# Find the linear approximation of the function f(x) = √4-x at a = 0 and use it to approximate the numbers √3.9 and √3.99 ? (Round your answers to four decimal places.)

Dec 21, 2017

Use the Taylor form:

$f \left(x\right) = f \left(a\right) + \left(x - a\right) \cdot f ' \left(a\right) \ldots .$

$f \left(x\right) ' = \left(\sqrt{4 - x}\right) ' = \frac{1}{2} \frac{- 1}{\sqrt{4 - x}}$

$f \left(0\right) = \sqrt{4 - 0} = 2$

$f \left(0\right) ' = \frac{1}{2} \frac{- 1}{\sqrt{4}} = - \frac{1}{4}$

$\sqrt{3.9} = f \left(x\right) \approx 2 + 0.1 \left(- \frac{1}{4}\right) = 1.9750$

$\sqrt{3.99} = f \left(x\right) \approx 2 + 0.01 \left(- \frac{1}{4}\right) = 1.9975$