Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Describe the graph in three-space of each equation. (a) \(z=2\) (b) \(x=y\) (c) \(x y=0\) (d) \(x y z=0\) (e) \(x^{2}+y^{2}=4\) (f) \(z=\sqrt{9-x^{2}-y^{2}}\)

Step-by-Step Solution

Answer

(a) Horizontal plane, (b) Diagonal plane, (c) Two planes, (d) Three planes, (e) Cylinder, (f) Hemisphere.

Step 1: Horizontal Plane

The equation \(z=2\) describes a horizontal plane parallel to the \(xy\)-plane, located at a height of \(2\) units along the \(z\)-axis. All points on this plane have their \(z\)-coordinate equal to \(2\).

Step 2: Diagonal Plane

The equation \(x = y\) represents a diagonal plane in three-space, passing through the origin at an angle of 45 degrees to both the \(x\)- and \(y\)-axes. Every point on this plane has equal values for \(x\) and \(y\).

Step 3: Coordinate Planes

The equation \(xy = 0\) describes the union of two coordinate planes: the \(yz\)-plane (where \(x = 0\)) and the \(xz\)-plane (where \(y = 0\)). Points on these planes satisfy the condition that the product \(xy\) is zero.

Step 4: Intersection of Planes

The equation \(xyz = 0\) represents a union of three planes: the \(yz\)-plane (\(x = 0\)), the \(xz\)-plane (\(y = 0\)), and the \(xy\)-plane (\(z = 0\)). Points on any of these planes satisfy one of the variables being zero.

Step 5: Cylindrical Surface

The equation \(x^2 + y^2 = 4\) describes a cylinder with a radius of 2 centered along the \(z\)-axis. The cylinder extends infinitely along the \(z\)-direction with circular cross-sections in the \(xy\)-plane.

Step 6: Hemisphere

The equation \(z = \sqrt{9 - x^2 - y^2}\) forms a hemisphere with a radius of 3 centered at the origin. The hemisphere is above the \(xy\)-plane, forming a dome where \(z\)-values are non-negative.

Key Concepts

Horizontal Plane

In the realm of 3D graphs, a horizontal plane holds a special significance. This concept can be visualized using the equation \(z=2\). Here, the equation conveys a plane that is parallel to the \(xy\)-plane.
What makes it distinct is its positioning at a specific height of 2 units along the \(z\)-axis. In simpler terms, this plane is "floating" in space, above the origin. Consequently, all the points on this horizontal plane share a common trait: their \(z\)-coordinate is always 2.

• This means that if you know the \(x\) and \(y\) coordinates, you can automatically know the \(z\) coordinate is 2.
• The plane extends infinitely in the directions of the \(x\) and \(y\) axes.

Such planes are a basic yet vital component in visualizing 3D objects and concepts in calculus.

Diagonal Plane

The diagonal plane represented by the equation \(x = y\) introduces a fascinating feature in three-dimensional space. Unlike horizontal or vertical planes, a diagonal plane slices through the space at an angle.
In this scenario, any given point on this plane has the same value for its \(x\)-coordinate and \(y\)-coordinate. Essentially, it passes through the origin diagonally, making an angle of 45 degrees with both the \(x\)- and \(y\)-axes.

• Consider points like (1,1) and (2,2); they lie on this plane because their \(x\) and \(y\) coordinates are equal.
• Unlike a single line, this plane isn't restricted to a single dimension; it stretches infinitely in the space defined by the equality of \(x\) and \(y\).

This concept enhances the understanding of how different planes intersect and project in a 3D environment.

Coordinate Planes

In 3D space, equations like \(xy = 0\) often denote coordinate planes. Specifically here, it signifies the union of two distinct planes.
These are the \(yz\)-plane, where \(x = 0\), and the \(xz\)-plane, where \(y = 0\). This means any point residing on these planes results in the product \(xy\) being zero.

• On the \(yz\)-plane, since \(x=0\), the points could be like (0,1,2) where only \(y\) and \(z\) have values.
• Similarly, for the \(xz\)-plane, a point could be (3,0,-1), implying values for \(x\) and \(z\) only.

This reflects how the interaction of different planes in space can be represented through these straightforward equations.

Cylindrical Surface

The notion of a cylindrical surface is an intriguing element in calculus, described in this instance by the equation \(x^2 + y^2 = 4\). This translates to a cylinder centered along the \(z\)-axis.
Think of this cylinder as comprising circles of radius 2 in the \(xy\)-plane that stack vertically throughout the \(z\)-dimension. It continues endlessly along the \(z\)-axis, forming a tubular structure.

• The center of each circular cross-section sits at the origin, making them symmetric around the \(z\)-axis.
• Regardless of the \(z\)-coordinate, all horizontal cross-sections are identical circles.

This visualization aids in comprehending surfaces that protrude through spaces without ending, essentially building a foundation in mapping complex surfaces.

Hemisphere in 3D

A hemisphere is a half of a sphere, and it is visually captivating when depicted in 3D space, exemplified by the equation \(z = \sqrt{9 - x^2 - y^2}\). This represents a hemisphere with a radius of 3.
Centered at the origin, it rests above the \(xy\)-plane, creating a dome-like structure.

• The equation ensures that the \(z\)-values are non-negative, reflecting the upper half of a sphere.
• When \(x\) and \(y\) are zero, \(z\) reaches its peak value at 3, the radius.

By delving into this concept, you're introduced to a variety of ways surfaces form in space, bolstering a more comprehensive perception of 3D figures in calculus.