Mathematics I

Ocorrência: 2022/2023 - 1S

Ciclos de Estudo/Cursos

Língua de trabalho

Objetivos

Resultados de aprendizagem e competências

1. 
   
   Identify the properties of real variable real functions, interpret the limit concept and compute function limits.
   
   


2. 
   
   Analyze the continuity of a function and apply the Bolzano and Weierstrass theorems.
   
   


3. 
   
   Characterize the inverse of a trigonometric function.
   
   


4. 
   
   Interpret the concept of derivative and compute the derivative of a function at a point by definition.
   
   


5. 
   
   Analyze the differentiability of a function on an open interval and apply the rules of derivation, Rolle, Lagrange and Cauchy theorems.
   
   


6. 
   
   Apply the Taylor formula of a function at a point.
   
   


7. 
   
   Recognize and determine antiderivatives.
   
   


8. 
   
   Integration by parts, by substitution and integration of rational functions.
   
   


9. 
   
   Understand the Riemann integral and apply the fundamental theorem of integral calculus.
   
   


10. 
    
    Compute definite integrals using Barrow's formula.
    
    


11. 
    
    Calculate areas and volumns using definite integrals.
    
    


12. 
    
    Calculate improper integrals with unbounded endpoints and with unbounded functions.
    
    

Modo de trabalho

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

Programa

1. Real functions of one real variable




1. Basic notions.


2. Limits; one-sides limits; properties.


3. Continuous functions, properties, continuous extension.


4. Intermediate-Value, Extreme-Value and inverse function theorem.


5. Inverse trigonometric functions.




2. Differential calculus in ℝ




1. Derivative and its interpretations; tangent and normal lines to a graph.


2. Differentiable function, properties; differentiation rules; chain rule and inverse function theorem; derivatives of inverse trigonometric functions;


3. Fundamental theorems: Rolle’s, Mean-Value and Cauchy’s Mean-Value theorem; Hospital’s rule.


4. Higher-order derivatives; Taylor’s and Maclaurin’s formulas (Lagrange’s remainder). Monotony, local extrema, concavities.




3. Integral calculus in ℝ




1. Antiderivatives; properties. Methods of integration: by parts, by substitution, of rational functions.


2. Riemann integral; properties. Indefinite integral; properties.


3. Fundamental Theorem of Calculus, Barrow’s formula.


4. Integration by parts and by substitution.


5. Areas and volumns of solids of revolution.


6. Improper integrals.

Bibliografia Obrigatória

Bibliografia Complementar

Métodos de ensino e atividades de aprendizagem

Tipo de avaliação

Componentes de Avaliação

Componentes de Ocupação

Obtenção de frequência

Fórmula de cálculo da classificação final

Avaliação especial (TE, DA, ...)

Melhoria de classificação

Observações

1. Rules for conducting assessment tests/exams:




1. For tests and exams, the student must register in Moodle in time. Students will be informed of deadlines and all information is published in Moodle.


2. 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).


3. A form is provided for consultation during the tests and will be made available in Moodle in advance.


4. During the assessment tests/exams, the presentation of an oficial identification document with a photograph will be mandatory.


5. 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.


6. Exams or questions written in pencil are not accepted, only blue or black pen.


7. It is not allowed to use a corrector pen in tests/exams.


8. 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 they were used.


9. The use of calculators is not allowed.


10. The rules for conducting remote assessments, if necessary, will be published in time.


11. 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 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.


12. 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.




2. Office hours will be published in Moodle.


3. 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.