https://file.aiquickdraw.com/mathbot/file/1722882277647vaz2xzdi.jpg

The graph of the function with serial number i) in column A has the serial number
in Column B.

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.

Understand how to match functions to graphs by analyzing their properties and shapes. Explore linear functions, paraboloids, Bessel functions, and Gaussian-like functions in multivariable calculus contexts. Suitable for advanced high school or college students.

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