Problem

(!) Given a differentiable function f, let when f′(x) ≠ 0. The function g is the function...

(!) Given a differentiable function f, let when f′(x) ≠ 0. The function g is the function that generates xn+1 from xn in Newton’s Method.

a) Verify that g(x) = x if and only if f(x) = 0.
b) When f(x) = x2 − 2, verify that .
c) Use (b) to show that when Newton’s Method is applied to x2 − 2 with x0 = 1, the value of x5 is within 2−31 of .
d) What can be said about Newton’s Method in general when a is a zero of f and |g(x) − a| ≤ c |x − a|2 for some constant c and for x near a?

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Solutions For Problems in Chapter 16

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