/*! 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 28 Describe the region \(R\) in the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 28

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ f(x, y)=y e^{1 / x} $$

Short Answer

Expert verified
The domain of the function \(f(x, y)=ye^{1/x}\) is \((x, y) : x \in \mathbb{R} \backslash {0}, y \in \mathbb{R}\), which signifies that \(x\) can be any real number except zero while \(y\) can be any real number.

Step by step solution

01

Determine Domain from the Equation

From the geometric point of view, the domain is the set of \(x\) and \(y\) coordinates in the plane for which the function \(f(x, y)=ye^{1/x}\) is defined. At first glance, the function seems to be defined for all values of \(x\) and \(y\), but there's a factor \(1/x\) up in the exponent of \(e\). The function \(e^{1/x}\) is defined for all \(x\) except at \(x = 0\). For \(y\), there seems to be no restrictions as any real number \(y\) multiplied by an exponential number is still a real number.
02

Formulate the Domain

Therefore, the region \(R\) in the \(xy\)-plane corresponding to the domain of the function \(f(x, y)=ye^{1/x}\) is all real numbers except when \(x = 0\). Formally, this can be expressed as \((x, y) : x \in \mathbb{R} \backslash {0}, y \in \mathbb{R}\).

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

The population density (in people per square mile) for a coastal town can be modeled by \(f(x, y)=\frac{120,000}{(2+x+y)^{3}}\) where \(x\) and \(y\) are measured in miles. What is the population inside the rectangular area defined by the vertices \((0,0)\), \((2,0),(0,2)\), and \((2,2) ?\)

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{1} \int_{y}^{\sqrt{y}} x^{2} y^{2} d x d y $$

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=e^{x+y}\\\ &R: \text { triangle with vertices }(0,0),(0,1),(1,1) \end{aligned} $$

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x^{2}+y^{2}\\\ &R: \text { square with vertices }(0,0),(2,0),(2,2),(0,2) \end{aligned} $$

Use a symbolic integration utility to evaluate the double integral. $$ \int_{1}^{2} \int_{y}^{2 y} \ln (x+y) d x d y $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.