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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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