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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:

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

An upper bound for the error of this term is

$R(x) = \dfrac{f'' (b)}{2} (x-a)^2, b\in [a,x]$

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

$3.03315$ .

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