Answer
$$3.317$$
Work Step by Step
$$\eqalign{
& {\text{We can note that }}\sqrt {11} {\text{ is a solution of the equation }}{x^2} - 11 = 0.{\text{ }} \cr
& {\text{Let }}f\left( x \right) = {x^2} - 11,\,\,\,\,\,\,f'\left( x \right) = 2x \cr
& {\text{Since 3 < }}\sqrt {11} < 4,{\text{ we can use }}{c_1} = 3{\text{ as the first approximation }} \cr
& {\text{Then using Newton's method we have}}: \cr
& {c_2} = {c_1} - \frac{{f\left( {{c_1}} \right)}}{{f'\left( {{c_1}} \right)}} = 3 - \frac{{{{\left( 3 \right)}^2} - 11}}{{2\left( 3 \right)}} = 3.3333 \cr
& {c_3} = {c_2} - \frac{{f\left( {{c_2}} \right)}}{{f'\left( {{c_2}} \right)}} = 3.3333 - \frac{{{{\left( {3.3333} \right)}^2} - 11}}{{2\left( {3.3333} \right)}} = 3.3166 \cr
& {\text{In the same way}} \cr
& {c_4} = 3.3166 - \frac{{{{\left( {3.3166} \right)}^2} - 11}}{{2\left( {3.3166} \right)}} = 3.3166 \cr
& {\text{Since }}{c_3} = {c_4} = 3.316,{\text{ to the nearest thousand}}{\text{, }}\sqrt {11} = 3.317 \cr} $$