Root of 37 to three decimals by Newton's method

Without a calculator, estimate $\sqrt{37}$ correct to three decimal places.

$6.083$ (more precisely $6.08276$). Apply Newton's method to $f(x)=x^2-37$, i.e. $x_{k+1}=\tfrac12(x_k+37/x_k)$, from $x_0=6$: $x_1=6+\tfrac1{12}\approx6.0833$, and the next step changes it only in the fifth decimal. The quick estimate $\sqrt{a^2+b}\approx a+\tfrac{b}{2a}$ gives the same first step.