How can we find some radius of circle so $-xarctan(x)+0.2x-yarctan(y)+0.9y=0$ will be fully inside this...
$begingroup$
If he have this region
$$
begin{align}
-xarctan(x)+0.2x-yarctan(y)+0.9y=0\
end{align}
$$
How can we find some $R$ (maybe minimum) so this region will fully inside this circle $(x-a)^2+(y-b)^2=R^2$.
In WolframAlpha I've found for example $(x-0.2/2)^2+(y-0.9/2)^2=0.8$ so $R=sqrt0.8$ fits but have no idea how to prove that and how to find minimum radius.
inequality maxima-minima implicit-function
asked Dec 3 '18 at 12:46
TagTag
696
$endgroup$
add a comment |
$begingroup$
If he have this region
$$
begin{align}
-xarctan(x)+0.2x-yarctan(y)+0.9y=0\
end{align}
$$
How can we find some $R$ (maybe minimum) so this region will fully inside this circle $(x-a)^2+(y-b)^2=R^2$.
In WolframAlpha I've found for example $(x-0.2/2)^2+(y-0.9/2)^2=0.8$ so $R=sqrt0.8$ fits but have no idea how to prove that and how to find minimum radius.
inequality maxima-minima implicit-function
asked Dec 3 '18 at 12:46
TagTag
696
$endgroup$
add a comment |
$begingroup$
If he have this region
$$
begin{align}
-xarctan(x)+0.2x-yarctan(y)+0.9y=0\
end{align}
$$
How can we find some $R$ (maybe minimum) so this region will fully inside this circle $(x-a)^2+(y-b)^2=R^2$.
In WolframAlpha I've found for example $(x-0.2/2)^2+(y-0.9/2)^2=0.8$ so $R=sqrt0.8$ fits but have no idea how to prove that and how to find minimum radius.
inequality maxima-minima implicit-function
asked Dec 3 '18 at 12:46
TagTag
696
$endgroup$
If he have this region
$$
begin{align}
-xarctan(x)+0.2x-yarctan(y)+0.9y=0\
end{align}
$$
How can we find some $R$ (maybe minimum) so this region will fully inside this circle $(x-a)^2+(y-b)^2=R^2$.
In WolframAlpha I've found for example $(x-0.2/2)^2+(y-0.9/2)^2=0.8$ so $R=sqrt0.8$ fits but have no idea how to prove that and how to find minimum radius.
inequality maxima-minima implicit-function
inequality maxima-minima implicit-function
asked Dec 3 '18 at 12:46
TagTag
696
asked Dec 3 '18 at 12:46
TagTag
696
asked Dec 3 '18 at 12:46
TagTag
696
asked Dec 3 '18 at 12:46
TagTag
696
asked Dec 3 '18 at 12:46
TagTag
696
696
add a comment |
add a comment |
1 Answer
1
active
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$begingroup$
There will be hardly an exact solution to find the minimal radius. If you want to find the maximal and minimal height of the curve, differentiate in respect to $x$, set $y'=0$ and you'll get
$$arctan(x)+frac{x}{1+x^2}+0.2=0.$$
From here you'll find only a numerical approximation of the solutions.
answered Dec 3 '18 at 16:10
Michael HoppeMichael Hoppe
10.8k31834
$endgroup$
$begingroup$
But what about not minimal radius? I don't need minimal radius for my problem. Can we just find some radius without numerical approximation and be sure that region will be inside this circle?
$endgroup$
– Tag
Dec 3 '18 at 16:17
$begingroup$
Well, obviously you've found one.
$endgroup$
– Michael Hoppe
Dec 3 '18 at 16:20
$begingroup$
Yes, but it's not proved, I've found it only by attempt, can we make some simple estimation which will show that it's correct?
$endgroup$
– Tag
Dec 3 '18 at 16:22
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
There will be hardly an exact solution to find the minimal radius. If you want to find the maximal and minimal height of the curve, differentiate in respect to $x$, set $y'=0$ and you'll get
$$arctan(x)+frac{x}{1+x^2}+0.2=0.$$
From here you'll find only a numerical approximation of the solutions.
answered Dec 3 '18 at 16:10
Michael HoppeMichael Hoppe
10.8k31834
$endgroup$
$begingroup$
But what about not minimal radius? I don't need minimal radius for my problem. Can we just find some radius without numerical approximation and be sure that region will be inside this circle?
$endgroup$
– Tag
Dec 3 '18 at 16:17
$begingroup$
Well, obviously you've found one.
