Find the equations of the level curves of the function at the indicated values of *c*.

*g*(*x*, *y*) = *x*2 + *y*2;  *c* = 4, 36

*c* = 4
 
*c* = 36
 
Sketch the level curves to help visualize the function.

### Problem Explanation:
The level curves of a function are curves where the function takes on a constant value \( c \). For the function \( g(x, y) = x^2 + y^2 \), the level curves represent the points \( (x, y) \) where \( g(x, y) \) is equal to a constant \( c \).

We are asked to find the equations of the level curves for the function:
\[
g(x, y) = x^2 + y^2
\]
at the given values of \( c = 4 \) and \( c = 36 \).

### Step-by-Step Solution:

1. **For \( c = 4 \):**

   The level curve equation is obtained by setting \( g(x, y) = 4 \):
   \[
   x^2 + y^2 = 4
   \]
   This represents a circle centered at the origin \( (0, 0) \) with radius \( r = \sqrt{4} = 2 \).

2. **For \( c = 36 \):**

   Similarly, for \( c = 36 \):
   \[
   x^2 + y^2 = 36
   \]
   This represents another circle centered at the origin \( (0, 0) \), but with radius \( r = \sqrt{36} = 6 \).

### Summary of Level Curves:
- For \( c = 4 \), the level curve is a circle with radius 2.
- For \( c = 36 \), the level curve is a larger circle with radius 6.

### Sketch the Level Curves:
To visualize the level curves:
- Draw two concentric circles centered at the origin.
  - The first circle has a radius of 2 (for \( c = 4 \)).
  - The second circle has a radius of 6 (for \( c = 36 \)).

I can create a sketch of these level curves for you as well. Would you like a sketch of these level curves to help visualize the function?

### Follow-up Questions:
1. What do the level curves of a general quadratic function look like?
2. How would the level curves change if \( g(x, y) \) included mixed terms (like \( xy \))?
3. What do the level curves of \( g(x, y) = x^2 - y^2 \) look like?
4. How would adding a linear term (e.g., \( g(x, y) = x^2 + y^2 + x \)) affect the shape of the level curves?
5. How do you find the level curves of a 3D function like \( g(x, y, z) \)?

### Tip:
Level curves can be useful to visualize the "height" of a surface in 2D space. Each curve represents points with the same "height" on the surface.

Find the equations of the level curves of the function at the indicated values of *c*. g(x, y) = x^2 + y^2; c = 4, 36

Learn how to find the level curves of the function g(x, y) = x^2 + y^2 for c = 4 and c = 36. This problem covers multivariable calculus concepts and explores the geometry of circles. Suitable for high school and early college students.

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