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Numerade Educator

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Problem 30 Medium Difficulty

(a) Apply Newton's method to the equation $ 1/x - a = 0 $ to derive the following reciprocal algorithm:
$$ x_{x + 1} = 2x_n - ax{_n}^2 $$
(This algorithm enables a computer to find reciprocals without actually dividing.)
(b) Use part (a) to compute $ 1/1.6984 $$ correct to six decimal places.

Answer

a. $2 x_{n}-a x_{n}^{2}$
b. $\frac{1}{1.6984} \approx 0.588789$

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Video Transcript

in question 30. I'm starting with a function, um, one over X minus a. They're for my crying equals one of rights were and my exit and plus one, My next operation, People's ex of end minus effort, Exit end over Pride Exit End, which is in this case, picks up in yes, minus one over X minus a divided by one over X where So now I'm going to just simplify it a little bit? No, I have minus one over exit in minus a ex of end over X of end liking Combine like terms my new greater than gives me and over over x squared n my numerator is now one minus a and I'm good at in bird fly x weird that now becomes except in times one minus a except in or x a n minus a ex were And my first federation is approximately 1.94 part B Into that formula A equals 1.689 four. Therefore, except one. It's approximately when I exit to our point by I get 57 by four. When I plug in fights 0.5754 in that region, I get 0.5 a or a. Why, and I put that in export gives me a 789 I put that e get the same thing back, and that would be like