## FANDOM

1,025 Pages

Linear approximation (or linearization) is a method of estimating a value on a function by placing that value on a tangent line of the function at a nearby value. This approximation can be found with the formula:

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

with $a$ being the nearby value, $x$ being the exact value, and $f(x)$ being the approximation. Since a tangent line is being used, accuracy decreases the further apart $x$ and $a$ are. A linear approximation is a Taylor approximation of the first degree.

## Example

To find $\sqrt{9.2}$, the steps would be as follows:
$f(x) = \sqrt{x} = x^{\frac{1}{2}}$ (This is the operation being applied)
$f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$ (The derivative of the function)
$f(9.2) \approx f(9) + f'(9)(9.2 - 9)$ (All values substituted into the formula, $a$ is given the nearby value of $9$)
$f(9.2) \approx \sqrt{9} + \frac{1}{2}9^{-\frac{1}{2}}(9.2 - 9)$
$f(9.2) \approx 3 + (\frac{1}{2})(\frac{1}{3})(0.2)$
$f(9.2) \approx 3 + (\frac{1}{6})(0.2)$
$f(9.2) \approx 3 + (\frac{1}{30})$
$f(9.2) \approx \frac{91}{30} \approx 3.03333$
This value is quite close to the actual square root of $9.2$ which is approximately $3.03315$.