C3 Coursework


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  • Created on: 19-03-12 10:25
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C3 Coursework
I have chosen the equation 2x3+2x2-3.6x+1=0. The function is f(x) =2x3+2x2-3.6x+1, illustrated as a
graph above. I shall attempt to solve this equation and thus find the roots of the function.
Change of sign
To find a root, I shall use the Decimal Search Method to find a change of sign. This will find roughly
where the root is.
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There is a change of sign between x=-3 and x=-2. This is noted as [-3,-2] and suggests a root being
present between these values. Remember that a root will always have the value y=0, which is
between a positive value and a negative value.
I shall now continue using the Decimal Search Method to find the root to 3 decimal places, using
smaller steps each time to "home in" on the root.…read more

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The solution to this is to use smaller steps. This finds a change of sign in the region [0, 0.5]
However, on closer inspection you can see that there are not one, but two roots in this region.
Without the graph for analysis you would fail to see one of the roots.
Newton-Raphson MethodTM
I will now carry out the Newton-Raphson MethodTM on a new equation.…read more

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The Newton-Raphson MethodTM relies on the general iterative formula:
For this function, it becomes:
Starting at 2, the method quickly converges to the root
4 | Page…read more

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The root is found to be between 2.0557 and 2.0556. This x value can be written as:
2.0556<x<2.0557, or as a value with error bounds: -2.055695±0.0000005
The Newton-RaphsonTM method will not work if we start at x=0 as it encounters a division by 0 in its
process, when xn is 0:
x1 = 0 - 0
(Green line represents method working)
The Method will draw a tangent at 1, and then try to find the x axis.…read more

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Rearrangement of f(x) =0 to give x=g(x)
I will now carry out the Rearrangement of f(x)=0 to give x=g(x) on a new function and equation.
New Function:
F(x) =0.2x3+1.4x2+2.5x+2.5
New Equation:
0.2x3+1.4x2+2.5x+2.…read more

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Rearranging f(x) gives x = -2.5-0.2x -1.4x
This now provides our iterative formula:
xn+1 = 2.5
This is called g(x) and is displayed as the blue curve:
The function y=x is also added to the graph. Starting at x=0, a line is drawn vertically to meet the g(x)
line. A horizontal line is drawn from this meeting point to the line y=x. This is repeated until the
method converges to a root.…read more

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Page 9

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After 17 iterations the method converges to -5.00005±0.000005.
However, if we choose a different rearrangement of f(x), we can get:
f(x)= -2.5-0.2x2-1.4x2
0.…read more

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Clearly, the method will continue to infinity, not providing a solution, even though the lines
y=x and y=g(x) cross.
The magnitude of g'(x) (the gradient of g(x)) must be less than 1 (the gradient of y=x), otherwise the
series will be divergent and never reach a solution.
Comparison of Methods
Now I will carry out all three methods on one equation.
F(x) =0.2x3+0.…read more



What grade/mark did you get for this? Thanks :)

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