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Of the infinitely many lines that are tangent to the curve y = −6 sin x and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places.

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4 answers

  1. Naturally, the tangent with the largest slope will be the one closest to the origin.

    So, we want

    -6cosx = -6sinx/x
    x = tanx
    solve that using Newton's method, and you have x = 4.93

    so, your line is

    y = -6cos(4.93)(x-4.93)+6sin(4.93)
    or
    y-5.98 = 0.52(x-4.93)

    See the graphs at

    http://www.wolframalpha.com/input/?i=plot+y+%3D+-6sin%28x%29%2C+y%3D0.52%28x-4.93%29%2B5.98+where+0%3C%3Dx%3C%3D2pi

    if you tighten your accuracy, you should get a better tangent.

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  2. Wasn't your tangent line supposed to pass through the origin?

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  3. Reiny, yes it does pass through the origin.

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  4. I sure flubbed that one. Your later solution was spot on, Reiny.

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