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Hero

  • 2 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
    • 2 years ago
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    <--i hate math

  2. Hero
    • 2 years ago
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    Are you going to try it or not?

  3. lgbasallote
    • 2 years ago
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    nope. no idea

  4. mathmate
    • 2 years ago
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    @Hero , why don't you try it, is there a problem?

  5. Hero
    • 2 years ago
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    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
    • 2 years ago
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    I'm pretty sure this isn't the first time you've seen users post "challenges"

  7. mathmate
    • 2 years ago
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    Oh, I see!

  8. mathmate
    • 2 years ago
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    It probably will converge, but to which root? Are you looking for a particular one?

  9. Hero
    • 2 years ago
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    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
    • 2 years ago
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    So you want us to blindly find a root, and you don't care which of the three we give you?

  11. Hero
    • 2 years ago
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    I have to warn you though.... Challenges are usually not "easy"

  12. mathmate
    • 2 years ago
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    Why start with 10 pi, so far from the roots?

  13. Hero
    • 2 years ago
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    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
    • 2 years ago
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    Because that's part of the "challenge" of course.

  15. mathmate
    • 2 years ago
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    I call it blindly when we don't have any judgment to make, or stick to our preferences! :)

  16. Hero
    • 2 years ago
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    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
    • 2 years ago
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    Hey, everything is close when compared to infinity!

  18. Hero
    • 2 years ago
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    Okay, so are you going to solve this challenge or not?

  19. Hero
    • 2 years ago
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    Right now, you're just teasing

  20. mathmate
    • 2 years ago
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    Just wanted to find out what you're after! I'll be back.

  21. Hero
    • 2 years ago
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    Well, you better hurry up before someone else figures it out! lol

  22. mathmate
    • 2 years ago
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    :)

  23. Hero
    • 2 years ago
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    @asnaseer, you're more than welcome to contribute

  24. asnaseer
    • 2 years ago
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    -1.8955 is what I get (approx)

  25. Hero
    • 2 years ago
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    See what I mean @mathmate

  26. asnaseer
    • 2 years ago
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    in 13 iterations

  27. Hero
    • 2 years ago
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    Impressive.

  28. Hero
    • 2 years ago
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    What tool did you use to calculate it?

  29. asnaseer
    • 2 years ago
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    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
    • 2 years ago
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    until it converged

  31. Hero
    • 2 years ago
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    Wow, only 13 iterations is impressive.

  32. asnaseer
    • 2 years ago
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    well - wolf did most of the hard slog here :)

  33. Hero
    • 2 years ago
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    I did it in less than 13

  34. asnaseer
    • 2 years ago
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    sorry - it took 12 iterations not 13 :)

  35. asnaseer
    • 2 years ago
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    for 4 decimal place accuracy that is

  36. Hero
    • 2 years ago
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    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
    • 2 years ago
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    I did it using TI-Nspire in the same manual manner as you and got it.

  38. asnaseer
    • 2 years ago
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    I assume TI-Nspire is some sort of scientific calculator?

  39. Hero
    • 2 years ago
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    You don't know what TI-Nspire is?

  40. asnaseer
    • 2 years ago
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    nope :)

  41. Hero
    • 2 years ago
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    You should look it up

  42. asnaseer
    • 2 years ago
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    I have a Mac - why would I also need a calculator?

  43. Hero
    • 2 years ago
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    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
    • 2 years ago
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    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
    • 2 years ago
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    Good for you. Maybe you can look into it for your kids who might want one some day.

  46. asnaseer
    • 2 years ago
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    good point - I will - I guess from the manner in which you are promoting it, it must be a good calculator?

  47. Hero
    • 2 years ago
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    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
    • 2 years ago
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    there seem to be lots of variants - is there a particular model that ou would recommend?

  49. Hero
    • 2 years ago
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    The latest model. TI-Nspire CAS models. CX is the latest version

  50. Hero
    • 2 years ago
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    But I would recommend you try out the student software just to get the hang of the usage.

  51. asnaseer
    • 2 years ago
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    and where do I get this software from?

  52. Hero
    • 2 years ago
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    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.

  53. asnaseer
    • 2 years ago
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    yes please

  54. Hero
    • 2 years ago
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    Are you using the Mac or Windows at the moment?

  55. asnaseer
    • 2 years ago
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    I use both - but I am on the Mac at the moment

  56. asnaseer
    • 2 years ago
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    thanks Hero - greatly appreciated! :)

  57. Hero
    • 2 years ago
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    The homepage of the site is simply ti.com

  58. asnaseer
    • 2 years ago
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    ok

  59. Hero
    • 2 years ago
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    It's the best calculator ever, that's why I'm surprised you never heard of it.

  60. asnaseer
    • 2 years ago
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    us old fogeys don't always keep up with the latest gadgets! :D

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