Problem 24 - 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. $$ \begin{array}{l}{\text { a. } z=1-y, \quad \text { no restriction on } x} \\\ {\text { b. } z=y^{3}, \quad x=2}\end{array} $$

Step-by-Step Solution

Answer

Equation a represents a plane; Equation b forms a cubic curve in the plane where \( x = 2 \).

Step 1: Analyzing Equation a

The equation given is \( z = 1 - y \) with no restriction on \( x \). This represents a plane in 3-dimensional space. Since there is no constraint on the value of \( x \), for every point that satisfies \( z = 1 - y \), \( x \) can take any real number value. This means the plane extends infinitely in the x-direction.

Step 2: Describe Equation a Set of Points

The set of points represented by the equation \( z = 1 - y \) forms a plane inclined with respect to the \( yz \)-plane. Any point \( (x, y, z) \) satisfying this equation has coordinates that fulfill the condition \( z \) is exactly one unit less than \( y \), for any real number \( x \).

Step 3: Analyzing Equation b

The given conditions include the equation \( z = y^3 \) and a fixed plane where \( x = 2 \). Setting \( x = 2 \) indicates that we are examining this equation in a vertical plane where \( x \) does not change.

Step 4: Describe Equation b Set of Points

Within the vertical plane at \( x = 2 \), the points satisfy \( z = y^3 \). This indicates a curve in this plane because as \( y \) changes, \( z \) changes according to the cubic relationship. The set of points forms a cubic curve confined to the plane where \( x = 2 \).

Key Concepts

Plane Equation

In 3D geometry, a plane can be thought of as a flat, two-dimensional surface that extends infinitely in every direction within its own plane. It is typically given by an equation involving three variables, like the one in the exercise:

• Equation: The exercise uses the equation \( z = 1 - y \) without restriction on \( x \).
• Implication: This equation tells us that for any chosen \( y \) value, \( z \) is exactly one less. Since \( x \) has no constraint, the plane can extend infinitely along the \( x \)-axis, making it span across the whole 3D space in one dimension.

This plane exists everywhere the condition \( z = 1 - y \) holds true. It creates a surface inclined relative to the \( yz \)-plane. Understanding and visualizing this can help in perceiving complex spatial relationships.

Cubic Curve

A cubic curve in 3D geometry arises when there is a cubic relationship between the coordinates, often looking something like \( z = y^3 \). This kind of relationship gives rise to a curve rather than a flat plane.

• Given Equation: For \( z = y^3 \) with \( x = 2 \), the cubic relationship is confined to a specific plane where \( x \) is constantly 2.
• Visual Representation: As \( y \) changes, \( z \) varies following the cube of the \( y \) value, creating a curved line within the plane.

This means on the plane where \( x = 2 \), we get a dynamic, curving graph representing how \( z \) depends on \( y \). Understanding cubic curves helps in visualizing complex forms not easily represented by straight lines or flat surfaces.

Coordinates Analysis

Coordinates analysis in 3D space involves understanding how equations define surfaces, curves, and shapes using three values—typically \( x \), \( y \), and \( z \).

• Plane Analysis: When looking at equations like \( z = 1 - y \), observe how each variable affects the other. \( z \) always being one less than \( y \) with unrestricted \( x \) leads to a plane.
• Curve Analysis: In \( z = y^3 \) and \( x = 2 \), exploring how \( z \) is a function of \( y \) while \( x \) stays constant allows visualization of a curved trajectory.

By breaking down how each coordinate varies or remains fixed, one gains deeper insight into the geometric structure and behavior of points in space. This kind of analysis is fundamental in 3D geometry, opening up ways to model and understand physical shapes and paths in the real world.