Problem 30 - Calculus 3

Question

For the following exercises, find the vertical traces of the functions at the indicated values of \(x\) and \(y,\) and plot the traces. $$ z=4-x-y ; x=2 $$

Step-by-Step Solution

Answer

The vertical trace at \(x=2\) is the line \(z = 2 - y\) in the \(yz\)-plane.

Step 1: Identify Given Equation

The given function is \( z = 4 - x - y \). We need to find vertical traces at the specified values of \(x\) and \(y\). First, let's substitute \(x = 2\) into the equation.

Step 2: Substitute for x=2

Substituting \(x = 2\) into the equation \(z = 4 - x - y\), we have:\[ z = 4 - 2 - y \] Simplifying gives us:\[ z = 2 - y \].

Step 3: Plotting the Vertical Trace

The equation \( z = 2 - y \) represents a line in the \(yz\)-plane. - When \( y = 0 \), \( z = 2 - 0 = 2 \).- When \( z = 0 \), \( 2 - y = 0 \) which implies \( y = 2 \).These points (\(0, 2\)) and (\(2, 0\)) can be plotted in the \(yz\)-plane. Draw a line through these points.

Key Concepts

Multivariable Functions

A multivariable function is a type of mathematical function that involves more than one input variable. Unlike single-variable functions that have a single independent variable and one dependent variable, multivariable functions map two or more input variables to a single output. For example, in the function \( z = 4 - x - y \), \( x \) and \( y \) are the input variables that determine the value of \( z \), which is the output. This is a simple linear example with two variables, but multivariable functions can be more complex with many variables and intricate relationships.
Understanding multivariable functions involves examining how changes in one variable affect others. This is commonly done through different techniques like partial differentiation, which can show how the function changes with respect to one variable while keeping others constant. Another useful technique is finding traces, such as vertical traces, which help visualize the function on specific planes by keeping one variable constant.

Coordinate Planes

Coordinate planes are the two-dimensional spaces where we plot the graphs of functions. For multivariable functions, coordinate planes like the \( xy \)-plane, \( yz \)-plane, and \( xz \)-plane are essential for visualizing different aspects of the function. Each plane shows a different view based on which variables are being plotted.

• In the \( xy \)-plane, we generally plot points where \( z \) is constant.
• The \( yz \)-plane is where we observe how \( z \) changes with \( y \) when \( x \) is fixed, as seen in vertical traces.
• The \( xz \)-plane similarly observes how \( z \) varies with \( x \) when \( y \) is constant.

The use of coordinate planes simplifies the process of understanding multivariable functions by breaking down the function into more familiar 2D relations, making the visualization and analysis more manageable.

Plotting Graphs

Plotting graphs for multivariable functions can be tricky, but it becomes easier when broken down into simpler steps using techniques like tracing. Plotting vertical traces is one such method that aids in simplifying 3D data into 2D representations. For example, when we set \( x = 2 \) in the function \( z = 4 - x - y \), we derive the equation \( z = 2 - y \).
This equation is plotted in the \( yz \)-plane as a straight line, marking points where specific values of \( y \) correspond to specific values of \( z \). The typical steps to follow include:

• Identify key points by setting \( y \) and \( z \) to 0 alternately.
• Plot these points on the respective coordinate plane.
• Draw a line through the plotted points to represent the trace.

Visualization using graphs not only helps in understanding the behavior of the function but also aids in interpreting how variables influence the outcome within a defined space.