24. Let f(x, y) = x
2 + y
2
.
a) Sketch a contour plot for f, including labelled level curves at heights
9, 4, 1, 0 and −1.
b) Sketch the level curves you drew in part (a) as they would appear in
xyz-space as part of the graph of f.
c) Based on the picture you drew in part (b), sketch a crude graph of f.

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.

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### 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?

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### 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.

24. Let f(x, y) = x^2 + y^2. a) Sketch a contour plot for f, including labelled level curves at heights 9, 4, 1, 0 and −1. b) Sketch the level curves you drew in part (a) as they would appear in xyz-space as part of the graph of f. c) Based on the picture you drew in part (b), sketch a crude graph of f.

Learn how to sketch contour plots and graphs for the function f(x, y) = x^2 + y^2. This problem includes drawing level curves at different heights, visualizing the function in xyz-space, and understanding the paraboloid structure. Ideal for undergraduate calculus students.

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