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Use Newton's method to find all solutions of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

6 cos(x) = x + 1
can somebody explain to me step by step on how to solve this problem?

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2 answers

  1. I assume x is in radians
    we want 0 = x + 1 - 6 cos x
    let f(x) = x + 1 - 6 cos x
    find where f(x) = 0

    pick a point, say x = 1
    then
    f(1) = 1 + 1 - 6 cos 1 = 2 - 6*0.54
    = -1.24

    plot that point (1, -1.24) on a graph
    what is the slope of f(x) at that point?
    f'(x) = 1 + 6 sin x
    = 6*.84 = 5.05

    draw that as tangent to curve of f(x) at (1,-1.24). It is aimed up toward the x axis. Where does it hit the x axis?

    slope = 5.05 = 1.24/delta x
    delta x = 1.24/5.05 = .246

    so new x = 1 + .246 = 1.246
    new f(x) = 1.246 + 1 - 6 cos 1.246
    = .331

    so now we are at (1.246, .331)
    well .331 is not zero so repeat using that point
    keep repeating until you get close to zero

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  2. There are several handy online calculators you can use to confirm your results.

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