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AP calculus, implicit derivatives

  1. Nov 16, 2007 #1
    1. The problem statement, all variables and given/known data
    1. Given the curve x² - xy + y² = 9
    (a) Write a general expression for the slope of the curve
    (b) find the coordinates of the points on the curve where the tangents are vertical
    (c) at the point (0,3) find the rate of change in the slope of the curve with respect to x.

    2. Relevant equations



    3. The attempt at a solution
    No problems with a or b i believe:
    (a)2x - x(dy/dx) - y + 2y (dy/dx) = 0
    Put all the terms containing dy/dx to one side and everything else on the other:
    (2y-x) (dy/dx) = y-2x
    dy/dx = (y-2x) / (2y-x)

    (b) (2y)^2 - (2y)y + y^2 = 9
    4y^2 - 2y^2 + y^2 = 9
    y^2 = 3
    y = +- sqrt(3) so x = +- 2sqrt(3)
    So the points are (2sqrt(3),sqrt(3)) and (-2sqrt(3),-sqrt(3)).

    (c) i'm not exactly sure how to do this one, im thinking to just plug (0,3) into the first derivative? or do i need to take the 2nd derivative?
    Help is appreciated
     
  2. jcsd
  3. Nov 16, 2007 #2

    Dick

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    Science Advisor
    Homework Helper

    "slope" is derivative. So "rate of change of slope" is the derivative of the derivative. Yes, you need to find a second derivative.
     
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