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

6.012 + 5.972 + 6.982
. (round your answer to five decimal places.)

f(6.01, 5.97, 6.98) ≈

See below in bold.

Step-by-step explanation:

f(x, y, z) = x^2 + y^2 + z^2 at (6, 6, 7)

Taking partial derivatives

f'x = 2x + y^2 + z^2 --> f'x(6,6,7) =  12+36+49= 97

f'y = x^2 + 2y + z^2---> f'y(6,6,7) = 36 + 12 + 49 = 97

f'z = x^2 + y^2 + 2z > f'z(6,6,7) =  36 + 36 + 14 = 86.

f(6,6,7) =  36 + 36 + 49 = 121

So our linear approximation is:

L(x,y,z) = 121 + 97(x - 6) + 97(y - 6) + 86(z - 7)

Substituting the given values

f(6.012, 5.972,6.982)

= L(6.012, 5.972, 6.982) ≈ 121 + 97(6.012 - 6) + 97(5.972 - 6) + 86(6.982 - 7)

= 117.9.

answer: 40.95

step-by-step explanation: