Find E if E = Eix:P(x)]:P(x)0,0029 0 03190 1418 0 3157 0 3513 0 1564E=(Simplify your answer Round to four decimal places as needed )...

A decimal representation of $e$ Find $e$ to as many decimal places as your calculator allows by solving the equation $\ln x=1$ using Newton's method in Section $4.6 .$

For 91 first Newton's Method Number one. Guess first approximation toe a solution off the equation. F X equals zero number two use. Use the first approximation to get the second and second to get a feared and so on. Using the formula it's in plus one equal six in minus If you fix me over. If dish off fix and if X in eight days off, examine not equal zero. So make f of X equal, then X minus one. They're full. If dish off X equal one over X now, let first approximation toe a solution. All F of X equals you B x one equal 2.7 Hence sweet we get x two equal x one minus f of x one over if dish or fix one equal 2.7 minus minus 4.4675 over 4.37 037 It will equal approximately equal two point 71 eight to to right 018 and accessory equal 2.7 18 to 250 one eight minus, if off, 2.71 mhm to to 5018 over If Dish 2.71 Eat toe fight 018 Approximately equal 2.71 Eat 281 8 to 8. It's full equal 2.718 to eat one. Eat to eat minus if off. 2.718 to eat. Born. Eat to eat over. If Daschle, 2.718 to eat one. Eat to eat it approximately equal 2.71 eat to eat 18 toe Eat so x equal. 2.718 to 8. One eat to eat is equal toe E to nine. This women places yes.

You want to find E to as many desolate places as we can by solving the equation. Natural Log of X is equal to one using new its method, and we learned in section 4.7. Now, the reason why this will work is cause. Remember, if we were to solve for X, we would exponentially ate each side and we just get X is equal to eat. So that's why this works. But remember, for Newton's method, this is for solving for zeros. So we'd want to subtract one over, and I'm going to call F of X equal to the natural log of X minus one, and this is what we're going to start using everything for. So we want to used this equation right here to help us all. And you can do this as many times as you want until you feel really comfortable with your answer. But I'm just going to go ahead and do it only two or three times. And if you want to do it more, you can, because otherwise we'll be here forever. Just watching me do algebra. Yeah, now we need to find our derivative, though, so that's C F prime of X. Well, that should just be one over X. Now let's plug everything into here so we can kind of quickly evaluate everything. So we still have xon and then minus natural log Oh, X in minus one all over that. It's one over x in. So we can actually reciprocate that now. And it looks like it would give us exit, then negative accent times the natural log times the natural log of X in minus are actually plus, except when we distribute that negative. So the ex ends, we'll add together, actually get factor all the accents out. Now we have X in times to minus natural log of X. So this is going to be equal to X N plus one. Now what? What should our initial guess? B will. We know that e is approximately equal to 2.71 So maybe we just start with that value. So we say X zero is equal to 2.71 So x one is going to be equal to Well, let's see, we'd have 2.71 times two minus natural log of 2.71 and now plugging this in should give us approximately. So natural log of 2.71 is about 0.99 and two minus. That is about 0.1 of three. And multiply that by a two point seven one Is, uh, something around 2.718269 199 And then let's just do it a couple more times. Just see if we could get anything better. So we're gonna plug this number in now, and so I'm just gonna call it X one, so we don't have to keep plugging it in. So you'd have x one to minus natural log of x one. And doing that should give us. So it'll be x one times two minus the natural log. Well, that and this gives us something around 2.718 to 8. 18 to 8. And let's just do it one more time and then if you want, you can keep on going past this. But I think I'll go ahead and stop after this here. And so this is approximately equal to. And actually, when I plug this and I get the same thing again for my answer and now Let's actually see what e is and how it compares, actually and E is approximate 2.718 to 8. 1828 at least when I plug it into my calculator. So this right here seems like a good place to stop.

Hello and welcome to this video where we're gonna solve the equation. E. To the negative two X plus one is equal to eight. And then we're gonna round our answer to four places. Uh decimal places. That is all right. So the first thing we want to do when we're given an equation like this is make sure that we have E all by itself and we do everything is up in the exponent so we can't do anything with it quite yet. And so the way we get to the stuff in the exponents is we do the inverse of raising or of using E. Which is to take the natural log of both sides. And so we're gonna take the natural log of the left hand side that cancels out this stuff and that gives us negative two X plus one Is equal to the natural log of eight. Now we can go about continuing to solve our equation. Um And we just treat the natural log of it like a number. So I'm going to subtract one from both sides and I have negative two. X. Is equal to the natural log of eight minus one, divide everything by negative two. And then I will have solved for X. Now here's what I'm gonna use my calculator and I'm gonna figure out what the natural log of eight is. Subtract one from it and divide by negative two. And when I do that I get negative 0.5397207708. And so the final step is just around it to the fourth spot. So they'll be negative 0.5397. There we go. That's how we solved that equation.

Eat two decimal places is just too 0.7 two.