**Finding Approximate position of roots of f(x)**

You need to show that you can prove numerically where an

We know a

If you can't work out x from this then you can find an approximation

You have a function

Prove that it has a root between

There is a

**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**The Iteration Formula**

Iteration formula is a formula that creates a

It comes from the

f(x) is transformed into x = g(x)

The iteration formula here is

**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)****Finding Root**

Iteration formulae can be

They can also find the other root of the function sometimes.

In an exam you are most likely to be

If

You are looking for:

x6 = 1.204

x7 = 1.204

x8 = 1.204

They round to 1.2043

If

To prove that it is a root use

If the signs are different then

**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****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***a***three consecutive results**with the same result to**one decimal place more**than the**is given to***a*If

**a = 1.204**You are looking for:

x6 = 1.204

**2**7x7 = 1.204

**3**4x8 = 1.204

**2**9They 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