x = g(x) Iteration (AQA A-Level Mathematics): Quizzes

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x = g(x) Iteration
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What type of equations can be solved using fixed point iterative methods?

Simultaneous equations where one is y=xy = x

What is the first step in the fixed point iterative method?

Choose a start value for xx

In fixed point iteration, which two equations are solved simultaneously?

y=xy = x and y=g(x)y = g(x)

What does a change of sign in f(a)f(a) and f(b)f(b) indicate for a continuous function?

A root exists between aa and bb

For change of sign to work, how must the equation be written?

One side must equal 0

Using xn+1=3−xn34x_{n+1} = \frac{3-x_n^3}{4} with x1=0.5x_1 = 0.5, what is x2x_2 to 3 dp?

0.7190.719

Rearranging x3+4x−3=0x^3 + 4x - 3 = 0 to form x=g(x)x = g(x) gives which equation?

x=3−x34x = \frac{3-x^3}{4}

When should the iterative process stop?

When suitably close to a root

Which log property is used to rearrange 3x=x+33^x = x + 3 to xln⁡(3)=ln⁡(x+3)x\ln(3) = \ln(x+3)?

ln⁡(ab)=bln⁡(a)\ln(a^b) = b\ln(a)

Using xn+1=ln⁡(xn+3)ln⁡3x_{n+1} = \frac{\ln(x_n+3)}{\ln 3} with x2=1.140x_2 = 1.140, what is x3x_3 to 2 sf?

1.31.3

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