$endgroup$
– Michael Hoppe
Dec 3 '18 at 16:20
$begingroup$
Yes, but it's not proved, I've found it only by attempt, can we make some simple estimation which will show that it's correct?
$endgroup$
– Tag
Dec 3 '18 at 16:22
add a comment |
$begingroup$
There will be hardly an exact solution to find the minimal radius. If you want to find the maximal and minimal height of the curve, differentiate in respect to $x$, set $y'=0$ and you'll get
$$arctan(x)+frac{x}{1+x^2}+0.2=0.$$
From here you'll find only a numerical approximation of the solutions.
answered Dec 3 '18 at 16:10
Michael HoppeMichael Hoppe
10.8k31834
$endgroup$
$begingroup$
But what about not minimal radius? I don't need minimal radius for my problem. Can we just find some radius without numerical approximation and be sure that region will be inside this circle?
$endgroup$
– Tag
Dec 3 '18 at 16:17
$begingroup$
Well, obviously you've found one.
$endgroup$
– Michael Hoppe
Dec 3 '18 at 16:20
$begingroup$
Yes, but it's not proved, I've found it only by attempt, can we make some simple estimation which will show that it's correct?
$endgroup$
– Tag
Dec 3 '18 at 16:22
add a comment |
$begingroup$
There will be hardly an exact solution to find the minimal radius. If you want to find the maximal and minimal height of the curve, differentiate in respect to $x$, set $y'=0$ and you'll get
$$arctan(x)+frac{x}{1+x^2}+0.2=0.$$
From here you'll find only a numerical approximation of the solutions.
answered Dec 3 '18 at 16:10
Michael HoppeMichael Hoppe
10.8k31834
$endgroup$
There will be hardly an exact solution to find the minimal radius. If you want to find the maximal and minimal height of the curve, differentiate in respect to $x$, set $y'=0$ and you'll get
$$arctan(x)+frac{x}{1+x^2}+0.2=0.$$
From here you'll find only a numerical approximation of the solutions.
answered Dec 3 '18 at 16:10
Michael HoppeMichael Hoppe
10.8k31834
answered Dec 3 '18 at 16:10
Michael HoppeMichael Hoppe
10.8k31834
answered Dec 3 '18 at 16:10
Michael HoppeMichael Hoppe
10.8k31834
answered Dec 3 '18 at 16:10
Michael HoppeMichael Hoppe
10.8k31834
10.8k31834
$begingroup$
But what about not minimal radius? I don't need minimal radius for my problem. Can we just find some radius without numerical approximation and be sure that region will be inside this circle?
$endgroup$
– Tag
Dec 3 '18 at 16:17
$begingroup$
Well, obviously you've found one.
$endgroup$
– Michael Hoppe
Dec 3 '18 at 16:20
$begingroup$
Yes, but it's not proved, I've found it only by attempt, can we make some simple estimation which will show that it's correct?
$endgroup$
– Tag
Dec 3 '18 at 16:22
add a comment |
$begingroup$
But what about not minimal radius? I don't need minimal radius for my problem. Can we just find some radius without numerical approximation and be sure that region will be inside this circle?
$endgroup$
– Tag
Dec 3 '18 at 16:17
$begingroup$
Well, obviously you've found one.
$endgroup$
– Michael Hoppe
Dec 3 '18 at 16:20
$begingroup$
Yes, but it's not proved, I've found it only by attempt, can we make some simple estimation which will show that it's correct?
$endgroup$
– Tag
Dec 3 '18 at 16:22
$begingroup$
But what about not minimal radius? I don't need minimal radius for my problem. Can we just find some radius without numerical approximation and be sure that region will be inside this circle?
$endgroup$
– Tag
Dec 3 '18 at 16:17
$begingroup$
But what about not minimal radius? I don't need minimal radius for my problem. Can we just find some radius without numerical approximation and be sure that region will be inside this circle?
$endgroup$
– Tag
Dec 3 '18 at 16:17
$begingroup$
Well, obviously you've found one.
$endgroup$
– Michael Hoppe
Dec 3 '18 at 16:20
$begingroup$
Well, obviously you've found one.
$endgroup$
– Michael Hoppe
Dec 3 '18 at 16:20
$begingroup$
Yes, but it's not proved, I've found it only by attempt, can we make some simple estimation which will show that it's correct?
$endgroup$
– Tag
Dec 3 '18 at 16:22
$begingroup$
Yes, but it's not proved, I've found it only by attempt, can we make some simple estimation which will show that it's correct?
$endgroup$
– Tag
Dec 3 '18 at 16:22
add a comment |
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