5

Problcm Two: Calculate f(3.5) with order through using the below Newton - divided differences.FrSt Second [hird 2.5625 0.588533 0083367 0145827 8.4453 0.54685 0.375...

Question

Problcm Two: Calculate f(3.5) with order through using the below Newton - divided differences.FrSt Second [hird 2.5625 0.588533 0083367 0145827 8.4453 0.54685 0.37502 0.145842 33516 073436 0.13543 1875 09315 5.4375Fourth 26905E-06

Problcm Two: Calculate f(3.5) with order through using the below Newton - divided differences. FrSt Second [hird 2.5625 0.588533 0083367 0145827 8.4453 0.54685 0.37502 0.145842 33516 073436 0.13543 1875 09315 5.4375 Fourth 26905E-06



Answers

In Exercises $11-14,$ approximate to three decimalplacesusing Newton's Method and compare with the value from a calculator.
$$
5^{1 / 3}
$$

Okay Swimming to use Newton's method to approximate to to the seven thirds power. First, let's notice that to to the seven thirds power is to to the third power times four. So if we need the function where this is a root, we get that the function is equal to X cubed minus two. Hey, we, when we plug it into a calculator to to the seven third is approximately so it's four times two to the third, so we really need to approximate to to the third, which is how we get this formula here. So we plug this into a calculator. You should get five point 0397 which is four times 1.2599 Okay, so we have our function. We know what the value and a calculator will yield. Let's use Newton's method. So the formula for Newton's method is X been. Plus one is to go to exit been minus a FedEx been over prime. Excellent. So our formula becomes excellent minus X cubed minus two. The derivative of X cubed is three x squared. The derivative of negative two is zero. So let's let since we found that it is 1.2599 Let's say ex not is equal toe one. Okay, so in exit been when an zero excellent is one plugging that in when n is one at 1.333 when n is too at one point 26 four and when n is three at 1.25 99 which is what we expected.

We need to approximate three to the negative 1/4 power using Newton's method. So if we have three to the negative 1/4 power, this is a root of the function f of X equals X to the negative fourth minus three. If we plug three to the negative 1/4 power into a calculator 0.75 983 to use a few more decimal places than three so we can see what's going on. Feel freed around if you need thio. Okay, so let's let our X of not B 0.8 as a good guess. So the equation for Newton's method is X n plus one equals xn minus f xn over F private axon. So we have in an X N we said, When in is zero X and is 0.8. When an is one, you will get 0.7 54 My name is to you will get zero point 759 seven. And when N is three, you were getting zero point 75 nine. It's which is what we expected

I guess this pump has a several pieces of information on. First thing we know is that our time is 35 seconds. We know our initial speed from rest is from rest is gonna be zero. Um, we don't know. Takeoff speed the distance required. So it gives it to this is 1.5 kilometers going to write that as 1500 meters just to kind of keep everything. And the units for Nunes can so heart a with just that information, it says that the mass is going to be 1.7 times 10 to the fear 1.70 times to 10 to the fifth kilograms. Okay, what force is needed for takeoff? Okay, so we don't know our final come speed. But we do know that our final position is to be 1500 meters in 35 seconds. Somebody is the Kinnah Matic equation of our delta acts equals the not times t uh, which do not a zero times 1/2 a t squared. Okay, So our Delta X was 1500 meters and this could be 1/2 times 35 squared times. A good to sort of solve this with 1500 times two divided, about 35 squared. So I am just gonna put all that in my calculator. My acceleration is going to be 2.45 meters per second squared. Okay, so I wanted to know the force. So all we have to do is multiply our acceleration by our mass. Um, no, You just do that Times 1.7 times 10 to the death. So our total brust our total force, um, is gonna be 416. Approximately 416,000 newtons. Okay, if we're keeping to significant, it's okay for part B. It has asked us to use this strategy. Eso this. I mean, basically, just use my Kinnah Mannix equation to know that if I have a plane, uh, bad drawing but playing and I know that it has to go a distance acts I can use my kinetic equation where X equals 1/2 a t squared. Was he not tea plus X? Not and so that. And I said, this is my position equation to find my acceleration because I had my mass. So all I needed was acceleration and calculate the Newtons of force. Okay, thank you very much.

Hello. The value are going to solve a problem. Number 11 for Newton Method, which is one of the methods to solve nonlinear nonlinear equation. That's meaning we can get the route off the main equation or the zeroes off that equation. So we can we have to follow some step number one. We have to get initial solution or start with initial solution, which will be Devon in the problem, or we can assume it. Assume it according toe the graph or the nearest value toe. Get zero off the question. Okay, Number two, we have to get the derivative off the main function of dash off X and number three substitute at the Newton formula X n plus one equal X and minus F X and over F dash off X and okay so way have, uh, our problem, which is contained, um f of X Equal X was a power 3.3 point nine x squared oh plus 4.79 x minus 1.881 So the derivative of that function, if they should x equal three x squared minus seven point eight eggs lost four 0.79 so we can before, before the before the forming off the table toe solve. We can assume that curve off that function. So it will be an that that, uh, that square or that the right off the destination off the two off access so we can assume explode almost equal 4.5 or try with value nears for it. So we can assume x x n s the initial contribution with point five So f xn equal negative 4.3 36 on DSO on toe. Finally to get the X n plus one equals 2.7 on that can trial. Second trial will be give us all 0.8266 and the third one will be equal Will be get a 10.87 on dso on so we can continue toe get the the value off x n plus one it almost equal on DNO. No, no change between the true value so we can get final results off the zero or approximate off zero off off f equal. Almost equal All point mine. Okay, Yeah, okay, the other zero. Similarly, we can try with different initial solution. So I try trial with 1.1. So we can, uh, end plus one will be equal 1.1. So the zero it will be equal 1.1 as the second zero or the second truth. Okay. And the third truth will be equal. I assume X and equal 1.9. We can try with 1.5 or 1.6 and get the X M plus one. So finally we can get that with 1.9. So they're zeros. Or the third day he rose is 1.9. Sorry. So the zeroes off the function or the solution will be equal. All point mine on 1.1 on one point. Mine. So thanks for watching.


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