Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 13: Problem 1
Let \(f(x, y)=(x-y)^{2} .\) Determine the equations and shapes of the cross- sections when \(x=0, y=0, x=y,\) and describe the level curves. Use a three- dimensional graphing tool to graph the surface.
Short Answer
Expert verified
Cross-sections: Parabolas for \(x=0\) and \(y=0\), horizontal line for \(x=y\). Level curves: Parallel lines.
Step by step solution
01
Evaluate the Cross-Section when x = 0
To find the cross-section when \(x = 0\), substitute \(x = 0\) in the function \(f(x, y) = (x-y)^2\). This gives:\[f(0, y) = (0 - y)^2 = y^2.\]This is a quadratic function in terms of \(y\), describing a parabola that opens upwards.
02
Evaluate the Cross-Section when y = 0
To find the cross-section when \(y = 0\), substitute \(y = 0\) in the function \(f(x, y) = (x-y)^2\). This gives:\[f(x, 0) = (x - 0)^2 = x^2.\]This is a quadratic function in terms of \(x\), describing a parabola that opens upwards.
03
Evaluate the Cross-Section when x = y
Substitute \(x = y\) in the function \(f(x, y) = (x-y)^2\). This gives:\[f(y, y) = (y-y)^2 = 0.\]This indicates a cross-section that is a horizontal line at height 0.
04
Describe the Level Curves
A level curve is a set where \(f(x, y) = c\) for some constant \(c\). Setting \((x-y)^2 = c\), we get:\[x-y = \pm\sqrt{c}.\]Each level curve is a pair of parallel lines in the plane, equidistant from the line \(x=y\), and the distance between the lines increases with \(c\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross-Sections
Cross-sections are essentially slices of a three-dimensional object at a particular value of one of its variables. When you visualize a 3D surface, like the one described by the function \( f(x, y) = (x-y)^2 \), cross-sections can be thought of as the shadows or profiles we see when a flashlight shines from a specific direction across the surface. They help us understand the shape and behavior of a surface in a simpler, more two-dimensional context.
- For the cross-section when \( x = 0 \), substitute \( x = 0 \) into the function. This gives you \( f(0, y) = y^2 \). The cross-section is a parabola opening upwards in the \( y \)-axis direction.
- When \( y = 0 \), substitute \( y = 0 \) into the function to get \( f(x, 0) = x^2 \). Similarly, this cross-section is also a parabola opening upwards, but in the \( x \)-axis direction.
- For \( x = y \), plug in \( x = y \) into the function. The result \( f(y, y) = 0 \) represents a horizontal line along \( z = 0 \).
Level Curves
Level curves are crucial tools in multivariable calculus. They represent sets of points where the function takes on a constant value, creating 2D snapshots of the 3D surface. For the function \( f(x, y) = (x-y)^2 \), level curves give unique insights into the surface's incline and form.
Level curves for the equation are found by setting \( f(x, y) = c \) which simplifies to \( (x-y)^2 = c \). Solving this, we find \( x - y = \pm \sqrt{c} \).
Level curves for the equation are found by setting \( f(x, y) = c \) which simplifies to \( (x-y)^2 = c \). Solving this, we find \( x - y = \pm \sqrt{c} \).
- This equation describes pairs of parallel lines. The lines are equidistant from the line \( x = y \), making these level curves symmetric around it.
- The distance between these parallel lines increases with the value of \( c \). Larger \( c \) values result in lines further apart, reflecting that the surface rises more steeply.
3D Graphing
3D graphing is an essential technique in understanding functions like \( f(x, y) = (x-y)^2 \). Visualizing a function in three dimensions allows us to interpret its shape, saddles, and peaks.
To graph the surface of \( f(x, y) \), you would generally use a 3D graphing tool. This would render a vibrant visual display of the function, highlighting:
To graph the surface of \( f(x, y) \), you would generally use a 3D graphing tool. This would render a vibrant visual display of the function, highlighting:
- The parabolas, as deduced from earlier cross-sections, showing how the function behaves for different values of \( x \) and \( y \).
- The level curves, depicted as parallel lines, revealing how the surface elevates or dips as we analyze across different areas of the \( xy \)-plane.
Parabolas
A parabola is a symmetric curve that appears as you slice through a 3D surface along certain planes. It is a familiar shape in geometry, identified by its U-like form. This arises frequently in multivariable calculus, including in scenarios like in the function \( f(x, y) = (x-y)^2 \).
- In this task, when cross-sections at \( x = 0 \) and \( y = 0 \) were evaluated, parabolas emerged. Each represented a quadratic dependency on the remaining variable, either \( y \) or \( x \).
- Opening upwards, these parabolas provide a visual clue to the function's curvature, mapping out how quickly the height of the 3D surface rises or falls as you move along the axes.
