October 20, 2020 ## Newton’s Method

what is the true solution to 2 in 4x=2 in 8 This is a topic that many people are looking for. voteyesons.org is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, voteyesons.org would like to introduce to you Newtons Method. Following along are instructions in the video below:
This lesson. Were gonna focus on newtons method for approximating zeros of a a function so lets say if we have this function f of x. Ands equal to x.
Cubed. Minus. 4x squared plus.
1. Now there could be at least three solutions to this equation. Were going to focus on finding just one of them so we have to guess a value for x.
We have to pick a value to start with so lets see what the function value is when x is 0. So f of 0. Is going to be 1 and now lets pick the next best number lets try it one.
So. This is gonna be 1 to the 3rd minus 4. Times.
1. Squared. Plus.
1. So thats 1 minus. 4.
Plus. 1. 1.
Minus. 4. Is negative.
3 plus. 1. This is negative.
2.

So going from 0 to 1 notice that the y value changes from positive to negative and since this is a continuous function at some point it has to cross 0. So one solution to this equation. That is if we set it equal to 0 x.
Has to be somewhere between 0. 1. Theres some value of x.
Where y will equal 0. So lets solve this equation. So if x.
Cubed. Minus. 4x.
Squared plus 1. Is equal to 0. What is the value of x.
So pick a number between 0 1. Were going to start with point. 5.
Thats going to be our first iteration now using newtons method. It helps you to calculate the next 0. Given the first one so x raised to the or x sub n.
Plus 1 is equal to x sub n. Minus f. Of x sub.
N divided by f prime of x. 7 so using the first 05. This equation will help us to get a more accurate 0 or an more accurate solution to the equation.
So if n. Is 1 n. Plus.
1.

Is going to be 2. So we have this expression so. X.
2 is going to equal x. 1. Well.
X. 1 is 05. And then minus f of point.
5. Divided by f. Prime of 25.
So lets figure out what f of point. 5. Is so this is going to be.
Point five raised to the third power minus four times. 05. Squared.
Plus. One so lets use a calculator for that so this is equal to point one two five now we need to find the first derivative the derivative of x cube is to reiax squared and the derivative of 4x squared is 8x. So now with the first derivative.
We need to plug in. Point. Five as.
Well. So this is gonna be three times 05. Squared minus eight times point five so this will give you negative three point two five now lets plug in everything into this equation.
So f of point five. Thats point one two five and f prime of point five is negative three point two five so the two negative signs will cancel so its 05. Plus point one two five divided by three point.
Two five and so this will give us x.

Two. And so thats going to be point five three eight five. So now we need to do another iteration.
So im going to rewrite the first derivative here because were going to use. It again so x. 3.
Is going to equal x 2. Minus f. Of.
X. 2 over. F prime of x.
2 so x. 2 in this example is 053 a 5 so lets evaluate the function at x2 so f of 05. 3.
8. 5. Lets plug it in into this equation.
So go ahead and type it in your calculator. And lets see what thats gonna give us point. 5.
3. A. 5 raised to the 3rd power.
Minus 4. Times point. 5.
3. A. 5 squared plus 1.
Make sure you type in everything correctly.

If you make one mistake. The whole problem is ruined so this is negative point zero zero three seven seven four you might see it as three point seven seven four times 10. So minus three its the same and now lets do the same for the first derivative.
So were gonna plug it in to this equation. So its going to be three times point five three eight five squared minus eight times point five three eight five so you should get negative three point four three eight one so now we can plug these values into that expression. So extreme is going to be point five three eight five minus.
And then f of point five three eight five thats a negative point zero zero three seven seven four and then we need to divide that by negative three point four three eight one now notice that this answer is close to 0. That means x. The solution that were looking for where.
The function has a y value of 0. Is close to this number so x. Three shouldnt be too far away from x2 now these two negative signs will cancel so overall.
Well still have a negative sign so this is going to be point five three eight five when zero zero three seven seven four divided by three point four three eight. One. So you should get this answer point.
Five three seven four which is very close to x2. So chances are because the answer is so close that means that this is a good estimation and lets check in what were going to do is plug it into this expression. And see if we get an answer thats a very very close to zero.
If we do then we could say this is the solution to the equation or at. Least its one of the. Solutions so lets evaluate f of.
05. 374 lets see if its equal to zero. So go ahead and type that in to your calculator.
So i got five point four one three six times ten to the minus six. So thats very small thats basically point zero zero zero zero zero five four one three six. So we could say that is approximately zero.
Which means the solution is point five three seven four thats one of the solutions to this equation. And so thats how you can use the newtons method to solve functions you can approximate zeros of the function just by picking a value thats close to the actual zero. If the value is too far away you may have some issues with newtons method.
But if you want to pick a number thats close to the actual zero and using multiple iterations with newtons method each time you do it youre gonna get an answer. Thats even closer to the real answer so if you do it two or three times. Its usually good and as you can see we only have to do it two times for this example so hopefully this gave you a better understanding of how to use newtons method to solve zeros or find zeros of a function.
So thats it for this video. Thanks again for watching and have a good day .