Calculus (3rd Edition)

$$10.05,\ \ \ 10.049875 ,\ \ 1.244\times 10^{-4}$$
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$$|10.049875 - 10.05|=1.244\times 10^{-4}$$