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

Answer:

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

Step-by-step explanation:

The partial derivatives of the function are, respectively:

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

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

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

The linear approximation of the function follows the following model:

$f(x,y,x) \approx f(9, 2, 6) + \frac{\partial f(9,2,6)}{\partial x}\cdot (9.01 - 9) + \frac{\partial f(9,2,6)}{\partial y}\cdot (1.98 - 2) + \frac{\partial f(9,2,6)}{\partial z}\cdot (5.972 - 6)$

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

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

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