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 March 20th, 2017, 05:40 PM #1 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 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.... March 20th, 2017, 06:47 PM   #2
Senior Member

Joined: Jan 2017
From: Toronto

Posts: 209
Thanks: 3

Would the following correct?

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

Quote:
 Originally Posted by zollen 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.... Tags application, derivative, distance, partial, shortest Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zollen Calculus 5 March 14th, 2017 05:21 AM bracke Algebra 3 May 10th, 2012 10:44 AM mikeportnoy Algebra 5 May 16th, 2010 03:56 PM gaziks52 Algebra 4 April 11th, 2009 12:58 PM arun Algebra 6 March 4th, 2007 10:01 AM

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