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(a) Apply Newton's method to the equation $ x^2 - a = 0 $ to derive the following square-root algorithm (used by the ancient Babylonians to compute $ \sqrt{a} $):

$$ x_{n + 1} = \dfrac{1}{2} \left( x_n + \dfrac{a}{x_n} \right) $$

(b) Use part (a) to compute $ \sqrt{1000} $ correct to six decimal places.

a -$\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)$

b -31.622777

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question 29. I have the function That's weird. Minus A which makes my rhyme Two weeks. So then when I have my and one pulls extra x of n minus and of x were crying of X I get it exit in minus X where with minus a ever two x And which gives me when I I am except in minus X Weird aim over too necks of n plus Because subtracting your negative, eh? Hey, two ex of n which is really, uh I'm gonna combine my life terms Ex of n minus 1/2 quite person Gonna X weird becomes an ex now I combined my like terms and I have ex of n minus 1/2 ex of end So I get a positive one has mix of end nuts A to X So when I backed throughout my one Yeah, my ex of end minus a over Except then that would be court, eh? Or B is cute This word of 1000 which was that formula. You came up with the six decimal places Their core eh would equal 1000. The closest that I have the 1000 is 900 so my except one would be this weird of 900. Therefore, Exit one equals 30. Except to when I put 30 into, um, my 1/2 that clip in mine. It's a over X. Evan. I get approximately 31 0.6. Repeating I quit three when I quit. 31. Into that equation gives me 31.6 22807 and four gives me 31 0.622777 which is the same thing as X, and that's my

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