Problem 47 - University Calculus: Early Transcendentals

Question

Write inequalities to describe the sets. The half-space consisting of the points on and below the \(x y\) -plane

Step-by-Step Solution

Answer

The inequality is \(z \leq 0\).

Step 1: Understand the XY Plane

The XY plane is defined by all points where the z-coordinate is zero. In a 3D coordinate system, any point on the XY plane has the form \((x, y, 0)\).

Step 2: Define the Condition for 'On or Below'

The points that are on the XY plane have a z-coordinate of 0. Points below the XY plane will have a z-coordinate less than zero. Hence, the condition for points on and below the XY plane can be expressed by the inequality \(z \leq 0\).

Step 3: Write the Inequality

Combine the information from the previous step to write an inequality that describes all the points below and on the XY plane. This is represented simply by the inequality \(z \leq 0\).

Step 4: Express the Set of Solutions

The set of points satisfying the condition can be written in set notation: \(\{(x, y, z) | z \leq 0 \}\). This expresses all (x, y, z) points that are part of the half-space on or below the XY plane.

Key Concepts

XYZ coordinate system

The XYZ coordinate system is crucial in understanding how objects are placed and interact in three-dimensional space. This system is often visualized as three perpendicular lines (axes) that intersect at a single point called the origin. Each line represents one direction: the x-axis, y-axis, and z-axis.

• X-axis: Runs horizontally from left to right.
• Y-axis: Runs vertically from bottom to top.
• Z-axis: Runs from front to back.

Coordinates in this system are written as \((x, y, z)\), where each value represents a point's position along the respective axis. The z-coordinate often determines if a point is in front of or behind the XY plane. Understanding this coordinate system is essential for working with 3D graphs and geometry.

half-space

The concept of half-space in geometry is a way to describe a region in space. Specifically, in a three-dimensional context, a half-space is one of the two regions into which a plane divides the surrounding space.
For example, if you have the XY-plane, it divides space into two halves: one where the z-coordinates are positive (above the plane) and one where they are non-positive (on or below the plane). In this exercise, the half-space we are examining includes all points on and below the XY-plane, represented by the inequality \( z \leq 0 \).

• Points on the plane have \(z = 0\).
• Points below the plane have \(z < 0\).

This half-space concept helps visualize and solve problems involving sections of space.

set notation

Set notation is a mathematical language used to describe collections of objects or numbers that satisfy certain conditions. In the context of this exercise, set notation is used to represent the collection of points forming the half-space below and on the XY-plane.
The set can be written as \(\{(x, y, z) | z \leq 0\}\). This notation provides a concise way to specify that we're interested in all points \(x, y, z\) where the condition \(z \leq 0\) holds.

• The curly braces \(\{ \}\) denote the set.
• The vertical bar \(|\) means "such that" or "where".
• The expression to the right of the bar gives the condition on the elements of the set.

Mastering set notation is key to writing and understanding mathematical solutions succinctly.

inequality notation

Inequality notation is a way of representing conditions that show how one value relates to another. It is frequently used in mathematics to build expressions that describe sets of numbers or points. In this problem, inequality notation helps express that we are interested in points \(x, y, z\) that satisfy a specific spatial condition.
The inequality \( z \leq 0 \) is used to indicate that points can have a z-coordinate of zero or less, thereby describing the region of space on and below the XY-plane.
Understanding inequality signs is essential:

• \(\leq\): less than or equal to
• \(\geq\): greater than or equal to
• \(<\): strictly less than
• \(>\): strictly greater than

Using inequalities effectively allows for the precise definition of conditions in mathematical and real-world applications.