Find the linear approximation of the function f(x)=\sqrt{1-x} at a=0 and use it to approximate the numbers \sqrt{0.9} and \sqrt{0.99}.

Question

Find the linear approximation of the function $f (x) = \sqrt{1 - x}$ at $a = 0$ and use it to approximate the numbers $\sqrt{0.9}$ and $\sqrt{0.99}$ .

Answer 1

The formula for linearization L(x) is
$L (x) = f (a) + f^{'} (x) (x - a)$
we have
$f (x) = \sqrt{1 - x}$ with $a = 0$
$f (a) = \sqrt{1 - 0} = 1$
$f^{'} (x) = (\sqrt{1 - x})^{'} = (1 - x)^{'} (\sqrt{1 - x})^{'} = - \frac{1}{2} \frac{1}{\sqrt{1 - x}}$
$f^{'} (a) = - \frac{1}{2} \frac{1}{\sqrt{1 - 0}} = - \frac{1}{2}$
$L (x) = 1 - \frac{1}{2} (x - a) = 1 - \frac{1}{2} x$
$\sqrt{1 - x} = \sqrt{0.99}$ so $x = 0.01$
$L (0.01) = 1 - \frac{1}{2} 0.01 = 0.95$
$\sqrt{1 - x} = \sqrt{0.999}$ so $x = 0.001$
$L (0.01) = 1 - \frac{1}{2} 0.001 = 0.995$

Find the linear approximation of the function f(x)=\sqrt{1-x} at a=0 and use it to approximate the numbers \sqrt{0.9} and \sqrt{0.99}.

Expert Answer

Not exactly what you’re looking for?

You might be interested in

New questions