Challenge: Use Newton's Method to approximate the zero of the following function using $10 \pi$ as the initial value. And yes, it DOES coverge.

$$f(x) = \frac{1}{2} + \frac{x^2}{4} - x \sin(x) - \frac{\cos(2x)}{2}$$

Question

Challenge: Use Newton's Method to approximate the zero of the following function using $10 \pi$ as the initial value. And yes, it DOES coverge. 

$$f(x) = \frac{1}{2} + \frac{x^2}{4} - x \sin(x) - \frac{\cos(2x)}{2}$$

lgbasallote · Answer

<--i hate math

hero · Answer

Are you going to try it or not?

lgbasallote · Answer

nope. no idea

mathmate · Answer

@Hero , why don't you try it, is there a problem?

hero · Answer

I posted this as a challenge. Do you know what that means? It means I already know the answer and I'm challenging others to try it as well.

hero · Answer

I'm pretty sure this isn't the first time you've seen users post "challenges"

mathmate · Answer

Oh, I see!

mathmate · Answer

It probably will converge, but to which root?  Are you looking for a particular one?

hero · Answer

All you have to do is use $10 \pi$ as the initial root and see what it converges to. When you find the number, post it on here.

mathmate · Answer

So you want us to blindly find a root, and you don't care which of the three we give you?

hero · Answer

I have to warn you though....
Challenges are usually not "easy"

mathmate · Answer

Why start with 10 pi, so far from the roots?

hero · Answer

Not "blindly". The only thing you need to use Newton's method are the following:

1. Newton's Formula
2. f(x)
3. f'(x)
4. The initial value

hero · Answer

Because that's part of the "challenge" of course.

mathmate · Answer

I call it blindly when we don't have any judgment to make, or stick to our preferences!  :)

hero · Answer

The bit about how far $10 \pi$ is from the root is only relative. It is pretty close to one of the roots compared to infinity.

mathmate · Answer

Hey, everything is close when compared to infinity!

hero · Answer

Okay, so are you going to solve this challenge or not?

hero · Answer

Right now, you're just teasing

mathmate · Answer

Just wanted to find out what you're after!  I'll be back.

hero · Answer

Well, you better hurry up before someone else figures it out! lol

mathmate · Answer

:)

hero · Answer

@asnaseer, you're more than welcome to contribute

asnaseer · Answer

-1.8955 is what I get (approx)

hero · Answer

See what I mean @mathmate

asnaseer · Answer

in 13 iterations

hero · Answer

Impressive.

hero · Answer

What tool did you use to calculate it?

asnaseer · Answer

I calculates the derivative, then plugged it into the Newton-Raphson equation and entered that into Wolfram as this: http://www.wolframalpha.com/input/?i=y%3Dx%2B%281%2Bx^2%2F2-2*x*sin%28x%29-cos%282x%29%29%2F%28x-2sin%28x%29%29%282cos%28x%29-1%29+for+x%3D10pi this gave a value for y, which I then plugged back into x in wolfram and continued this iteration

asnaseer · Answer

until it converged

hero · Answer

Wow, only 13 iterations is impressive.

asnaseer · Answer

well - wolf did most of the hard slog here :)

hero · Answer

I did it in less than 13

asnaseer · Answer

sorry - it took 12 iterations not 13 :)

asnaseer · Answer

for 4 decimal place accuracy that is

hero · Answer

Funny thing is, if you use mathematica, maple, or any ready-made program to do it, it will say that it doesn't converge.

hero · Answer

I did it using TI-Nspire in the same manual manner as you and got it.

asnaseer · Answer

I assume TI-Nspire is some sort of scientific calculator?

hero · Answer

You don't know what TI-Nspire is?

asnaseer · Answer

nope :)

hero · Answer

You should look it up

asnaseer · Answer

I have a Mac - why would I also need a calculator?

hero · Answer

Well, I guess if you are not still in school, it won't be of very much use to you. I just like to play around with it. Plus you can program all kinds of stuff on it.

asnaseer · Answer

I left school (and Uni) a loooong time ago my friend - and I use the Mac at home and a windows PC at work to program in. so I don't really need a calculator as such these days. :)

hero · Answer

Good for you. Maybe you can look into it for your kids who might want one some day.

asnaseer · Answer

good point - I will - I guess from the manner in which you are promoting it, it must be a good calculator?

hero · Answer

I don't recommend stuff that isn't impressive. I think you should at least try out the student software. It's something you can download onto your computer and play around with.

asnaseer · Answer

there seem to be lots of variants - is there a particular model that ou would recommend?

hero · Answer

The latest model. TI-Nspire CAS models. CX is the latest version

hero · Answer

But I would recommend you try out the student software just to get the hang of the usage.

asnaseer · Answer

this one? http://www.amazon.co.uk/Texas-Instruments-Graphic-Calculator-Science/dp/B0052RU8TO/ref=sr_1_1?ie=UTF8&qid=1347150004&sr=8-1

asnaseer · Answer

and where do I get this software from?

hero · Answer

Yeah, I was just about to mention that you should go to TI's site to get the software. I can post a link to that.

asnaseer · Answer

yes please

hero · Answer

Are you using the Mac or Windows at the moment?

asnaseer · Answer

I use both - but I am on the Mac at the moment

hero · Answer

http://education.ti.com/calculators/downloads/US/Software/Download/en/6770/8184/TI-Nspire_CAS_Student_Software-3.2.0.1219.zip

asnaseer · Answer

thanks Hero - greatly appreciated! :)

hero · Answer

The homepage of the site is simply ti.com

asnaseer · Answer

ok

hero · Answer

It's the best calculator ever, that's why I'm surprised you never heard of it.

asnaseer · Answer

us old fogeys don't always keep up with the latest gadgets! :D

amilapsn · Answer

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