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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Racetrack principle ： ウィキペディア英語版 Racetrack principle In calculus, the racetrack principle describes the movement and growth of two functions in terms of their derivatives. This principle is derived from the fact that if a horse named Frank Fleetfeet always runs faster than a horse named Greg Gooseleg, then if Frank and Greg start a race from the same place and the same time, then Frank will win. More briefly, the horse that starts fast and stays fast wins. In symbols: :if $f\text{'}(x)>g\text{'}(x)$ for all $x>0$, and if $f(0)=g(0)$, then $f(x)>g(x)$ for all $x>0$. or, substituting ≥ for > produces the theorem :if $f\text{'}(x)\; \backslash ge\; g\text{'}(x)$ for all $x>0$, and if $f(0)=g(0)$, then $f(x)\; \backslash ge\; g(x)$ for all $x>0$. which can be proved in a similar way ==Proof== This principle can be proven by considering the function h(x) = f(x) - g(x). If we were to take the derivative we would notice that for x>0 :$h\text{'}=\; f\text{'}-g\text{'}>0.$ Also notice that h(0) = 0. Combining these observations, we can use the mean value theorem on the interval (x ) and get :$h\text{'}(x\_0)=\; \backslash frac=\; \backslash frac>0.$ Since x > 0 for the mean value theorem to work then we may conclude that f(x) - g(x) > 0. This implies f(x) > g(x). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Racetrack principle」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.