Consider the function 𝑧 = 𝑓(𝑥, 𝑦) = 𝑥2 + 4𝑦2.
a. Sketch level curves for 𝑘 = 0, 2, 4, 6
b. Sketch the surface

The given function is \( z = f(x, y) = x^2 + 4y^2 \). This is a quadratic function representing an elliptic paraboloid. 

### Part (a): Sketch Level Curves

Level curves are the curves formed when the function \( z = f(x, y) \) is set to a constant value, i.e., \( f(x, y) = k \). 

For this function, the level curves are obtained by setting \( z = k \), which gives the equation:

\[
x^2 + 4y^2 = k
\]

This represents ellipses in the \( xy \)-plane for different values of \( k \). The equation can be rewritten as:

\[
\frac{x^2}{k} + \frac{y^2}{k/4} = 1
\]

This is the standard form of an ellipse, with the semi-major axis along the \( x \)-axis and the semi-minor axis along the \( y \)-axis. The axes lengths depend on the value of \( k \).

- **For \( k = 0 \)**: The equation becomes \( x^2 + 4y^2 = 0 \), which means \( x = 0 \) and \( y = 0 \), so the level curve is a point at the origin.
- **For \( k = 2 \)**: The equation becomes \( x^2 + 4y^2 = 2 \), which describes an ellipse.
- **For \( k = 4 \)**: The equation becomes \( x^2 + 4y^2 = 4 \), which is another ellipse, but larger.
- **For \( k = 6 \)**: The equation becomes \( x^2 + 4y^2 = 6 \), describing an even larger ellipse.

Thus, the level curves for different values of \( k \) represent ellipses with increasing size as \( k \) increases.

### Part (b): Sketch the Surface

The surface described by \( z = x^2 + 4y^2 \) is an elliptic paraboloid. Here's what it looks like:

- The cross-sections parallel to the \( xy \)-plane are ellipses.
- The paraboloid opens upward because both \( x^2 \) and \( 4y^2 \) are positive and increase as \( x \) or \( y \) increase.
- The \( x^2 \)-term indicates that the surface stretches faster along the \( x \)-axis, while the \( 4y^2 \)-term shows that it stretches more slowly along the \( y \)-axis (due to the factor of 4).

Let me know if you would like to see a graph of these level curves and the surface.

Would you like a graphical sketch, any further details, or a deeper explanation?

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### 5 Related Questions:
1. What does the factor of 4 in \( 4y^2 \) indicate geometrically about the surface?
2. How would the level curves change if the equation were \( z = x^2 + y^2 \)?
3. Can we generalize the surface to higher dimensions, e.g., \( z = f(x, y, w) \)?
4. How does this surface compare with a hyperbolic paraboloid?
5. How would the surface change if negative terms were introduced, such as \( z = x^2 - 4y^2 \)?

### Tip:
When visualizing surfaces, always check how the function behaves along key axes like the \( x \)-axis, \( y \)-axis, and along planes where \( x = y \) or \( x = -y \) to understand the overall shape better.

Consider the function z = f(x, y) = x^2 + 4y^2. a. Sketch level curves for k = 0, 2, 4, 6 b. Sketch the surface

Explore how to sketch level curves and the surface for the quadratic function z = f(x, y) = x^2 + 4y^2. Learn the geometry of level curves in the form of ellipses and the surface represented by an elliptic paraboloid. Suitable for Grades 11-12 and college-level calculus students.

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