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March 20th, 2017, 04:40 PM   #1
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Partial Derivative Application for shortest distance

Find the shortest distance from the point (0, 0, b) to the paraboloid z = x^2 + y^2.

D = sqrt(( x - 0 )^2 + ( y - 0 )^2 + ( x^2 + y^2 - b )^2)

D = sqrt( x^2 + y^2 + x^4 + 2x^2y^2 - 2bx^2 - 2by^2 +b^2 + y^4 )

∂D/∂x = 2x + 4x^3 + 4xy^2 - 4bx = 1 + 2x^2 + 2y^2 - 2b = 0

∂D/∂y = 2y + 4y^3 + 4x^2y - 4by = 1 + 2y^2 + 2x^2 - 2b = 0

2 unknowns but with one equation, I am stuck!

Any tips would be much appreciated....
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March 20th, 2017, 05:47 PM   #2
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Would the following correct?

1 - 4x^2 - 2b = 0
4x^2 = 2b - 1
x^2 = (2b - 1)/4
x = sqrt((2b - 1) /4)


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Originally Posted by zollen View Post
Find the shortest distance from the point (0, 0, b) to the paraboloid z = x^2 + y^2.

D = sqrt(( x - 0 )^2 + ( y - 0 )^2 + ( x^2 + y^2 - b )^2)

D = sqrt( x^2 + y^2 + x^4 + 2x^2y^2 - 2bx^2 - 2by^2 +b^2 + y^4 )

∂D/∂x = 2x + 4x^3 + 4xy^2 - 4bx = 1 + 2x^2 + 2y^2 - 2b = 0

∂D/∂y = 2y + 4y^3 + 4x^2y - 4by = 1 + 2y^2 + 2x^2 - 2b = 0

2 unknowns but with one equation, I am stuck!

Any tips would be much appreciated....
zollen is offline  
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