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Chapter 9: Problem 21

Sketch the graph of the function. \(f(x, y)=3\)

Short Answer

Expert verified
The graph is a horizontal plane at \( z = 3 \) parallel to the \( xy \)-plane.

Step by step solution

01

Understand the Function

The function given is a function of two variables, denoted as \( f(x, y) = 3 \). This means for every point \( (x, y) \), the function value \( f(x, y) \) equals 3. Our goal is to sketch this in a three-dimensional space.
02

Recognize the Graph Type

The equation \( f(x, y) = 3 \) represents a plane, specifically a horizontal plane. Since \( z = 3 \) for all values of \( x \) and \( y \), it implies that the plane is parallel to the \( xy \)-plane and is located at a height of 3 above it.
03

Sketch the Axes in 3D Space

Draw three perpendicular axes to represent a three-dimensional coordinate system. Typically, the horizontal axis represents \( x \), the vertical one represents \( z \), and the axis coming out of the page (or turning into paper) represents \( y \).
04

Draw the Horizontal Plane

Since the function defines a plane at \( z = 3 \), sketch a horizontal line parallel to the \( xy \)-plane at \( z = 3 \). Imagine or draw a large, flat sheet that spans infinitely along the \( x \) and \( y \) directions at this height.
05

Caption and Label the Sketch

Label the axes as \( x \), \( y \), and \( z \). Indicate the height of the plane as \( z = 3 \) on the \( z \)-axis. Ensure that the plane is clearly marked, either using shading or parallel lines to distinguish it from the coordinate grid itself.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3D graphing
Three-dimensional graphing is a way to visually represent functions involving variables that define coordinates in three-dimensional space.
  • In 3D graphing, we typically deal with three axes: the x-axis, the y-axis, and the z-axis.
  • These axes are mutually perpendicular to each other, creating a coordinate system that lets us locate points in space.
When graphing a function of two variables, such as our equation \( f(x, y) = 3 \), you can think of it as a surface or a plane in this 3D space.
This specific function results in a flat plane, which we'll discuss in further detail in the next section.
horizontal plane
A horizontal plane in three-dimensional space is a plane that extends infinitely in two directions, forming a flat surface parallel to the ground.
In our example, the function \( f(x, y) = 3 \) describes a horizontal plane. This means it does not slope upward or downward but sits flat.
  • The function equals a constant value, 3, so for every combination of \( x \) and \( y \), the height, \( z \), remains at 3.
  • It is parallel to the \( xy \)-plane, indicating it has no incline.
This plane can be visualized as a blanket spread infinitely on a table, standing precisely 3 units above the \( xy \)-plane.
function of two variables
Functions of two variables are mathematical expressions where two variables interact to produce a third value: typically a height or depth.
  • In our function \( f(x, y) = 3 \), both \( x \) and \( y \) are independent variables, meaning they can take any value without restriction from each other or \( z \).
  • The resulting value, 3, is constant, showing that any point \( (x, y) \) on our graph results in a plane at a constant height.
These functions are fundamental in multivariable calculus as they allow us to explore how surfaces behave and change in 3D space.
For our exercise, it's important to understand that such a function creates a plane rather than a curve, revealing relationships in more than one direction.

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Most popular questions from this chapter

Determine the set of points at which the function is continuous. \(G(x, y)=\ln \left(x^{2}+y^{2}-4\right)\)

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Find the first partial derivatives of the function. $$w=\ln (x+2 y+3 z)$$
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