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Parametric Equations finding largest radius

  1. May 17, 2013 #1
    1. The problem statement, all variables and given/known data

    Suppose that r = f (θ) defines a polar graph. Find an expression for dx/dθ. It should not involve the letter r. Explain a procedure to determine the farthest that the graph r = f (θ) extends to the left and to the right (Hint: If x = x0 is the x - value of the point that is farthest left, how does x0 compare with all other x-values?).

    2. Relevant equations



    3. The attempt at a solution
    The wording of the question seems a bit off especially the hint "is the x - value of the point that is farthest left".
    Here is how I started teh problem. I know that x=r*cos(θ) so i subsituded r for f(θ) to get
    x=f(θ)cos(θ). My guess is that when the derivative of dx/dθ is equal to 0 then that means that the polar graph is turning back around so I take the derivative of x to get dx/dθ =-f(θ)sin(θ) +f'(θ)cos(θ)
    then I set it equal to 0 to get
    -f(θ)sin(θ) +f'(θ)cos(θ)=0
    f(θ)sin(θ) =f'(θ)cos(θ)
    tan(θ)=f'(θ)/f(θ)
    tan(θ)=f'(θ)/(x/cos(θ))
    x=(f'(θ) tan(θ)cos(θ))

    I don't really know what I'm doing at this point. How should I approach this problem?
     
  2. jcsd
  3. May 18, 2013 #2

    haruspex

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    It all looks fine to here. In practice, you would know f(), so could then solve the above equation to find the theta of interest, then compute x from x = f(θ) cos(θ). There's no point in bringing x back into the equation first unless you can eliminate theta and get an equation in x only - which you can't without knowing what f is.
     
  4. May 19, 2013 #3
    I see. That makes sense. I guess I was expecting a bit more out of the problem. Is there any point to the hint that was provided? I feel like I'm still missing a big chunk of the question.
     
  5. May 19, 2013 #4

    haruspex

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    You used the hint when you looked for the extremum of x. One thing you have not done is show how to determine it's a leftmost value, not a rightmost one.
     
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