Problem $2.$

Approximate $\sqrt[\phantom{2}]{24\mathrm{.}82}$ by considering the function $\sqrt[\phantom{2}]{x^2+y^2},$ for which $(3\mathrm{.}1,3\mathrm{.}9)$ satisfies $(3\mathrm{.}1{)}^2+(3\mathrm{.}9{)}^2=24\mathrm{.}82,$ and finding its linearization $L(x,y)$ at $(3,4).$