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164 5. Approximation and Location of Zeros

0)

(ii)

(iii) Y = P(X)

*X

6. Let c # 1. Show that Newton’s Method applied to the polynomial
   tc - 1 and any first approximation yields the zero c-l in one step.
   With a diagram, explain why this occurs.


7. (a) Let c > 0. Sh ow that Newton’s Method applied to the poly-
   nomial t2 - c yields from any positive approximation al, the
   sequence {a,,} of successive approximations to 4, where

1

a,+1 = --(a, 2 + c/a,) (n 2 1).

(b) Find an expression which relates the difference ai+r - c to the

difference u: - c. Argue that, as n becomes larger, a, gets closer

and closer to JE. Show that {a,.,} is decreasing for n 2 2.

8. Determine to three decimal places all of the real zeros of the polyno-
   mial 3t4 - 2t3 - t2 - 3t + 1. Use all of the methods discussed in these
   exercises.


9. Newton’s Method for successive approximations to the root of an
   equation is a particular instance of a more general approach:

Given an equation p(t) = 0, derive from it an equation of

theformt = f(t) with which it shares a solution. Pick a first