Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 7-16, sketch the graph of \(f\). $$ f(x, y)=6 $$

Step-by-Step Solution

Answer

Graph is a plane parallel to the xy-plane at z=6.

Step 1: Understand the Function

The function given is a constant function, where \(f(x, y) = 6\). This means that no matter what values \(x\) and \(y\) take, the function \(f(x, y)\) will always equal 6.

Step 2: Recognize the Graph Type

Since \(f(x, y) = 6\) is a constant function, the graph of this function in three-dimensional space is a plane parallel to the \(xy\)-plane.

Step 3: Determine the Position of the Plane

The plane represented by \(f(x, y) = 6\) is positioned such that all points on the plane have a \(z\)-value of 6, i.e., \((x, y, 6)\). This plane lies parallel to and 6 units above the \(xy\)-plane.

Step 4: Sketch the Graph

To sketch the graph, draw a plane parallel to the \(xy\)-plane. Mark this plane at \(z = 6\) to illustrate that every point on this plane has a \(z\)-coordinate of 6.

Key Concepts

Understanding Constant Functions

A constant function in multivariable calculus is quite simple yet powerful. It is defined as a function where the output value is the same, no matter the input values. Mathematically, if \(f(x, y) = c\), then it means that for any value of \(x\) and \(y\), the result of the function is always the constant \(c\). This remains true regardless of changes in \(x\) or \(y\).
There are key characteristics of constant functions to remember:

• Their graphs are always flat surfaces in 3D space.
• The output value, \(c\), is repeated uniformly throughout the input domain.

These functions offer a straightforward introduction to more complex multivariable calculus concepts, serving as a stable base without the complications of dependent variable changes.

The Basics of 3D Graphing

Graphing in three dimensions adds depth and complexity to visualizations. Traditional graphs are often two-dimensional, with axes that accommodate two variables. However, 3D graphs introduce a third axis, typically the \(z\)-axis, which adds another layer to our understanding.
Key aspects of 3D graphing include:

• The three axes, \(x\), \(y\), and \(z\), each represent a different dimension.
• Every point in the graph is represented by a coordinate set \((x, y, z)\).
• 3D graphs are useful for representing complex relationships between variables.

By mastering 3D graphing, you can visualize how functions behave in spaces beyond the usual two-dimensional view. Such graphs help clarify the behavior of functions like the constant function \(f(x, y) = 6\), where the output is represented as a static plane level in the 3D space.

Exploring Planes in Three Dimensions

In multivariable calculus, a plane is a two-dimensional surface extending infinitely in three dimensions. When dealing with constant functions, such as \(f(x, y) = 6\), the graph forms a horizontal plane.
Details about planes in 3D include:

• A plane is determined by a fixed coordinate, such as a constant \(z\)-value in the function \(f(x, y) = z\).
• Planes are usually parallel to one of the primary coordinate planes: \(xy\), \(xz\), or \(yz\).
• Every point on a plane has a consistent coordinate value along the axis corresponding to the function's constant value.

For the function \(f(x, y) = 6\), we see a plane that is parallel to the \(xy\)-plane. It's specifically positioned six units above the \(xy\)-plane, indicating that wherever you go on this plane, the \(z\)-value remains consistently 6. Understanding how planes fit into three-dimensional graphs is key to visualizing multivariable functions properly.