Problem 17 - Calculus 3

Question

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ g(x, y)=4-x-y ; c=0,4 $$

Step-by-Step Solution

Answer

Level curves: for \(c=0\) is \(x+y=4\); and for \(c=4\) is \(x+y=0\).

Step 1: Understand level curves

Level curves of a function are the set of points \(x, y\) where the function takes a constant value \(c\). For a function \(g(x, y)=4-x-y\), we are asked to find the level curves at \(c=0\) and \(c=4\).

Step 2: Set up equation for level curve at c=0

For \(c=0\), set \(g(x, y) = 0\). Therefore, solve the equation \(4-x-y = 0\). Simplifying, we get: \(-x - y = -4\) or \(x + y = 4\).

Step 3: Set up equation for level curve at c=4

For \(c=4\), set \(g(x, y) = 4\). Therefore, solve the equation \(4-x-y=4\). Simplifying, we get: \(-x - y = 0\) or \(x + y = 0\).

Step 4: Plot the level curves

The level curve for \(c=0\) is the line \(x+y=4\), and the level curve for \(c=4\) is the line \(x+y=0\). These are straight lines by combining all \(x, y\) that satisfy these equations.

Key Concepts

Functions of Two Variables

A function of two variables is like a formula that takes two inputs, often named \(x\) and \(y\), and gives back a result. You can think of it as a machine where you put in \(x\) and \(y\) and get out some number.
These functions are often written as \(g(x, y)\), where \(g\) signifies the function itself, and \(x\) and \(y\) are the variables. They help us understand how two different elements can work together to produce outcomes.
Functions of two variables are very powerful because they allow mathematicians and scientists to model real-world relationships, like the temperature at different places (latitude and longitude) on Earth.
Let's look at the function from the exercise, \(g(x,y)=4-x-y\). Here, for each combination of \(x\) and \(y\), the function computes the value of \(4-x-y\). This specific function can be visualized with level curves, which we'll explore next.

Constant Value

A constant value in the context of level curves is the fixed output the function provides. Imagine you are tuning a radio; when perfectly tuned, the sound stays the same no matter how you move around the room. That's similar to a constant value.
For our function \(g(x, y)=4-x-y\), a constant value \(c\) dictates the equation of the level curve. Level curves are lines where the function's output does not change—it stays constant. For example, if \(c=0\), the equation becomes \(x+y=4\) because \(4-x-y=c\) when rearranged.
Changing the constant value will produce different curves. For \(c=4\), rearranging gives \(x+y=0\), showing a different set of values for \(x\) and \(y\) that keep the function's value constant.

Graphical Representation

Graphical representation of functions and their level curves gives us a visual way to understand what is happening. Just like a map helps you see mountains and valleys, a graph helps us see how the values of two variables interact.
In the exercise, we have two level curves derived from the function \(g(x, y)=4-x-y\). For \(c=0\), the line \(x+y=4\) is plotted—this represents one level curve. For \(c=4\), the line \(x+y=0\) is another.

• The line \(x+y=4\) is like drawing all points on a map with exactly the same height above sea level—here, the height is the value \(c=0\).
• Similarly, the line \(x+y=0\) represents all points where the function's value is constantly \(c=4\).

Visualizing these lines on a two-dimensional graph helps us see how small changes in \(x\) and \(y\) affect the function. It's a handy way of understanding complex relationships at a glance.