| 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 classes will be theoretical-practical. The teacher presents the concepts and enunciates 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. Autonomous work will be encouraged, mainly consisting of exercises, problem-solving and mathematical proofs. In the interaction with the teacher, it will be promoted the use of mathematical language, in oral and written form.
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Language |
Portuguese. Tutorial support is available in English.
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