Let (a) Approximate ln(1 − ) with a third degree Taylor polynomial expanded about = 0 Using this approximation, what value should you assign to (0)? (b) Using MATLAB, plot () for −10−15 ≤ ≤ 10−15....

Let

(a) Approximate ln(1 −
) with a third degree Taylor polynomial expanded about
= 0 Using this approximation, what value should you assign to
(0)?

(b) Using MATLAB, plot
() for −10⁻¹⁵
≤

≤ 10−¹⁵. What value does MATLAB assign to
() for
very near zero? The interval where the function is zero is not symmetric about

= 0. Why? Also, does this also explain why there are more oscillations on the right (
> 0) than on the left (
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(c) Using MATLAB, plot ln(1−) for −5×10⁻¹⁵
≤

≤ 5×10⁻¹⁵
. The graph should resemble steps with the step containing

= 0 corresponding to the value of ln(1) = 0. As

increases from
= 0, what determines the value of the first nonzero step? How do these steps explain the oscillations seen in the plot for part (b)?