Linear approximation

Tangent line at (a, f(a))

In mathematics, a linear approximation is an approximation of a general function using a linear transformation.

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 $a$. Here, the linear transformation is $Df(a)$.

Examples

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

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

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

3. According to linear approximation

   $f(25)\approx f(27)+f\ '(27)(25-27)=3-2/27.$

4. The result, 2.926, lies fairly close to the actual value 2.924…