/*! 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 6 Describe the graph of \(x=z^{2}\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 6

Describe the graph of \(x=z^{2}\) in \(\mathbb{R}^{3}\).

Short Answer

Expert verified
Describe the graph of the equation \(x=z^2\) in \(\mathbb{R}^3\). The graph of the equation \(x=z^2\) in \(\mathbb{R}^3\) represents a parabolic surface extending infinitely along the y-axis. The surface has a "U" shape, with its vertex at the origin (0,0,0) and opening in the positive x-direction. The graph is symmetrical with respect to the z-axis, as any point on the surface has coordinates (z^2, y, z), where both y and z can be any real value.

Step by step solution

01

Understand the given equation in 3D space

Given the equation \(x=z^2\), we are in a 3-dimensional space, with x, y, and z as the coordinates. The equation does not involve the y-coordinate, which means that our graph will not depend on the value of y; it can be any real number. Therefore, the graph will extend infinitely along the y-axis.
02

Analyze and visualize x and z relationship

The equation given is \(x=z^2\). From this equation, we can see that the z-coordinate is squared, and that determines the x-coordinate. This means that, for every positive value of z, there will be a corresponding value of x. In addition, since z is squared, the value of x will never be negative. To visualize this graph properly, suppose we slice the space with a plane parallel to the xz-plane at a particular y-coordinate. Each of these slices will show the graph of the equation \(x=z^2\) in 2-dimensional space. The graph in 2D space would look like the graph of a quadratic function opening to the right.
03

Describe the 3D graph

Considering the full 3-dimensional space, the graph of the equation \(x=z^2\) would look like a parabolic or a "U" shaped surface that extends infinitely along the y-axis. To be more specific, any point on this surface has the coordinates (z^2, y, z) where both y and z can take any real value. The surface will be symmetrical with respect to the z-axis since squaring z will result in the same value for x, regardless of the sign of z. In summary, the graph of the equation \(x=z^2\) in \(\mathbb{R}^3\) represents a parabolic surface extending infinitely along the y-axis, with its vertex at the origin (0,0,0) and opening in the positive x-direction, symmetrical with respect to the z-axis.

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!

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

Use the definition of the gradient (in two or three dimensions), assume that \(f\) and \(g\) are differentiable functions on \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3},\) and let \(c\) be a constant. Prove the following gradient rules. a. Constants Rule: \(\nabla(c f)=c \nabla f\) b. Sum Rule: \(\nabla(f+g)=\nabla f+\nabla g\) c. Product Rule: \(\nabla(f g)=(\nabla f) g+f \nabla g\) d. Quotient Rule: \(\nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}}\) e. Chain Rule: \(\nabla(f \circ g)=f^{\prime}(g) \nabla g,\) where \(f\) is a function of one variable

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=(x+y+z) e^{x y z}$$

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{x^{2}+x y+y^{2}}$$

Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form \(P=f(K, L)=C K^{a} L^{1-a},\) where K represents capital, L represents labor, and C and a are positive real numbers with \(0

Ideal Gas Law Many gases can be modeled by the Ideal Gas Law, \(P V=n R T,\) which relates the temperature \((T,\) measured in Kelvin (K)), pressure ( \(P\), measured in Pascals (Pa)), and volume ( \(V\), measured in \(\mathrm{m}^{3}\) ) of a gas. Assume that the quantity of gas in question is \(n=1\) mole (mol). The gas constant has a value of \(R=8.3 \mathrm{m}^{3} \cdot \mathrm{Pa} / \mathrm{mol} \cdot \mathrm{K}.\) a. Consider \(T\) to be the dependent variable and plot several level curves (called isotherms) of the temperature surface in the region \(0 \leq P \leq 100,000\) and \(0 \leq V \leq 0.5.\) b. Consider \(P\) to be the dependent variable and plot several level curves (called isobars) of the pressure surface in the region \(0 \leq T \leq 900\) and \(0< V \leq 0.5.\) c. Consider \(V\) to be the dependent variable and plot several level curves of the volume surface in the region \(0 \leq T \leq 900\) and \(0 < P \leq 100,000.\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

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.