AS and A Level: Core & Pure Mathematics
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Fixed point iteration: Rearrangement method explained
g(x) = 3? (9x+112x2). The steps I followed to derive this were: f(x) = x3 +2x2 ? 9x ? 11 0 = x3 +2x2 ? 9x ? 11 9x + 11 ? 2x2 = x3 So: 3? (9x+112x2) Now I will need to create a table to see if my ?x? values and g(x) values converge, where they do is the value of a root. Hence the ?x? values should be the same as the g(x) values or extremely close. In this case I will quote my answer to 5 s.f. I need to repeat my iterations until x and g(x)
 Word count: 683

Fractals. In order to create a fractal, you will need to be acquainted with complex numbers. Complex numbers on a graph are characterized by the coordinates of (x,y)
An example of a complex number can be: (5 + 4i). Complex numbers look ambitious, but they are not. In order to add complex numbers, you just need to add like terms:. In order to multiply complex numbers, you need to use the distributive law: . If you are faced with i2, please note that it is equal to 1! We shall start with Mandelbrot?s set. The Mandelbrot set was popularized due to its aesthetic appeal to many, and the simple rules that were applied in order to generate a complex structure.
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Solving Equations Using Numerical Methods
This allowed me to work out that the three roots were between the integers of 4 and 1. These three roots are: 4 < ? > 3 1 < ? > 0 0 < ? > 1 For one root, I worked out where the change of sign took place to 5 sig.fig. The first step to complete this process was to start with entering 4 to 3 (in 0.1?s) into Excel and the formula that I was using into the column next to it to show me where the change of sign was ? between 3.5 and 3.4.
 Word count: 872