How to find fixed point without graph approach?
$begingroup$
The function tan(x)=x has
1. Unique fixed point
2. No fixed point
3. Infinite many fixed points
4. More than one but finitely many fixed points.
My attempt: I have solved this problem by graphical approach. The intersecting points of the function y=tan(x) and the line y=x are fixed points that I obtained which are infinite. But my problem is that how to solve it without graphical approach and how to discard options. Is there any general approach to tackle this type of problems?
fixedpoints
asked Dec 24 '18 at 22:17
MathforjobMathforjob
164
$endgroup$
add a comment |
$begingroup$
The function tan(x)=x has
1. Unique fixed point
2. No fixed point
3. Infinite many fixed points
4. More than one but finitely many fixed points.
My attempt: I have solved this problem by graphical approach. The intersecting points of the function y=tan(x) and the line y=x are fixed points that I obtained which are infinite. But my problem is that how to solve it without graphical approach and how to discard options. Is there any general approach to tackle this type of problems?
fixedpoints
asked Dec 24 '18 at 22:17
MathforjobMathforjob
164
$endgroup$
1
$begingroup$
There's no general approach. It all depends on the function. Sometimes, setting $f(x)=x$ leads to a nice algebraic solution, or sometimes rote argumentation is needed. Other times, a combination of the two works. In your case, I would simply argue that since $tan(x)$ takes on all real values continuously on the domain $(npi-fracpi2,npi+fracpi2),;ninmathbb Z$, that shows that there is exactly one fixed point per value in $mathbb Z$.
$endgroup$
– Don Thousand
Dec 24 '18 at 22:23
$begingroup$
If x is non negative real number and it is not an odd multiple of pi/2 then what about fixed points.
$endgroup$
– Mathforjob
Dec 24 '18 at 22:45
add a comment |
$begingroup$
The function tan(x)=x has
1. Unique fixed point
2. No fixed point
3. Infinite many fixed points
4. More than one but finitely many fixed points.
My attempt: I have solved this problem by graphical approach. The intersecting points of the function y=tan(x) and the line y=x are fixed points that I obtained which are infinite. But my problem is that how to solve it without graphical approach and how to discard options. Is there any general approach to tackle this type of problems?
fixedpoints
asked Dec 24 '18 at 22:17
MathforjobMathforjob
164
$endgroup$
The function tan(x)=x has
1. Unique fixed point
2. No fixed point
3. Infinite many fixed points
4. More than one but finitely many fixed points.
My attempt: I have solved this problem by graphical approach. The intersecting points of the function y=tan(x) and the line y=x are fixed points that I obtained which are infinite. But my problem is that how to solve it without graphical approach and how to discard options. Is there any general approach to tackle this type of problems?
fixedpoints
fixedpoints
asked Dec 24 '18 at 22:17
MathforjobMathforjob
164
asked Dec 24 '18 at 22:17
MathforjobMathforjob
164
asked Dec 24 '18 at 22:17
MathforjobMathforjob
164
asked Dec 24 '18 at 22:17
MathforjobMathforjob
164
asked Dec 24 '18 at 22:17
MathforjobMathforjob
164
164
1
$begingroup$
There's no general approach. It all depends on the function. Sometimes, setting $f(x)=x$ leads to a nice algebraic solution, or sometimes rote argumentation is needed. Other times, a combination of the two works. In your case, I would simply argue that since $tan(x)$ takes on all real values continuously on the domain $(npi-fracpi2,npi+fracpi2),;ninmathbb Z$, that shows that there is exactly one fixed point per value in $mathbb Z$.
$endgroup$
– Don Thousand
Dec 24 '18 at 22:23
$begingroup$
If x is non negative real number and it is not an odd multiple of pi/2 then what about fixed points.
$endgroup$
– Mathforjob
Dec 24 '18 at 22:45
add a comment |
1
$begingroup$
There's no general approach. It all depends on the function. Sometimes, setting $f(x)=x$ leads to a nice algebraic solution, or sometimes rote argumentation is needed. Other times, a combination of the two works. In your case, I would simply argue that since $tan(x)$ takes on all real values continuously on the domain $(npi-fracpi2,npi+fracpi2),;ninmathbb Z$, that shows that there is exactly one fixed point per value in $mathbb Z$.
$endgroup$
– Don Thousand
Dec 24 '18 at 22:23
$begingroup$
If x is non negative real number and it is not an odd multiple of pi/2 then what about fixed points.
$endgroup$
– Mathforjob
Dec 24 '18 at 22:45
1
1
$begingroup$
There's no general approach. It all depends on the function. Sometimes, setting $f(x)=x$ leads to a nice algebraic solution, or sometimes rote argumentation is needed. Other times, a combination of the two works. In your case, I would simply argue that since $tan(x)$ takes on all real values continuously on the domain $(npi-fracpi2,npi+fracpi2),;ninmathbb Z$, that shows that there is exactly one fixed point per value in $mathbb Z$.
$endgroup$
– Don Thousand
Dec 24 '18 at 22:23
$begingroup$
There's no general approach. It all depends on the function. Sometimes, setting $f(x)=x$ leads to a nice algebraic solution, or sometimes rote argumentation is needed. Other times, a combination of the two works. In your case, I would simply argue that since $tan(x)$ takes on all real values continuously on the domain $(npi-fracpi2,npi+fracpi2),;ninmathbb Z$, that shows that there is exactly one fixed point per value in $mathbb Z$.
$endgroup$
– Don Thousand
Dec 24 '18 at 22:23
$begingroup$
If x is non negative real number and it is not an odd multiple of pi/2 then what about fixed points.
$endgroup$
– Mathforjob
Dec 24 '18 at 22:45
$begingroup$
If x is non negative real number and it is not an odd multiple of pi/2 then what about fixed points.
$endgroup$
– Mathforjob
Dec 24 '18 at 22:45
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051688%2fhow-to-find-fixed-point-without-graph-approach%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051688%2fhow-to-find-fixed-point-without-graph-approach%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
There's no general approach. It all depends on the function. Sometimes, setting $f(x)=x$ leads to a nice algebraic solution, or sometimes rote argumentation is needed. Other times, a combination of the two works. In your case, I would simply argue that since $tan(x)$ takes on all real values continuously on the domain $(npi-fracpi2,npi+fracpi2),;ninmathbb Z$, that shows that there is exactly one fixed point per value in $mathbb Z$.
$endgroup$
– Don Thousand
Dec 24 '18 at 22:23
$begingroup$
If x is non negative real number and it is not an odd multiple of pi/2 then what about fixed points.
$endgroup$
– Mathforjob
Dec 24 '18 at 22:45