Near the point (6, 6, 3), we have the approximation
[tex]f(x,y,z)\approx f(6,6,3)+f_x(6,6,3)(x-6)+f_y(6,6,3)(y-6)+f_z(6,6,3)(z-3)[/tex]
where the subscripted [tex]f[/tex]'s refer to the corresponding partial derivatives.
[tex]f(x,y,z)=x^2+y^2+z^2[/tex]
[tex]\implies\begin{cases}f_x(x,y,z)=2x\\f_y(x,y,z)=2y\\f_z(x,y,z)=2z\end{cases}[/tex]
So
[tex]f(6.02,5.97,2.99)\approx80.8288[/tex]
Compare to more precise value,
[tex]f(6.02,5.97,2.99)\approx80.8214[/tex]
