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Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.$e^{-x}=2+x$

This is only one zero, and that is $-0.44285440$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 6

Newton's Method

Derivatives

Differentiation

Applications of the Derivative

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so the roots of the equation e to the negative X is equal. Chew two plus X can be estimated using well, Nunes method of approximation. Now, Nunes method says that well X sub and plus one is equal to while Xa ben minus f of x A been so the function evaluated at X Urban divided by well, f prime of x, a band. So the derivative of the function of crime evaluated at excellent. Okay, um, so we have we can have what we that basically just let after backs be equal to, well, set this equal to zero. Also, we have e to the negative X um minus X minus two. Okay. And then therefore we have we have to find, um, the derivative F prime of X so f prime of X is therefore equal to While did root of eat the negative x is well negative e to negative x so negative e to the negative X and then derivative of negative X is just minus one end of 1982 is zero. So the derivative prime of X is well negative E to the negative X minus one. Okay, and then we can go ahead and substitute these terms into the formula for Newton's method. So we have that X sub and plus one is equal to well X samen minus well minus. Um um Ex urban minus will e to the minus X sub in minus X urban minus two. And that is divided by negative E to the negative x of men minus one. Okay, um, so what is this equal? Well, this equals negative Xa ben e to the negative x of an minus e to the negative x saman plus two, um, all divided by negative e to the negative x seven minus one. Okay. And, well, that is equal chew, while the quantity x Saban plus one times e to the negative x samen plus two all divided by e to the negative x sub in, uh, plus one. Okay, now, well, to get, um, an idea about where the possible routes of the function lies, we can graph the function to get, um, negative 0.4. Okay, so we use X sub one. Okay. If you wanted to put this into you, let's say dez most or computer as much a system to graph it. We can get that are X of one, right. Our initial approximation would then be equal to That's a negative 0.4. Okay, so then therefore X up to is equal Chew. Well, um, except one plus one times e to the negative. X seven plus two, all divided by E chew The, um negative x sub one that's excellent here so easily. Accept one, right? This is X one. Um right. So divided by native e to the name negative except one plus one. Okay. And you work this out, you get that, except to is equal to negative 0.4 43 for one 208 Okay. And similarly confined more x of three by just using his iterated process. Fine. Except three Chu, where I'll be equal to Except two plus one times e to the negative. X up to plus two all divided by e to the negative. X up to plus one and we get for X up three to be equal. Chu um negative. 0.44 28 Uh, 54 50 Okay, I m continuous. Same method. We get X sub four, get, except for to be equal to? Well, negative. Um, zero point for four, 28 544 zero. Okay, every do except five. Get. This is an exact value. So therefore, the root of the function. Approximately eight decimal places, right? Once you do accept five, you basically don't change up to eight decimal places. So you want approximation? That's, um, good. Up to eight decimal places. So here we go. Um, right. Negative. 0.44285440 If you do the fifth generation except five get the same thing. So therefore the function approximated to eight different places is given right here. All right, take it.

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