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.