We need to describe and sketch the graphs of the following equations in three-space: Equation 18: $y = \cos x$ Equation 20: $z = \sqrt{x^2 + y^2 + 1}$

1. Problem Description

We need to describe and sketch the graphs of the following equations in three-space:

Equation 18: $y = \cos x$

Equation 20: $z = \sqrt{x^2 + y^2 + 1}$

2. Solution Steps

Equation 18: $y = \cos x$

In the $xy$-plane, $y = \cos x$ is the standard cosine curve. Since there is no restriction on $z$, $z$ can take on any value. Thus, in three-space, the graph is a surface formed by translating the cosine curve along the $z$-axis. The surface is a cosine cylinder.

Equation 20: $z = \sqrt{x^2 + y^2 + 1}$

Square both sides to get $z^2 = x^2 + y^2 + 1$.

Rearranging, we have $z^2 - x^2 - y^2 = 1$.

Since $z = \sqrt{x^2 + y^2 + 1}$, $z$ is always non-negative ($z \ge 0$).

This is the equation of a hyperboloid of two sheets, but because of the square root, we only consider the upper sheet where $z \ge 0$. The equation can also be interpreted as a hyperboloid of one sheet, with its axis aligned along z axis, with its lower part truncated at z =

1.

3. Final Answer

Equation 18: $y = \cos x$ is a cosine cylinder.

Equation 20: $z = \sqrt{x^2 + y^2 + 1}$ is the upper sheet of a hyperboloid of two sheets (or a hyperboloid of one sheet truncated at $z=1$).