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Abstract(s)
Neste trabalho lançámos o desafio de estudar derivadas de Dini, o porquê do seu aparecimento
e algumas aplicações.
Para abordarmos este assunto de uma forma coerente foi necessário traçar um caminho no
qual tivemos de recordar alguns conceitos lecionados no Ensino Secundário, como por exemplo:
sucessões e subsucessões, limite, função contínua, função diferenciável, monotonia e extremos
de uma função; assim como os resultados relacionados. Mas foi também necessário introduzir
assuntos que vão além do Ensino Secundário, como limite superior e limite inferior, funções
semicontínuas.
Para aplicar as derivadas de Dini recordamos os Teoremas de Rolle e de Lagrange, para os quais
apresentamos uma generalização envolvendo as derivadas de Dini.
Tal como em qualquer curso de Cálculo, depois do cálculo diferencial surge a integração, pois
isso no Capítulo final consta os conhecidos integrais de Riemann e Lebesgue e a construção do
integral de Henstock-Kurzweil.
In this work we launched the challenge of studying derivatives of Dini, the reason for its appearance and some applications. In order to approach this subject in a coherent way, it was necessary to draw a path in which we had to remember some concepts taught in Secondary Education, such as: sequences and subsequences, limit, continuous function, differentiable function, monotony and extremes of a function; as well as related results. But it was also necessary to introduce subjects that go beyond Secondary Education, as upper limit and lower limit, semicontinuous functions. To apply the Dini derivatives we recall the Rolle and Lagrange Theorems, for which we present a generalization involving the Dini derivatives. As in any Calculus course, after the differential calculus arises integration, for this in the final Chapter consists of the well-known integrals of Riemann and Lebesgue and the construction of the Henstock-Kurzweil integral.
In this work we launched the challenge of studying derivatives of Dini, the reason for its appearance and some applications. In order to approach this subject in a coherent way, it was necessary to draw a path in which we had to remember some concepts taught in Secondary Education, such as: sequences and subsequences, limit, continuous function, differentiable function, monotony and extremes of a function; as well as related results. But it was also necessary to introduce subjects that go beyond Secondary Education, as upper limit and lower limit, semicontinuous functions. To apply the Dini derivatives we recall the Rolle and Lagrange Theorems, for which we present a generalization involving the Dini derivatives. As in any Calculus course, after the differential calculus arises integration, for this in the final Chapter consists of the well-known integrals of Riemann and Lebesgue and the construction of the Henstock-Kurzweil integral.
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Keywords
Derivada Derivadas de Dini Integral Integral de Henstock-Kurzweil