Question

In Exercises 27-32, use the function $$f(x, y)=3-\frac{x}{3}-\frac{y}{2}$$ Sketch the graph of \(f\) in the first octant and plot the point (3,2,1) on the surface.

Short answer

The function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\) is represented in the first octant as a descending plane from the point (0,0,3). The specific point (3,2,1) lies on this plane and is represented within the three dimensional space.

Step-by-step solution

Understand the function

The first thing one needs to do is understand the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\). This function gives the z-coordinate for every point (x,y), and therefore represents a surface in the 3-dimensional space.

Evaluate the function for various (x,y) pairs

Start at (0,0) and evaluate the function for different values of x and y in the first octant. Remember that in the first octant both x and y are positive. For example, for (x, y) = (0, 0), the function evaluates to \(f(0,0) = 3\). For (x, y) = (3, 0), the function evaluates to \(f(3,0) = 2\). For (x, y) = (0, 2), the function evaluates to \(f(0,2) = 2\). Continue this process to generate several points.

Plot the function

With several points in hand, plot the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\) in 3D space. Place x on the horizontal axis, y on the vertical axis, and z going up. Draw the 2D contours of the function, i.e. lines were the function assumes constant values. This gives a clearer picture of the function and shows its variation in the first octant.

Plot the specific point

Finally, plot the specific point (3,2,1) by going over 3 units on the x-axis, up 2 on the y-axis and up 1 on the z-axis. Mark this point on the surface of the graph.

Key concepts

Multivariable Functions

Multivariable functions, as the name suggests, are mathematical functions that have more than one independent variable. In the context of our problem, we are considering a function of two independent variables, namely x and y. The function given is

\[ f(x, y)=3-\frac{x}{3}-\frac{y}{2} \]
This type of function is essential in visualizing phenomena in many fields such as engineering, physics, and economics where the output depends on multiple factors.

When dealing with multivariable functions, each combination of input values (x, y) gives a unique output z, which can be represented as a point in 3D space. To fully understand these functions, it's critical to explore their graphical representations, which lead us to the concept of 3D surface plotting.

Function Graphing

Graphing is a powerful tool for visualizing the behavior of functions. For multivariable functions, graphing enables us to see the form of the surface they create in three dimensions. The function \[ f(x, y)=3-\frac{x}{3}-\frac{y}{2} \]
, represents a plane in 3D space, and each point on this plane corresponds to a specific set of (x, y, z) coordinates.

By evaluating the function at various points, as we saw in the step-by-step solution, we build up a collection of points that define the shape of the function. These points, when connected, form a surface. Typically, this is done using either computer software or by carefully plotting points and drawing contours by hand on graph paper. It's crucial, especially when first learning, to take the time to plot several points to understand how the function behaves across different areas of the x-y plane.

First Octant

The first octant in a three-dimensional coordinate system is where all three coordinates (x, y, z) are positive. In other words, it is a sub-section of the 3D space that we are particularly interested in this exercise.

For functions like \[ f(x, y)=3-\frac{x}{3}-\frac{y}{2} \], plotting the graph in the first octant allows us to focus on the part of the graph where x and y contribute positively to the z-value. This can simplify our understanding of the graph's behavior since we are limiting our view to where the function increases or decreases as x and y increase. Remember, in the context of the exercise, to verify if and how the surface intersects with the axes in the first octant, as this can provide additional insights into the function's geometry.

3D Space Coordinates

The 3D space coordinates refer to a system where any point can be located by three numbers, each representing a position along the x-axis (horizontal), y-axis (vertical), and z-axis (perpendicular to both x and y). This system is crucial for plotting the surface created by our multivariable function.

When given a point like (3,2,1), it is important to interpret it correctly: move 3 units along the x-axis, 2 units along the y-axis, and then 1 unit up along the z-axis. This coordinate system is fundamental in geometry, physics, engineering, and computer graphics, helping to describe the structure of objects and spaces in the physical and virtual worlds.

For students, understanding how to navigate and plot points in this system is a foundational skill that will be applicable in a wide range of problems and practical applications.