Answer
$$10.05,\ \ \ 10.049875 ,\ \ 1.244\times 10^{-4}$$
Work Step by Step
Given $$\sqrt{101} $$
Consider $f(x)=x^{1 /2}, a=100,$ and $\Delta x=1$, since
\begin{align*}
f^{\prime}(x)&=\frac{1}{2}x^{-1/2}\\
f^{\prime}(100)&=\frac{1}{20}
\end{align*}
Then the linear approximation is given by
\begin{align*}
L(x)&=f^{\prime}(a)(x-a)+f(a)\\
&= \frac{1}{20}(x-100)+ 10\\
&=\frac{x}{20}+5\\
L(101)&\approx \frac{101}{20}+5=10.05
\end{align*}
By using a calculator, $\sqrt{101} = 10.049875$ and the error in the linear approximation is$$
|10.049875 - 10.05|=1.244\times 10^{-4}
$$
