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.