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Math Help - Determining points on a plane

  1. #1
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    Lightbulb Determining points on a plane

    consider S defined by

    z = f(x,y) = x^2 + 4(y - 1)^2


    1) Graph the level curves of f in the (x,y) plane. So will this be a 2D drawing, if/if not how do i do this? What will it look like?

    2) Find /_\f {upside down triangle in front of 'f'} and show where it is on graph.

    so fx = 2x
    fy = 8y - 8

    3) Determine the directional derivative of 'f' at (1, 1) in the direction of (1, 1).

    so fx = 2x
    fy = 8y - 8

    sub in (1, 1) and i get (2, 0)

    is directional vector 1/(1^2 + 1^2)= 1/2 ?

    Where do i go from here?

    4) Determine the maximum and minimum values of 'f' in the domain and show were the values occur on the graph.
    {(x, y) | -1 <= x <= 1, 0 <= y <= 2}


    Thank you, working would be much appreciated.
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  2. #2
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    Re: Determining points on a plane

    Quote Originally Posted by Brennox View Post
    consider S defined by

    z = f(x,y) = x^2 + 4(y - 1)^2


    1) Graph the level curves of f in the (x,y) plane. So will this be a 2D drawing, if/if not how do i do this? What will it look like?
    Taking z= C a constant, you have x^2+ 4(y-1)^2= C. What kind of conic section is that?

    2) Find /_\f {upside down triangle in front of 'f'} and show where it is on graph.
    "Upside down triangle"? Doesn't your text book tell you what the "gradient" is? And I doubt that the problem really says "show where it is on the graph". \nabla f= grad f exists at every point on the graph. I expect the problem asks you to draw \nabla at a specific point on the level curves. It is a vector perpendicular to the level curve at point.

    so fx = 2x
    fy = 8y - 8

    3) Determine the directional derivative of 'f' at (1, 1) in the direction of (1, 1).

    so fx = 2x
    fy = 8y - 8

    sub in (1, 1) and i get (2, 0)

    is directional vector 1/(1^2 + 1^2)= 1/2 ?
    No, it is not. For one thing, that is NOT a vector! Didn't it occur to you that the directional vector should be a vector?
    The length of the directional vector, in the direction of the unit vector u, is \nabla f\cdot\u. That's almost what you have except that (1, 1) has length \sqrt{2}, not 1, so it is not a unit vector. After you have corrected that, multiply by the unit vector in that direction.

    [quoteWhere do i go from here?

    4) Determine the maximum and minimum values of 'f' in the domain and show were the values occur on the graph.
    {(x, y) | -1 <= x <= 1, 0 <= y <= 2}[/quote]
    You have one ' outside the italics and one inside. Do you meam max and min values of f?
    Think about this geometrically. What does the graph of this function look like? Do you see that f(x,y) is never negative? Where is f(x,y)= 0?

    Thank you, working would be much appreciated.
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