Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface.1. z=2x^2+3y^2
2. z=x^2+y^2
3. z=sqrt(x^2+y^2)
4. z=sqrt((25-x^2-y^2))
5. z=2x+3y
6. z= 1/(x-1)
7. z=xy
A. a collection of equally spaced concentric circles
B. two straight lines and a collection of hyperbolas
C. a collection of unequally spaced parallel lines
D. a collection of unequally spaced concentric circles
E. a collection of concentric ellipses
F. a collection of equally spaced parallel lines

Let's analyze the surfaces to match them with the verbal descriptions:

1. **\( z = 2x^2 + 3y^2 \)**  
   - This represents an **elliptic paraboloid**. The level curves \( z = c \) are of the form \( 2x^2 + 3y^2 = c \), which are ellipses.  
   - **Verbal description**: E. a collection of concentric ellipses.

2. **\( z = x^2 + y^2 \)**  
   - This represents a **circular paraboloid**. The level curves \( z = c \) are of the form \( x^2 + y^2 = c \), which are circles.  
   - **Verbal description**: D. a collection of unequally spaced concentric circles.

3. **\( z = \sqrt{x^2 + y^2} \)**  
   - This represents a **cone**. The level curves \( z = c \) are of the form \( \sqrt{x^2 + y^2} = c \), which are circles.  
   - **Verbal description**: A. a collection of equally spaced concentric circles.

4. **\( z = \sqrt{25 - x^2 - y^2} \)**  
   - This is a portion of a **sphere**. The level curves \( z = c \) are circles with radius dependent on \( c \).  
   - **Verbal description**: D. a collection of unequally spaced concentric circles.

5. **\( z = 2x + 3y \)**  
   - This represents a **plane**. The level curves \( z = c \) are of the form \( 2x + 3y = c \), which are straight lines.  
   - **Verbal description**: F. a collection of equally spaced parallel lines.

6. **\( z = \frac{1}{x-1} \)**  
   - This is a **hyperbolic surface** with vertical asymptotes. The level curves will be vertical lines and hyperbolas.  
   - **Verbal description**: B. two straight lines and a collection of hyperbolas.

7. **\( z = xy \)**  
   - This represents a **saddle surface**. The level curves \( z = c \) are of the form \( xy = c \), which are hyperbolas.  
   - **Verbal description**: B. two straight lines and a collection of hyperbolas.

### Final matching:
1. **E**  
2. **D**  
3. **A**  
4. **D**  
5. **F**  
6. **C**  
7. **B**

Would you like more details on any of these concepts? Here are 5 related questions to explore further:

1. How do you derive level curves for a given surface?
2. What is the difference between hyperbolic and parabolic surfaces?
3. How do concentric circles differ in spacing for different surfaces?
4. Can you explain the geometric meaning of a saddle surface?
5. How do elliptical level curves arise from quadratic equations?

**Tip**: Level curves can often reveal the symmetry and nature of a surface; plotting them helps in visualizing complex 3D surfaces!

Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface. 1. z=2x^2+3y^2 2. z=x^2+y^2 3. z=sqrt(x^2+y^2) 4. z=sqrt((25-x^2-y^2)) 5. z=2x+3y 6. z=1/(x-1) 7. z=xy A. a collection of equally spaced concentric circles B. two straight lines and a collection of hyperbolas C. a collection of unequally spaced parallel lines D. a collection of unequally spaced concentric circles E. a collection of concentric ellipses F. a collection of equally spaced parallel lines

Explore how to match level curves with their corresponding surface equations, covering conic sections, elliptic paraboloids, hyperbolic surfaces, and more. This detailed explanation is suitable for advanced high school students (Grades 11-12) and helps visualize complex 3D surfaces.

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