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## Hero 3 years ago 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. $f(x) = \frac{1}{2} + \frac{x^2}{4} - x \sin(x) - \frac{\cos(2x)}{2}$

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1. lgbasallote

<--i hate math

2. Hero

Are you going to try it or not?

3. lgbasallote

nope. no idea

4. mathmate

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

5. Hero

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.

6. Hero

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

7. mathmate

Oh, I see!

8. mathmate

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

9. Hero

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.

10. mathmate

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

11. Hero

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

12. mathmate

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

13. Hero

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

14. Hero

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

15. mathmate

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

16. Hero

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.

17. mathmate

Hey, everything is close when compared to infinity!

18. Hero

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

19. Hero

Right now, you're just teasing

20. mathmate

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

21. Hero

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

22. mathmate

:)

23. Hero

@asnaseer, you're more than welcome to contribute

24. asnaseer

-1.8955 is what I get (approx)

25. Hero

See what I mean @mathmate

26. asnaseer

in 13 iterations

27. Hero

Impressive.

28. Hero

What tool did you use to calculate it?

29. asnaseer

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

30. asnaseer

until it converged

31. Hero

Wow, only 13 iterations is impressive.

32. asnaseer

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

33. Hero

I did it in less than 13

34. asnaseer

sorry - it took 12 iterations not 13 :)

35. asnaseer

for 4 decimal place accuracy that is

36. Hero

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

37. Hero

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

38. asnaseer

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

39. Hero

You don't know what TI-Nspire is?

40. asnaseer

nope :)

41. Hero

You should look it up

42. asnaseer

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

43. Hero

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.

44. asnaseer

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. :)

45. Hero

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

46. asnaseer

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

47. Hero

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.

48. asnaseer

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

49. Hero

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

50. Hero

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

51. asnaseer
52. asnaseer

and where do I get this software from?

53. Hero

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.

54. asnaseer

yes please

55. Hero

Are you using the Mac or Windows at the moment?

56. asnaseer

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

57. Hero
58. asnaseer

thanks Hero - greatly appreciated! :)

59. Hero

The homepage of the site is simply ti.com

60. asnaseer

ok

61. Hero

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

62. asnaseer

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

63. amilapsn

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