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Math Help - linear approximation

  1. #1
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    Question linear approximation

    Using the linear approximation to f(x) = cos x at the point x = 1 to estimate
    the value of cos(1.03) works well because cos x looks like a line when you zoom
    in on it.

    true/false
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  2. #2
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    I say it's false since you don't know an exact value for cos(1), and since there is a closer exact value near 1.03, such as \frac{\pi}{3}.

    I may have answered this in a different way than the question expects you to answer.
    Last edited by Chop Suey; September 23rd 2008 at 10:23 PM. Reason: Correction
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by thecount View Post
    Using the linear approximation to f(x) = cos x at the point x = 1 to estimate
    the value of cos(1.03) works well because cos x looks like a line when you zoom
    in on it.

    true/false
    The linear approximation takes on the form L(x)=-\sin(1)(x-1)+\cos(1)\implies L(x)=-.841x+1.382

    So, is L(1.03) a good approximation to \cos(1.03)?

    L(1.03)=-.841(1.03)+1.382\approx .515

    Now, \cos(1.03)\approx .5148

    This is a pretty good approximation, with an error of about 10^{-3}.

    If you were to zoom in on the cosine function at about 1, it appears linear:



    Here's a graph of the function with its linear approximation:



    (the blue line is \cos(x) and the purple is L(x)

    So I would say its true.

    --Chris
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  4. #4
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    Quote Originally Posted by Chop Suey View Post
    I say it's false since you don't know an exact value for cos(1), and since there is a closer exact value near 1.03, such as \frac{\pi}{3}.

    I may have answered this in a different way than the question expects you to answer.
    I have made a mistake and written pi/4 instead of pi/3.
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