Mixing
Find the number of points of discontinuity
$begingroup$
Question Let $f(x)=[sin x+cos x]$ where $x in left(0,2piright)$ and $left[cdotright]$ denotes the greatest integer function.The number of points
of discontinuity of $f(x)$ is
$left(aright)6$
$left(bright)5$
$left(cright)4$
$left(dright)$3
My Approach $f(x)=left[sin x + cos xright] = left[sqrt{2}sinleft(x+frac{pi}{4}right)right]$
calculus continuity
asked Oct 30 '17 at 4:42
Kislay TripathiKislay Tripathi
447224
$endgroup$
|
show 3 more comments
$begingroup$
Question Let $f(x)=[sin x+cos x]$ where $x in left(0,2piright)$ and $left[cdotright]$ denotes the greatest integer function.The number of points
of discontinuity of $f(x)$ is
$left(aright)6$
$left(bright)5$
$left(cright)4$
$left(dright)$3
My Approach $f(x)=left[sin x + cos xright] = left[sqrt{2}sinleft(x+frac{pi}{4}right)right]$
calculus continuity
asked Oct 30 '17 at 4:42
Kislay TripathiKislay Tripathi
447224
$endgroup$
1
$begingroup$
What do you know about the discontinuities of the greatest integer function?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 4:54
2
$begingroup$
@астонвіллаолофмэллбэрг greatest integer functions are discont. at integral values of x
$endgroup$
– Kislay Tripathi
Oct 30 '17 at 4:56
1
$begingroup$
A little comment about MathJax: theleftandrightcommands aren't necessary for every bracket. They are useful for adjusting the size of the brackets to what's in them. For example, $$(frac{a}{b})$$ is the result of(frac{a}{b})whereas $$left(frac{a}{b}right)$$ is the result ofleft(frac{a}{b}right)
$endgroup$
– Theo Bendit
Oct 30 '17 at 4:59
1
$begingroup$
Ok, so where does this function attain integral values? There you shall find discontinuities, right?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:00
2
$begingroup$
@KislayTripathi If you are correct, then discontinuities must occur at precisely these points. How many of them are there?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:04
|
show 3 more comments
$begingroup$
Question Let $f(x)=[sin x+cos x]$ where $x in left(0,2piright)$ and $left[cdotright]$ denotes the greatest integer function.The number of points
of discontinuity of $f(x)$ is
$left(aright)6$
$left(bright)5$
$left(cright)4$
$left(dright)$3
My Approach $f(x)=left[sin x + cos xright] = left[sqrt{2}sinleft(x+frac{pi}{4}right)right]$
calculus continuity
asked Oct 30 '17 at 4:42
Kislay TripathiKislay Tripathi
447224
$endgroup$
Question Let $f(x)=[sin x+cos x]$ where $x in left(0,2piright)$ and $left[cdotright]$ denotes the greatest integer function.The number of points
of discontinuity of $f(x)$ is
$left(aright)6$
$left(bright)5$
$left(cright)4$
$left(dright)$3
My Approach $f(x)=left[sin x + cos xright] = left[sqrt{2}sinleft(x+frac{pi}{4}right)right]$
calculus continuity
calculus continuity
asked Oct 30 '17 at 4:42
Kislay TripathiKislay Tripathi
447224
asked Oct 30 '17 at 4:42
Kislay TripathiKislay Tripathi
447224
edited Dec 1 '17 at 6:21
user99914
asked Oct 30 '17 at 4:42
Kislay TripathiKislay Tripathi
447224
asked Oct 30 '17 at 4:42
Kislay TripathiKislay Tripathi
447224
asked Oct 30 '17 at 4:42
Kislay TripathiKislay Tripathi
447224
447224
1
$begingroup$
What do you know about the discontinuities of the greatest integer function?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 4:54
2
$begingroup$
@астонвіллаолофмэллбэрг greatest integer functions are discont. at integral values of x
$endgroup$
– Kislay Tripathi
Oct 30 '17 at 4:56
1
$begingroup$
A little comment about MathJax: theleftandrightcommands aren't necessary for every bracket. They are useful for adjusting the size of the brackets to what's in them. For example, $$(frac{a}{b})$$ is the result of(frac{a}{b})whereas $$left(frac{a}{b}right)$$ is the result ofleft(frac{a}{b}right)
$endgroup$
– Theo Bendit
Oct 30 '17 at 4:59
1
$begingroup$
Ok, so where does this function attain integral values? There you shall find discontinuities, right?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:00
2
$begingroup$
@KislayTripathi If you are correct, then discontinuities must occur at precisely these points. How many of them are there?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:04
|
show 3 more comments
1
$begingroup$
What do you know about the discontinuities of the greatest integer function?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 4:54
2
$begingroup$
@астонвіллаолофмэллбэрг greatest integer functions are discont. at integral values of x
$endgroup$
– Kislay Tripathi
Oct 30 '17 at 4:56
1
$begingroup$
A little comment about MathJax: theleftandrightcommands aren't necessary for every bracket. They are useful for adjusting the size of the brackets to what's in them. For example, $$(frac{a}{b})$$ is the result of(frac{a}{b})whereas $$left(frac{a}{b}right)$$ is the result ofleft(frac{a}{b}right)
$endgroup$
– Theo Bendit
Oct 30 '17 at 4:59
1
