/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Sketch the graph of \(f\). $$ ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 37

Sketch the graph of \(f\). $$ f(x, y)=x^{2}-y^{2} $$

Short Answer

Expert verified
The graph of \(f(x,y)=x^2-y^2\) is a hyperbolic paraboloid with lines \(y=x\) and \(y=-x\) as asymptotes.

Step by step solution

01

Analyze the Function

First, note that the function is a type of hyperbolic function given by \(f(x,y) = x^2 - y^2\). This tells us it's a saddle surface, known as a hyperbolic paraboloid.
02

Identify Intersections with Coordinate Planes

For the xy-plane where \(z=0\), we have \(x^2 - y^2 = 0\), which simplifies to \(x^2 = y^2\). This gives the lines \(y = x\) and \(y = -x\) (asymptotes along the xy-plane).
03

Analyze Cross-Sections in One Variable

Set \(x = c\) to find cross-sections parallel to the yz-plane, giving \(f(c, y) = c^2 - y^2\), which are parabolas opening downwards. Similarly, for cross-sections parallel to the xz-plane, set \(y = c\) and find \(f(x, c) = x^2 - c^2\), which gives parabolas opening upwards.
04

Determine the Origin's Behavior

At the origin \((0,0)\), the function \(f(x,y) = x^2 - y^2\) evaluates to \(0\). This serves as a saddle point, where the surface changes curvature.
05

Sketch the Graph

Combine all the above observations to draw the graph. Use the key lines \(y = x\) and \(y = -x\) as guides that separate the different quadrants. In quadrant 1 (+,+) and 3 (-,-), the surface rises, and in quadrant 2 (+,-) and quadrant 4 (-,+), the surface dips, forming a saddle shape.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Paraboloid
The hyperbolic paraboloid is a unique and intriguing surface in three-dimensional space. It is described by the equation \( f(x, y) = x^2 - y^2 \). This kind of surface is known as a saddle surface due to its distinct shape that curves upwards in one direction and downwards in a perpendicular direction, much like the seat of a saddle.
  • This surface is doubly ruled, meaning it can be constructed by two families of straight lines.
  • The hyperbolic paraboloid is used in various architectural designs due to its aesthetic and structural properties.
The saddle shape occurs because as you move away from the origin, the values of the function \(x^2\) grow positively along the \(x\)-axis, while the values of \(-y^2\) grow negatively along the \(y\)-axis. Displaying symmetry, this balanced, opposing curvature gives us the prominent saddle-like structure.
Cross-Sections
Cross-sections are essential for understanding the behavior of complex surfaces like the hyperbolic paraboloid. Imagine slicing through the surface with a plane. Where this plane intersects the surface, you get cross-sections, and they can be analyzed to gain deeper insights:
  • When you set a constant \(x = c\), the cross-section parallel to the \(yz\)-plane becomes \(f(c, y) = c^2 - y^2\). It results in parabolas that open downward.
  • Conversely, setting a constant \(y = c\) creates cross-sections parallel to the \(xz\)-plane, given by \(f(x, c) = x^2 - c^2\). Here, parabolas open upward.
These differing directions of opening reflect the saddle-like nature of the hyperbolic paraboloid. The intersecting curves along these planes contribute to both the aesthetic and the mathematical interest of this surface.
Coordinate Planes
Coordinate planes like the \(xy\)-plane, \(yz\)-plane, and \(xz\)-plane are vital reference points for analyzing three-dimensional surfaces. When studying the hyperbolic paraboloid \(f(x, y) = x^2 - y^2\), coordinate planes help us understand how the function behaves as it approaches zero or any other fixed value.
  • On the \(xy\)-plane \(z = 0\), the points of intersection given by \(x^2 - y^2 = 0\) result in the lines \(y = x\) and \(y = -x\). These lines act as the asymptotes and guide as a symmetry axis of the surface.
  • The plane intersections at the \(yz\) and \(xz\) planes give valuable insights into how cross-sections behave and influence the curvature and direction of the surface.
Understanding these interactions with coordinate planes is a critical aspect of visualizing and sketching the hyperbolic paraboloid, providing a framework for recognizing its structure and behavior.

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Most popular questions from this chapter

Solve using Lagrange multipliers. Suppose that the temperature at a point \((x, y)\) on a metal plate is \(T(x, y)=4 x^{2}-4 x y+y^{2}\). An ant, walking on the plate, traverses a circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant?

Find the directional derivative of \(f\) at \(P\) in the direction of a vector making the counterclockwise angle \(\theta\) with the positive \(x\) -axis. $$ f(x, y)=\sqrt{x y} ; \quad P(1,4) ; \theta=\pi / 3 $$

Find the gradient of \(f\) at the indicated point. $$ f(x, y)=5 x^{2}+y^{4} ;(4,2) $$

Solve using Lagrange multipliers. Find the point on the plane \(4 x+3 y+z=2\) that is closest to \((1,-1,1)\).

(a) Show that the second partials test provides no information about the critical points of the function \(f(x, y)=x^{4}+y^{4}\) (b) Classify all critical points of \(f\) as relative maxima, relative minima, or saddle points.
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