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Math Help - Free response question help

  1. #1
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    Free response question help

    I would appreciate some help with this problem, I have some ideas how to approach it but I'm confused on what to do on some parts.

    Consider the curve given by y^2= 2+xy

    1) show that dy/dx= y/2y-x (im pretty sure i can do this if its just the derivative)

    2) find all points (x,y) on the curve where the line tangent to the curve has slope 1/2 (how do u find a tangent line with a specific slope, and how would you get the points)

    3) show that there are no points (x,y) on the curve where the line tangent to the curve is horizontal (this isn't horizontal asymptote right?)

    4) let x and y be functions of time t that are related by the equation y^2= 2+xy. At time t=5, the value of y is 3 and dy/dt= 6. Find the value of dx/dt at time t=5 (where exactly does t go in that equation and how do i apply it?)
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  2. #2
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    Quote Originally Posted by wisezeta View Post
    I would appreciate some help with this problem, I have some ideas how to approach it but I'm confused on what to do on some parts.

    Consider the curve given by y^2= 2+xy

    1) show that dy/dx= y/2y-x (im pretty sure i can do this if its just the derivative)

    2) find all points (x,y) on the curve where the line tangent to the curve has slope 1/2 (how do u find a tangent line with a specific slope, and how would you get the points)

    3) show that there are no points (x,y) on the curve where the line tangent to the curve is horizontal (this isn't horizontal asymptote right?)

    4) let x and y be functions of time t that are related by the equation y^2= 2+xy. At time t=5, the value of y is 3 and dy/dt= 6. Find the value of dx/dt at time t=5 (where exactly does t go in that equation and how do i apply it?)

    2) slope = m = \frac{dy}{dx}. find all point that satisfy \frac{dy}{dx} = 0.5

    3) show that there is no point satisfy \frac{dy}{dx} = 0

    4) use implicit differentiation with respect to t
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  3. #3
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    i don't understand how to do the implicit differentiation for t
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  4. #4
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    Quote Originally Posted by wisezeta View Post
    i don't understand how to do the implicit differentiation for t
    \frac{d}{dt}[y^2 = 2+xy]

    2y \cdot \frac{dy}{dt} = x \cdot \frac{dy}{dt} + y \cdot \frac{dx}{dt}
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  5. #5
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    so what happens to t=5?
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  6. #6
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    Quote Originally Posted by wisezeta View Post
    so what happens to t=5?
    let x and y be functions of time t that are related by the equation y^2= 2+xy. At time t=5, the value of y is 3 and dy/dt= 6. Find the value of dx/dt at time t=5
    they told you that at time t = 5, y = 3 and dy/dt = 6.

    you have enough information to find dx/dt.

    (the t = 5 is inconsequential)
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  7. #7
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    Ok I understand now. Thank you very much
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  8. #8
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    Quick question, for 2 and 3 do I set the derivative equal to the given number, and if so don't I need a second equation in order to get the points? Would that be the original equation?
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  9. #9
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    yes and yes.
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