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Greatest Rate of Angle Change

  1. Feb 28, 2013 #1
    So I have been thinking of this problem... what is the greatest rate of angle change for a function? As in, what is the point in which a function achieves its greatest rate of angle change....

    Well, the angle of a function can be determined by arctan(y')

    The Rate of Angle change is (arctan(y'))', which equals [itex]\frac{y'}{(y')^2 +1}[/itex]

    So the greatest rate of angle change is the derivative of that set to zero, which is equal to

    [itex]\frac{y''' - 2(y')^2y' + y'''(y')^2}{((y')^2 +1)} = 0[/itex]

    Which , you can simplify to....

    [itex]y''' - 2(y')^2y' + y'''(y')^2= 0[/itex]

    Is there a way that this differential equation can be solved? (This is not for homework, this is just a general question that I would like to know the answer to)
  2. jcsd
  3. Feb 28, 2013 #2


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    Science Advisor

    Or you could use the fact that ##\frac{x}{x^2+1}## attains it's maximum at x=1.
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