Discipline details

Course details

Recommended Prior Knowledge

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Objectives

The goal is to carry on developing the mathematical reasoning initiated in highschool education, in order to be able to meet the demands of other curricular units. On completing this curricular unit, students should have acquired the necessary skills in differential calculus and integration of functions of one variable, including the fundamental theorems of calculus.

Not applicable
Teaching Methods

Theoretical classes with lecturing periods with application examples followed by small tasks to be done by the students in order to consolidate the contents previously taught. Practical classes dedicated to problem solving, individually or in small groups.
Internship(s)

Não
Syllabus

Limits and continuity: Basics on real valued functions. Exponential and logarithmic function. Trigonometric inverse functions. Continuity and limit. Mean Value and Weierstrass Theorems.
Differential calculus: Derivative concept, rules; differentiability and continuity; higher order derivatives, applications. Rolle’s, Lagrange’s, Cauchy’s and L’Hôpital’s Theorem. Taylor’s formula and its applications. Integral calculus: Antiderivatives by inspection, by parts, by substitution and integration of rational functions. Integral calculus of real functions. Integrability conditions; properties of integrable functions. Indefinite integral, derivative of an indefinite integral, Fundamental Theorem of Calculus, Barrow’s formula. Integration by parts and by substitution. Application of integral calculus to the computation of area, volume of revolution solids and curve length. Moments, center of mass and centroids.
Content Explanation

The syllabus of the curricular unit allows the student to be familiarized with differentiation and integration techniques of functions of one variable, to be able to use them when requested in other specific curricular units of the course.
Methodology Explanation

The teaching methodology, rather focused on autonomous reasoning training as well as exercise-problem solving, fulfils the purpose of giving the students the ability of applying calculus techniques that will be useful in other contexts.
Responsible Lecturer(s)

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Bibliography

Departamento de Matemática do IST; Exercícios de Análise Matemática I e II, IST Press, 2005. ISBN: 978-989-8481-83-2
T. Apostol,; Calculus, Vol. I, second edition, Wiley, 1967
M. Ferreira e I. Amaral, ; Matemática, Exercícios, Primitivas, Integrais, edições sílabo, 1996
J. Campos Ferreira; Introdução à Análise Matemática, Fundação Gulbenkian, 8a ed., 2005
Larson, Hostetler e Edwards; Cálculo, Vol. 1, 8a edição, McGraw-Hill, 2006
B. Demidovitch; Problemas e Exercícios de Análise Matemática, Editora Mir, 1997
N. Piskounov; Cálculo Diferencial e Integral, Vol. I, Lopes da Silva Editora, 1997
C. Sarrico; Análise Matemática, Leitura e exercícios, 1a edição, Gradiva, 1997