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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 Exponential function ： ウィキペディア英語版 Exponential function In mathematics, an exponential function is a function of the form : $f(x)\; =\; b^x.\; \backslash ,$ The input variable ''x'' occurs as an exponent – hence the name. A function of the form ''ƒ''(''x'') = ''b''''x'' ± c is also considered an exponential function, and a function of the form ''ƒ''(''x'') = ''a''·''b''''x'' can be re-written as ''ƒ''(''x'') = ''b''''x'' ± c by the use of logarithms and so is an exponential function. In contexts where the base ''b'' is not specified, especially in more theoretical contexts, the term exponential function is almost always understood to mean the natural exponential function : $x\; \backslash mapsto\; e^x,\; \backslash ,$ also written as : $x\; \backslash mapsto\; \backslash exp(x)$ where ''e'' is Euler's number, a transcendental number approximately 2.718281828. The reason this number ''e'' is considered the "natural" base of exponential functions is that this function is its own derivative. Every exponential function is directly proportional to its own derivative, but only when the base is ''e'' does the constant of proportionality equal 1. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The function is often written as exp(''x''), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics. The graph of is upward-sloping, and increases faster as ''x'' increases. The graph always lies above the ''x''-axis but can get arbitrarily close to it for negative ''x''; thus, the ''x''-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its ''y'' coordinate at that point. The inverse function is the natural logarithm ln(''x''); because of this, some old texts refer to the exponential function as the antilogarithm. In general, the variable ''x'' can be any real or complex number or even an entirely different kind of mathematical object; see the formal definition below. ==Formal definition== (詳細はpower series: : $e^x\; =\; \backslash sum\_^\; =\; 1\; +\; x\; +\; +\; +\; +\; \backslash cdots$ Using an alternate definition for the exponential function leads to the same result when expanded as a Taylor series. Less commonly, ''e''''x'' is defined as the solution ''y'' to the equation : $x\; =\; \backslash int\_1^y\; \backslash mathrmt$ It is also the following limit:〔Eli Maor, ''e: the Story of a Number'', p.156.〕 : $e^x\; =\; \backslash lim\_\; \backslash left(1\; +\; \backslash frac\backslash right)^n$ 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「Exponential function」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.