| Code |
13898
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| Year |
1
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| Semester |
S1
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| ECTS Credits |
7,5
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| Workload |
TP(75H)
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| Scientific area |
Mathematics
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Entry requirements |
Not applicable.
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Learning outcomes |
i) To understand, to relate and to apply concepts and basic results of calculus in one variable;
ii) To apply the concepts of limit, derivative and integral of a real function of one variable;
iii) To analyze and understand mathematical proofs;
iv) To communicate using mathematical language, written and orally;
v) To formulate and to solve problems related to one variable real functions.
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Syllabus |
1. Real numbers
1.1. Axiomatics of the real numbers
1.2. Natural numbers: induction
1.3. Sequences
1.4. Cauchy sequences
1.5. Topological notions
2. Real functions of a real variable
2.1. Domain, range and graph.
2.2. Limits; lateral limits; infinite limits and limits at infinity
2.3. Asymptotes
2.4. Continuity
2.5. Uniform co ntinuity
2.6. Bolzano’s and Weierstrass’ theorems
3. Differential calculus
3.1. Derivative: geometric interpretation; lateral derivatives
3.2. Differentiability; differentiation rules
3.3. Derivatives of a composition and of t he inverse
3.4. Theorems of Fermat, Rolle, Lagrange and Cauchy
3.5. Cauchy’s rule and indeterminations
3.6. Higher order derivatives and Taylor formula
3.7. Extremes and convexity
4. Integral calculus
4.1. Riemann integral; integrability
4.2. Fundamental Theorem of Calculus
4.3. Techniques of primitivation and integration
4.4. Applications
4.5. Improper integrals.
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Main Bibliography |
- Conway, J. B. (2017). A First Course in Analysis. Cambridge University Press.
- Ferreira, J. C. (2008). Introdução à Análise Matemática. (9ª edição). Lisboa: Fu ndação Calouste Gulbenkian.
- Figueira, M. (2011). Fundamentos de Análise Infinitesimal (5ª edição). Textos de Matemática. Faculdade de Ciências da Universidade de Lisboa.
- Lages Lima, E. (1992). Cur so de Análise, vol. 1. (7ª edição). IMPA.
- Lages Lima, E. (2017). Análise Real, vol. 1. (12ª edição). IMPA.
- Sarrico, C. (2017). Análise Matemática - Leituras e Exercícios. (8.ª edição). Gradiva.
- Tao, T. (2016). Analysis I, Texts and Readings in Mathematics. (3rd edition). Springe r.
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Language |
Portuguese. Tutorial support is available in English.
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