Problem 21 - Calculus Volume 3

Question

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$g(x, y)=\frac{x}{x+y} ; c=-1,0,2$$

Step-by-Step Solution

Answer

Level curves: \(2x + y = 0\) for \(c = -1\), \(x = 0\) for \(c = 0\), and \(y = -\frac{1}{2}x\) for \(c = 2\).

Step 1: Understanding Level Curves

A level curve for a function of two variables, such as \( g(x, y) \), is the set of points \((x, y)\) in the plane at which the function takes a constant value, \( c \). For the function \( g(x, y)=\frac{x}{x+y} \), we want to find the sets of points that make \( g(x, y)=c \). So we solve the equation \( \frac{x}{x+y} = c \).

Step 2: Solving for Level Curve with c = -1

Set the equation \( \frac{x}{x+y} = -1 \). This implies the equation \( x = -1(x+y) \). Simplify to get \( x = -x - y \). Rearranging terms, we obtain \( 2x + y = 0 \), which represents a line.

Step 3: Solving for Level Curve with c = 0

Set the equation \( \frac{x}{x+y} = 0 \). This implies the equation \( x = 0(x+y) \), which simplifies directly to \( x = 0 \). Therefore, the level curve is the line \( x = 0 \), which is the y-axis.

Step 4: Solving for Level Curve with c = 2

Set the equation \( \frac{x}{x+y} = 2 \). Solving, we have \( x = 2(x+y) \), which simplifies to \( x = 2x + 2y \). Rearranging gives \( x - 2x = 2y \), or \( -x = 2y \). Simplifying this, we have \( y = -\frac{1}{2}x \).

Key Concepts

Multivariable Functions

Multivariable functions are mathematical functions that depend on more than one variable. In simple terms, they take several inputs and return a single output. These are common in calculus where we often work with functions having two inputs, like in our example with function \( g(x, y) = \frac{x}{x+y} \).
When dealing with multivariable functions, you are considering a surface in three-dimensional space. Each point on this surface corresponds to a particular combination of \(x\) and \(y\) and gives a specific function value.
Understanding how these functions behave and interact with each variable is crucial for applications such as physics and engineering, where the relationship between multiple factors needs to be analyzed.

Equations of Curves

When constructing equations of curves for multivariable functions like \( g(x, y) = \frac{x}{x+y} \), you deal with setting the function equal to a constant. Let's consider the problem described in the exercise where one needs to find level curves of \( g(x, y) \) for various constants \( c \).
Level curves are specific types of curves that represent where this function has constant values when plotted on a 2D plane.

• For \( c = -1 \), the equation becomes \( x = -1(x+y) \), simplifying to a linear equation \( 2x + y = 0 \).
• When \( c = 0 \), it becomes \( x = 0 \), representing the y-axis.
• For \( c = 2 \), it simplifies to \( y = -\frac{1}{2}x \), another line, albeit with a different slope.

Each value of \( c \) yields a different curve and provides insights into the function's behavior at those levels.

Visualizing Functions

Visualizing functions, especially those with multiple variables, helps us grasp how variables influence function values. Often, graphing level curves provides a clear picture of a function's structure.
When graphing level curves from our example, you essentially draw lines on a plane where each line corresponds to a set value of \( c \), showcasing how function values spread across the \(x\) and \(y\) axes.

• Assessing \( 2x + y = 0 \) for \( c = -1 \) shows a line crossing the origin and inclined in a certain direction.
• The line \( x = 0 \) for \( c = 0 \) simply traces the entire y-axis.
• \( y = -\frac{1}{2}x \) for \( c = 2 \) introduces a different slope, diverging from the origin.

These visual elements provide an intuitive understanding of how the function behaves for different input combinations, aiding in deeper conceptual comprehension.