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.


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.