Mixing
Am I doing this correctly? ( Tangent plane to a surface )
$begingroup$
I have this surface, and I'm asked to find the tangent plane at the point
$(x,y,z) = (-alpha, alpha-1, -alpha)quad alpha>0$
The surface is this one:
$(x,y,z)inmathbb{R}^3quadrvertquad y+1 = sqrt{ z^2 +2(x + alpha)^2 }$
I did this way: I wrote this as
$sqrt{ z^2 +2(x + alpha)^2 } - y - 1 = 0$
and found the partial derivatives with respect to each direction:
The derivative with respect to $x,y,z$ turned out to be
$frac{2(x+alpha)}{ sqrt{ z^2 +2(x + alpha)^2} }quad-1quadfrac{z}{sqrt{ z^2 +2(x + alpha)^2}}$
Evaluating them at the given point I found $(0, -1, -1)$
To end with it the expression of the tangent plane should be
$pi: 0(x + alpha) -1[y - (alpha - 1)] - 1(z +alpha) = 0 to \pi: y + z + 1 = 0$
Is this correct? (I'm surprised it doesn't depend on $alpha$). Where can I find more exercises like this one?
calculus tangent-spaces
asked Jan 28 at 15:18
Baffo rastaBaffo rasta
14311
$endgroup$
add a comment |
$begingroup$
I have this surface, and I'm asked to find the tangent plane at the point
$(x,y,z) = (-alpha, alpha-1, -alpha)quad alpha>0$
The surface is this one:
$(x,y,z)inmathbb{R}^3quadrvertquad y+1 = sqrt{ z^2 +2(x + alpha)^2 }$
I did this way: I wrote this as
$sqrt{ z^2 +2(x + alpha)^2 } - y - 1 = 0$
and found the partial derivatives with respect to each direction:
The derivative with respect to $x,y,z$ turned out to be
$frac{2(x+alpha)}{ sqrt{ z^2 +2(x + alpha)^2} }quad-1quadfrac{z}{sqrt{ z^2 +2(x + alpha)^2}}$
Evaluating them at the given point I found $(0, -1, -1)$
To end with it the expression of the tangent plane should be
$pi: 0(x + alpha) -1[y - (alpha - 1)] - 1(z +alpha) = 0 to \pi: y + z + 1 = 0$
Is this correct? (I'm surprised it doesn't depend on $alpha$). Where can I find more exercises like this one?
calculus tangent-spaces
asked Jan 28 at 15:18
Baffo rastaBaffo rasta
14311
$endgroup$
add a comment |
$begingroup$
I have this surface, and I'm asked to find the tangent plane at the point
$(x,y,z) = (-alpha, alpha-1, -alpha)quad alpha>0$
The surface is this one:
$(x,y,z)inmathbb{R}^3quadrvertquad y+1 = sqrt{ z^2 +2(x + alpha)^2 }$
I did this way: I wrote this as
$sqrt{ z^2 +2(x + alpha)^2 } - y - 1 = 0$
and found the partial derivatives with respect to each direction:
The derivative with respect to $x,y,z$ turned out to be
$frac{2(x+alpha)}{ sqrt{ z^2 +2(x + alpha)^2} }quad-1quadfrac{z}{sqrt{ z^2 +2(x + alpha)^2}}$
Evaluating them at the given point I found $(0, -1, -1)$
To end with it the expression of the tangent plane should be
$pi: 0(x + alpha) -1[y - (alpha - 1)] - 1(z +alpha) = 0 to \pi: y + z + 1 = 0$
Is this correct? (I'm surprised it doesn't depend on $alpha$). Where can I find more exercises like this one?
calculus tangent-spaces
asked Jan 28 at 15:18
Baffo rastaBaffo rasta
14311
$endgroup$
I have this surface, and I'm asked to find the tangent plane at the point
$(x,y,z) = (-alpha, alpha-1, -alpha)quad alpha>0$
The surface is this one:
$(x,y,z)inmathbb{R}^3quadrvertquad y+1 = sqrt{ z^2 +2(x + alpha)^2 }$
I did this way: I wrote this as
$sqrt{ z^2 +2(x + alpha)^2 } - y - 1 = 0$
and found the partial derivatives with respect to each direction:
The derivative with respect to $x,y,z$ turned out to be
$frac{2(x+alpha)}{ sqrt{ z^2 +2(x + alpha)^2} }quad-1quadfrac{z}{sqrt{ z^2 +2(x + alpha)^2}}$
Evaluating them at the given point I found $(0, -1, -1)$
To end with it the expression of the tangent plane should be
$pi: 0(x + alpha) -1[y - (alpha - 1)] - 1(z +alpha) = 0 to \pi: y + z + 1 = 0$
Is this correct? (I'm surprised it doesn't depend on $alpha$). Where can I find more exercises like this one?
calculus tangent-spaces
calculus tangent-spaces
asked Jan 28 at 15:18
Baffo rastaBaffo rasta
14311
asked Jan 28 at 15:18
Baffo rastaBaffo rasta
14311
asked Jan 28 at 15:18
Baffo rastaBaffo rasta
14311
asked Jan 28 at 15:18
Baffo rastaBaffo rasta
14311
asked Jan 28 at 15:18
Baffo rastaBaffo rasta
14311
14311
add a comment |
add a comment |
1 Answer
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$begingroup$
That is correct.
Here are some practice problems.
https://math.la.asu.edu/~surgent/mat267/Tangent_Planes_Px.pdf
answered Jan 28 at 16:27
Michael LeeMichael Lee
5618
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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oldest
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active
oldest
votes
$begingroup$
That is correct.
Here are some practice problems.
https://math.la.asu.edu/~surgent/mat267/Tangent_Planes_Px.pdf
answered Jan 28 at 16:27
Michael LeeMichael Lee
5618
$endgroup$
add a comment |
$begingroup$
That is correct.
Here are some practice problems.
https://math.la.asu.edu/~surgent/mat267/Tangent_Planes_Px.pdf
answered Jan 28 at 16:27
Michael LeeMichael Lee
5618
$endgroup$
add a comment |
$begingroup$
That is correct.
Here are some practice problems.
https://math.la.asu.edu/~surgent/mat267/Tangent_Planes_Px.pdf
answered Jan 28 at 16:27
Michael LeeMichael Lee
5618
$endgroup$
That is correct.
Here are some practice problems.
https://math.la.asu.edu/~surgent/mat267/Tangent_Planes_Px.pdf
answered Jan 28 at 16:27
Michael LeeMichael Lee
5618
answered Jan 28 at 16:27
Michael LeeMichael Lee
5618
answered Jan 28 at 16:27
Michael LeeMichael Lee
5618
answered Jan 28 at 16:27
Michael LeeMichael Lee
5618
5618
add a comment |
add a comment |
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