Content deleted Content added
Tag: Reverted |
Tag: Reverted |
||
Line 14:
This is a good approximation when <math>x</math> is close enough to <math>a</math>; 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 <math>f</math> at <math>(a,f(a))</math>. For this reason, this process is also called the '''tangent line approximation'''.
If <math>f</math> is [[concave down]] in the interval between <math>x</math> and <math>a</math>, the approximation will be an overestimate (since the derivative is decreasing in that interval). If <math>f</math> is [[convex function|convex]], the approximation will be an underestimate.<ref>{{cite web|title=12.1 Estimating a Function Value Using the Linear Approximation|url=http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html|access-date=3 June 2012}}</ref>
Linear approximations for [[vector (geometric)|vector]] functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the [[Jacobian matrix and determinant|Jacobian]] matrix. For example, given a differentiable function <math>f(x, y)</math> with real values, one can approximate <math>f(x, y)</math> for <math>(x, y)</math> close to <math>(a, b)</math> by the formula
| |||