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The contour plot displayed represents a surface, and the problem asks for a cross-section through \( x = -1 \). To determine which graph corresponds to this cross-section, we need to analyze the surface at this particular \( x \)-value.

Here's the general approach:

1. **Contour Plot**: This plot shows level curves (contours) where the surface has constant height (or value) at different points in the \( x, y \)-plane. Each contour represents a specific value of the function (seen in the color scale to the right).
2. **Cross-section**: A cross-section through \( x = -1 \) means we fix \( x = -1 \) and examine how the height (or value) of the function changes with \( y \). Essentially, this is a 2D slice where the surface is reduced to a line plot of \( f(-1, y) \).

### Analyzing \( x = -1 \):

- On the plot, \( x = -1 \) corresponds to the vertical line at \( x = -1 \), which cuts through the contours.
- To determine the behavior of the function along this slice, observe how the colors and contour lines change as you move up or down along \( y \) at \( x = -1 \).
- At \( y = 0 \), near \( x = -1 \), the contours suggest a minimum, since the contour lines are closely packed, indicating a dip in the surface. Moving up or down in \( y \), the surface rises slightly and then falls again (indicated by the contour pattern).

Thus, the graph of this cross-section should reflect this "dip" at \( y = 0 \) with a symmetric rise and fall around it.

Let me know if you'd like more details, or have any questions about the steps. Here are 5 related questions to expand on this topic:

1. What is the significance of closely spaced contours in a contour plot?
2. How can we compute the function values for specific points on a contour plot?
3. How would the cross-section change if we took \( x = 0 \) instead of \( x = -1 \)?
4. What types of surfaces produce symmetric cross-sections like this one?
5. How would a saddle point appear in a contour plot and its cross-section?

**Tip**: In a contour plot, when the lines are closer together, the surface is steeper, while widely spaced lines indicate a flatter region.

Which one of the following graphs below corresponds to the cross-section of this surface through x = -1?

Learn how to analyze a surface represented by a contour plot and determine the cross-section through x = -1. This problem involves multivariable calculus, contour plots, and understanding how cross-sections relate to level curves. Suitable for undergraduate students studying calculus.

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