/*! 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 43 Verify that \(f_{x y}=f_{y x}\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 43

Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=e^{x+y}$$

Short Answer

Expert verified
Question: Verify that the mixed partial derivatives of the function \(f(x, y) = e^{x+y}\) are equal. Answer: After calculating the mixed partial derivatives \(f_{xy}\) and \(f_{yx}\), we found that both are equal to \(e^{x+y}\). Therefore, we have verified that the mixed partial derivatives of the given function \(f(x, y) = e^{x+y}\) are equal.

Step by step solution

01

Compute \(f_x\) and \(f_y\)

First, let's compute the first partial derivatives of f(x, y) with respect to x and y. To find \(f_x\), we'll differentiate f with respect to x while treating y as a constant: $$ f_x = \frac{\partial}{\partial x} (e^{x+y}) = e^{x+y} $$ Likewise, to find \(f_y\), we'll differentiate f with respect to y while treating x as a constant: $$ f_y = \frac{\partial}{\partial y} (e^{x+y}) = e^{x+y} $$
02

Compute \(f_{xy}\) and \(f_{yx}\)

Now we will compute the second partial derivatives \(f_{xy}\) and \(f_{yx}\). To compute \(f_{xy}\), we'll differentiate \(f_x\) with respect to y while treating x as a constant: $$ f_{xy} = \frac{\partial}{\partial y} (e^{x+y}) = e^{x+y} $$ To compute \(f_{yx}\), we'll differentiate \(f_y\) with respect to x while treating y as a constant: $$ f_{yx} = \frac{\partial}{\partial x} (e^{x+y}) = e^{x+y} $$
03

Verify \(f_{xy} = f_{yx}\)

Comparing the expressions we have obtained for \(f_{xy}\) and \(f_{yx}\), we notice that they are both equal to \(e^{x+y}\). Since these expressions are equal, we can conclude that: $$ f_{xy} = f_{yx} $$ Therefore, we have verified that the mixed partial derivatives of the given function \(f(x, y) = e^{x+y}\) are equal.

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 flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1\). $$u(x, t)=e^{-t}(2 \sin x+3 \cos x)$$

Suppose that in a large group of people, a fraction \(0 \leq r \leq 1\) of the people have flu. The probability that in \(n\) random encounters you will meet at least one person with flu is \(P=f(n, r)=1-(1-r)^{n} .\) Although \(n\) is a positive integer, regard it as a positive real number. a. Compute \(f_{r}\) and \(f_{n}.\) b. How sensitive is the probability \(P\) to the flu rate \(r ?\) Suppose you meet \(n=20\) people. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.1\) to \(r=0.11(\text { with } n \text { fixed }) ?\) c. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.9\) to \(r=0.91\) with \(n=20 ?\) d. Interpret the results of parts (b) and (c).

Let \(h\) be continuous for all real numbers. a. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{x}^{y} h(s) d s\). b. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{1}^{x y} h(s) d s\).

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. \(u(x, t)=A f(x+c t)+B g(x-c t),\) where \(A\) and \(B\) are constants and \(f\) and \(g\) are twice differentiable functions of one variable

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the x y-, x z^{-}, \text {and } y z-\text {traces, when they exist. c. Sketch a graph of the surface. $$-\frac{x^{2}}{6}-24 y^{2}+\frac{z^{2}}{24}-6=0$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.