To match the functions of two variables in Column A with their corresponding graphs in Column B, we need to analyze the given functions and understand their general shapes and properties.

1. $f(x, y) = 2x + 3y$:

   • This is a linear function. It represents a plane in three dimensions.

2. $f(x, y) = x^2 + y^2$:

   • This is a parabolic surface known as a paraboloid. The graph opens upwards and has a minimum point at the origin (0,0).

3. $f(x, y) = \frac{\sin(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}}$:

   • This is a Bessel function. The graph will have a series of peaks and valleys, resembling ripples.

4. $f(x, y) = xe^{-(x^2 + y^2)}$:

   • This function resembles a Gaussian function modulated by $x$. It will have a peak at the origin and rapidly decay to zero away from the origin.

Now, let's match each function with its graph:

1. $f(x, y) = 2x + 3y$ is a plane, corresponding to graph 2.
2. $f(x, y) = x^2 + y^2$ is a paraboloid, corresponding to graph 1.
3. $f(x, y) = \frac{\sin(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}}$ is the ripple-like Bessel function, corresponding to graph 3.
4. $f(x, y) = xe^{-(x^2 + y^2)}$ is a Gaussian-like function with a peak at the origin, corresponding to graph 4.

Therefore, the correct matching is:

• i) with 2
• ii) with 1
• iii) with 3
• iv) with 4

Do you want details or have any questions?

Here are 8 relative questions that expand on this information:

1. What is a contour plot, and how does it relate to the functions given?
2. How do you find the critical points of the function $f(x, y) = x^2 + y^2$?
3. Can you derive the equation of the tangent plane to $f(x, y) = 2x + 3y$ at a given point?
4. What are the practical applications of Bessel functions in engineering and physics?
5. How do you calculate the gradient vector of $f(x, y) = xe^{-(x^2 + y^2)}$?
6. What is the significance of the Gaussian function in statistics?
7. How do you visualize the function $f(x, y) = \frac{\sin(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}}$ using software tools like MATLAB or Mathematica?
8. What are the properties of the exponential function $e^{-(x^2 + y^2)}$ and its relevance in different scientific fields?

Tip: Always label your axes when plotting functions to clearly represent the variables and their ranges.