Use differentials to approximate the number sqrt(2.012^2 + 5.972^2 + 8.982^2) . (Round your answer to five decimal places.)

To approximate the value of \( \sqrt{2.012^2 + 5.972^2 + 8.982^2} \) using differentials, we can define the function:

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

This is essentially the distance formula for three-dimensional coordinates. Now, we can use the idea of differentials to approximate small changes in this function.

### Step 1: Choose a nearby point
We will approximate \( f(x, y, z) \) near the point \( (x_0, y_0, z_0) \), which is close to the values given for \( x = 2.012 \), \( y = 5.972 \), and \( z = 8.982 \). It's useful to choose round values that are nearby for easier calculations, so let’s choose \( x_0 = 2.000 \), \( y_0 = 6.000 \), and \( z_0 = 9.000 \).

### Step 2: Evaluate \( f(x_0, y_0, z_0) \)
First, we calculate the value of \( f(x_0, y_0, z_0) \):

\[
f(2.000, 6.000, 9.000) = \sqrt{2.000^2 + 6.000^2 + 9.000^2} = \sqrt{4 + 36 + 81} = \sqrt{121} = 11.000
\]

### Step 3: Compute the partial derivatives
Next, we find the partial derivatives of \( f(x, y, z) \) with respect to \( x \), \( y \), and \( z \).

\[
\frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + y^2 + z^2}}, \quad \frac{\partial f}{\partial y} = \frac{y}{\sqrt{x^2 + y^2 + z^2}}, \quad \frac{\partial f}{\partial z} = \frac{z}{\sqrt{x^2 + y^2 + z^2}}
\]

At the point \( (2.000, 6.000, 9.000) \), we can plug in the values:

\[
\frac{\partial f}{\partial x} \bigg|_{(2.000, 6.000, 9.000)} = \frac{2.000}{11.000}, \quad \frac{\partial f}{\partial y} \bigg|_{(2.000, 6.000, 9.000)} = \frac{6.000}{11.000}, \quad \frac{\partial f}{\partial z} \bigg|_{(2.000, 6.000, 9.000)} = \frac{9.000}{11.000}
\]

These simplify to:

\[
\frac{\partial f}{\partial x} = \frac{2}{11}, \quad \frac{\partial f}{\partial y} = \frac{6}{11}, \quad \frac{\partial f}{\partial z} = \frac{9}{11}
\]

### Step 4: Calculate the differential
Now we compute the changes \( \Delta x = 2.012 - 2.000 = 0.012 \), \( \Delta y = 5.972 - 6.000 = -0.028 \), and \( \Delta z = 8.982 - 9.000 = -0.018 \).

Using the differential approximation:

\[
df = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + \frac

Use differentials to approximate the number sqrt(2.012^2 + 5.972^2 + 8.982^2). (Round your answer to five decimal places.)

Learn how to approximate sqrt(2.012^2 + 5.972^2 + 8.982^2) using differentials and multivariable calculus. This solution breaks down the process of using partial derivatives and the differential approximation theorem, ideal for college-level calculus students.

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