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In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. ==Definition== Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case states that : where is the remainder term. The linear approximation is obtained by dropping the remainder: :. This is a good approximation for when it is close enough to ; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of at . For this reason, this process is also called the tangent line approximation. If is concave down in the interval between and , the approximation will be an overestimate (since the derivative is decreasing in that interval). If is concave up, the approximation will be an underestimate.〔(【引用サイトリンク】url=http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html )〕 Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix. For example, given a differentiable function with real values, one can approximate for close to by the formula : The right-hand side is the equation of the plane tangent to the graph of at In the more general case of Banach spaces, one has : where is the Fréchet derivative of at . 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「linear approximation」の詳細全文を読む スポンサード リンク
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