$begingroup$
Ok, so where does this function attain integral values? There you shall find discontinuities, right?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:00
2
$begingroup$
@KislayTripathi If you are correct, then discontinuities must occur at precisely these points. How many of them are there?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:04
1
1
$begingroup$
What do you know about the discontinuities of the greatest integer function?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 4:54
$begingroup$
What do you know about the discontinuities of the greatest integer function?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 4:54
2
2
$begingroup$
@астонвіллаолофмэллбэрг greatest integer functions are discont. at integral values of x
$endgroup$
– Kislay Tripathi
Oct 30 '17 at 4:56
$begingroup$
@астонвіллаолофмэллбэрг greatest integer functions are discont. at integral values of x
$endgroup$
– Kislay Tripathi
Oct 30 '17 at 4:56
1
1
$begingroup$
A little comment about MathJax: the
left and right commands aren't necessary for every bracket. They are useful for adjusting the size of the brackets to what's in them. For example, $$(frac{a}{b})$$ is the result of (frac{a}{b}) whereas $$left(frac{a}{b}right)$$ is the result of left(frac{a}{b}right)$endgroup$
– Theo Bendit
Oct 30 '17 at 4:59
$begingroup$
A little comment about MathJax: the
left and right commands aren't necessary for every bracket. They are useful for adjusting the size of the brackets to what's in them. For example, $$(frac{a}{b})$$ is the result of (frac{a}{b}) whereas $$left(frac{a}{b}right)$$ is the result of left(frac{a}{b}right)$endgroup$
– Theo Bendit
Oct 30 '17 at 4:59
1
1
$begingroup$
Ok, so where does this function attain integral values? There you shall find discontinuities, right?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:00
$begingroup$
Ok, so where does this function attain integral values? There you shall find discontinuities, right?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:00
2
2
$begingroup$
@KislayTripathi If you are correct, then discontinuities must occur at precisely these points. How many of them are there?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:04
$begingroup$
@KislayTripathi If you are correct, then discontinuities must occur at precisely these points. How many of them are there?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:04
|
show 3 more comments
1 Answer
1
active
oldest
votes
$begingroup$
So you have written that $sin x + cos x = sqrt 2 sin(x + frac pi 4)$.
When does this take integral values? Well, we know that $- sqrt 2 leq sqrt 2 sin(x + frac pi 4) leq sqrt 2$, so it can take precisely three integral values, namely $0, pm 1$. There is a discontinuity whenever it takes one of these values.
We note that it takes the value $1$ at $x=0,2pi$, but these are outside our interval. Next, if $sin(x+ frac pi 4) = frac{1}{sqrt 2}$, then $x = frac {pi}{2}$. If $sin(x + frac pi 4) = 0$, then $x = frac{3pi}{4}, frac{7pi}{4}$, and finally, if $sin(x+ frac pi 4) = frac 1{sqrt 2}$, then $x = pi,frac{3pi}{2}$. So ,there are five values at which discontinuities exist, and I will confirm this by getting you the graph between the points:
where you can see that there are five discontinuities at the points of mention.
answered Oct 30 '17 at 5:22
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.8k33477
$endgroup$
$begingroup$
Your link is broken. I imagine it pointed to something like this.
$endgroup$
– mephistolotl
Dec 1 '17 at 6:30
add a comment |
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%2f2496050%2ffind-the-number-of-points-of-discontinuity%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
So you have written that $sin x + cos x = sqrt 2 sin(x + frac pi 4)$.
When does this take integral values? Well, we know that $- sqrt 2 leq sqrt 2 sin(x + frac pi 4) leq sqrt 2$, so it can take precisely three integral values, namely $0, pm 1$. There is a discontinuity whenever it takes one of these values.
We note that it takes the value $1$ at $x=0,2pi$, but these are outside our interval. Next, if $sin(x+ frac pi 4) = frac{1}{sqrt 2}$, then $x = frac {pi}{2}$. If $sin(x + frac pi 4) = 0$, then $x = frac{3pi}{4}, frac{7pi}{4}$, and finally, if $sin(x+ frac pi 4) = frac 1{sqrt 2}$, then $x = pi,frac{3pi}{2}$. So ,there are five values at which discontinuities exist, and I will confirm this by getting you the graph between the points:
where you can see that there are five discontinuities at the points of mention.
answered Oct 30 '17 at 5:22
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.8k33477
$endgroup$
$begingroup$
Your link is broken. I imagine it pointed to something like this.
$endgroup$
– mephistolotl
Dec 1 '17 at 6:30
add a comment |
$begingroup$
So you have written that $sin x + cos x = sqrt 2 sin(x + frac pi 4)$.
