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Thread: Polar curves Maximum and minimum.

  1. #1
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    Polar curves Maximum and minimum.

    I do not understand how to find the maximum and minimum.

    Find the maximum and minimum point form the pole.

    a) r=6cos(x)

    b)r=3-6sin(x)

    I can find the answer on my calculator I believe but do not know how to do this by hand. Please help.

    Thank You.
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  2. #2
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    Hey,

    First realize that the functions you are dealing with are the cos/sin functions. Do the following (i haven't specified an interval):

    1) Differentiate the function, i.e. find r'(x)

    2) Solve the equation r'(x)=0. You may get several x-values, depending on the interval you are considering.

    3) Find r'(x) for values greater and smaller than the x value found in 2). From the found r'(x) in this part you can establish where the function increases and where it decreases.

    4) Hence if the function increases up until x=k and then starts decreasing you'll know that at x=k you have the maximum function-value.
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  3. #3
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    I haven't learned how to differentiate yet? Is there another way?
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  4. #4
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    Hello, IDontunderstand!

    By the way, polar expressions are always written in terms of $\displaystyle \,r$ and $\displaystyle \,\theta.$


    Find the maximum and minimum points from the pole.

    . . $\displaystyle a)\;r\:=\:6\cos\theta$

    . . $\displaystyle b)\;r\:=\:3-6\sin\theta$

    Without Calculus? .It can be done, but it takes a bit of Thinking.
    Can you handle that?


    $\displaystyle (a)\;r \:=\:6\cos\theta$

    We know that $\displaystyle \cos\theta$ ranges between -1 and +1.

    The maximum value of $\displaystyle \,r$ occurs when $\displaystyle \cos\theta = +1$
    . . That is, when $\displaystyle \theta = 0.$
    Hence, a maximum occurs at $\displaystyle (6,\,0)$

    The minimum value of $\displaystyle \,r$ occurs when $\displaystyle \cos\theta = \text{-}1.$
    . . That is, when $\displaystyle \theta = \pi.$
    Hence, a minimum occurs at $\displaystyle (-6,\,\pi)$



    $\displaystyle (b)\;r \:=\:3 - 6\sin\theta$

    We know that $\displaystyle \sin\theta$ ranges between -1 and +1.

    The maximum value of $\displaystyle \,r$ occurs when $\displaystyle \sin\theta = \text{-}1$
    . . That is, when $\displaystyle \theta = \frac{3\pi}{2}$
    Hence, a maximum occurs at $\displaystyle \left(9,\:\frac{3\pi}{2}\right)$

    The minimum value of $\displaystyle \,r$ occurs when $\displaystyle \sin\theta = +1.$
    . . That is, when $\displaystyle \theta = \frac{\pi}{2}$
    Hence, a minimum occurs at $\displaystyle \left(-3,\:\frac{\pi}{2}\right)$

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