Problem 18 - University Calculus: Early Transcendentals

Question

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. a. \(0 \leq x \leq 1\) b. \(0 \leq x \leq 1, \quad 0 \leq y \leq 1\) c. \(0 \leq x \leq 1, \quad 0 \leq y \leq 1, \quad 0 \leq z \leq 1\)

Step-by-Step Solution

Answer

(a) Line segment; (b) Square; (c) Cube.

Step 1: Describe the Set for Part (a)

The inequality \(0 \leq x \leq 1\) represents all points along the x-axis that lie between \(x = 0\) and \(x = 1\), inclusive. In other dimensions (y and z), there are no restrictions. Therefore, this set describes a line segment on the x-axis extending from 0 to 1.

Step 2: Describe the Set for Part (b)

The inequalities \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\) define a region in the xy-plane. It includes all points that lie within or on the boundaries of a square where both x and y range from 0 to 1. Since there is no restriction on z, this set extends vertically through the entire z-axis.

Step 3: Describe the Set for Part (c)

The inequalities \(0 \leq x \leq 1\), \(0 \leq y \leq 1\), and \(0 \leq z \leq 1\) define a cube in 3D space. The cube lies in the positive octant and includes all points where x, y, and z each independently range between 0 and 1.

Key Concepts

Inequalities in Equations

Inequalities are mathematical expressions that show the relationship between two values, where one is either smaller or larger than the other, instead of being equal. In the given exercise, the inequalities such as \(0 \leq x \leq 1\) represent a range that the variable \(x\) can take. This inequality indicates that \(x\) can be any value between 0 and 1, including the endpoints.

• Closed Intervals: The use of "\(\leq\)" indicates a closed interval, meaning the boundary values themselves are included.
• Variable Freedom: When only one variable is restricted, others maintain their freedom, affecting how we interpret the shape or region.

These inequalities are foundational in solving problems within coordinate systems, helping define specific regions or segments in space.

Geometric Interpretation

The geometric interpretation of inequalities is the visualization of these constraints within a coordinate system. In part (a), \(0 \leq x \leq 1\) defines a line segment along the x-axis from 0 to 1. There are no limits on the y or z-axes, creating an infinite plane in those directions.For part (b), the inequalities \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\) constrains both x and y, which outlines a square in the xy-plane. This region extends infinitely along the z-axis because no bounds are given for z.In part (c), when the inequalities \(0 \leq x \leq 1\), \(0 \leq y \leq 1\), and \(0 \leq z \leq 1\) all apply together, they describe a cube in 3D space. All axes are limited similarly, forming a defined, finite 3-dimensional shape. This cube is a direct consequence of limiting x, y, and z to values between 0 and 1, enshrining all points within that range.

Graphing Inequalities

Graphing inequalities is a practical skill that aids in visualizing the regions defined by those inequalities. Each inequality corresponds to a region in the space it is applied to. Here's how it's interpreted:

• 1D Graph: In one dimension, inequalities like \(0 \leq x \leq 1\) are represented as line segments on an axis.
• 2D Graph: For two variables, such as \(x\) and \(y\), the inequalities form shapes like squares or rectangles in a plane. The area included aligns with the inequality's bounds.
• 3D Graph: Introducing a third variable adds a new dimension, pulling a 2D shape into a 3D volume, like a cube in this exercise.

To graph these, you start by plotting each point determined by the inequalities' bounds and then connecting them to form the complete shape. Exploring graphing tools or software can simplify the process, allowing real-time visualization of the inequalities and how they form recognizable geometric shapes.