Answer
$$\eqalign{
& f\left( {x + \Delta x} \right) \approx 9.97 \cr
& \sqrt {99.4} \approx 9.969954 \cr} $$
Work Step by Step
$$\eqalign{
& \sqrt {99.4} \cr
& {\text{Using }}f\left( x \right) = \sqrt x \cr
& {\text{Differentiating}} \cr
& f'\left( x \right) = \frac{1}{{2\sqrt x }} \cr
& {\text{Using the formula}} \cr
& f\left( {x + \Delta x} \right) \approx f\left( x \right) + f'\left( x \right)dx \cr
& {\text{We can write,}} \cr
& f\left( {x + \Delta x} \right) \approx \sqrt x + \frac{1}{{2\sqrt x }}dx \cr
& f\left( {x + \Delta x} \right) = \sqrt {99.4} \cr
& {\text{Now, choosing }}x = 100{\text{ and }}dx = - 0.6,{\text{ and substituting }} \cr
& f\left( {x + \Delta x} \right) = \sqrt {99.4} \approx \sqrt {100} + \frac{1}{{2\sqrt {100} }}\left( { - 0.6} \right) \cr
& f\left( {x + \Delta x} \right) \approx 10 + \frac{1}{{20}}\left( { - 0.6} \right) \cr
& f\left( {x + \Delta x} \right) \approx 9.97 \cr
& \cr
& {\text{*Using a calculator to obtain }}\sqrt {99.4} \cr
& \sqrt {99.4} \approx 9.969954 \cr} $$
