Problem 18 - Thomas Calculus

Question

In Exercises \(17-24,\) describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ \begin{array}{ll}{\text { a. } 0 \leq x \leq 1} & {\text { b. } 0 \leq x \leq 1, \quad 0 \leq y \leq 1} \\ {\text { c. } 0 \leq x \leq 1,} & {0 \leq y \leq 1, \quad 0 \leq z \leq 1}\end{array} $$

Step-by-Step Solution

Answer

a. Line segment on x-axis; b. Infinite plane in xy space; c. Unit cube in 3D space.

Step 1 - Understanding Inequality a

The inequality \(0 \leq x \leq 1\) describes the set of points in space where the x-coordinate ranges from 0 to 1. This forms a line segment along the x-axis, stretching from (0, 0, 0) to (1, 0, 0). There are no restrictions on the y and z coordinates.

Step 2 - Analyzing Inequality b

The inequalities \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\) define a region in the xy-plane. This is a square with vertices at (0, 0), (0, 1), (1, 0), and (1, 1). Since there is no restriction on the z-coordinate, the set of points are located on an infinite plane parallel to the xy-plane, extending above and below it.

Step 3 - Explaining Inequality c

The inequalities \(0 \leq x \leq 1\), \(0 \leq y \leq 1\), and \(0 \leq z \leq 1\) define a three-dimensional cube. This cube has its lower corner at (0, 0, 0) and its upper corner at (1, 1, 1). It includes all points within and on the surface of this cube, which exists entirely within the first octant of the coordinate system.

Key Concepts

Inequalities

Inequalities are mathematical expressions that describe the relative size or order of two values. They are crucial in defining ranges of values for different variables.
When we express inequalities, such as \(0 \leq x \leq 1\), we're describing all the possible values for \(x\) that lie between 0 and 1, inclusive.
This means that 0 and 1 are included in the set of solutions for \(x\).

• In coordinate geometry, inequalities help to define regions in space, such as lines, planes, and volumes, by specifying the limits of one or more dimensions.
• Multiple inequalities can be combined to describe complex shapes and regions. For instance, intersecting multiple ranges on the \(x\), \(y\), and \(z\) axes forms bounded regions in 2D or 3D space.

It's useful to visualize inequalities as constraints or boundaries that limit the expanse of a solution.
In the context of 3D coordinates, they often help articulate the boundaries of geometric shapes, like line segments or cubes.

3D Shapes

Three-dimensional shapes (3D shapes) are geometric figures that have length, width, and height. They exist in a three-dimensional space and can be described using three coordinates.
In the exercise, the inequalities define different stages of 3D shapes:

• A *line segment* on the x-axis entails fixing one dimension while allowing freedom in the others.
• A *plane* in the xy-region establishes a surface with fixed x and y boundaries, open to movement in the z direction.
• A *cube* in 3D is defined by fixed x, y, and z boundaries, giving it a bounded, finite volume.

Each point in a 3D shape can be identified with a triplet (x, y, z), representing its position along the Cartesian axes.
Visualizing these shapes geometrically can help understand how inequalities can determine their structures.
This visualization also aids in recognizing how adjustments in inequalities shift or transform the corresponding geometric shape.

Cartesian Coordinate System

The Cartesian Coordinate System is a mathematical framework that enables the precise description of points in space using coordinates.
This system uses horizontal and vertical lines, called axes, to establish a reference grid in 2D or 3D.

• The 2D Cartesian plane consists of an x-axis (horizontal) and a y-axis (vertical), forming a flat surface on which any point can be plotted.
• In 3D, a z-axis is added, perpendicular to both the x and y-axes, allowing for the plotting of points in space.
• Each point in space is defined by its coordinates: \((x, y)\) in 2D or \((x, y, z)\) in 3D, showing its position relative to the origin \((0, 0, 0)\).

Using the Cartesian system, it becomes easier to define shapes and solve problems involving geometry and algebra.
In exercise problems similar to the given one, the use of inequalities in conjunction with the Cartesian system allows for the precise description and visualization of shapes and regions in 3D space.