Question

Find the linear approximation of the function f(x, y, z) = x2 + y2 + z2 at (9, 2, 6) and use it to approximate the number 9.012 + 1.982 + 5.972 . (Round your answer to five decimal places.) f(9.01, 1.98, 5.97) ≈

Answer 1

`f(9.012,1.982,5.972)\approx 120.76400`

Explanation:

The partial derivatives of the function are, respectively:

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

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

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

The linear approximation of the function follows the following model:

`f(x,y,x) \approx f(9, 2, 6) + (\partial f(9,2,6))/(\partial x)\cdot (9.01 - 9) + (\partial f(9,2,6))/(\partial y)\cdot (1.98 - 2) + (\partial f(9,2,6))/(\partial z)\cdot (5.972 - 6)`

`f(9.012,1.982,5.972)\approx 121 + 18\cdot (9.01-9) + 4\cdot (1.98-2)+12\cdot (5.972-6)`

`f(9.012,1.982,5.972)\approx 121 + 18\cdot (9.01-9) + 4\cdot (1.98-2)+12\cdot (5.972-6)`

`f(9.012,1.982,5.972)\approx 120.76400`