Consider the function
g (x) = 12x + 4 - cos x. Given
g (x) = 0 has a unique solution
x = b in the interval (−1/2, 0), and you can use this
without justification.
(a) Show that Newton's method of starting point
x0
= 0 gives a number sequence with
b <··· <xn+1
<xn
<··· <x1
<x0
= 0
(The word "curvature" should be included in the argument!)
(b) Calculate
x1
and x2.
Use theorem 2 in section 4.2 (page 229-
Error bounds for Newtons Method) or
intermediate value theorem to show that
x2
- b <0.0000003.
(c) Find a suitable function that you can perform fix piont
iteration to find
b. Calculate and compare the results of calculating the
first three iterations of Newton's method and fix point
iteration