Problem 22 - Calculus Early Transcendentals

Question

Use what you learned about surfaces in Section 1 to sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function. $$f(x, y)=3 x-6 y+18.$$

Step-by-Step Solution

Answer

Answer: The graph of the function $$f(x, y) = 3x - 6y + 18$$ represents a plane. The domain of the function is $$(-\infty, \infty) \times (-\infty, \infty)$$ for $$x$$ and $$y$$, and the range is $$(-\infty, \infty)$$ for $$f(x, y)$$.

Step 1: Rewrite the equation in standard form

Rewrite the given function $$f(x, y) = 3x - 6y + 18$$ as follows: $$3x - 6y = -18 + f(x, y)$$ Now, divide by 3 to simplify the equation: $$x - 2y = -6 + \frac{1}{3}f(x, y)$$

Step 2: Identify the surface type and sketch the graph

The equation $$x - 2y = -6 + \frac{1}{3}f(x, y)$$ represents a plane in the 3-dimensional space. Let's rewrite the equation as $$\frac{1}{3}f(x, y) = x - 2y + 6$$ to clearly see that $$f(x, y)$$ is a linear expression of $$x$$ and $$y$$. A plane is a flat, two-dimensional surface that extends infinitely far in all directions. To sketch the graph, we can find some points on the plane that satisfy the equation and then connect them to visualize the surface. Let's find the points where the plane intersects the coordinate axes: 1. When $$x = 0$$ and $$y = 0$$, we have $$f(0, 0) = 18$$. 2. When $$y = 0$$ and $$f(x, y) = 0$$, we have $$x = 2$$. 3. When $$x = 0$$ and $$f(x, y) = 0$$, we have $$y = -3$$. 4. When $$y = 0$$ and $$x = -2$$, we have $$f(-2, 0) = -18$$. Plot these points, and connect them to visualize the plane.

Step 3: Determine the domain and range

The domain of a function is the set of all possible input values (in this case, $$x$$ and $$y$$) that result in a valid output value (in this case, $$f(x, y)$$). Since the function represents a plane that extends infinitely far in all directions, the domain of the function is $$(-\infty, \infty) \times (-\infty, \infty)$$ for $$x$$ and $$y$$. The range of a function is the set of all possible output values (in this case, $$f(x, y)$$) that result from input values in the domain. Since the plane extends infinitely far in all directions, the range of the function is also $$(-\infty, \infty)$$ for $$f(x, y)$$. In conclusion, the graph of the function $$f(x, y) = 3x - 6y + 18$$ is a plane, with the domain $$(-\infty, \infty) \times (-\infty, \infty)$$ for $$x$$ and $$y$$, and the range $$(-\infty, \infty)$$ for $$f(x, y)$$.

Key Concepts

plane

A plane in mathematics is considered a flat, two-dimensional surface. The equation of a plane takes a linear form, which often has a variable relationship in three-dimensional space. Here, each point on the plane is described by three coordinates:

• (x, y, f(x, y))

For example, the function given, \(f(x, y) = 3x - 6y + 18\), is a linear equation because there are no squared terms involving \(x\) or \(y\).
This means the surface represented by this function is a plane.
Planes can extend infinitely in all directions within their dimension, illustrating why they are described as two-dimensional.
A plane does not bend or curve, maintaining a constant slope throughout. When graphing such a function, choose a few strategic points to help visualize the flat surface it forms.
By connecting these points in a three-dimensional space, you will find the infinite stretch of the plane that forms from the simple, linear equation.

domain and range

In multivariable calculus, understanding the domain and range of a function is crucial. The domain refers to all the possible input values, represented often as coordinates in a space.

• For the function \(f(x, y) = 3x - 6y + 18\), both \(x\) and \(y\) can take on any real number since there are no restrictions such as square roots or denominators that would limit the inputs.

As a result, the domain of \(f(x, y)\) is the entire set of real numbers across the two axes, expressed as \((-fty, fty) \times (-fty, fty)\).
The range represents all potential output values the function can produce.
In our linear plane, just like the domain, the outputs span all real numbers without bounds, giving the range \((-fty, fty)\).
Understanding these concepts helps in creating accurate graphical representations and predictions about behaviors of multivariable functions.

graphing functions

Graphing functions in three-dimensional space is a fundamental skill in multivariable calculus. Doing so allows us to capture the behavior and properties of functions visually.
Initially, we identify the kind of surface the equation represents; in this case, our function \(f(x, y) = 3x - 6y + 18\) forms a plane.
To illustrate this, start by plotting strategic points that the plane passes through.

• For example, when \(x = 0\) and \(y = 0\), the function results in \(f(x, y) = 18\), giving a point at (0, 0, 18).
• Continue with different combinations, such as setting \(y = 0\) with varying \(x\), or vice versa.

Connecting these points helps form the angled surface of the plane, extending outward in every direction.
This hands-on graphing process gives a sense of the three-dimensional space these equations inhabit, solidifying understanding and providing a clear view of multivariable relationships.
Practice is essential, using simple plots to gradually understand more complex dimensional interactions.