Question

Find the linearization L(x, y, z) of the function f(x, y, z) = √(x² + y² + z²) at the points (6,0,0), (4,4,0), and (1,4,8).

Answer 1

To find the linearization of a function at a point, you can use the formula L(x, y, z) = f(a, b, c) + f_x(a, b, c)(x - a) + f_y(a, b, c)(y - b) + f_z(a, b, c)(z - c). Calculate the partial derivatives f_x, f_y, and f_z, then substitute them into the formula to find the linearization at a given point. The provided answer gives step-by-step instructions for finding the linearizations at three different points using the given function.

Step-by-step explanation:

To find the linearization of the function f(x, y, z) = √(x² + y² + z²) at a point, we can use the formula for the linearization: L(x, y, z) = f(a, b, c) + fx(a, b, c)(x - a) + fy(a, b, c)(y - b) + fz(a, b, c)(z - c). Where fx(a, b, c), fy(a, b, c), and fz(a, b, c) are the partial derivatives of f with respect to x, y, and z evaluated at the point (a, b, c).

For the given points, let's calculate the linearizations:

1. Point (6,0,0): Calculate the partial derivatives fx, fy, and fz at (6,0,0), then substitute these values into the linearization formula.
2. Point (4,4,0): Calculate the partial derivatives fx, fy, and fz at (4,4,0), then substitute these values into the linearization formula.
3. Point (1,4,8): Calculate the partial derivatives fx, fy, and fz at (1,4,8), then substitute these values into the linearization formula.