Question

Find the linear approximation of the function f(x, y, z) = x2 + y2 + z2 at (6, 6, 3) and use it to approximate the number 6.022 + 5.972 + 2.992 .

Answer 1

Near the point (6, 6, 3), we have the approximation

$f(x,y,z)\approx f(6,6,3)+f_x(6,6,3)(x-6)+f_y(6,6,3)(y-6)+f_z(6,6,3)(z-3)$

where the subscripted $f$'s refer to the corresponding partial derivatives.

$f(x,y,z)=x^2+y^2+z^2$
$\implies\begin{cases}f_x(x,y,z)=2x\\f_y(x,y,z)=2y\\f_z(x,y,z)=2z\end{cases}$

So

$f(6.02,5.97,2.99)\approx80.8288$

Compare to more precise value,

$f(6.02,5.97,2.99)\approx80.8214$