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 the derivative at a point is replaced by the Jacobian matrix. For example, given a differentiable function $f (x, y)$ with real values, one can approximate $f (x, y)$ for $(x, y)$ close to $(a, b)$ by the formula

$f\left(x,y\right)\approx f\left(a,b\right)+\frac{\partial f}{\partial x}\left(a,b\right)\left(x-a\right)+\frac{\partial f}{\partial y}\left(a,b\right)\left(y-b\right).$

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

In the more general case of Banach spaces, one has

$f(x) \approx f(a) + Df(a)(x - a)$

where $D f (a)$ is the Fréchet derivative of $f$ at $a$ .

[edit] Examples

To find an approximation of $\sqrt[3]{25}$ one can do as follows.

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

$f\ '(x)= 1/3x^{-2/3}.$
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…