/*! 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 35 Sketch the surfaces in Exercises... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 35

Sketch the surfaces in Exercises \(13-44\) $$y=-\left(x^{2}+z^{2}\right)$$

Short Answer

Expert verified
The surface is a downward-opening paraboloid centered at the origin.

Step by step solution

01

Identify the Shape of the Surface

The given equation is \(y = -(x^2 + z^2)\). Recognize that this equation represents a paraboloid that opens downwards along the y-axis, since \(y\) is negating the terms \(x^2 + z^2\), both squared terms.
02

Analyze Cuts in Coordinate Planes

To better understand the surface's structure, let's analyze cross-sections:- **Plane \(x = k\):** The equation becomes \(y = -(k^2 + z^2)\), which is a parabola in the \(yz\)-plane.- **Plane \(z = k\):** The equation is \(y = -(x^2 + k^2)\), another parabola in the \(xy\)-plane.Both will intersect on a parabola opening downward along the negative y-axis.
03

Sketch the Surface

To sketch the surface:1. Draw the y-axis going negative, as the maximum value of y is 0, occurring when \(x = 0\) and \(z = 0\).2. Draw parabolas that open downward, centered at the origin on both the \(yz\) and \(xy\)-planes at several slices. These parabolas should show the surface folding downwards uniformly along all directions defined by x and z.

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.

Coordinate Planes
Coordinate planes are essential in visualizing and analyzing three-dimensional surfaces and objects. The three main coordinate planes in a three-dimensional Cartesian coordinate system are the xy-plane, yz-plane, and xz-plane.
  • The xy-plane is where the z-coordinate is zero, and it extends infinitely in the x and y directions.
  • The yz-plane is defined by x = 0, extending infinitely in the y and z directions.
  • The xz-plane is the plane where y = 0, and it extends infinitely along the x and z axes.
Understanding these coordinate planes allows us to dissect and visualize the behavior of complex surfaces like paraboloids. For the given paraboloid equation, slicing it along these planes helps in understanding its orientation and shape.
Cross-sections
Cross-sections are a powerful tool to understand the three-dimensional shapes by taking two-dimensional slices of them. Imagine taking a knife and cutting through a loaf of bread to see the pattern inside; this is similar to analyzing cross-sections.
For the paraboloid described by the equation \( y = -(x^2 + z^2) \):
  • In the plane x = k, the equation becomes \( y = -(k^2 + z^2) \). This results in a parabolic cross-section in the yz-plane. The parabola opens downward along the y-axis, maximizing at the origin.
  • Similarly, in the plane z = k, the equation is \( y = -(x^2 + k^2) \), which gives another parabolic cross-section in the xy-plane, also opening downward.
These cross-sections provide a clear view of the structure and orientation of the surface. They confirm that regardless of the cut, the parabola always opens downwards along the y-axis, illustrating a consistent shape.
Surface Sketching
Surface sketching is a method of visual representation to better understand and interpret the shape and features of mathematical surfaces. Sketching begins by plotting key features like vertices, axes, and notable cross-sections, and combining them into a coherent image.
When sketching the paraboloid described by \( y = -(x^2 + z^2) \), several steps are useful:
  • Begin by drawing the y-axis, prominently showing the negative direction since y decreases indefinitely.
  • Mark the key point of the surface, which is the origin (0,0,0), where \( y \) reaches its highest value, 0.
  • Use vertical slices to draw parabolas on both the yz and xy-planes. Each parabola appearing as a downward "U", shows the shape curving downwards from the origin.
  • Connecting these parabolas smoothly helps visualize the complete surface creating a dome-like appearance, emphasizing the downward opening nature.
Sketching can reveal insights about symmetry and critical points of a surface, providing a deeper understanding beyond algebraic expressions.

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

Find the vector from the origin to the point of intersection of the medians of the triangle whose vertices are $$ A(1,-1,2), \quad B(2,1,3), \quad \text { and } \quad C(-1,2,-1) $$

The graph of \((x / a)+(y / b)+(z / c)=1\) is a plane for any nonzero numbers \(a, b,\) and \(c .\) Which planes have an equation of this form?

Find a formula for the distance from the point \(P(x, y, z)\) to the a. \(x\) -axis. b. \(y\) -axis. c. \(z\) -axis.

Find the point in which the line meets the plane. $$x=1+2 t, \quad y=1+5 t, \quad z=3 t ; \quad x+y+z=2$$

Sketch the surfaces in Exercises \(13-44\) $$y^{2}+z^{2}-x^{2}=1$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.