Can someone explain how the error “M” was found in this question?
$begingroup$
I'm doing error in linear approximation rn, and the equation is $M over2 $ $(x-a)^2$. To find M, I have to assume $f$ is such that $f''(x) leq M$, for each a. I'm confused on what M is and how to find it. In this example, the error for the approximation between $1 leq x leq 6$ is shown for $f(x) = sqrt x$.
So I'm confused about this line here:
$f''(x) = {-1 over 4x^{3 over 2}}$ $leq {1 over 4}$ $leq$ $M$.
Did they plug in $1$ inside of $x$ because that's the minimum?
calculus
asked Dec 11 '18 at 0:09
mingming
3365
$endgroup$
add a comment |
$begingroup$
I'm doing error in linear approximation rn, and the equation is $M over2 $ $(x-a)^2$. To find M, I have to assume $f$ is such that $f''(x) leq M$, for each a. I'm confused on what M is and how to find it. In this example, the error for the approximation between $1 leq x leq 6$ is shown for $f(x) = sqrt x$.
So I'm confused about this line here:
$f''(x) = {-1 over 4x^{3 over 2}}$ $leq {1 over 4}$ $leq$ $M$.
Did they plug in $1$ inside of $x$ because that's the minimum?
calculus
asked Dec 11 '18 at 0:09
mingming
3365
$endgroup$
$begingroup$
Yes, the second derivative $f''$ is maximized (in absolute value) at $x = 1$, when $x^{3/2}$ is minimized.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:09
add a comment |
$begingroup$
I'm doing error in linear approximation rn, and the equation is $M over2 $ $(x-a)^2$. To find M, I have to assume $f$ is such that $f''(x) leq M$, for each a. I'm confused on what M is and how to find it. In this example, the error for the approximation between $1 leq x leq 6$ is shown for $f(x) = sqrt x$.
So I'm confused about this line here:
$f''(x) = {-1 over 4x^{3 over 2}}$ $leq {1 over 4}$ $leq$ $M$.
Did they plug in $1$ inside of $x$ because that's the minimum?
calculus
asked Dec 11 '18 at 0:09
mingming
3365
$endgroup$
I'm doing error in linear approximation rn, and the equation is $M over2 $ $(x-a)^2$. To find M, I have to assume $f$ is such that $f''(x) leq M$, for each a. I'm confused on what M is and how to find it. In this example, the error for the approximation between $1 leq x leq 6$ is shown for $f(x) = sqrt x$.
So I'm confused about this line here:
$f''(x) = {-1 over 4x^{3 over 2}}$ $leq {1 over 4}$ $leq$ $M$.
Did they plug in $1$ inside of $x$ because that's the minimum?
calculus
calculus
asked Dec 11 '18 at 0:09
mingming
3365
asked Dec 11 '18 at 0:09
mingming
3365
asked Dec 11 '18 at 0:09
mingming
3365
asked Dec 11 '18 at 0:09
mingming
3365
asked Dec 11 '18 at 0:09
mingming
3365
3365
$begingroup$
Yes, the second derivative $f''$ is maximized (in absolute value) at $x = 1$, when $x^{3/2}$ is minimized.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:09
add a comment |
$begingroup$
Yes, the second derivative $f''$ is maximized (in absolute value) at $x = 1$, when $x^{3/2}$ is minimized.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:09
$begingroup$
Yes, the second derivative $f''$ is maximized (in absolute value) at $x = 1$, when $x^{3/2}$ is minimized.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:09
$begingroup$
Yes, the second derivative $f''$ is maximized (in absolute value) at $x = 1$, when $x^{3/2}$ is minimized.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:09
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%2f3034680%2fcan-someone-explain-how-the-error-m-was-found-in-this-question%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%2f3034680%2fcan-someone-explain-how-the-error-m-was-found-in-this-question%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
$begingroup$
Yes, the second derivative $f''$ is maximized (in absolute value) at $x = 1$, when $x^{3/2}$ is minimized.
$endgroup$
– T. Bongers
Dec 11 '18 at 0:09