Problem

(a) Apply Newton's method to the equation $1 / x-…

08:34

Problem 23 Hard Difficulty

(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 $\sqrt{a}$ ):
$$x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)$$
(b) Use part (a) to compute $\sqrt{1000}$ correct to six decimal places.

Answer

(a) $=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right) \quad$ factor out 1$/ 2$
(b) 31.622777

Related Courses

Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 6

Newton's Method

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Video Transcript

we have the equation here. X squared minus a is equal to zero. Hey, you can let well let after backs right be equal to world This function here x squared minus a um okay. And then we get while the derivative of F f Prime of X is just of course Well, two x Okay, so, um, that's so if the first approximation right is xom one, then we get the while the n plus n plus one approximation or um x sub and plus one right by Nunes. Method is just equal to well X men minus the function evaluated at X urban So f of X men divided by the derivative evaluated at X urban so f prime of X some and Okay, um and, um likewise, when a second approximation. Right? Um right. So forgiven. Right? Except one Than accept and plus one right into this, um, x sub. If we're so again, if our initial approximation is x one and X sub to right is equal to well except one. Um, in an hour, um, given this function right, X up two is equal to accept one minus well x sub one squared minus a divided by the derivative there. So is two times X somewhat. Okay, um or, um, except to is equal to one. You have a 1/2 outfront. Here's a 1/2 times while two times except one minus x of one squared, plus a divided by while x of one. Okay, Or this is also equivalent shoe. So x so, too is equal to just 1/2 times well X sub one plus a over except one. Okay. And similarly, we get the third approximation, Um, X up three to be equal to, well, 1/2 times except two plus a over X up to okay. And, um such as, Well, the end approximation. So the end approximation x ub n right is just equal to well X sub. And plus one, um, right is equal. So the will, the ends. The confirmation is, um, I have equals here, but his equals well is, except M plus one, which is equal to, well, just 1/2 times x up and plus a over X up at so, um, right here would be the answer to party. Right. It's just 1/2 times x urban plus a over X event to find the the, um, and plus once approximation. Okay, Um then for part B, part B, his part B we've got Well, let's, um, x be equal to the square root of 1000. Okay, then, if this is true, then we have x squared. Minus 1000 is equal to you. Zero. Right? Because you let x be or two square root of 1000 squared 1000 Mind 2000 0 So the square 2000 is a route to this equation here. And we know that, um, 30 square. Well, that's equal to 900 right? So therefore, the value, the value of the square root of 1000 right? Should be near 30. Right? So he takes or good first approximation here. Right. Says that 30 squared is 900 close with 1000. So you think your this the answer here should be a bigger than 30. But a good first approximation is the let X sub one be equal to 30. Okay, so now we have while the n plus once approximation, remember is X up end plus one is equal to 1/2 times x up end, plus a over ex upend. So, um, when n equals one. Right? Then we have, um, ex up to is equal to, well, 1/2 times except one plus a over except one. Right where here? A is equal to 1000. Um, and X of one is equal to 30. Okay. Or like wise, you can say that well, except to is equal to well, 1/2 times 30 plus 1000 over 30. Okay. And that is approximately equal to 31 point 666 667 All right, um, and a similarly we find, except three. Right. The same way. Here we get. A except three is approximately equal to 31.6 to to, um eight. Ah, 070 Okay. And we confined, except for looks, except for in the same fashion. And accept fours approximately equal to 31.6 ah, to to 77 66 And if we went one more further, and if we get X sub five, it except five is approximately equal to this exact same thing, right? 31.6227766 So we see that X afford except five agree to six decimal places. So therefore the root of the equation. The root of the equation. Um, X squared minus 1000 equals zero is well, approximately equal to or just said 31 point six 227766 Right, You have it. All right, take care.

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