/*! 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 67 Find an equation for the family ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 67

Find an equation for the family of level surfaces pe corresponding to \(f .\) Describe the level surfaces.$$f(x, y, z)=x^{2}+y^{2}-z.$$

Short Answer

Expert verified
Answer: The level sets of the function \(f(x,y,z)=x^2+y^2-z\) represent a family of paraboloids.

Step by step solution

01

Set the function equal to a constant

To find the level surfaces of the given function \(f(x,y,z)=x^2+y^2-z\), set it equal to a constant \(c\) to form the equation: $$x^2+y^2-z=c.$$
02

Solve for z

Solve the equation for z to better visualize the form of the level surfaces: $$z=x^2+y^2-c.$$
03

Describe the level surfaces

The equation \(z=x^2+y^2-c\) represents a paraboloid. The value of \(c\) determines the position of the paraboloid along the z-axis, and the level surfaces for different values of \(c\) will be a family of paraboloids. The paraboloids open upwards and their vertex is translated along the z-axis as \(c\) changes.

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 surface that resembles a parabola when sliced along specific planes. In the exercise, the equation \( z = x^2 + y^2 - c \) describes a paraboloid. This specific form of paraboloid is an elliptic paraboloid because the cross-sections parallel to the \( xy \)-plane are ellipses, though they appear circular when \( x^2 + y^2 \) has equal coefficients, as is the case here.
  • The paraboloid opens upwards along the \( z \)-axis.
  • The equation can be rearranged to visualize the situation: \( z = x^2 + y^2 - c \).
  • The parameter \( c \) shifts the vertex of the paraboloid up or down the \( z \)-axis.
Understanding paraboloids is crucial in multivariable calculus, as they represent simple, yet vital, three-dimensional shapes.
Multivariable Calculus
Multivariable calculus extends the concepts of calculus to functions of multiple variables, often involving 2, 3, or more interdependent variables. It is a powerful tool to analyze and describe changes within dimensions beyond the typical graphable functions of basic calculus.
  • Functions like \( f(x, y, z) = x^2 + y^2 - z \) require methods from multivariable calculus for analysis.
  • Such studies involve concepts like partial derivatives, gradients, and examining surfaces like paraboloids.
  • Level surfaces, as seen in this exercise, are a particular feature of multivariable functions.
In this exercise, using multivariable calculus allows us to better understand and visualize the family of surfaces represented by paraboloids.
Equation of Surfaces
Equations of surfaces describe the mathematical relationship defining the points on a surface in space. Here, for \( f(x, y, z) = x^2 + y^2 - z \), setting \( f \) equal to a constant \( c \) offers an equation of a surface: \( x^2 + y^2 - z = c \).
  • This particular equation is a form of equation highlighting level surfaces.
  • Each surface for different \( c \) values describes varying positions of the same type of shape, a paraboloid.
  • These surfaces assist in visualizing how certain functions delineate three-dimensional space.
Recognizing and manipulating the equations of surfaces is a foundational skill in multivariable calculus, allowing one to comprehend complex spatial interactions.

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 differentiability to prove that the following functions are differentiable at \((0,0) .\) You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\) with the required properties. $$f(x, y)=x y$$

Prove that for the plane described by \(f(x, y)=A x+B y,\) where \(A\) and \(B\) are nonzero constants, the gradient is constant (independent of \((x, y)\) ). Interpret this result.

Identify and briefly describe the surfaces defined by the following equations. $$y=4 z^{2}-x^{2}$$

A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example, \(f(x)=x^{2}\) ). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on \(\mathbb{R}^{2}\). a. \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) b. \(f(x, y)=\left(2 y^{2}-y^{4}\right)\left(e^{x}+\frac{1}{1+x^{2}}\right)-\frac{1}{1+x^{2}}\) This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum.

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 75 about Steiner's problem.)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

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.