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Unformatted text preview: he sequence
repeated over and over again, making those digits useless. For one of the
problems, she has been asked to compute A3/4 for a few different integer values of A. Mal knows
that Newton’s Method can be used to compute the square root or the cube root of an integer A. So
as a ﬁrst step in computing A3/4 , Mal wants to use Newton’s Method to compute A1/4 . She
then plans to use that information to compute A3/4 .
(a) [5 points] Mal decides to use the function f (x) = x4 − A, because it has a root
at x = A1/4 . Use Newton’s Method on f (x) to generate a formula for computing
increasingly accurate estimates of A1/4 . In other words, give a formula for the
more accurate estimate xi+1 in terms of a less accurate estimate xi . The formula you
construct must use only addition, subtraction, multiplication, and division. (You do
not need to simplify the formula.)
Solution: If we take the derivative of f (x), we get f (x) = 4x3 . Hence, Newton’s
Method says that we have
xi+1 = xi − x4 − A
xi · xi · xi · xi − A
i
.
= xi −
3
4xi
4 · xi · xi · x i 6.006 Quiz 2...
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 Fall '11
 ErikDemaine

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