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Let's break down the problem step by step.

We are given the function \( g(x, y, z) = e^{-\left(x^2 + (y - 2)^2 + z^2\right)} \). This is a Gaussian function describing level surfaces in 3D space, which are level sets of this function, defined as \( g(x, y, z) = c \), where \( c \) is a constant. Let's address each part of the question:

### Part (a)
You are asked to find the values of \( c \) for which the level surfaces are defined, i.e., \( e^{-\left(x^2 + (y - 2)^2 + z^2\right)} = c \).

1. The function \( e^{-A} \), where \( A = x^2 + (y - 2)^2 + z^2 \geq 0 \), will always produce a value between 0 and 1, because the exponential function decays to 0 as \( A \to \infty \) and equals 1 when \( A = 0 \).
   
Thus, the possible values of \( c \) are in the range \( (0, 1] \). This gives the interval:
\[
c \in (0, 1]
\]

### Part (b)
We are asked to pick a level surface that intersects the \( xz \)-plane and find the value of \( c \). To find this, we set \( y = 0 \) (the condition for the \( xz \)-plane).

1. The equation becomes:
   \[
   e^{-\left(x^2 + (0 - 2)^2 + z^2\right)} = c
   \]
   Simplifying the expression:
   \[
   e^{-\left(x^2 + 4 + z^2\right)} = c
   \]
   Taking the natural logarithm on both sides:
   \[
   -\left(x^2 + 4 + z^2\right) = \ln(c)
   \]
   Rearranging:
   \[
   x^2 + z^2 = -\ln(c) - 4
   \]
   
To ensure that the equation is valid, the right-hand side must be positive, so \( c \) must be small enough such that \( -\ln(c) - 4 \geq 0 \). Solving for \( c \):
\[
-\ln(c) \geq 4 \quad \Rightarrow \quad \ln(c) \leq -4 \quad \Rightarrow \quad c \leq e^{-4}
\]

Thus, we can choose a value of \( c \) such as \( c = e^{-4} \), and the equation for the intersection would be:
\[
x^2 + z^2 = 0
\]
At \( c = e^{-4} \), the level surface intersects the \( xz \)-plane at the point \( (0, 0) \).

---

Do you have any questions about these steps or would you like further explanation?

### Five Relative Questions:
1. Why does the function \( g(x, y, z) \) decay to 0 as the distance from the origin increases?
2. How would the shape of the level surfaces change as \( c \) increases from 0 to 1?
3. Can we find the intersections of these level surfaces with the \( yz \)-plane or the \( xy \)-plane similarly?
4. What is the significance of setting \( y = 2 \) instead of \( y = 0 \) when finding the intersection?
5. How does the equation \( x^2 + z^2 = -\ln(c) - 4 \) compare to the equation of a circle?

### Tip:
When working with Gaussian functions like \( g(x, y, z) \), always remember that the level surfaces are ellipsoids in 3D space, centered around specific points.

Describe in words the level surfaces of g(x, y, z) = e^{-(x^2 + (y - 2)^2 + z^2)}. Think what the intersections of these surfaces with the three coordinate planes look like. (a) In symbolic form, these level surfaces are given by e^{-(x^2 + (y - 2)^2 + z^2)} = c where c is a constant. For what values of c are level surfaces defined? (b) Pick a level surface that intersects the xz-plane. What is your value of c?

Explore the level surfaces of the function g(x, y, z) = e^{-(x^2 + (y - 2)^2 + z^2)} and understand how they intersect with the xz-plane. This problem covers key concepts in multivariable calculus, including exponential functions and level surfaces.

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