2 hours of practicallaboratory class (PL). In the current context, it is expected that all classes will be carried out in person.
In TP classes, the fundamental concepts of the different topics of the syllabus will be presented and the main results demonstrated. Exercises that illustrate the topics covered are also solved. In this type of classes, students should acquire an overview of the themes and their interconnections, accompanied by a correct and objective formulation of mathematical definitions, the precise enunciation of propositions and the practice of deductive reasoning.
In PL classes, students will carry out, under the guidance of the teacher, a set of exercises, with a view to a deeper understanding of the topics covered and a greater consolidation of knowledge.
It will be up to the student, a posteriori, to carry out an autonomous study on the topics covered and deepen their knowledge, using the study material recommended in the CU bibliography and the support of the CU teachers in the respective office hours.
The CU will have all the information and specific materials (slides where the contents of the CU are displayed, exercise sheets and suggestions for videos that address the different topics of the syllabus), in the Moodle platform. To consolidating knowledge, students will be able to take 3 formative multiplechoice tests on this platform.
Observation:
the theoreticalpractical classes of the day classes are in person;
 the theoreticalpractical class in afterwork hours will have distance learning classes, on the Microsoft Teams platform, in the Mathematics Virtual Class I team.
Tipo de avaliação
Distributed evaluation with final exam
Componentes de Avaliação
Designation 
Peso (%) 
Exame 
100,00 
Total: 
100,00 
Componentes de Ocupação
Designation 
Tempo (Horas) 
Estudo autónomo 
85,00 
Frequência das aulas 
75,00 
Total: 
160,00 
Obtenção de frequência
If there is no indication to the contrary by the competent bodies, the tests and exams will be in person. If this is not possible, these assessments will be carried out online and the assessment rules and the procedures will be updated and timely disclosed on the CU page in Moodle.
The achievement of Mathematics I can be obtained through two processes of assessment in person: Continuous Assessment and Assessment by Exam.
Continuous evaluation
Continuous assessment is based on two tests (with grades rounded to the nearest tenth). The conditions for passing the continuous assessment are as follows:
 If the average (rounded to the units) of the test classifications is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to that average, provided that in any of the tests the grade has been greater than or equal to 6.5 points;
 If the average (rounded to the units) of the test classifications is greater than or equal to 18, the student will have to take an oral exam, the final grade being the average of these two grades. If you do not attend the oral exam, the final classification will be 17 points.
Recovery of one of the tests
The conditions for its realization will be as follows:
• 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 6.5), a student who has a classification greater than or equal to 8.0 in one of the tests has the option to perform the recovery of one and only one of the tests, on the same day and time of the regular period exam;
• a student who has scored less than 8.0 values 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;
• Successful students cannot take a test recovery with a view to improving their grade.
Exam assessment
The assessment based on the completion of an exam will follow the usual rules, that is, students who choose not to perform continuous assessment, or who, having opted for it, have not been approved, may attend the regular exam periods.
The approval conditions are as follows:
 If the exam grade (rounded to the units) is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to the exam grade (rounded to the units);
 If the exam grade (rounded to the units) is greater than or equal to 18, the student will have to take an oral exam, obtaining as a final grade the average of the marks of the referred oral exam and the exam. If you do not attend the oral exam, the final classification will be 17 points.
Observations:
The tests last 2 hours and the exams 2 hours and 30 minutes, being rated on a scale from 0 to 20.
Fórmula de cálculo da classificação final
Continuous Evaluation
Let T1 and T2 be the test classifications rounded to tenths and CF=0.5T1x0.5T2 (rounded to units).
 If the average (rounded to the units) of the test classifications is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to that average, provided that in any of the tests the grade has been greater than or equal to 6.5 points;
 If the average (rounded to the units) of the test classifications is greater than or equal to 18, the student will have to take an oral exam, the final grade being the average of these two grades. If you do not attend the oral exam, the final classification will be 17 points.
Exam Assessment
 If the exam grade (rounded to the units) is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to the exam grade (rounded to the units);
 If the exam grade (rounded to the units) is greater than or equal to 18, the student will have to take an oral exam, obtaining as a final grade the average of the marks of the referred oral exam and the exam. If you do not attend the oral exam, the final classification will be 17 points.
Avaliação especial (TE, DA, ...)
Students with special rights referred to in the Regulation of Academic Activities and Guidelines for Assessment and School Performance of IPS Students must, by the second week of the beginning of the semester, contact the person responsible for the curricular unit, via email at patricia.ribeiro@estsetubal .ips.pt, to present their relevant specificities, under the terms provided for in the respective diplomas, otherwise they cannot be executed due to lack of objective conditions.
Melhoria de classificação
According to Article 11 of the Regulation of Academic Activities and Guidelines for the Assessment of Academic Performance of IPS Students, the improvement of classification may occur in the academic year of enrollment, at the time of appeal, or in the academic year following the of approval, in any period of assessment by examination, with the exception of the special period.
Observações
 Rules for conducting assessment tests/exams:
 For tests and exams, it will be necessary to register in Moodle up to one week before each assessment (students will receive information via email about the opening and deadline of these registrations).
 Students must take the tests/exams in their own IPS notebooks, for which they must first acquire a notebook from reprography (with 1 cover and 4 continuation sheets).
 The consultation of an A4 handwritten sheet by the student will be allowed.
 During the assessment tests/exams, the presentation of an identification document with a photograph will be mandatory.
 Leaving the room can only take place one hour after the start of the test and will imply the final delivery of the test. In the case of exams, leaving the room is only allowed after an hour and a half from the beginning of the exam and will imply the final delivery of the exam.
 Tests or questions written in pencil are not accepted.
 It is not allowed to use a corrector in tests/exams.
 The handling or display of mobile phones, or any other means of remote communication, during the test is not allowed, which is sufficient reason for the cancellation of the assessment test, regardless of whether or not they were used.
 The use of calculators is not allowed.
 The rules for conducting remote assessments, if necessary, will be published in due time.
 In case of suspicion of fraud or other circumstance that leads to the need to confirm the knowledge evidenced in the resolution of a test, the student may be called to a facetoface 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.
 According to the IPS rules, whenever there is a situation of proven fraud in assessment tests carried out in person or at a distance in this UC, it will be canceled and the student will be subject to the application of the Disciplinary Regulation for Students of the IPS.
 Office hours will be published in Moodle and carried out remotely on the MSTeams/Zoom platform. However, the student may request facetoface support.
In any case, the student must notify the teacher in advance.
 Communication with students is made exclusively to the institutional address (@estudantes.ips.pt). It is up to the student to consult their area in the information system, on the course page in Moodle, as well as to manage and periodically consult their email account in the IPS domain, using it to communicate with the IPS services.