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Cálculo I

Código 14630
Ano 1
Semestre S1
Créditos ECTS 6
Carga Horária TP(60H)
Área Científica Matemática
Mode of delivery Face-to-face
Work placements N/A.
Learning outcomes In this Curricular Unit the students will obtain the basic knowledge of Differential and Integral Calculus of real-valued functions of a real variable.

At the end of this Curricular Unit the students should
- solve inequalities involving rational expressions and absolute values;
- determine domains and sketch the graph of functions;
- compute limits of functions;
- study the continuity of functions;
- compute derivatives of functions;
- know how to approximate functions by Taylor's polynomials;
- apply the derivatives to compute maximums and minimums and sketch the graph of functions;
- integrate functions;
- apply integrals to compute plane areas, to compute the length of curves and to compute areas of surfaces and volumes of solids generated by revolution;
- determine whether a numerical series is convergent or divergent;
- compute the interval of convergence of a power series.
Syllabus 1. Real Numbers: order relations; absolute value. 2. real functions of real variable: definition and examples; composition of functions; inverse of a function; graphic representation; exponential and logarithmic function; trigonometric and inverse functions; hyperbolic functions; continuous functions and fundamental properties. 3. Differential calculation in R: definition of derivative; derivation rules; derived function; higher-order derivatives; Rolle, Lagrange, Cauchy and Taylor theorems; calculation of limits; monotony and local extremes; inflection points and concavities; optimization problems; asymptotes; Taylor and Maclaurin polynomials. 4. Integral calculation in R; primitivation; Riemann integral: definitions and examples; properties of integrable functions; fundamental theorem of calculus; integration by substitution and integration by parts; geometric applications. 5. Numerical Series and Power Series: convergence criteria; Power series; Taylor series.
Teaching Methodologies and Assessment Criteria The evaluation will be periodic.
There is a minimum score of 6 values for the student to be admitted to the exam.
They will carry out 3 practical tests (T1, T2, T3).
The quotation for the various tests will be:
First test (T1): 8 values.
Second test (T2): 8 values.
Third test (T3): 4 values.
The teaching-learning (CEA) classification will be:
CEA = T1 + T2 + T3
The final classification (CF) will be:
CF = Not allowed if CEA <6
CF = E if 6 <= CEA <10
CF = CEA if CEA >= 10
being E the exam classification.
Main Bibliography Main Bibliography:
Alberto Simões, Apontamentos de Cálculo I, UBI.
F. R. Dias Agudo, Análise Real, Vol. I, Escolar Editora, 1989.
J. Campos Ferreira, Introdução à Análise Matemática, 6ª Edição, Fundação Calouste Gulbenkian, 1995.
Tom M. Apostol, Cálculo I, Editorial Reverté, 1994.
Supplementary Bibliography:
James Stewart, Cálculo - 5ª edição, volume 1 e volume 2, CENGAGE Learning, 2008.
Lang, S., A first course in Calculus, 5th edition,Undergraduate texts in Mathematics, Springer.
Language Portuguese. Tutorial support is available in English.

Regente

Imagem d@ Alberto Manuel Tavares Simões [Ficheiro Local]
 Alberto Manuel Tavares Simões

Curso

Engenharia Civil
Data da última atualização: 2025-01-06
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