Three Methods to Find Roots of Equations
There are many different kind of methods which can be used to find the
roots of equations which can not be sold algebraically. In this
coursework we are going to analyse the use of three of these methods
which are called the: change of sign, Newton-Raphson and the
rearrangement method and are going to use them to find roots of
different equations.
Change of sign method
A root of an equation (where the graph crosses the x-axis) can be
detected by finding where the solution of a formula changes sign from
positive to negative. Where we find a change of sign on a graph using
omnigraph we take the range of the numbers it is in and divide it by
ten to find where it now changes sign. This procedure is then repeated
to the required level of accuracy.
Here is a graphical representation of the systematic decimal search:
[IMAGE]
X 1
X 10-1
X10-2
Using change of sign method on excel
First you look up between 2 values (e.g. 1 and 2, or 4 and 5) where
the graph crosses the x-axis. Then you take that as your range and
divide it up into 10 equal segments. For example, if between 1 and 2..
you use 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, etc. you work out the formula
for each of these and where the change of sign occurs you repeat the
process between that range.
A
B
C
D
E
E
1
X1
f(A1)
A2+0.01
f(C1)
C1+0.001
f(C1)
2
A1+0.1
f(A2)
C1+0.01
f(C2)
E1+0.001
f(C2)
3
A2+0.1
f(A3)
C2+0.01
f(C3)
E2+0.001
f(C3)
4
A3+0.1
f(A4)
C3+0.01
f(C4)
E3+0.001
f(C4)
5
This equation shifts from the parent function based on the equation f(x) = k+a(x-h) . In this equation, k shifts the parent function vertically, up or down, depending on the value of k. The h value shifts the parent function to the left or right. If h equals 1, it goes to the right 1 unit, if it is negative 1, it goes to the left 1 unit. If a is negative, the parent function is reflected on the x-axis. If x is negative, the parent function is reflected on the y-axis.
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order to do this you really need to try to put together an explanation of
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Show your work. Note that your answer will probably not be an even whole number as it is in the examples, so round to the nearest whole number.
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in exponential form. For instance, in a base 2 system, 4 can be written as 2
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