Newton's method (introduction & example)

Using Newton’s method to solve a quintic equation! Newton's method is one of the must-know topics in calculus 1 and the concept is just based on using the equations of the tangent lines.

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#bprp

0:00 opening story
2:05 deriving Newton's method
10:07 using Newton's method to "solve" the quintic equation
19:49 check out Brilliant to learn more calculus!
20:47 Fun fact, x^5-5x+3 is actually factorable

[tonmoisingh]

man i absolutely hate it when someone jumps out of the blue and asks me to solve a quintic equation :/

[fantiscious]

15:19 "Everything is doable if you have patience. But I have a calculator"

*Proof mathematicians are not accountants*

[ronaldrosete]

"Everything is doable if we have patience."
-BlackPenRedPen, 2022 15:19

[barobabu]

The other two solutions are
x = -1.618034 and
x = 0.618034
And I see why you said the answers would be interesting !!

[bradryan]

Just taught this to my grade 11's this week. They loved how we can finally solve equations involving different classes of functions, where no closed form solutions are possible. The best part of Newton's Method is that with a good "First Guess" the solution converges very quickly to remarkable accuracy. We also discussed and I showed them graphically, what happens when you do not start off with a good initial value.

[okaro]

Newton's method is easy to do with a calculator. First enter the initial guess and press = (or Exe on some models). Then enter the x-f(x)/f'(x) using Ans in place of x. Then you just hit = until the value does not change.

Many calculators have built in Newton's method. I tried to trick it by giving initial value where f'(x)=0 but it did not fool. They likely use some small difference in x instead of a true derivate.

The nice thing in Newton's method is that even if you make an error it likely just slows you down, you will not get a wrong answer.

[diogoprudente]

im almost graduating at mathematics and i cant express how much i love your yt channel

[ipcheng]

I just did a tutorial about newton method for UG students last week

[flowingafterglow]

OK, is there any way to set this up to solve x as the limit of when xn - xn-1 goes to 0?

I guess for the other solutions start at 0 and -2. That should get there, right? There is a solution between 0 (pos) and 1 (neg) and a solution between -1 (pos) and -2 (neg), but -1 will blow up so start at -2

[troyshrauger]

Literally just co-wrote a paper on Galois. I love this video so much.

[templateaccount]

We had some weird calculus aasignment in school where you were supposed to prove that the integral of -ax^2+b from root to root always was 3/2. I thought it didnt make any sense

[onlythefacts]

Newton's method is also one of the easiest (or at least most well known) methods for solving Kepler's equation for the eccentric anomaly.

[ampisiades]

Congrats for the 903k followers. You will get a million just fast!!!

[letstalksciencewithshashwa]

He's the guy Steven's dad compares his son with

[ehoumanjohann]

Thanks for this video, I have a better un comprehension of this optimization’s method

[watching]

This is best video about newton's method
I didn't understand how got formula y-f(x_1) = f`(x_1) (x-x_1)
Some else mentioned right angle to drive the equation, maybe that video explained how derive equation.
He really used decimal value 1.67 instead of 1 2/3 or 5/3

[georget]

Actually, when I was taught this method, it was named the Newton-Raphson method.

[Infinium]

I’m in the middle of making a video on the Newton-Raphson method! ❤️

[PatrickxJames]

Finding all solutions here is simple: just find x values where the derivative is zero and run Newtons method at most once on both sides of the x values found. For instance, in this example we have f’(x) = 5x^4 - 5. Setting f’(x)=0 gives x=1 and x=-1 as solutions where the slope is 0. This means that there will be 3 solutions, since there are 3 different intervals where the slope does not equal 0: [-inf, -1), (-1, 1), and (1, +inf). Now just pick points within each interval and run Newton’s method to find the 3 solutions.

[tobybartels]

As you noted, Newton's Method doesn't always work, but there _is_ a theorem that sometimes guarantees that it will. It's fairly intuitive if you think about it this way:

First of all, if the derivative of the function is zero, then of course Newton's method doesn't work at all, as you noted. On the other hand, if the _value_ of the function is zero, then you already have the solution, and Newton's method finishes after 0 steps. In general, if the derivatives are small in absolute value, then that's bad; while if the values are small in absolute value, then that's good. Putting these together, if the difference between two successive approximations (which is a value divided by a derivative) is small, then that's a good sign. A little less obvious, but also true, is that if the _second_ derivatives are small in absolute value, then that is good too. (One way to see this is to imagine that you're travelling along a tangent line that closely follows the curve; if the second derivative is rather large, then the curve will quickly pull away from your tangent line, and you will not be heading towards a solution after all.)

So here's the theorem: If you start with one guess x0 and use Newton's method to get a second guess x1, then you can look at the interval J that has x0 at one endpoint and x1 in the middle. If on that interval J, the absolute value of f″(x) is always at most the absolute value of f′(x1) divided by the length of the interval J (in other words, |f″(x)| ≤ ½|f′(x1)f′(x0)/f(x0)|), then you're guaranteed that Newton's Method, starting from x0 as you did, will converge to a solution in the interval J. (And then this is the _only_ solution in that interval.)

There are even more general versions of this, where the function doesn't have to be twice differentiable, and where you have a function of several variables. With some additional assumptions, you can also guarantee that Newton's Method converges _quickly_ (which is again something that usually happens but not always). Look up Kantorovich's Theorem if you want to see the fancy versions.