Problem 23 - Calculus Early Transcendentals

Question

Sketch the graph of the function. $$f(x, y)=10-4 x-5 y$$

Step-by-Step Solution

Answer

The graph of \(f(x, y) = 10 - 4x - 5y\) is a plane, slanting downward due to negative coefficients.

Step 1: Understand the function type

The function given is of the form \(f(x, y) = c - ax - by\), which is a plane in three-dimensional space. This means that the graph is a flat surface.

Step 2: Find the intercepts

To find the intercepts, set \(x\) and \(y\) to zero one at a time.\- When \(x = 0\) and \(y = 0\), \( f(0, 0) = 10 \). \- For the \(x\)-intercept, set \(y = 0\), so \(0 = 10 - 4x\). Solving this gives \(x = 2.5\). \- For the \(y\)-intercept, set \(x = 0\), so \(0 = 10 - 5y\). Solving this gives \(y = 2\). \These intercepts help to plot the plane.

Step 3: Set the function for level curves

To see how the surface behaves, consider setting \( f(x, y) = k \) where \( k \) is a constant. This creates a level curve equation: \(4x + 5y = 10 - k\), which is a line for each value of \( k \).

Step 4: Plot key level curves

Consider \(k = 0\), \(k = 5\), and \(k = 10\) to create level curves.\- For \(k = 0\), the line is \(4x + 5y = 10\). \- For \(k = 5\), it becomes \(4x + 5y = 5\). \- For \(k = 10\), it's \(4x + 5y = 0\). \Plot these lines in the xy-plane as part of visualizing the graph's elevation.

Step 5: Sketch the three-dimensional graph

Using the intercepts and level curves, sketch a 3D surface. The intercepts give the boundary points on the \(z = 0\) plane. The level curves show slope, illustrating the surface tilts downward in both the \(x\) and \(y\)-directions due to the negative coefficients \(-4\) and \(-5\) respectively.

Key Concepts

Three-Dimensional Plane

The concept of a three-dimensional plane can be compared to a flat sheet of paper floating in space. In this context, the function given, \( f(x, y) = 10 - 4x - 5y \), represents a flat surface within a three-dimensional coordinate space. Instead of relying solely on \(x\) and \(y\), this function introduces a third dimension, \(z = f(x,y)\), to create depth.
This plane is defined by constants \(-4\) and \(-5\), which dictate the slope's steepness in the \(x\) and \(y\) directions. No matter where you are on this surface, the slope remains consistent, unlike curved surfaces in three-dimensional space. The function therefore describes a perfectly straight, endless sheet.
Understanding this flatness is crucial, as it indicates that any section of this plane between levels can be assembled by parallel lines. Imagining this plane helps you visualize how such surfaces function and interact with the axes.

Level Curves

Level curves are a helpful tool to showcase changes in height within a three-dimensional graph. These are lines within the \(xy\)-plane that show where the function \( f(x, y) \) is constant. By setting the function equal to a constant, \( f(x, y) = k \), we obtain equations like \(4x + 5y = 10 - k\). Each substitution of \(k\) results in a different line representative of altitude on the graph.
These lines present a unique way to solve real-world problems by imagining slices through the 3D shape at constant heights. When multiple level curves are plotted, they provide a preview of the changes in elevation across the surface. Visualizing a series of these helps clarify how the surface looks between values and can show the gradient or slope of the plane.
Selecting various values for \(k\) lets you understand how these cuts allow visualization of different heights and slopes of the three-dimensional plane.

Intercepts

Intercepts are points where the plane intersects the axes, helping us anchor and understand the orientation of the plane in three-dimensional space. For the function \( f(x, y) = 10 - 4x - 5y \), we identify these by setting either variable to zero and solving for the others.
When both \(x = 0\) and \(y = 0\), \( f(0, 0) \) equals 10, providing the \(z\)-intercept. This is where the plane crosses the \(z\)-axis.

• To find the \(x\)-intercept, set \(y = 0\). This results in \(x = 2.5\).
• For the \(y\)-intercept, set \(x = 0\) to find \(y = 2\).

These intercepts form essential points needed to plot the plane in the three-dimensional space. They grant insight into specific locations where the surface meets or crosses axes, giving practical starting points for graphical representation.

Graphing Techniques

In order to graph a function like \( f(x, y) = 10 - 4x - 5y \), a combination of intercepts and level curves comes into play. First, establish the intercepts; these provide boundary points helpful in sketching the plane.
Then, utilize level curves by setting \(k\) values to distinct numbers and drawing these curves on the \(xy\)-plane. Consider the lines such as \(4x + 5y = 10, \;\) \(4x + 5y = 5, \) and \(4x + 5y = 0\). These straight lines allow students to grasp plane orientation and gradient.
Finally, bring together all this data to visualize the 3D graph. By locating points from intercepts and learning the plane's slope from curves, sketching an accurate three-dimensional representation of the function becomes possible. Thus, graphs serve as a straightforward way to comprehend three-dimensional functions and enrich learning by making the visual component simple to digest.