Let's solve this step by step.

Problem Breakdown:

Given the function $f(x, y) = x^2 + y^2$, this is the equation of a circle in terms of $x$ and $y$, with the function representing the sum of squares.

---

Part (a): Contour Plot and Level Curves

1. Contour Plot is a 2D plot where we fix different levels of $f(x, y)$, i.e., we sketch circles of constant radius.
   • $f(x, y) = x^2 + y^2$ represents concentric circles in the $xy$-plane.
   • For different level values $f(x, y) = c$, we get circles with radius $\sqrt{c}$.

Level Curves for Different Heights:

• $f(x, y) = 9$ → $x^2 + y^2 = 9$ → This is a circle of radius 3.
• $f(x, y) = 4$ → $x^2 + y^2 = 4$ → Circle of radius 2.
• $f(x, y) = 1$ → $x^2 + y^2 = 1$ → Circle of radius 1.
• $f(x, y) = 0$ → $x^2 + y^2 = 0$ → A single point at the origin (0,0).
• $f(x, y) = -1$ → This is not possible, as $x^2 + y^2 \geq 0$ (non-negative), so there is no level curve for $-1$.

The contour plot will consist of concentric circles centered at the origin with radii $3, 2, 1$, and a point at the origin for the level curve $f(x, y) = 0$.

Part (b): Level Curves in XYZ-space

Now we translate the 2D contour plot into 3D space.

• The level curves from part (a) (which are circles) will appear as horizontal cross-sections of the surface $z = f(x, y) = x^2 + y^2$.
• Each horizontal slice corresponds to a particular height ($z$), and for each height $z = c$, the cross-section is a circle with radius $\sqrt{c}$.

Thus, for:

• $z = 9$, you will see a circle with radius 3.
• $z = 4$, a circle with radius 2.
• $z = 1$, a circle with radius 1.
• $z = 0$, a single point (origin).
• There is no level curve for $z = -1$ because the function is always non-negative.

Part (c): Sketch of the Graph of $f(x, y)$

• The function $f(x, y) = x^2 + y^2$ is a paraboloid.
• The graph of $f$ is a surface that opens upwards from the origin in all directions.
• The contour lines are horizontal cross-sections of this surface, with the radii of the cross-sections increasing as you go higher in $z$.
• The graph looks like a 3D bowl, where the base is at $z = 0$ and expands outwards as $z$ increases.

Summary:

• (a) The contour plot consists of concentric circles with radii $3, 2, 1$, centered at the origin.
• (b) In xyz-space, these circles correspond to horizontal cross-sections at different heights.
• (c) The graph of $f(x, y) = x^2 + y^2$ is a paraboloid, with the surface opening upwards.

Would you like me to create or assist with the visual sketches? Or do you need further details?

---

Related Questions:

1. What is the general shape of the graph of functions involving $x^2 + y^2$?
2. How would the contour plot change if the function were $f(x, y) = x^2 - y^2$?
3. What happens to the level curves if we change the function to $f(x, y) = \sqrt{x^2 + y^2}$?
4. How can the contour plot be used to find the gradient of a function?
5. What would the level curves look like for functions of the form $f(x, y) = ax^2 + by^2$?

Tip:

The graph of $f(x, y) = x^2 + y^2$ can be related to cylindrical coordinates, where the distance from the origin on the xy-plane corresponds to the height in z.