| Code |
14757
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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 |
- Lages Lima, E. (2017). Análise Real, vol. 1. (12ª edição). IMPA.\\
- Ferreira, J. C. (2008). Introdução à Análise Matemática. (9ª edição). Lisboa: Fundação Calouste Gulbenkian.
- Lages Lima, E. (1992). Curso de Análise, vol. 1. (7ª 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). Springer.
- Stewart, James, {\it C\'alculo} - Volume I, $7^a$ edição, Cengage Learning, 2014.
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Teaching Methodologies and Assessment Criteria |
Classes will be theoretical-practical. The teacher presents the concepts, states the results, demonstrating many of them. It also illustrates the theory with examples and applications. The student is encouraged to participate in classes, interacting with the teacher and sometimes solving exercises and problems. Independent work is also encouraged, consisting mainly of exercises, problems and mathematical demonstrations. Interaction with the teacher will promote the improvement of the use of written and oral mathematical language.
The assessment will consist of three individual lists of exercises (E1, E2 and E3, on a scale of 0 to 1 value), a work to be presented in class (A, on a scale of 0 to 2 values) and two written tests (T1 and T2 , presented on a scale of 0 to 20 values). The final classification will be given by rounding the result obtained by the following calculation to the units:
CF =0.75T+E+A,
where T= (T1+T2)/2, E=E1+E2+E3.
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Language |
Portuguese. Tutorial support is available in English.
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