Voskoglou М. [voskoglou@teiwest.gr]
Graduate Technological Educational Institute of Western Greece, Patras, Greece
Download in PDF: http://fmo-journal.fizmatsspu.sumy.ua/journals/2017-v2-12/2017_2-12-Voskoglou_Scientific_journal_FMO.pdf

A NOTE ON THE GRAPHICAL REPRESENTATION OF THE DERIVATIVES

Abstract. In the article at hands an alternative definition of the concept of the derivative is presented, which makes no use of limits. This definition is based on an old idea of Descartes for calculating the slope of the tangent at a point of a curve and holds for all the algebraic functions. Caratheodory extended this definition to a general definition of the derivative in terms of the concept of continuity. However, although this definition has been used successfully by many German mathematicians, it is not widely known in the international literature, nor it is used in the school book texts. After presenting Caratheodory’s definition, the article closes by describing methods for calculating the derivative at a point of a function y = f(x) with the help of a suitably chosen table of values of f(x), and for designing of the graph of the derivative function f΄(x) given the graph, but not the formula, of f(x). These methods are based on the graphical representation of the derivative, which should be reclaimed better in general for teaching purposes.
Key words: teaching/learning the derivatives, tangent at a point of the graph of a function, algebraic and transcendental functions, Caratheodory’s definition of the derivative, graphs of the derivative functions, Maple software.

ПРО ГРАФІЧНЕ ПРЕДСТАВЛЕННЯ ПОХІДНИХ
Воскоглой М. Гр.
Вищий навчально-технологічний інститут Західної Греції, Патрас, Греція

Анотація. У статті дано альтернативне визначення поняття похідної, в якому не використовується поняття границі. Це визначення ґрунтується на ідеї Декарта для обчислення нахилу дотичної до кривої в точці і справедливе для всіх алгебраїчних функцій. Каратеодорі розширив це визначення до загального визначення похідної в термінах неперервності. Проте, хоча це визначення успішно використовувалося багатьма німецькими математиками, воно не було широко відомим у міжнародній літературі і не використовувалося у шкільних підручниках. Також у статті описано методи обчислення похідної функції y = f (x) в точці за допомогою підібраної таблиці значень функції y = f (x) та методи побудови графіка похідної функції y = f΄(x) за графіком функції y = f (x), а не за її формулою. Ці методи ґрунтуються на графічному представленні похідної, що можна активно використовувати у навчанні.
Ключові слова: вивчення похідної, дотична до графіка функції в точці, алгебраїчні і трансцендентні функції, визначення похідної за Каратеодорі, графік похідної функцій, програма Maple.

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