## Calculus (3rd Edition)

$\sqrt {2}-\sqrt {1}\approx\frac{1}{2}$, the actual value is 0.41421356 (to 8 decimal places). $\sqrt {101}-\sqrt {100}\approx0.05$, the actual value is 0.0498756 (to 6 decimal places).
Given: $f(x)=\sqrt x$ and $h=1$. As $h=1$, we can use Equation (3) which is $f(x_{0}+1)-f(x_{0})\approx f'(x_{0})$ When $x_{0}=1$, we get $\sqrt {2}-\sqrt {1}\approx f'(1)=\frac{1}{2\sqrt 1}=\frac{1}{2}$ The actual value is 0.41421356 (to 8 decimal places) When $x_{0}=100$, we have $\sqrt {101}-\sqrt {100}\approx\frac{1}{2\sqrt {100}}=0.05$ The actual value is 0.0498756 (to 6 decimal places)