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high school inequality |(√(sinx)+1)^2−(√(sina)+1)^2|<b

solakis

Active member
Dec 9, 2012
304
Given 0<a<π/2 , b>0 find a c>0 such that :

for all ,x : if 0<x<π/2 and |x-a|<c ,then \(\displaystyle |(\sqrt sinx +1)^2-(\sqrt sin a +1)^2|<b\)
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,684
Given $0<a<\pi/2$, $b>0$, find a $c>0$ such that :

for all $x$ : if $0<x<\pi/2$ and $|x-a|<c $, then \(\displaystyle |(\sqrt {\sin x} +1)^2-(\sqrt {\sin a} +1)^2|<b\).
First step: $|\sin x - \sin a| < |x-a|$. That is essentially the Mean Value theorem, which says that $\sin x - \sin a = (x-a)\cos y$ for some $y$ between $a$ and $x$.

Next, $|(\sqrt {\sin x} +1)^2-(\sqrt {\sin a} +1)^2| = |\sin x + 2\sqrt{\sin x} + 1 - (\sin a + 2\sqrt{\sin a} + 1)| = |(\sin x -\sin a) + 2(\sqrt{\sin x} - \sqrt{\sin a})| \leqslant |(\sin x -\sin a)| + 2|\sqrt{\sin x} - \sqrt{\sin a}|.$

To get an estimate for $\sqrt{\sin x} - \sqrt{\sin a}$, multiply top and bottom by $\sqrt{\sin x} + \sqrt{\sin a}$: $$\sqrt{\sin x} - \sqrt{\sin a} = \frac{(\sqrt{\sin x} - \sqrt{\sin a})(\sqrt{\sin x} + \sqrt{\sin a})}{\sqrt{\sin x} + \sqrt{\sin a}} = \frac{\sin x - \sin a}{\sqrt{\sin x} + \sqrt{\sin a}},$$ from which \(\displaystyle |\sqrt{\sin x} - \sqrt{\sin a}| < \frac{|x - a|}{ \sqrt{\sin a}}.\)

Then \(\displaystyle |(\sqrt {\sin x} +1)^2-(\sqrt {\sin a} +1)^2| \leqslant |(\sin x -\sin a)| + 2|\sqrt{\sin x} - \sqrt{\sin a}| < |x-a|\left(1 + \frac1{ \sqrt{\sin a}}\right).\)

We want this to be less than $b$. So take \(\displaystyle c = \frac b{1 + \frac1{ \sqrt{\sin a}}}\). Then $$|\sqrt{\sin x} - \sqrt{\sin a}| < |x-a|\left(1 + \frac1{ \sqrt{\sin a}}\right) < c\left(1 + \frac1{ \sqrt{\sin a}}\right) = b.$$
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,033
Given 0<a<π/2 , b>0 find a c>0 such that :

for all ,x : if 0<x<π/2 and |x-a|<c ,then \(\displaystyle |(\sqrt sinx +1)^2-(\sqrt sin a +1)^2|<b\)
Hi solakis,

Is this a question that you want help with or know the answer to already and want to challenge others? Either way is fine but if it's the former then I'll move the thread to a different part of the forum.
 

solakis

Active member
Dec 9, 2012
304
Hi solakis,

Is this a question that you want help with or know the answer to already and want to challenge others? Either way is fine but if it's the former then I'll move the thread to a different part of the forum.

This is a challenge question
 

solakis

Active member
Dec 9, 2012
304
First step: $|\sin x - \sin a| < |x-a|$. That is essentially the Mean Value theorem, which says that $\sin x - \sin a = (x-a)\cos y$ for some $y$ between $a$ and $x$.

Next, $|(\sqrt {\sin x} +1)^2-(\sqrt {\sin a} +1)^2| = |\sin x + 2\sqrt{\sin x} + 1 - (\sin a + 2\sqrt{\sin a} + 1)| = |(\sin x -\sin a) + 2(\sqrt{\sin x} - \sqrt{\sin a})| \leqslant |(\sin x -\sin a)| + 2|\sqrt{\sin x} - \sqrt{\sin a}|.$

To get an estimate for $\sqrt{\sin x} - \sqrt{\sin a}$, multiply top and bottom by $\sqrt{\sin x} + \sqrt{\sin a}$: $$\sqrt{\sin x} - \sqrt{\sin a} = \frac{(\sqrt{\sin x} - \sqrt{\sin a})(\sqrt{\sin x} + \sqrt{\sin a})}{\sqrt{\sin x} + \sqrt{\sin a}} = \frac{\sin x - \sin a}{\sqrt{\sin x} + \sqrt{\sin a}},$$ from which \(\displaystyle |\sqrt{\sin x} - \sqrt{\sin a}| < \frac{|x - a|}{ \sqrt{\sin a}}.\)

Then \(\displaystyle |(\sqrt {\sin x} +1)^2-(\sqrt {\sin a} +1)^2| \leqslant |(\sin x -\sin a)| + 2|\sqrt{\sin x} - \sqrt{\sin a}| < |x-a|\left(1 + \frac1{ \sqrt{\sin a}}\right).\)

We want this to be less than $b$. So take \(\displaystyle c = \frac b{1 + \frac1{ \sqrt{\sin a}}}\). Then $$|\sqrt{\sin x} - \sqrt{\sin a}| < |x-a|\left(1 + \frac1{ \sqrt{\sin a}}\right) < c\left(1 + \frac1{ \sqrt{\sin a}}\right) = b.$$

This supose to be a high school inequality :

|sinx-sina|=\(\displaystyle 2|cos\frac{x+a}{2}|.|sin\frac{x-a}{2}|
\leq 2|sin\frac{x-a}{2}|\).............because \(\displaystyle |cost|\leq 1\forall t\)


\(\displaystyle \leq 2|\frac{x-a}{2}|=|x-a|\)...............since \(\displaystyle |sint|\leq |t|\forall t\).But i am wondering do they learn the inequality \(\displaystyle |sint|\leq | t|\forall t \) at high school ?
 
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