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

(a) Prove that the equation has at least one real…

05:22

Question

Answered step-by-step

Problem 57 Hard Difficulty

(a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that contains a root.

$ \cos x = x^3 $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Oswaldo Jiménez
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Oswaldo Jiménez

Numerade Educator

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

More Answers

05:47

Daniel Jaimes

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 5

Continuity

Related Topics

Limits

Derivatives

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Grace He
Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Join Course
Recommended Videos

0:00

(a) Prove that the equatio…

02:51

(a) Prove that the equatio…

05:22

(a) Prove that the equatio…

01:04

$51-52$ (a) Prove that the…

08:14

$55-56$ (a) Prove that the…

09:14

$43-44=$ (a) Prove that th…

03:34

Consider the following cos…

03:26

(a) Prove that the equatio…

01:07

$51-52$ (a) Prove that the…

01:33

(a) Prove that the equatio…

04:03

$57-58$ (a) Prove that the…

04:24

(a) Prove that the equatio…

08:00

$55-56$ (a) Prove that the…

Watch More Solved Questions in Chapter 2

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

Video Transcript

you park A. We're going to prove that the equation X cubed equals co sign of X has at least one real root barbie. We assure the calculator to find an interval of length 0.01 that contains a route. So we are going to define for that the function F equal X cube minus co sign of X. And we are going to consider that function. So first about before saying that to function less say the following. We know from this we want to final a root of this equation. That is Value Acts for which ethical zero. Mhm. So you say that If the function is secret zero we can say that X cube is equal to co sign of accessories. A root of this function is a solution to the given equation. But in fact what is important here is to notice that this implies that the absolute value of X cubed is equal to the absolute value of cosine of X. And then the answer value cube of X. Okay the sequence of the absolute value of coastline of eggs and we know that the delivery of coastline is less than or equal to one. This will implies that the absolute value of X cube is less than or equal to one. And from these taking spirit of size we know that this implies that the absolute value of X is less than or equal to one. That is X belongs to the interval negative 11 close both and points. And that means that the solutions to this equation here can only be in this interval. So we we restrict our attention and this interval. So that's why we say that. Uh huh. That the function F is defined inbred interval. Okay. Because there only there we can find a solution of the aggression. That's one thing. The other thing we can say before given a solution to the problem is that if we sketch the graph of these functions that is the function X cubed and co sign effects, you know, excuse is something like this. Good and co sign is something like this. And here we can have some kind of doubt about the behavior but there's no doubt really because we can say something here as you can see the function Co sign is positive between zero and the value by half here and is decreasing between Syrian by half. The function x cube is increasing is positive and increasing between zero and plus infinity. It means they cut in one point. They intersect in one point at one point here let's say this point here and it seems to be the only positive solution to that equation that point here. But from the left side we can say something here. Is that the function X cube is negative while the co sign is positive up to negative by half. So if we look at the value of the function X cuba by half then we can say that the graph will be way below the co sign of faith and that's the case because if you calculate um that is by health cube, I have cube predicting here. He's about -38 75 3.875. And means that at this point where cause and begins to be negative The function excuse has a value of negative 3.8 because go sign can be uh Up to the value -1. That cannot be lower than that. It's impossible for the to your curves to intersect on the negative values of X. So there is no solution in danger of -10. So we know Moreover that the solution is only interval 01. So we can say this here uh four X belonging to a natty 10 which is a negative part of the interval. We know there lies the solution for X. In that interval we know that co sign of X is positive because that's another way of saying different from what I see here that is, we know that -1 which is uh the minimum value that can attain the solution of the equation at that point. Co sign is a positive value has a positive value. That's because the co sign, I'd see it again here if you want here we have negative plus by half and negative by half about 1.57 That is -1 is like this and there we have co sign with the positive value. So being cause I'm positive on this interval and X cube negative and that interval mhm. These two things together imply that X cubed equals co sign of eggs has no solution on zero negative 10 mm. Mhm Mhm. And then knowing that the solution can only be on the interval negative 11. And that there is no solution on the interval negative 10, then the equation X cubed equals co sign of eggs can have solution only in sierra one. The solution or solutions if there is more than one lies there. Okay so all this we have said without calculating anything that is we are only analyzing the functions through the definitions of the function in this case using the properties of coz i the actual values less than or equal to one. And here is in the science of the functions on the intervals With only that we can we have constrained our attention to the interval 01 now because if this continues. So f from 01 to the real numbers, the finest X cubed minus cosine of X is continuous. What we need to do is to find an interval where the function changes sign but the interval will be the same interval here. We need a close intervals of course and this close interval and let's see that the function changes sign there. F at zero is equal to zero cubed minus go sign of zero, that is 0 -1 that is -1 which is negative. an effort. One is one cubed minus co sign of one. That is one minus co sign of one. And co sign of one. We know is positive number but less than one. If we see the graph here of co sign here here is the the maximum value of course in which is one. The interval native by health by half and uh mhm we can say that there you hear is probably half and we see clearly that At the value one which is decide here, which is less than by half. We know that the co sign of that one is positive is positive and it's less than one because Co sign is only one at 0. So this number here is a positive number Being close in one. Less than one. So the function changes sign at the end points of the club of the close interval and the function is continuous. We know from that that the functions can have zero from the internal zero water. And uh moreover, the route is inside the intervals not the endpoints because at the end points The functions in the zero. So let's write it here. So F is continuous. Only closed interval. Yeah. And changes sign on the endpoints of these interval. More precisely At zero. The functions negative and a one The function is bust. Then there exist at least a value eggs on the open interval such that F of x equals zero. That is x cubed minus cosine of X equals zero. That is exc your equal co sign of exit is there is at least one solution of the equation which we wanted to prove. So that the main result. Yeah. So we have proved the first part. That is there is at least one real wood for the equation, the human equation. And we know more over that. The solution is on the Internet, In fact, if we draw, if we sketch the graphs of the functions together, we can see that the solution is uh huh, Closer to one than 0 mm But now let's calculate that. We got to find an interval of length mhm And an interval of length 0.01 that contains a route to the equation. So for that we do kind of by section method which consists in the following we can we start with the interval 01 cutting in house by the mid 0 .1. And we know that we can apply the same theorem here for these two interval studies, we evaluate the function at the end points in this case. We have evaluated already at zero already at one. We know that at zero remember is negative and that one is positive And now we evaluated zero points uh 0.5. We do that you get yeah -0.75 or less. So at 0.1 the function is negative and so the change of sign now happened on the interval 0.5 one. Yeah. So applying the same theory, we know that there is a route Now in the interval 0.51 and we do the same again. We could the interval zero point I want in the half And that's a midpoint, there is 0.75. And you always look at the interval where the sign changes and going for this way and this call by section method and with that we do that until the interval which contains the solution is office size we want that is the precision is given by the length of the interval containing the world. In this case if we do this procedure uh sufficiently large or for many durations we can find yeah very good estimation of the route. And in this case we we can find the following value. So I root for the equation equals zero is zero point 86 54 74 033 10 16 13. Here we have used a very good precision that is. We stop the process when the interval world Containing the route was about the length of 10-2014. So it's very good approximation. And knowing that the route is about that value, we can say that an interval containing the route And of land, 0.01 is the following. So the interval 0.86, contains the route because these values a little bit greater than this and less than this and The length of this interval is 0.01. We always can do that. That is we do by section procedure or method up to a very good precision. And then we find an interval with give a measure that contains the route. We can give even and even smaller than this interval that contains the room because we have here in fact Opposition of 10 to the -14 About that. So we can do we can give intervals of of smaller size and this containing Taru. And as a final remark, we can say that this is the only road of this equation On the Interval 01. We can verify that for example, graphically if we sketch the graph of X cubed and co sign of eggs over 01. And we can verify that is the only world

