Problem 6 - Calculus Early Transcendentals

Question

Describe the graph of \(x=z^{2}\) in \(\mathbb{R}^{3}\).

Step-by-Step Solution

Answer

Describe the graph of the equation \(x=z^2\) in \(\mathbb{R}^3\). The graph of the equation \(x=z^2\) in \(\mathbb{R}^3\) represents a parabolic surface extending infinitely along the y-axis. The surface has a "U" shape, with its vertex at the origin (0,0,0) and opening in the positive x-direction. The graph is symmetrical with respect to the z-axis, as any point on the surface has coordinates (z^2, y, z), where both y and z can be any real value.

Step 1: Understand the given equation in 3D space

Given the equation \(x=z^2\), we are in a 3-dimensional space, with x, y, and z as the coordinates. The equation does not involve the y-coordinate, which means that our graph will not depend on the value of y; it can be any real number. Therefore, the graph will extend infinitely along the y-axis.

Step 2: Analyze and visualize x and z relationship

The equation given is \(x=z^2\). From this equation, we can see that the z-coordinate is squared, and that determines the x-coordinate. This means that, for every positive value of z, there will be a corresponding value of x. In addition, since z is squared, the value of x will never be negative. To visualize this graph properly, suppose we slice the space with a plane parallel to the xz-plane at a particular y-coordinate. Each of these slices will show the graph of the equation \(x=z^2\) in 2-dimensional space. The graph in 2D space would look like the graph of a quadratic function opening to the right.

Step 3: Describe the 3D graph

Considering the full 3-dimensional space, the graph of the equation \(x=z^2\) would look like a parabolic or a "U" shaped surface that extends infinitely along the y-axis. To be more specific, any point on this surface has the coordinates (z^2, y, z) where both y and z can take any real value. The surface will be symmetrical with respect to the z-axis since squaring z will result in the same value for x, regardless of the sign of z. In summary, the graph of the equation \(x=z^2\) in \(\mathbb{R}^3\) represents a parabolic surface extending infinitely along the y-axis, with its vertex at the origin (0,0,0) and opening in the positive x-direction, symmetrical with respect to the z-axis.