To graph the level curves for the function $f(x, y) = x - y$ at $k = -2, -1, 0, 1, 2$, we need to set $f(x, y) = k$ and solve for $y$ in terms of $x$ for each value of $k$. The equation becomes:

$f(x, y) = x - y = k \quad \Rightarrow \quad y = x - k$

Now, let's calculate the level curves for each given $k$:

1. For $k = -2$, the equation becomes: $y = x + 2$

2. For $k = -1$, the equation becomes: $y = x + 1$

3. For $k = 0$, the equation becomes: $y = x$

4. For $k = 1$, the equation becomes: $y = x - 1$

5. For $k = 2$, the equation becomes: $y = x - 2$

Each of these equations represents a straight line, with a slope of 1 and different y-intercepts. The level curves are just straight lines, each shifted up or down depending on the value of $k$.

Now, I'll plot these level curves on the same graph.

Let me generate that plot for you. numpy as np import matplotlib.pyplot as plt

Define the x range for the graph

x = np.linspace(-10, 10, 400)

Define the level curves for k = -2, -1, 0, 1, 2

k_values = [-2, -1, 0, 1, 2] y_values = [x + k for k in k_values]

Create the plot

plt.figure(figsize=(8, 8))

Plot each level curve

for k, y in zip(k_values, y_values): plt.plot(x, y, label=f'k = {k}')

Set labels and title

plt.xlabel('x') plt.ylabel('y') plt.title('Level Curves for the Function f(x, y) = x - y')

Add a legend

plt.legend()

Display the grid

plt.grid(True)

Show the plot

plt.show()Here is the graph of the level curves for the function $f(x, y) = x - y$ with $k = -2, -1, 0, 1, 2$. Each line corresponds to one of the values of $k$, and you can see that they are all straight lines with a slope of 1, but they shift up or down depending on the value of $k$.

Would you like more details on the graph or any other concepts explained?

Here are some related questions for further exploration:

1. How would the graph change if the function was $f(x, y) = x + y$?
2. How do we interpret the level curves geometrically?
3. What is the relationship between the slope of a level curve and the gradient of the function?
4. How can we use level curves to estimate the behavior of a function at specific points?
5. How would you graph the level curves for a more complex function, like $f(x, y) = x^2 + y^2$?

Tip: Level curves represent constant values of a function, and each curve represents a set of points where the function has the same value. For functions in two variables, level curves are particularly useful for visualizing how the function behaves spatially.