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
126
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
62
Hosted by: Alonso M
See More

Related Topics

Limits

Derivatives

Top Calculus 1 / AB Educators
Grace He

Numerade Educator

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Join Course
Recommended Videos

0:00

(a) Prove that the equation has at least one real root. (b) Use your calculato…

02:51

(a) Prove that the equation has at least one real root. (b) Use your calculato…

05:22

(a) Prove that the equation has at least one real root. (b) Use your calculato…

01:04

$51-52$ (a) Prove that the equation has at least one real root. (b) Use your ca…

08:14

$55-56$ (a) Prove that the equation has at least one real root. (b) Use your ca…

09:14

$43-44=$ (a) Prove that the equation has at least one real root (b) Use your c…

03:34

Consider the following cos(x) = x3 (a) Prove that the equation has at least one…

03:26

(a) Prove that the equation has at least one real root. (b) Use your graphing …

01:07

$51-52$ (a) Prove that the equation has at least one real root. (b) Use your ca…

01:33

(a) Prove that the equation has at least one real root. (b) Use your graphing d…

04:03

$57-58$ (a) Prove that the equation has at least one real root. (b) Use your ca…

04:24

(a) Prove that the equation has at least one real root. (b) Use your graphing …

08:00

$55-56$ (a) Prove that the equation has at least one real root. (b) Use your ca…
Additional Mathematics Questions

03:21

Test: 1 1 TEST
1 { 1

07:52

An evergreen nursery usually sells type of shrub after years of growth and s…

01:05

8 ft
6 f
1 9

12:51

company maintains three offices
certain region, each staffed by two emplo…

06:47

In this question, you will estimate the value of the integral 6 ce 1 dx 3 us…

04:37

A woman rides a bicycle for 2 hours and travels 32 kilometers (about 20 mile…

03:32

Evaluate each of the following standard values exactly (No calculators this …

02:16

The chi-square distribution used find confidence interval for one population…

00:58

Solve for X in the following equation: @o ogzo(1Ox2) = 5

03:26

For the subspace below; (a) find a basis for the subspace, and (b) state the…

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