We are tasked with finding the linearization $L(x, y, z)$ of the function

$f(x, y, z) = \sqrt{x^2 + y^2 + z^2}$

at the given points: (1, 0, 0), (5, 1, 0), and (1, 2, 2).

Step 1: General formula for linearization

The linearization of a function $f(x, y, z)$ around a point $(x_0, y_0, z_0)$ is given by:

$L(x, y, z) = f(x_0, y_0, z_0) + f_x(x_0, y_0, z_0)(x - x_0) + f_y(x_0, y_0, z_0)(y - y_0) + f_z(x_0, y_0, z_0)(z - z_0)$

Where:

• $f_x$, $f_y$, and $f_z$ are the partial derivatives of $f$ with respect to $x$, $y$, and $z$, respectively.

Step 2: Compute partial derivatives of $f(x, y, z)$

Given $f(x, y, z) = \sqrt{x^2 + y^2 + z^2}$, we calculate the partial derivatives.

1. Partial derivative with respect to $x$:

$f_x = \frac{\partial}{\partial x} \sqrt{x^2 + y^2 + z^2} = \frac{x}{\sqrt{x^2 + y^2 + z^2}}$

2. Partial derivative with respect to $y$:

$f_y = \frac{y}{\sqrt{x^2 + y^2 + z^2}}$

3. Partial derivative with respect to $z$:

$f_z = \frac{z}{\sqrt{x^2 + y^2 + z^2}}$

Step 3: Compute linearizations at the given points

a. At $(1, 0, 0)$:

• $f(1, 0, 0) = \sqrt{1^2 + 0^2 + 0^2} = 1$
• $f_x(1, 0, 0) = \frac{1}{1} = 1$
• $f_y(1, 0, 0) = \frac{0}{1} = 0$
• $f_z(1, 0, 0) = \frac{0}{1} = 0$

The linearization is:

$L(x, y, z) = 1 + 1(x - 1) + 0(y - 0) + 0(z - 0)$ $L(x, y, z) = 1 + (x - 1) = x$

b. At $(5, 1, 0)$:

• $f(5, 1, 0) = \sqrt{5^2 + 1^2 + 0^2} = \sqrt{26}$
• $f_x(5, 1, 0) = \frac{5}{\sqrt{26}}$
• $f_y(5, 1, 0) = \frac{1}{\sqrt{26}}$
• $f_z(5, 1, 0) = 0$

The linearization is:

$L(x, y, z) = \sqrt{26} + \frac{5}{\sqrt{26}}(x - 5) + \frac{1}{\sqrt{26}}(y - 1) + 0(z - 0)$ $L(x, y, z) = \sqrt{26} + \frac{5}{\sqrt{26}}(x - 5) + \frac{1}{\sqrt{26}}(y - 1)$

c. At $(1, 2, 2)$:

• $f(1, 2, 2) = \sqrt{1^2 + 2^2 + 2^2} = 3$
• $f_x(1, 2, 2) = \frac{1}{3}$
• $f_y(1, 2, 2) = \frac{2}{3}$
• $f_z(1, 2, 2) = \frac{2}{3}$

The linearization is:

$L(x, y, z) = 3 + \frac{1}{3}(x - 1) + \frac{2}{3}(y - 2) + \frac{2}{3}(z - 2)$ $L(x, y, z) = 3 + \frac{1}{3}(x - 1) + \frac{2}{3}(y - 2) + \frac{2}{3}(z - 2)$

Do you want more details on any part of this?

Here are five related questions:

1. How do we compute the gradient of a multivariable function?
2. What is the geometric meaning of a function's linearization?
3. How does the linearization differ from a tangent plane?
4. Can we generalize this approach to higher dimensions?
5. What role do partial derivatives play in optimization problems?

Tip: When computing partial derivatives, always remember that the other variables are treated as constants while differentiating with respect to one variable.