An Investigation of the Algebraic Curve y[superscript 3]

Autor/in	Kermond, John
Titel	An Investigation of the Algebraic Curve y[superscript 3] - 3y + 2x = 0
Quelle	In: Australian Senior Mathematics Journal, 21 (2007) 1, S.31-45 (15 Seiten) PDF als Volltext (1); PDF als Volltext kostenfreie Datei (2) Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0819-4564
Schlagwörter	Geometric Concepts; Intervals; Mathematics Instruction; Mathematical Concepts; Mathematical Formulas; Algebra; Mathematical Logic; Validity; Equations (Mathematics) + Suchen Sie Ihr Suchwort? Elementare Geometrie; Mathematics lessons; Mathematikunterricht; Mathematische Formel; Mathematical logics; Mathematische Logik; Gültigkeit; Equations; Mathematics; Gleichungslehre
Abstract	In this paper, the author investigates the algebraic curve defined by the relation y[superscript 3] - 3y + 2x = 0. Treating this relation as a reduced cubic in the variable y, he uses a procedure first discovered by the mathematician Scipione del Ferro (Nahin, 1998, pp. 8-10) to obtain an expression for y in terms of x, namely y = (-x + [square root](x[superscript 2]-1))[superscript 1/3] - (x + [square root](x[squared]-1))[superscript 1/3]. By applying de Moivre's theorem to each term on the right hand side of the second equation, he obtains three different branches of y and uses them to show that the domain of the curve is the set of all real numbers. He finds the derivative using implicit differentiation and uses it to determine additional properties and features of these branches. Although it might seem that a graphics calculator can be used to draw a graph of the curve from the second equation, there is a problem in that such graphs have a break over the interval -1 less than x less than 1. This suggests (incorrectly) that the interval -1 less than x less than 1 is not in the domain of the curve. He explains the reason for this problem in the last part of the paper. (Contains 3 figures.) (ERIC).
Anmerkungen	Australian Association of Mathematics Teachers (AAMT). GPO Box 1729, Adelaide, 5001, South Australia. Tel: +61-8-8363-0288; Fax: +61-8-8362-9288; e-mail: office@aamt.edu.au; Web site: http://www.aamt.edu.au
Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2017/4/10