The given function is $f(x, y) = x^2 + 4y^2$. To sketch the level curves, we set $f(x, y) = c$ for various values of $c$. This gives us the equation for each level curve:

$x^2 + 4y^2 = c$

This is the equation of an ellipse centered at the origin. For each value of $c$, the equation describes a different ellipse (or point, if $c = 0$).

Step-by-step breakdown for each level:

1. For $c = 0$: $x^2 + 4y^2 = 0$ This only holds when $x = 0$ and $y = 0$, so the level curve is a point at the origin.

2. For $c = 1$: $x^2 + 4y^2 = 1$ Dividing by 1, we get: $\frac{x^2}{1} + \frac{y^2}{\frac{1}{4}} = 1$ This is an ellipse with semi-major axis 1 along the $x$-axis and semi-minor axis $\frac{1}{2}$ along the $y$-axis.

3. For $c = 4$: $x^2 + 4y^2 = 4$ Dividing by 4, we get: $\frac{x^2}{4} + \frac{y^2}{1} = 1$ This is an ellipse with semi-major axis 2 along the $x$-axis and semi-minor axis 1 along the $y$-axis.

4. For $c = 9$: $x^2 + 4y^2 = 9$ Dividing by 9, we get: $\frac{x^2}{9} + \frac{y^2}{\frac{9}{4}} = 1$ This is an ellipse with semi-major axis 3 along the $x$-axis and semi-minor axis $\frac{3}{2}$ along the $y$-axis.

5. For $c = 16$: $x^2 + 4y^2 = 16$ Dividing by 16, we get: $\frac{x^2}{16} + \frac{y^2}{4} = 1$ This is an ellipse with semi-major axis 4 along the $x$-axis and semi-minor axis 2 along the $y$-axis.

6. For $c = 25$: $x^2 + 4y^2 = 25$ Dividing by 25, we get: $\frac{x^2}{25} + \frac{y^2}{\frac{25}{4}} = 1$ This is an ellipse with semi-major axis 5 along the $x$-axis and semi-minor axis $\frac{5}{2}$ along the $y$-axis.

Summary of ellipses:

• $c = 0$: Point at the origin.
• $c = 1$: Ellipse with axes 1 (x-axis) and 0.5 (y-axis).
• $c = 4$: Ellipse with axes 2 (x-axis) and 1 (y-axis).
• $c = 9$: Ellipse with axes 3 (x-axis) and 1.5 (y-axis).
• $c = 16$: Ellipse with axes 4 (x-axis) and 2 (y-axis).
• $c = 25$: Ellipse with axes 5 (x-axis) and 2.5 (y-axis).

Would you like further details, such as a visual sketch of these curves?

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Here are some related questions to explore:

1. What is the general form of an ellipse equation?
2. How do changes in the coefficients of $x^2$ and $y^2$ affect the shape of the ellipse?
3. How can you find the area of each ellipse given the equation $f(x, y)$?
4. What are the differences between ellipses and circles in terms of their equations?
5. How would the level curves change if we had $x^2 + 2y^2$ instead?

Tip: In an ellipse equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $a$ and $b$ represent the lengths of the semi-major and semi-minor axes, respectively.