Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of \(\bar{f}\). $$ f(x, y)=6 $$

Step-by-Step Solution

Answer

The graph is a horizontal plane at \( z = 6 \).

Step 1: Understand the Given Function

The given function is a constant function, where the output value is the same for any point within the plane. In this case, the function is given as \( f(x, y) = 6 \). This means no matter what \( x \) or \( y \) is, the function's value will always be 6.

Step 2: Visualize the Graph in 3D

Since the function is of two variables \( x \) and \( y \), we can visualize it in 3D space as a surface. Each point \((x, y, z)\) on this surface will satisfy \( z = f(x, y) = 6 \). Therefore, the graph of this function is a horizontal plane parallel to the \( xy \)-plane at \( z = 6 \).

Step 3: Sketch the Graph

To sketch the graph, draw the \( x \)-axis and \( y \)-axis on a 2D plane, and the \( z \)-axis coming out perpendicularly. Then, draw a horizontal plane at a height \( z = 6 \) above the \( xy \)-plane. This entire plane represents the graph of the function \( f(x, y) = 6 \).

Key Concepts

3D graphing

3D graphing is a way to visualize functions that have three variables: two independent variables and one dependent variable. This can help us gain insight into how these variables interact and change in space. Imagine you are trying to understand the shape and position of a surface in three-dimensional space. This is what 3D graphing can illustrate.
In 3D space, we use three axes:

• The x-axis (usually running horizontally)
• The y-axis (running horizontally perpendicular to the x-axis)
• The z-axis (coming out perpendicular to both the x and y axes, representing height or depth)

When you have a function like \[ f(x, y) = z \], you're graphing a surface. Each point on the surface aligns with a particular value of \( z \).
In our exercise, the graph is a simple plane, which makes our job a lot easier! By plotting this plane in the 3D graph, you can easily visualize the relationship between the variables and better understand the entire function.

constant functions

A constant function is a unique type of function where the output value remains the same regardless of the input. For any inputs \( x \) and \( y \), the output of the function stays fixed. In the exercise, we have the function \( f(x, y) = 6 \). Here, no matter what values \( x \) and \( y \) take, the output will continuously be 6. This creates a uniform and simple structure—a flat, horizontal plane.
Constant functions are easy to identify and graph due to their stability and predictability. This means we don't have to worry about sudden changes or irregular behavior in the function's output. In essence, once you've identified a constant function, an entire class of problems becomes straightforward to solve.
These types of functions serve as excellent examples when learning about functions in higher dimensions because they simplify into easy visual models that help build a foundational understanding of more complex multivariable functions later on.

visualization in 3D space

Visualization in 3D space helps transform abstract mathematical concepts into something tangible and easier to comprehend. By viewing a function's output within three dimensions, we can not only see numerical relationships but also gain intuitive knowledge about how these components fit together.
Considering the function \( f(x, y) = 6 \), visualizing it in 3D space allows us to literally "see" that all points \( (x, y, 6) \) form a flat floor at level 6 above the x-y plane.

• This entails that any scenario, problem, or real-world phenomenon modeled by this function will be restricted to this plane.
• Understanding its geometry becomes straightforward: no peaks, no valleys, just a flat expanse.

Such visualizations empower learners to break down complex relationships into more manageable perceptions. Once you grasp this, you can more confidently approach intricate functions that introduce curves, slopes, and more variable interactions. By practicing with constant functions and their simple graphs, learners lay a solid foundation for the more challenging aspects of multivariable calculus.