When does this take integral values? Well, we know that $- sqrt 2 leq sqrt 2 sin(x + frac pi 4) leq sqrt 2$, so it can take precisely three integral values, namely $0, pm 1$. There is a discontinuity whenever it takes one of these values.
We note that it takes the value $1$ at $x=0,2pi$, but these are outside our interval. Next, if $sin(x+ frac pi 4) = frac{1}{sqrt 2}$, then $x = frac {pi}{2}$. If $sin(x + frac pi 4) = 0$, then $x = frac{3pi}{4}, frac{7pi}{4}$, and finally, if $sin(x+ frac pi 4) = frac 1{sqrt 2}$, then $x = pi,frac{3pi}{2}$. So ,there are five values at which discontinuities exist, and I will confirm this by getting you the graph between the points:
where you can see that there are five discontinuities at the points of mention.
answered Oct 30 '17 at 5:22
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.8k33477
$endgroup$
$begingroup$
Your link is broken. I imagine it pointed to something like this.
$endgroup$
– mephistolotl
Dec 1 '17 at 6:30
add a comment |
$begingroup$
So you have written that $sin x + cos x = sqrt 2 sin(x + frac pi 4)$.
When does this take integral values? Well, we know that $- sqrt 2 leq sqrt 2 sin(x + frac pi 4) leq sqrt 2$, so it can take precisely three integral values, namely $0, pm 1$. There is a discontinuity whenever it takes one of these values.
We note that it takes the value $1$ at $x=0,2pi$, but these are outside our interval. Next, if $sin(x+ frac pi 4) = frac{1}{sqrt 2}$, then $x = frac {pi}{2}$. If $sin(x + frac pi 4) = 0$, then $x = frac{3pi}{4}, frac{7pi}{4}$, and finally, if $sin(x+ frac pi 4) = frac 1{sqrt 2}$, then $x = pi,frac{3pi}{2}$. So ,there are five values at which discontinuities exist, and I will confirm this by getting you the graph between the points:
where you can see that there are five discontinuities at the points of mention.
answered Oct 30 '17 at 5:22
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.8k33477
$endgroup$
So you have written that $sin x + cos x = sqrt 2 sin(x + frac pi 4)$.
When does this take integral values? Well, we know that $- sqrt 2 leq sqrt 2 sin(x + frac pi 4) leq sqrt 2$, so it can take precisely three integral values, namely $0, pm 1$. There is a discontinuity whenever it takes one of these values.
We note that it takes the value $1$ at $x=0,2pi$, but these are outside our interval. Next, if $sin(x+ frac pi 4) = frac{1}{sqrt 2}$, then $x = frac {pi}{2}$. If $sin(x + frac pi 4) = 0$, then $x = frac{3pi}{4}, frac{7pi}{4}$, and finally, if $sin(x+ frac pi 4) = frac 1{sqrt 2}$, then $x = pi,frac{3pi}{2}$. So ,there are five values at which discontinuities exist, and I will confirm this by getting you the graph between the points:
where you can see that there are five discontinuities at the points of mention.
answered Oct 30 '17 at 5:22
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.8k33477
edited Jan 18 at 6:16
answered Oct 30 '17 at 5:22
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.8k33477
answered Oct 30 '17 at 5:22
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.8k33477
answered Oct 30 '17 at 5:22
астон вілла олоф мэллбэргастон вілла олоф мэллбэрг
38.8k33477
38.8k33477
$begingroup$
Your link is broken. I imagine it pointed to something like this.
$endgroup$
– mephistolotl
Dec 1 '17 at 6:30
add a comment |
$begingroup$
Your link is broken. I imagine it pointed to something like this.
$endgroup$
– mephistolotl
Dec 1 '17 at 6:30
$begingroup$
Your link is broken. I imagine it pointed to something like this.
$endgroup$
– mephistolotl
Dec 1 '17 at 6:30
$begingroup$
Your link is broken. I imagine it pointed to something like this.
$endgroup$
– mephistolotl
Dec 1 '17 at 6:30
add a comment |
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%2f2496050%2ffind-the-number-of-points-of-discontinuity%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
$begingroup$
What do you know about the discontinuities of the greatest integer function?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 4:54
2
$begingroup$
@астонвіллаолофмэллбэрг greatest integer functions are discont. at integral values of x
$endgroup$
– Kislay Tripathi
Oct 30 '17 at 4:56
1
$begingroup$
A little comment about MathJax: the
leftandrightcommands aren't necessary for every bracket. They are useful for adjusting the size of the brackets to what's in them. For example, $$(frac{a}{b})$$ is the result of(frac{a}{b})whereas $$left(frac{a}{b}right)$$ is the result ofleft(frac{a}{b}right)$endgroup$
– Theo Bendit
Oct 30 '17 at 4:59
1
$begingroup$
Ok, so where does this function attain integral values? There you shall find discontinuities, right?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:00
2
$begingroup$
@KislayTripathi If you are correct, then discontinuities must occur at precisely these points. How many of them are there?
$endgroup$
– астон вілла олоф мэллбэрг
Oct 30 '17 at 5:04