A student, asked to find a root of the equation
did not notice the solution x=8 but chose, instead, to use Newton's Method taking x=7.2 as the first approximation. He then calculated correctly, f(7.2)=-7.712, f'(7.2)=2.02, and deduced (again correctly) the second approximation 9.84. By means of a graph, or otherwise, explain why Newton's Method failed to give a better approximation in this case. Prove that using Newton's Method, a first approximation , for a value of in the interval , would give a second approximation which is closer to the root x=8 provided that and deduce that any value of in the above range exceeding would in fact give improvement.
[You may, if you wish, assume without proof that f'(x) is positive and increasing forx>7.2; also that .]
For the first part, I found
There are three reasons given in the examples of this book for why the method doesn't work.
1) h is too large
2) f'(a) is too small
3) f''(a) is too large
I thought it would be that f''(a) is too large, f''(7.2)=15.2, but in the next portion, f''(7.3), say, is 15.8, so that can't be the case.
f'(a)= 2.92, which isn't too small, the other explanation then is that h=0.8 is too large, the problem is I'm given in the book that 0<h<1.
This is where I'm stuck