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The concept of a surface intersection involves understanding how a plane and a surface interact in three-dimensional space. In this context, we are examining the intersection of a plane with the equation \( x = 3 \) and a surface represented by the equation \( z = f(x,y) \). When these two geometries intersect, they typically create a curve on the surface.

In our specific problem, the plane \( x = 3 \) slices through the surface \( z = f(x,y) \), and this interaction along the plane results in a curve. The curve is described by the equation \( z = y^2 \), which means, for every value of \( y \), there is a corresponding \( z \) value that satisfies this equation.

It's important to visualize this concept by imagining peeling away the surface along the plane. What remains is a "cut" or "trace" on the surface, represented mathematically by our curve equation. This is a critical step in analyzing and solving intersection problems in multivariable calculus.

The plane equation \( x = 3 \) is a simple mathematical representation of a two-dimensional flat surface within three-dimensional space. Here, the plane is perfectly vertical and parallel to the yz-plane, intersecting the x-axis at exactly \( x = 3 \).

Planes can be perceived as infinite flat sheets, and within our problem, this particular plane is straightforward because it only fixes the x-coordinate while leaving \( y \) and \( z \) unrestricted.

When looking at equations of planes, remember:

• A plane can be defined by fixing one of the variables (x, y, or z), like \( x = 3 \) in this case.
• Plane equations can be described in the general form \( ax + by + cz = d \), where \( a, b, \) and \( c \) are constants that determine the tilt and orientation of the plane.

Understanding plane equations helps in visualizing how planes appear and behave within a three-dimensional system. They are the backbone of geometric interpretations in calculus.

The curve equation \( z = y^2 \) in this scenario is a specific type of equation that provides a relationship between the variables \( y \) and \( z \) on the intersection curve. At each instance of the intersection plane \( x=3 \), this equation tells us how \( z \) varies with changes in \( y \).

For our exercise, when inserting a specific \( y \)-value into \( z = y^2 \), we derive a \( z \)-value that reflects the curve's position at that point on the surface. At the given point \( (3, 4, 16) \), substituting \( y = 4 \) gives us \( z = 16 \), which confirms the point is on the curve.

Key aspects of understanding curve equations include:

• They express how one variable depends mathematically on another within the framework of the problem.
• Differentiating the equation helps find rates of change or slopes, which is pivotal for calculus operations like finding tangents, normals, and verifying solutions—as done here by differentiating \( z = y^2 \) to determine \( f_y(3,4) = 8 \).

Mastering curve equations unlocks the ability to explore dynamic surfaces and their intersections further.