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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 Related rates ： ウィキペディア英語版 Related rates In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time. Because science and engineering often relate quantities to each other, the methods of related rates have broad applications in these fields. Because problems involve several variables, differentiation with respect to time or one of the other variables requires application of the chain rule.〔(【引用サイトリンク】url=http://www.whitman.edu/mathematics/calculus_online/section06.02.html )〕 Fundamentally, if a function $F$ is defined such that $F=f(x)$, then the derivative of the function $F$ can be taken with respect to another variable. (The Variable $t$ is frequently used as many Related Rates problems apply to finding changes with respect to time.) We assume $x$ is a function of $t$, i.e. $x=g(t)$. Then $F=f(g(t))$, so $F\text{'}=f\text{'}(g(t))\; \backslash cdot\; g\text{'}(t)$ Written in Liebnitz notation, this is: $\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac.$ The value of this is: if it is know how $x$ changes with respect to $t$, then we can determine how $F$ changes with respect to $t$ and vice versa. We can extend this application of the chain rule with the sum, difference, product and quotient rules of calculus, etc. e.g. If $F(x)=\; G(y)+\; H(z)$ then $\backslash frac\backslash cdot\; \backslash frac=\backslash frac\; \backslash cdot\; \backslash frac+\backslash frac\; \backslash cdot\; \backslash frac$. == Procedure == The most common way to approach related rates problems is the following: #Identify the known variables, including rates of change and the rate of change that is to be found. (Drawing a picture or representation of the problem can help to keep everything in order) #Construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found. #Differentiate both sides of the equation with respect to time (or other rate of change). Often, the chain rule is employed at this step. #Substitute the known rates of change and the known quantities into the equation. #Solve for the wanted rate of change. Errors in this procedure are often caused by plugging in the known values for the variables ''before'' (rather than after) finding the derivative with respect to time. Doing so will yield an incorrect result, since if those values are substituted for the variables before differentiation, those variables will become constants; and when the equation is differentiated, zeroes appear in places of all variables for which the values were plugged in. 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「Related rates」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.