Question

Use a graphing utility to graph six level curves of the function. $$ f(x, y)=|x y| $$

Short answer

To graph level curves of the function \(f(x,y) = |xy|\), we select values of \(c\) and substitute them into the equation \(|xy| = c\). The chosen values for the constants are 0, 1, 2, 3, 4, and 5. The plot of these level curves gives a visual representation of the function \(f(x, y) = |xy|\). Note that the result is a graphical representation that may turn out as concentric circles or ellipses, depending on the function and chosen values for \(c\).

Step-by-step solution

Understanding Level Curves

Firstly, it's crucial to understand what level curves are. For a function of two variables, \(f(x,y)\), the level curve is obtained by setting \(f(x,y) = c\), where \(c\) is a constant. Each level curve corresponds to a specific value of \(c\), commonly chosen to be a range of values. For this exercise, we are graphing six level curves, which means we will be choosing six values of \(c\) and substituting each value into the equation \(|xy| = c\).

Selecting Constant Values

The next step is to select the constant values. The convenient values to select would be \(c = 0, 1, 2, 3, 4, 5\). Substituting these values into the function \(f(x,y)\), we get \(|xy|=c\). By graphing these six equations, we will have six level curves of the function \(f(x,y)\).

Plotting the Level Curves

Now we plot these level curves using a graphing utility. Graphing a level curve involves plotting a number of points that satisfy the equation and then connecting them. As we do so for multiple values of \(c\), we will obtain the level curves of the function.

Key concepts

Graphing Utility

Graphing utilities are essential tools for visualizing complex mathematical functions, particularly those involving two or more variables. For students tackling multivariable calculus, this visual aid is invaluable as it transforms abstract concepts into tangible images. Imagine trying to draw a three-dimensional surface or a set of level curves by hand; it's quite challenging!

A graphing utility, such as a software program or graphing calculator, takes the equation you're working with—say, an absolute value function like \( f(x, y)=|xy| \)—and generates the graph for you. When using these utilities for level curves, you simply input the function and specify the values of \( c \) for which you want to see curves. The utility then calculates and displays these curves on the coordinate plane, making it easier to analyze the function's behavior.

• Efficiency: Graphing many points manually is time-consuming; graphing utilities speed up the process.
• Accuracy: They provide precise visual representations, minimizing human error.
• Comprehension: Visual graphs help students better understand the relationship between variables and constants in functions.

These utilities are not just limited to drawing; they can also perform complex calculations, find derivatives, and integrate, making them indispensable in a student's mathematical toolkit.

Absolute Value Functions

Absolute value functions have a distinctive V-shaped graph and are crucial in understanding how distances are always non-negative in mathematics. Recognizing the role of absolute value in multivariable calculus is crucial as it helps students comprehend how these functions behave in more than one dimension.

In the context of the provided exercise \( f(x, y)=|xy| \), the absolute value symbol indicates that the output will be the same regardless of the signs of \( x \) and \( y \) when multiplied together. This results in mirrored level curves across both the x-axis and y-axis.

• Mirror Patterns: Absolute value functions feature symmetry, often resulting in mirrored graphs across axes.
• Non-negative Output: The result of an absolute value function is always zero or positive, shaping the graph accordingly.
• Intersection Points: The points where the V-shaped graph intersects the axes are particularly worth noting when graphing such functions.

When students graph level curves for an absolute value function, they are plotting the set of points where the function's output is constant. This creates an intuitive understanding of how the function behaves at different points in the coordinate plane.

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus to functions with several variables. This branch of mathematics plays a pivotal role in understanding the behavior of phenomena where multiple factors are at play. Level curves, also known as contour lines, are a fundamental concept in this area. They represent a cross-section of a three-dimensional graph where the function takes on a constant value, akin to contour lines on a geographic map that indicate elevation.

Considering the exercise function \( f(x, y)=|xy| \) in a multivariable context, the level curves describe the locations where the product of \( x \) and \( y \) leads to the same absolute value. They are essential in visualizing how output values of a function change across different regions of the xy-plane. Moreover, understanding the nature of these curves can help anticipate the shape of the entire surface.

• Visualizing Surfaces: Level curves provide a way to visualize intricate surfaces created by functions in two or more dimensions.
• Function Behavior: They highlight areas where the function increases, decreases, remains constant, or changes abruptly.
• Real-world Applications: Contour maps derived from level curves are used in fields like meteorology, engineering, and economics to visualize data.

By mastering level curves in multivariable calculus, students can better comprehend complex interactions between variables and make informed predictions about function outputs.