Answer:
<em>f(3.022 , 5.992 , 5.972)=80.7</em>
Step-by-step explanation:
<u>Linearization of Multivariable Functions</u>
Let f be a function that depends on the independent variables (x,y,z) and assume the following partial derivatives exist:
![\displaystyle \frac{\partial f}{\partial x}\ ,\ \frac{\partial f}{\partial y}\ ,\ \frac{\partial f}{\partial z}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%5C%20%2C%5C%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%5C%20%2C%5C%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D)
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)+\frac{\partial f(x_o,y_o,z_o)}{\partial x} (x-x_o)+\frac{\partial f(x_o,y_o,z_o)}{\partial y} (y-y_o)+\frac{\partial f(x_o,y_o,z_o)}{\partial z} (z-z_o)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%28x%2Cy%2Cz%29%5Capprox%20f%28x_o%2Cy_o%2Cz_o%29%2B%5Cfrac%7B%5Cpartial%20f%28x_o%2Cy_o%2Cz_o%29%7D%7B%5Cpartial%20x%7D%20%28x-x_o%29%2B%5Cfrac%7B%5Cpartial%20f%28x_o%2Cy_o%2Cz_o%29%7D%7B%5Cpartial%20y%7D%20%28y-y_o%29%2B%5Cfrac%7B%5Cpartial%20f%28x_o%2Cy_o%2Cz_o%29%7D%7B%5Cpartial%20z%7D%20%28z-z_o%29)
Given
![f(x, y, z) = x^2 + y^2 + z^2](https://tex.z-dn.net/?f=f%28x%2C%20y%2C%20z%29%20%3D%20x%5E2%20%2B%20y%5E2%20%2B%20z%5E2)
Evaluating in (3,6,6)
![f(x, y, z) = 3^2 + 6^2 + 6^2=81](https://tex.z-dn.net/?f=f%28x%2C%20y%2C%20z%29%20%3D%203%5E2%20%2B%206%5E2%20%2B%206%5E2%3D81)
The partial derivatives are
![\displaystyle \frac{\partial f}{\partial x}=2x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D2x)
Evaluating at (3,6,6)
![\displaystyle \frac{\partial f}{\partial x}=2(3)=6](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D2%283%29%3D6)
![\displaystyle \frac{\partial f}{\partial y}=2y](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D2y)
Evaluating at (3,6,6)
![\displaystyle \frac{\partial f}{\partial y}=2(6)=12](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D2%286%29%3D12)
![\displaystyle \frac{\partial f}{\partial z}=2z](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D%3D2z)
Evaluating at (3,6,6)
![\displaystyle \frac{\partial f}{\partial z}=2(6)=12](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D%3D2%286%29%3D12)
The linearization of f is
![\displaystyle f(x,y,z)\approx 81+6 (x-3)+12 (y-6)+12 (z-6)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%28x%2Cy%2Cz%29%5Capprox%2081%2B6%20%28x-3%29%2B12%20%28y-6%29%2B12%20%28z-6%29)
Operating
![\displaystyle f(x,y,z)\approx 81+6 x-18+12y-72+12z-72](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%28x%2Cy%2Cz%29%5Capprox%2081%2B6%20x-18%2B12y-72%2B12z-72)
![\displaystyle f(x,y,z)\approx 6 x+12y+12z-81](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%28x%2Cy%2Cz%29%5Capprox%206%20x%2B12y%2B12z-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](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%283.022%20%2C%205.992%20%2C%205.972%29%5Capprox%206%20%283.022%29%2B12%285.992%29%2B12%285.972%29-81%3D80.70000)