Newton's Method & Error Analysis

[Physics197]

Homework Statement

(i) Use Newton’s Method and apparent convergence
to solve x ln(x) = 5 accurate to 3 and 4 significant figures. Start out with x0 = 2. (ii) Directly
approximate the absolute error on f, i.e. _f = f(x) − f(˜x). (iii) Use the difference between
the 4 significant figures and 3 significant figures results for x and the error formula to estimate
_f. You should find that they approximate each other.

Homework Equations

_f = ef'(x)

The Attempt at a Solution

I understand (i), I got 3.77 & 3.769.

For (ii), I don't understand what f(x) and f(^~x) are. I know x is the measured value and ^~x is the true value, but what are they in this case?

For (iii) do we take the difference to be the absolute error and the multiple it by the derivative at the true value?

 

[HallsofIvy]

x is the solution you get using Newton's method and ^~x is the "true" value. You can't know ~x, that's why they say "approximate".