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Image:800px-Tangent-calculus a.png In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).

For example, given a differentiable function f of one real variable, Taylor's theorem for n=1 states that

<math> 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> f(x) \approx f(a) + f\ '(a)(x - a)</math>

which is true for x close to a. The expression on the right-hand side is just the equation for the tangent line to the graph of f at (a, f(a)), and for this reason, this process is also called the tangent line approximation.

One can also use linear approximations for vector functions of a vector variable, in which case the derivative at a point is replaced by the 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>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 spaces, one has

<math> 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>.

[edit] Examples

To find an approximation of <math>\sqrt[3]{25}</math> one can do as follows.

1. Consider the function <math> f(x)= x^{1/3}.\,</math> Hence, the problem is reduced to finding the value of <math>f(25)</math>.
2. We have

   <math> f\ '(x)= 1/3x^{-2/3}.</math>

3. According to linear approximation

   <math> f(25) \approx f(27) + f\ '(27)(25 - 27) = 3 - 2/27.</math>

4. The result, 2.926, lies fairly close to the actual value 2.924…eo:Vikipedio:Projekto matematiko/Lineara proksimuma kalkulado

fr:Approximation linéaire zh:线性近似