/*! 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 50 At what points of \(\mathbb{R}^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 50

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$h(x, y)=\frac{\sqrt{x-y}}{4}$$

Short Answer

Expert verified
Question: Identify the points in the domain \(\mathbb{R}^{2}\) where the function \(h(x, y) = \frac{\sqrt{x-y}}{4}\) is continuous. Answer: The function is continuous at all points \((x, y)\) in \(\mathbb{R}^{2}\) such that \(x \geq y\).

Step by step solution

01

Identify problematic points for continuity

To do this, we start by finding where the function is undefined. The function is only undefined when the value within the square root is negative. We have: $$x - y < 0$$
02

Solve inequality for x and y

Now, we must solve the inequality for \(x\) and \(y\), which can be rearranged as: $$x < y$$ This inequality represents that the function is undefined if \(x\) is less than \(y\). It means that the points in the \(xy\)-plane above the line \(x=y\) are the problematic points for the continuity of the given function.
03

Determine the points for which the function is continuous

For the function to be continuous in the \(xy\)-plane (in \(\mathbb{R}^2\)), it must be true that the square root is never negative. This happens if: $$x - y \geq 0$$ By rearranging the inequality, we have: $$x \geq y$$ This inequality means that the function is continuous everywhere in the \(xy\)-plane below or on the line \(x=y\) (including the points on this line).
04

Write down the answer

In conclusion, the function \(h(x, y) = \frac{\sqrt{x-y}}{4}\) is continuous at all points \((x, y)\) in \(\mathbb{R}^{2}\) such that \(x \geq y\).

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

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

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. $$u(x, t)=5 \cos (2(x+c t))+3 \sin (x-c t)$$

Let \(E\) be the ellipsoid \(x^{2} / 9+y^{2} / 4+z^{2}=1, P\) be the plane \(z=A x+B y,\) and \(C\) be the intersection of \(E\) and \(P\). a. Is \(C\) an ellipse for all values of \(A\) and \(B\) ? Explain. b. Sketch and interpret the situation in which \(A=0\) and \(B \neq 0\) c. Find an equation of the projection of \(C\) on the \(x y\) -plane. d. Assume \(A=\frac{1}{6}\) and \(B=\frac{1}{2} .\) Find a parametric description of \(C\) as a curve in \(\mathbb{R}^{3}\). (Hint: Assume \(C\) is described by \(\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) and find \(a, b, c, d, e, \text { and } f .)\)

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y)=x y \cos x y$$

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) The data may be plotted as a scatterplot in the \(x y\) -plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Let the equation of the best-fit line be \(y=m x+b,\) where the slope \(m\) and the \(y\) -intercept \(b\) must be determined using the least squares condition. First assume that there are three data points \((1,2),(3,5),\) and \((4,6) .\) Show that the function of \(m\) and \(b\) that gives the sum of the squares of the vertical distances between the line and the three data points is $$ \begin{aligned} E(m, b)=&((m+b)-2)^{2}+((3 m+b)-5)^{2} \\ &+((4 m+b)-6)^{2} \end{aligned}. $$ Find the critical points of \(E\) and find the values of \(m\) and \(b\) that minimize \(E\). Graph the three data points and the best-fit line.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics 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.