Question

Find the linear approximation of the function f(x, y, z) = x2 + y2 + z2 at (3, 6, 6) and use it to approximate the number 3.022 + 5.992 + 5.972 . (Round your answer to five decimal places.)

Answer 1

f(3.022 , 5.992 , 5.972)=80.7

Explanation:

Linearization of Multivariable Functions

Let f be a function that depends on the independent variables (x,y,z) and assume the following partial derivatives exist:

`\displaystyle (\partial f)/(\partial x)\ ,\ (\partial f)/(\partial y)\ ,\ (\partial f)/(\partial z)`

The function f can be linearized around a known point (xo,yo,zo) by the equation:

`\displaystyle f(x,y,z)\approx f(x_o,y_o,z_o)+(\partial f(x_o,y_o,z_o))/(\partial x) (x-x_o)+(\partial f(x_o,y_o,z_o))/(\partial y) (y-y_o)+(\partial f(x_o,y_o,z_o))/(\partial z) (z-z_o)`

Given

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

Evaluating in (3,6,6)

`f(x, y, z) = 3^2 + 6^2 + 6^2=81`

The partial derivatives are

`\displaystyle (\partial f)/(\partial x)=2x`

Evaluating at (3,6,6)

`\displaystyle (\partial f)/(\partial x)=2(3)=6`

`\displaystyle (\partial f)/(\partial y)=2y`

Evaluating at (3,6,6)

`\displaystyle (\partial f)/(\partial y)=2(6)=12`

`\displaystyle (\partial f)/(\partial z)=2z`

Evaluating at (3,6,6)

`\displaystyle (\partial f)/(\partial z)=2(6)=12`

The linearization of f is

`\displaystyle f(x,y,z)\approx 81+6 (x-3)+12 (y-6)+12 (z-6)`

Operating

`\displaystyle f(x,y,z)\approx 81+6 x-18+12y-72+12z-72`

`\displaystyle f(x,y,z)\approx 6 x+12y+12z-81`

Using the linearization to find f(3.022 , 5.992 , 5.972)

`\displaystyle f(3.022 , 5.992 , 5.972)\approx 6 (3.022)+12(5.992)+12(5.972)-81=80.70000`