Sketch the level curves of f(x, y) = x^2 + 4y^2 at levels c = 0, 1,4,9,1,6,25

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.

Sketch the level curves of f(x, y) = x^2 + 4y^2 at levels c = 0, 1, 4, 9, 16, 25

Explore the level curves of the function f(x, y) = x^2 + 4y^2 for various values of c, including 0, 1, 4, 9, 16, and 25. Learn how to derive the equations of ellipses and understand their geometric properties. Suitable for students studying multivariable calculus or conic sections.

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