Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

The Fresnel function $ S $ was defined in Example…

06:52

Question

Answered step-by-step

Problem 70 Hard Difficulty

The error function $$ \displaystyle \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int^x_0 e^{-t^2} \, dt $$ is used in probability, statistics, and engineering.
(a) Show that $ \displaystyle \int^b_a e^{-t^2} \, dt = \frac{1}{2} \sqrt{\pi} [ \text{erf}(b) - \text{erf}(a) ] $.
(b) Show that the function $ y = e^{x^2} \text{erf}(x) $ satisfies the differential equation $ y' = 2xy + 2/\sqrt{\pi} $.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Chris Trentman
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Chris Trentman

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

01:57

Frank Lin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 5

Integrals

Section 3

The Fundamental Theorem of Calculus

Related Topics

Integrals

Integration

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Calculus 1 / AB Courses

Lectures

Video Thumbnail

05:53

Integrals - Intro

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

Video Thumbnail

40:35

Area Under Curves - Overview

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

Join Course
Recommended Videos

03:59

The error function
$$\o…

05:58

The error function

01:21

The error function
$$\o…

03:32

The error function, erf $(…

02:41

The error function, $\oper…

01:15

The error function erflx) …

02:44

point) The error function,…

Watch More Solved Questions in Chapter 5

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Problem 81
Problem 82
Problem 83
Problem 84
Problem 85
Problem 86

Video Transcript

It's like every were given the error function which is used in probability statistics and engineering. The error function is defined to be Two over the square root of pi times the integral from 0 to X of each. The negative T squared D T is I'm afraid of action. In part, they were asked to show that the integral from A to B of each of the negative T squared DT can be written in terms of error functions. So the integral from A to B. Eat a negative t squared DT. Well, who wants to the kids? If you wanted those two fellows for instance, Big favorite system Harleys. We can rewrite this as the integral from zero to be of E to the negative T squared DT plus the integral from a +20 of E to the negative T squared Bt. Using integral properties, we can rewrite this as everything I have Integral from 0 to be. In fact, I'll write this as route high over to here times two of a root pie. And the integral from zero to be of E. To the negative T square D. T minus two of the Root Pie. Times. In the girl from zero to a. Of each of the negative T squared DT. The notice that by definition this is rude pi over two times error function evaluated at B minus the error function evaluated at a. Yeah, this is what we wanted to show for part egg. Mhm. Yeah. Then in part B we're trying to lose weight from china for us to show that the function Y equals E. To the X squared times the error function evaluated at X satisfies the differential equation. Why prime equals two X one Plus two over the square root of Pi. Well, all we have to do is simply plugs our function into this differential equation on both sides. So first let's find the left hand side. Wide prime. Buy the product rule. This is a derivative of E. V. X squared which is uh two X times E. To the X squared times the error function evaluated at X plus E. To the X squared times the derivative of the air function which this is going to be to overrule pie. And then by the fundamental theorem of calculus, the derivative of the error function is two of the route high times each of the negative X squared. Since the function E. To the negative T squared is continuous for all values of T. Yeah. Mhm. Yeah. As he Hard that's hard. eight are they hard up? Here's the thing bigger animals actually lost. And sure now that's the left hand side. The only way to see used. Do you do you do? I'm up party. Yeah. So you know that saying the house you can join now joined right now by partly my sister. So actually could simplify first and this is equal to two X times E. To the X squared air function of X plus. And this is simply two of her route pie Times each of the zero which is the same as one. Now we can rearrange since error function is Y over E. To the X squared. This is two X times not during not during the ship wow. Where we're just either the X squared error function. X. This is why This is two x. Y Plus two Over Group I. No and this is the right hand side of the equation. Therefore the function is a solution to this differential equation. That's probably enterprise. No that's not. Oh it's coming. Oh I can tell you.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
143
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Integrals

Integration

Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Calculus 1 / AB Courses

Lectures

Video Thumbnail

05:53

Integrals - Intro

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

Video Thumbnail

40:35

Area Under Curves - Overview

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

Join Course
Recommended Videos

03:59

The error function $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e…

05:58

The error function $$\operatorname{erf}(\mathrm{x})=\frac{2}{\sqrt{\pi}} \in…

01:21

The error function $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e…

03:32

The error function, erf $(x),$ is defined by $\operatorname{erf}(x)=\frac{2}{\s…

02:41

The error function, $\operatorname{Erf}(x),$ is defined for $x \geq 0$ by $$\op…

01:15

The error function erflx) = Xf e-t2 dt is a special function from statistics. F…

02:44

point) The error function, or erf, is defied as erf(x) = 0 2 dt. The error func…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started