/*! 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 30 Graph several level curves of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 30

Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. $$z=x^{2}+y^{2} ;[-4,4] \times[-4,4]$$

Short Answer

Expert verified
Question: Graph the level curves of the function $$z = x^2 + y^2$$ within the window $$[-4, 4] \times [-4, 4]$$. Use the z-values $$1$$ and $$4$$ for the level curves. Answer: The level curves for $$z = 1$$ and $$z = 4$$ are circles with radii of 1 and 2 centered at the origin, respectively. The graph of these level curves can be found within the window $$[-4, 4] \times [-4, 4]$$.

Step by step solution

01

Choose z-values for level curves

Select two distinct z-values to find level curves (We could choose more, but for this exercise, we'll just graph two). For example, let's choose $$z = 1$$ and $$z = 4$$.
02

Solve for y in terms of x

For each chosen z-value, substitute the chosen value into the function's formula and solve for y in terms of x. For $$z = 1$$ and $$z = 4$$, we get: $$1 = x^2 + y^2$$ and $$4 = x^2 + y^2$$ Which gives us the equations for the level curves of: $$y^2 = 1 - x^2$$ and $$y^2 = 4 - x^2$$
03

Convert the equations into polar form

To graph these level curves, it's easier to convert the equations into polar form. Using the substitution $$x = r \cos \theta$$ and $$y = r \sin \theta$$, the level curve equations become: $$r^2 = 1$$ and $$r^2 = 4$$ which simplifies to: $$r = 1$$ and $$r = 2$$
04

Graph the level curves

In polar coordinates, the level curves are circles with radii of 1 and 2 centered at the origin. So, graph the circles in the given window $$[-4, 4] \times [-4, 4]$$.
05

Label the level curves

Label the graphed circles with their respective z-values. The circles with radii of 1 and 2 correspond to the level curves for $$z = 1$$ and $$z = 4$$, respectively.

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

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. (Source: Mathematics Magazine, May 1985, and Calculus and Analytical Geometry, 2nd ed., Philip Gillett, 1984)

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}}$$

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

Show that the following two functions have two local maxima but no other extreme points (therefore, there is no saddle or basin between the mountains). a. \(f(x, y)=-\left(x^{2}-1\right)^{2}-\left(x^{2}-e^{y}\right)^{2}\) b. \(f(x, y)=4 x^{2} e^{y}-2 x^{4}-e^{4 y}\)

An identity Show that if \(f(x, y)=\frac{a x+b y}{c x+d y},\) where \(a, b, c,\) and \(d\) are real numbers with \(a d-b c=0,\) then \(f_{x}=f_{y}=0,\) for all \(x\) and \(y\) in the domain of \(f\). Give an explanation.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.