linear approximation

Using a function's tangent line at a nearby point to estimate its value, since a smooth curve looks almost straight up close.

The linear approximation of $f$ near $x=a$ is $L(x) = f(a) + f'(a)(x-a)$ — the tangent line at $a$ , used as a stand-in for $f$ when $x$ is close to $a$ .

Examples

$\sqrt{4.1} \approx \sqrt4 + \dfrac{1}{2\sqrt4}(0.1) = 2.025$ , using $f(x)=\sqrt x$ at $a=4$ .