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    I think I've had a huge misunderstanding of this topic all year now (yikes) but now I think I've finally corrected myself, I just want to check if what I know is right.

    - You can use iteration and put an equation in the form x=f(x) to find a root. This is when you say x=f(x) and call it F(x), and you may be asked to plot F(x) and y=x to show cobwebbing and staircasing on a graph.

    -Alternative you can use Newton raphson and use x = x - f(x)/f'(x) and questions can ask you to show the convergence with tangents that get closer and closer to the gradient of the root.

    This is where I'm the most unsure about, especially bits in bold.

    -You can calculate the error en by using e(n) = x(n) - a where a is the true value of the root. For an iteration e(n)/e(n+1) = k = f'(a) but in the case f'(a)=0 you use (e(n))^2/e(n+1) which equals -1/2f''(a)

    -For Newton Raphson you ALWAYS use (e(n))^2/e(n+1)

    Corrections appreciated if I made a mistake, thanks.
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Updated: March 26, 2016
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