Given function:

$f(x, y) = xy$

(a) Determine several level curves of the given function for $c = -2, -1, 0, 1, 2$.

The level curves of a function $f(x, y)$ for a constant $c$ are the curves along which the function equals $c$, i.e., for each $c$, the equation becomes: $xy = c$

This equation represents a family of hyperbolas. Let's find the level curves for the given values of $c$:

1. For $c = -2$: $xy = -2 \quad \Rightarrow \quad y = \frac{-2}{x}$ This is a hyperbola opening in the 2nd and 4th quadrants.

2. For $c = -1$: $xy = -1 \quad \Rightarrow \quad y = \frac{-1}{x}$ This is another hyperbola, but it will be closer to the origin than the one for $c = -2$.

3. For $c = 0$: $xy = 0 \quad \Rightarrow \quad y = 0 \quad \text{or} \quad x = 0$ This corresponds to the coordinate axes (the x-axis and y-axis), where the function value is zero.

4. For $c = 1$: $xy = 1 \quad \Rightarrow \quad y = \frac{1}{x}$ This is a hyperbola opening in the 1st and 3rd quadrants.

5. For $c = 2$: $xy = 2 \quad \Rightarrow \quad y = \frac{2}{x}$ This is another hyperbola, opening in the 1st and 3rd quadrants, but farther from the origin than the one for $c = 1$.

Thus, the equations of the level curves are:

• For $c = -2$, $y = \frac{-2}{x}$
• For $c = -1$, $y = \frac{-1}{x}$
• For $c = 0$, $x = 0$ or $y = 0$
• For $c = 1$, $y = \frac{1}{x}$
• For $c = 2$, $y = \frac{2}{x}$

(b) Sketching the graph of $f(x, y) = xy$:

1. The graph consists of level curves that represent hyperbolas in various orientations (depending on whether the value of $c$ is positive or negative).
2. The level curve for $c = 0$ corresponds to the coordinate axes (x-axis and y-axis).
3. The hyperbolas get closer to the origin as the magnitude of $c$ decreases, and they move farther from the origin as $|c|$ increases.

To summarize the graph of $f(x, y)$:

• For positive values of $c$, the hyperbolas are in the 1st and 3rd quadrants.
• For negative values of $c$, the hyperbolas are in the 2nd and 4th quadrants.
• The axes $x = 0$ and $y = 0$ correspond to the level curve $c = 0$.

Would you like further details on how to sketch the graph step by step? Or would you like help with other functions? Here are a few related questions you might find interesting:

1. How does the graph of a function change when the function is multiplied by a constant?
2. What happens to the level curves if the function was $f(x, y) = x^2 + y^2$?
3. How can we determine the symmetry of a function based on its level curves?
4. What would the graph of a function $f(x, y) = x + y$ look like?
5. How does the equation $xy = c$ behave for very large or very small values of $c$?

Tip: Always visualize level curves by drawing a set of coordinate axes and sketching the curves for various $c$ values. It will help you understand the function’s behavior better.