Problem

Use Newton's method to approximate the given numb…

07:28

Problem 11 Medium Difficulty

Use Newton's method to approximate the given number correct to eight decimal places.
$\sqrt[5]{20}$

Answer

$\sqrt[5]{20} \approx 1.8205642$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 6

Newton's Method

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Video Transcript

Okay, so let's consider here the fifth root of 20. Okay, So considering the fifth root of 20 um and we want to well find. So basically, consider the equation. Um, X to the fifth minus 20 is equal to zero. Okay, I'm so solving right. The fifth root of 20 is equivalent. They're finding the root of the equation. Extra fifth minus 20 is equal to zero. Because solving for the value of X in, um, this equation while this would give us the fifth root of 20. Okay, So what? We're really trying to dio when we're trying to find a 15 to 20 is trying to find the roots of this equation here. Exited thrift minus 20 is equal to zero. Okay, so we take after backs is equal to exit of F minus 20 and, well, to use Nunes, Method would wanna find f prime of X. So therefore, F crime of acts is equal. Juel five acts to the fourth. Okay. Just you know, the five right, and then did decrease explained by one. So yes, so five x to the fourth is are derivative of X minus 20. And then according to Nunes Method So just reminding you what does Newton's method say? Where Nunes Method says that? Um, right, if we're given X amend to find X sub and plus one well, that is equal to X. A ban Manus F of X a man. So the function evaluated at X men divided by the derivative evaluated at Xidan so blinded by f crime of X seven. Okay, um, where, you know, exit one except to accept three. And so on is a sequence of approximations. Okay, so here we can substitute. Well, um, so f of x ub end right equals x, except end to the fifth minus 20. And, um, prime vex event equals x five times X events the force. So we get to find X sub and plus one while x sub n plus one is, um, equal to x sub. And to the fifth minus 20 divided by five Xa Ben to the fourth. Okay, so let's choose an initial approximation. Let's choose our approximation X up one to be equal to two. Okay, you just start with an initial approximation. Let's choose our initial except one, um, to be equal to two. Okay, You could You could you could call this except zero, except not sometimes, but I mean your initial could be mean. Let's just start at one. So it's a except one secret to Okay, so therefore, accept to write. Except n plus one is equal to what? What's equal to X urban, which is two minus FX event, which will be two to the fifth. Minus 20 um, divided by five times, two to the fourth. Okay, so what we get here is that except two is equal to Well, um, tu minus that. See, we have 30 to minus 20. So minus 12 of the numerator and then divided by or five times, um, to to the fourth. That's five times 16 which is 80. So to minus over 80 and that is approximately equal to so accepts. X sub two is approximately equal to one point a five. Okay. And then you just repeat right. Given that X up to is 1.85 well, except three that just do one more here. So, except three is equal to then, while except two, which is 1.85 um, minus well f of X, up to 1.85 to the fifth power, which is Ah, 21.67 minus 20. Divided by while five times. Except two, which is five times one point, in fact, of the fourth, which is 11.71 Okay. And you evaluate this and we get X sub three X up three to be approximately equal to 1.82 one for age 613 Okay. And then we're off to find, except four. Right? You just did the same process. Okay. Except four is equal. Except three minus. After Brexit 3 20 I'm divided by five times. So you get what you get. If you just picked through the exact same thing with except four we get, except for to be approximately equal to 1.8 to 05 um, six, 513 ish. Okay, um, and if I accept five or accept five is again except five is just to take your Except for now, my nancy, that this number minus well, that number to the fifth minus 20 times divided by five times this number. And we get X sub five to be, um, approximately equal toe one point. Eat ah, to um, 056 We're getting pretty close here. Uh, 0565 or 0564 20 Actually, so we chains the hopes for 20. Thanks. Worked pretty. Okay, Um, and then well, X up six again. Same exact thing for except six. Right. You know, you plug in for except six. So we take this number now minus, um, that number shoe the fifth power. But it's 20 divided by by times, this number to the fourth power. And we get except six to be approximately equal to 1.82 056 um, for 20. So since right ex of five and X sub sects agree to eight decimal places, then we can stop, right? Were asked to Ah, approximate the fifth root of 20 to 8 decimal places. Right? And since now, we both agree, um, to eight decimal places. We can stop right there. Stop right there. And, um yeah. So the fifth with 20 is approximately equal to 1.820564 to 0. All right, take care

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