Finding Approximate position of roots of f(x)
You need to show that you can prove numerically where an approximate root is.
We know a function has a root when f(x)=0
If you can't work out x from this then you can find an approximation
You have a function
f(x) = 3x^3 + 3x^2 + 5x - 7
Prove that it has a root between x = 1 and x= 0
f(0) = -7
f(1) = 4
There is a change in sign therefore the x axis must have been crossed between x = 0 and x = 1
We know a function has a root when f(x)=0
If you can't work out x from this then you can find an approximation
You have a function
f(x) = 3x^3 + 3x^2 + 5x - 7
Prove that it has a root between x = 1 and x= 0
f(0) = -7
f(1) = 4
There is a change in sign therefore the x axis must have been crossed between x = 0 and x = 1
The Iteration Formula
Iteration formula is a formula that creates a sequence that oscillates around a certain value.
It comes from the original function
f(x) is transformed into x = g(x)
The iteration formula here is xn+1 = g(xn)
It comes from the original function
f(x) is transformed into x = g(x)
The iteration formula here is xn+1 = g(xn)
Finding Root
Iteration formulae can be convergent - get closer to the root - or can be divergent - get further from the root.
They can also find the other root of the function sometimes.
In an exam you are most likely to be given x0 and told to use the iterative formula you just worked out to prove the root is a
a will normally have four to six decimal places. To prove this is the root you need can create the sequence the iterative formula forms until there are three consecutive results with the same result to one decimal place more than the a is given to
If a = 1.204
You are looking for:
x6 = 1.20427
x7 = 1.20434
x8 = 1.20429
They round to 1.2043
Another way is to choose two results on either side of a
If a = 1.204
To prove that it is a root use f(1.2035) and f(1.2045)
If the signs are different then 1.204 is a root
They can also find the other root of the function sometimes.
In an exam you are most likely to be given x0 and told to use the iterative formula you just worked out to prove the root is a
a will normally have four to six decimal places. To prove this is the root you need can create the sequence the iterative formula forms until there are three consecutive results with the same result to one decimal place more than the a is given to
If a = 1.204
You are looking for:
x6 = 1.20427
x7 = 1.20434
x8 = 1.20429
They round to 1.2043
Another way is to choose two results on either side of a
If a = 1.204
To prove that it is a root use f(1.2035) and f(1.2045)
If the signs are different then 1.204 is a root