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Math Help - calculus 3 help, not sure if I posted in right place

  1. #1
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    calculus 3 help, not sure if I posted in right place

    Hi guys,

    Like I said Im not sure If I posted this in the right spot, but if someone can take a look at these 3 problems and help me out I would appreciate it.

    1.) Find ther extreme values of the function subject to the given constraint. F(x,y,z)= x^3 + y^3+z^3, x^2+y^2+z^2=4


    2.)Write parametric equations for the tangent line to the curve of intersection of the surfaces. x+y^2+7z=9 amd x=1 at the point (1,1,1)

    3.) A normal line tot he paraboloid z=6x^2 +2y^2 also passes thorugh the point (26,25,73). Find the point on the paraboloid that the normal line passes through.


    any help would be greatly appreciated, this is a test review for our final exam and Ive got all the other material figured out, and have spent 3 days and a crap load of time and paper trying to solve these but am havin no luck.
    Last edited by mr fantastic; April 18th 2009 at 03:07 PM. Reason: Removed Caps Lock text (shouting)
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  2. #2
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    Quote Originally Posted by Noobie View Post
    Hi guys,

    Like I said Im not sure If I posted this in the right spot, but if someone can take a look at these 3 problems and help me out I would appreciate it.

    1.) FIND THE EXTREME VALUES OF THE FUNCTION SUBJECT TO THE GIVEN CONSTRAINT. F(x,y,z)= x^3 + y^3+z^3, x^2+y^2+z^2=4


    2.)Write parametric equations for the tangent line to the curve of intersection of the surfaces. x+y^2+7z=9 amd x=1 at the point (1,1,1)

    3.) A normal line tot he paraboloid z=6x^2 +2y^2 also passes thorugh the point (26,25,73). Find the point on the paraboloid that the normal line passes through.


    any help would be greatly appreciated, this is a test review for our final exam and Ive got all the other material figured out, and have spent 3 days and a crap load of time and paper trying to solve these but am havin no luck.
    Here's a bit on the first. Using Lagrange multipliers

    G = x^3+y^3+z^3 + \lambda \left( x^2+y^2+z^2-4\right)

    G_x = 3x^2 + 2 x \lambda, G_y = 3y^2 + 2 y \lambda,
    G_z = 3z^2 + 2 z \lambda, G_{\lambda} = x^2+y^2+z^2-4

    which we set all to zero. From the first three we obtain

    x = 0,\;\;\text{or}\;\;x = - \frac{2 \lambda}{3}
    y = 0,\;\;\text{or}\;\;y = - \frac{2 \lambda}{3}
    z = 0,\;\;\text{or}\;\;z = - \frac{2 \lambda}{3}

    which gives rise to the following three cases

    (i) one of x,\; y, \; \text{or}\; z is zero, the other two - \frac{2 \lambda}{3} or equal to each other
    (ii) two of x,\; y, \; \text{or}\; z is zero the other - \frac{2 \lambda}{3}
    (iii) x = y = z = - \frac{2 \lambda}{3} or all equal to each other

    (i) Say z = 0, so x = y and from the constraint x = y =\pm \sqrt{2} so F = \pm \frac{8}{\sqrt{2}}
    (ii) Say y = z = 0, so from the constraint x = \pm 2 so F = \pm 8
    (iii) x = y = z = \pm \frac{2}{\sqrt{3}} giving F = \pm \frac{8}{\sqrt{3}}

    You could also use

    x = 2 \cos \theta \sin \phi, y = 2 \sin \theta \sin \phi, z = 2 \cos \phi, as this satisfies the constaint but it might be a bit messy.
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    that actually does kinda help with the first one, thanks bro
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    bump
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