Editing Linear approximation (section)

==Definition==
Given a twice continuously differentiable function <math>f</math> of one [[real number|real]] variable, [[Taylor's theorem]] for the case <math>n = 1 </math> states that
<math display="block"> f(x) = f(a) + f'(a)(x - a) + R_2 </math>
where <math>R_2</math> is the remainder term. The linear approximation is obtained by dropping the remainder:
<math display="block"> f(x) \approx f(a) + f'(a)(x - a).</math>

This is a good approximation when <math>x</math> is close enough to {{nowrap|<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'''. Linear approximations in this case are further improved when the [[second derivative]] of a, <math> f''(a) </math>, is sufficiently small (close to zero) (i.e., at or near an [[inflection point]]).

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 [[concave up]], 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 |archive-date=3 March 2013 |archive-url=https://web.archive.org/web/20130303014028/http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html |url-status=dead }}</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 
<math display="block">f\left(x,y\right)\approx f\left(a,b\right) + \frac{\partial f}{\partial x} \left(a,b\right)\left(x-a\right) + \frac{\partial f}{\partial y} \left(a,b\right)\left(y-b\right).</math>

The right-hand side is the equation of the plane tangent to the graph of <math>z=f(x, y)</math> at <math>(a, b).</math>

In the more general case of [[Banach space]]s, one has
<math display="block"> f(x) \approx f(a) + Df(a)(x - a)</math>
where <math>Df(a)</math> is the [[Fréchet derivative]] of <math>f</math> at <math>a</math>.