Question

Find the linear approximation of the function f(x, y, z) = x2 + y2 + z2 at (6, 6, 7) and use it to approximate the number 6.022 + 5.992 + 6.972 . (Round your answer to five decimal places.)

Answer 1

The linear approximation to $f(x,y,z)$ about a point $(a,b,c)$ is

$L(x,y,z)=f(a,b,c)+\nabla f(a,b,c)\cdot\langle x-a,y-b,z-c\rangle$

We have

$f(x,y,z)=x^2+y^2+z^2$

$\implies\nabla f(x,y,z)=\langle2x,2y,2z\rangle$

so that near (6, 6, 7), we get $f(x,y,z)$ approximately equal to

$L(x,y,z)=6^2+6^2+7^2+\langle12,12,14\rangle\cdot\langle x-6,y-6,z-7\rangle$

$L(x,y,z)=12x+12y+14z-121$

Then

$f(6.02,5.99,6.97)\approx L(6.02,5.99,6.97)=\boxed{120.70000}$