Consider the following function.

f$($x$,$ y$,$ z$)=$

x$2+$ y$2+$ z$2$

Find

fx$($x$,$ y$,$ z$),$

fy$($x$,$ y$,$ z$),$

and

fz$($x$,$ y$,$ z$).$

fx$($x$,$ y$,$ z$)$

$=$

fy$($x$,$ y$,$ z$)$

$=$

fz$($x$,$ y$,$ z$)$

$=$

Find the linear approximation of the function

f$($x$,$ y$,$ z$)=$

x$2+$ y$2+$ z$2$

at

$(4,7,4).$

f$($x$,$ y$,$ z$)$

Use the linear approximation to approximate the number

$4\mathrm{.}022+6\mathrm{.}992+3\mathrm{.}972$

$.$

$($Round your answer to four decimal places.$)$

f$(4\mathrm{.}02,6\mathrm{.}99,3\mathrm{.}97)$