Means and the Mean Value Theorem

Autor/inn/en	Merikoski, Jorma K.; Halmetoja, Markku; Tossavainen, Timo
Titel	Means and the Mean Value Theorem
Quelle	In: International Journal of Mathematical Education in Science and Technology, 40 (2009) 6, S.729-740 (12 Seiten)Infoseite zur Zeitschrift PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0020-739X
Schlagwörter	Mathematics Instruction; Equations (Mathematics); Problem Solving; Mathematical Logic; Validity; College Mathematics; Undergraduate Study + Suchen Sie Ihr Suchwort? Mathematics lessons; Mathematikunterricht; Equations; Mathematics; Gleichungslehre; Problemlösen; Mathematical logics; Mathematische Logik; Gültigkeit; Grundstudium
Abstract	Let I be a real interval. We call a continuous function [mu] : I x I [right arrow] [Bold R] a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I [right arrow] [Bold R} be a differentiable and strictly convex or strictly concave function. If a, b [image omitted] I with a [not equal to] b, then there exists a unique number [Xi] between a and b such that f(b) - f(a) = f '([Xi])(b - a). We study under what conditions [Xi] is a proper mean of a and b, and what kind of means are obtained by applying certain f 's. We also study the converse problem: Given a proper mean [mu](a, b), does there exist f such that f(b) - f(a) = f '([mu](a, b))(b - a) for all a, b [element of] I with a [not equal to] b? (As Provided).
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Update	2017/4/10