| 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 |
The assessment will consist of three individual exercise lists (E1, E2, and E3, each graded on a scale from 0 to 1 point), a presentation to be delivered in class (A, graded on a scale from 0 to 2 points), and two written tests (T1 and T2, each graded on a scale from 0 to 20 points). The final grade will be the result of rounding to the nearest whole number the value obtained by the following calculation:
FG = 0.75T + E + A,
where T = (T1 + T2)/2 and E = E1 + E2 + E3.
If, after rounding, the grade is higher than 16 points, the student must take an oral exam. In this case, the final grade will be determined by the examination board and cannot be lower than 16 points.
All students are allowed to take the oral exam, with the final grade being decided by the examination board, taking into account the grades from the different assessment components and the performance in the oral exam.
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Language |
Portuguese. Tutorial support is available in English.
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