Answer
$$ -0.0005,\ \ \ -0.00049,\ \ \ 3.72 \times 10^{-6} $$
Work Step by Step
Given $$\frac{1}{\sqrt{101}}-\frac{1}{10}$$
Consider $f(x)= \dfrac{1}{\sqrt{x} }$, $a= 100$, $\Delta x= 1$, since
\begin{align*}
f'(x) &= \frac{-1}{2}x^{-3/2}\\
f'(100)&=-0.0005
\end{align*}
Then the linear approximation is given by
\begin{align*}
\Delta &f \approx f^{\prime}(a) \Delta x\\
&= (-0.0005)(1)\\
&= -0.0005
\end{align*}
and the actual change is given by
\begin{align*}
\Delta f&=f(a+\Delta x)-f(a)\\
&=f(101)-f(100) \\
&\approx -0.0004962809
\end{align*}
Hence the error is
$$ | -0.0005 +0.00049628 | \approx 3.72\times 10^{-6} $$
