f(x, y, z) = 1 − square root of x
2 + y
2 + z
2
,
B =
⎧
⎩
⎨(x, y, z)
|
x
2 + y
2 + z
2 ≤ 9, y ≥ 0, z ≥ 0⎫

To understand the function and region given, let’s break down each component:

1. **Function Definition**:
   \[
   f(x, y, z) = 1 - \sqrt{x^2 + y^2 + z^2}
   \]
   This function essentially subtracts the Euclidean distance (the square root term) from \(1\). Therefore, \(f(x, y, z)\) will vary depending on the distance from the origin to the point \((x, y, z)\).

2. **Region \(B\)**:
   \[
   B = \{(x, y, z) \ | \ x^2 + y^2 + z^2 \leq 9, \ y \geq 0, \ z \geq 0\}
   \]
   - \(x^2 + y^2 + z^2 \leq 9\) describes a sphere with a radius of \(3\).
   - The conditions \(y \geq 0\) and \(z \geq 0\) restrict this region to the portion of the sphere located in the first octant where both \(y\) and \(z\) are non-negative.

Therefore, \(B\) represents the quarter-sphere (due to the first-octant restriction) with radius \(3\).

---

Would you like a detailed example or further breakdown of how to evaluate \(f(x, y, z)\) for specific points in \(B\)? 

Here are five related questions to explore further:

1. How does the function \(f(x, y, z)\) behave as points move away from the origin?
2. What is the maximum and minimum value of \(f(x, y, z)\) within the region \(B\)?
3. How does the restriction \(y \geq 0\) and \(z \geq 0\) impact the geometry of the region?
4. Could we extend \(B\) by changing the inequality \(x^2 + y^2 + z^2 \leq 9\) to a different radius?
5. What is the gradient of \(f(x, y, z)\) and what does it imply about the function’s behavior?

**Tip**: When examining regions like \(B\) defined by inequalities, visualize the constraints individually (like the sphere and octant restrictions) before combining them for clarity.

f(x, y, z) = 1 − square root of x^2 + y^2 + z^2, B = {(x, y, z) | x^2 + y^2 + z^2 ≤ 9, y ≥ 0, z ≥ 0}

Explore the multivariable function f(x, y, z) = 1 − √(x^2 + y^2 + z^2) and the defined region B = {(x, y, z) | x^2 + y^2 + z^2 ≤ 9, y ≥ 0, z ≥ 0}. This problem introduces key calculus concepts related to spherical regions, Euclidean distance, and restricted domains in three-dimensional space. Suitable for undergraduate calculus students.

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