Question

Find the linear approximation of the function f(x, y, z) = x2 + y2 + z2 at (6, 3, 2) and use it to approximate the number 6.022 + 2.992 + 1.972 . (Round your answer to five decimal places.)

Answer 1

Answer:

$f(6.022,2.922,1.972)\approx 49.10400$

Step-by-step explanation:

The partial derivatives fo the function are:

$\frac{\partial f}{\partial x} = 2\cdot x$

$\frac{\partial f}{\partial y} = 2\cdot y$

$\frac{\partial f}{\partial z} = 2\cdot z$

The linear approximation of the function at given value is:

$f(6.022,2.992,1.972) \approx f(6,3,2) + \frac{\partial f(6,3,2)}{\partial x}\cdot (6.022-6) + \frac{\partial f(6,3,2)}{\partial y}\cdot (2.992-3) + \frac{\partial f(6,3,2)}{\partial z}\cdot (1.972-2)$

$f(6.022,2.922,1.972)\approx 49 + 12\cdot (6.022 - 6) + 6 \cdot (2.992-3)+ 4 \cdot (1.972-2)$

$f(6.022,2.922,1.972)\approx 49.10400$