/*! 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 7 Describe in words the level curv... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 7

Describe in words the level curves of the paraboloid \(z=x^{2}+y^{2}.\)

Short Answer

Expert verified
The level curves of the paraboloid \(z=x^{2}+y^{2}\) are circles centered at the origin with radius \(\sqrt{c}\), where \(c\) represents the height of the paraboloid. As the value of \(c\) increases, the radius of the circle increases, which means the level curves will become larger.

Step by step solution

01

Set up the level curve equation for arbitrary constant height (c)

Level curves are formed by intersecting the given surface (paraboloid) with a set of horizontal planes at different heights. Mathematically, this is done by setting \(z=c\), where \(c\) is a constant. In our case, we have: $$z = x^{2}+y^{2}$$ and we set \(z=c\) to get the level curve equation: $$c = x^{2}+y^{2}$$
02

Recognize the equation of a circle

The equation we derived in step 1, \(c = x^{2}+y^{2}\), is the equation of a circle with radius \(\sqrt{c}\) and center at the origin (0, 0).
03

Describe the level curves

From step 2, we know that the level curves are circles with radius \(\sqrt{c}\) and center at the origin (0, 0). The level curves of the paraboloid \(z=x^{2}+y^{2}\) are circles centered at the origin with radius \(\sqrt{c}\), where \(c\) represents the height of the paraboloid. As we increase the value of \(c\), the radius of the circle increases, which means the level curves will become larger.

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.

Paraboloid
A paraboloid is a three-dimensional shape resembling an elongated bowl, which opens upwards or downwards. Mathematically, it can be expressed with the equation \( z = x^2 + y^2 \). The shape is defined such that each cross-section parallel to the base is a circle, forming the surface of a paraboloid.

When we cut through the paraboloid horizontally at different heights, the slices reveal level curves. These level curves tell us the shape of horizontal sections through the paraboloid. Visualize a stack of concentric circles rising up or receding down the paraboloid's surface. This geometric structure is seen in applications ranging from optics to designing satellite dishes.
Equation of a Circle
The equation of a circle is a fundamental concept in geometry, defining a set of points in a plane that are equidistant from a fixed point known as the center. A standard circle equation is given by \( x^2 + y^2 = r^2 \), where \(r\) is the radius.

In the context of level curves on a paraboloid, when a horizontal plane cuts through, the resulting intersection is a circle. This emerges from setting \( z = x^2 + y^2 = c \), leading to \( x^2 + y^2 = c \). Here, \(c\) plays the role of \( r^2 \), and describes a circle's radius squared centered at the origin (0, 0) in the xy-plane.
Radius of a Circle
The radius of a circle is the distance from the center of the circle to any point on its boundary. It serves as a crucial measure when describing circles.

In the exercise, the equation \( x^2 + y^2 = c \) gives the radius as \( \sqrt{c} \). This results because the square of the radius \( r^2 \) equals the constant house height \( c \) from our set level surface. As \( c \) increases, the corresponding radius \( \sqrt{c} \) also increases, depicting the expansion of level curves on a paraboloid.
Center of Circle
The center of a circle is a pivotal point from which every point on the circle is equidistant, serving as a symmetrical focal spot for geometric shapes.

For the equation \( x^2 + y^2 = c \), the center of each corresponding circle is at the origin, (0, 0). This is because the paraboloid is centered at its vertex, where x and y are zero, hence making each cross-sectional level curve concentric to the center of the coordinate plane. Being repeatedly centered at the origin simplifies many mathematical models and is instrumental in discussions involving symmetry in circles and paraboloids.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=e^{a x} \cos a y, \text { for any real number } a$$

Batting averages in baseball are defined by \(A=x / y,\) where \(x \geq 0\) is the total number of hits and \(y>0\) is the total number of at bats. Treat \(x\) and \(y\) as positive real numbers and note that \(0 \leq A \leq 1.\) a. Use differentials to estimate the change in the batting average if the number of hits increases from 60 to 62 and the number of at bats increases from 175 to 180 . b. If a batter currently has a batting average of \(A=0.350,\) does the average decrease if the batter fails to get a hit more than it increases if the batter gets a hit? c. Does the answer to part (b) depend on the current batting average? Explain.

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) The data may be plotted as a scatterplot in the \(x y\) -plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Let the equation of the best-fit line be \(y=m x+b,\) where the slope \(m\) and the \(y\) -intercept \(b\) must be determined using the least squares condition. First assume that there are three data points \((1,2),(3,5),\) and \((4,6) .\) Show that the function of \(m\) and \(b\) that gives the sum of the squares of the vertical distances between the line and the three data points is $$ \begin{aligned} E(m, b)=&((m+b)-2)^{2}+((3 m+b)-5)^{2} \\ &+((4 m+b)-6)^{2} \end{aligned}. $$ Find the critical points of \(E\) and find the values of \(m\) and \(b\) that minimize \(E\). Graph the three data points and the best-fit line.

The equation \(x^{2 n}+y^{2 n}+z^{2 n}=1,\) where \(n\) is a positive integer, describes a flattened sphere. Define the extreme points to be the points on the flattened sphere with a maximum distance from the origin. a. Find all the extreme points on the flattened sphere with \(n=2\) What is the distance between the extreme points and the origin? b. Find all the extreme points on the flattened sphere for integers \(n>2 .\) What is the distance between the extreme points and the origin? c. Give the location of the extreme points in the limit as \(n \rightarrow \infty\). What is the limiting distance between the extreme points and the origin as \(n \rightarrow \infty ?\)

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y)=\ln \left(1+x^{2}+y^{2}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.