Linear approximation

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Tangent line at (a, f(a))

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

[ f(x) = f(a) + f\ '(a)(x - a) + R_2 ]

where [R_2] is the remainder term. The linear approximation is obtained by dropping the remainder:

[ f(x) \approx f(a) + f\ '(a)(x - a)]

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 [f\ '(a)] is the Jacobian matrix. The approximation is the equation of the tangent line, plane, or hyperplane. It also applies for complex functions of a complex variable.

In the more general case of Banach spaces, one has

[ f(x) \approx f(a) + Df(a)(x - a)]

where [Df(a)] is the Fréchet derivative of [f] at [a].

Examples

To find an approximation of [\sqrt[3]] one can do as follows.

Consider the function [ f(x)= x^.\,] Hence, the problem is reduced to finding the value of [f(25)].
We have
:[ f\ '(x)= 1/3x^.]
According to linear approximation
:[ f(25) \approx f(27) + f\ '(27)(25 - 27) = 3 - 2/27.]
The result, 2.926, lies fairly close to the actual value 2.924…

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