To draw level curves for each of the functions, we first need to understand the concept. A level curve for a function $f(x, y)$ is a curve where the function takes on a constant value. These curves are determined by solving the equation $f(x, y) = c$, where $c$ is a constant.

Let's explore each function and plot at least six level curves for them.

(a) $f(x, y) = 4 - x^2 - y$

The level curve equation is: $f(x, y) = 4 - x^2 - y = c$ Rearranging this gives: $y = 4 - x^2 - c$ This is a family of parabolas with vertex along the line $y = 4 - c$. The shape opens downward, and the $c$-values shift the parabolas up or down vertically. By selecting different values of $c$, we can generate different curves. Some possible level curves are:

• $c = 0$: $y = 4 - x^2$
• $c = 1$: $y = 3 - x^2$
• $c = 2$: $y = 2 - x^2$
• $c = 3$: $y = 1 - x^2$
• $c = -1$: $y = 5 - x^2$
• $c = -2$: $y = 6 - x^2$

(b) $f(x, y) = x + 2y - 1$

The level curve equation is: $f(x, y) = x + 2y - 1 = c$ Rearranging: $x + 2y = c + 1 \quad \text{or} \quad x = -2y + (c + 1)$ This is a family of straight lines with different slopes and intercepts. Some possible level curves are:

• $c = 0$: $x = -2y + 1$
• $c = 1$: $x = -2y + 2$
• $c = 2$: $x = -2y + 3$
• $c = -1$: $x = -2y$
• $c = -2$: $x = -2y - 1$
• $c = 3$: $x = -2y + 4$

(c) $f(x, y) = x^2$

The level curve equation is: $f(x, y) = x^2 = c$ This gives: $x = \pm \sqrt{c}$ These are vertical lines at $x = \pm \sqrt{c}$. Some possible level curves are:

• $c = 0$: $x = 0$ (a vertical line)
• $c = 1$: $x = \pm 1$
• $c = 4$: $x = \pm 2$
• $c = 9$: $x = \pm 3$
• $c = 16$: $x = \pm 4$
• $c = 25$: $x = \pm 5$

(d) $f(x, y) = \frac{y}{x^2}$

The level curve equation is: $f(x, y) = \frac{y}{x^2} = c$ Rearranging: $y = c x^2$ These are parabolas that open upwards for $c > 0$ and downwards for $c < 0$. Some possible level curves are:

• $c = 1$: $y = x^2$
• $c = 2$: $y = 2x^2$
• $c = 0.5$: $y = 0.5x^2$
• $c = -1$: $y = -x^2$
• $c = -0.5$: $y = -0.5x^2$
• $c = 3$: $y = 3x^2$

These curves reflect the different behavior of each function. I can also help with visualizing these curves if needed.

Would you like me to generate plots for these level curves or provide more details on any specific step?

Here are five related questions to expand your understanding:

1. How do the shapes of level curves relate to the gradients of functions?
2. How would you interpret level curves on a 3D surface plot?
3. Can you explore the behavior of level curves as $c \to \infty$ or $c \to -\infty$?
4. What role do contour intervals play in interpreting level curves?
5. How do linear and quadratic functions differ in the shapes of their level curves?

Tip: For quadratic functions, level curves often form conic sections like parabolas, ellipses, or hyperbolas, depending on the terms involved.