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Taylor series

  1. Apr 13, 2014 #1
    1. The problem statement, all variables and given/known data[/b]

    Determine the Taylor series for the function below at x=0 by computing P 5 (x)
    f(x)=cos(7x^2)

    2. Relevant equations

    I used to taylor series for cosx and replaced it with 7x^2
    so i used 1-49x^4/2! +2401x^8/4!... and so on.
    That should be correct, my attempt below :(

    3. The attempt at a solution

    1-(49x^4/2)+(2401x^8/24)-(117649x^12/720)+7^8x^16/40320
    I even tried it by adding one more
    7^10(x^18)/10!
    Can someone tell me where I went wrong? It's nothing with the formatting because entering it like this into my homework showed a preview and it showed up like it should have :( what did I do wrong? Please advise. Thanks in advance.

    I know we're not supposed to upload pictures of the answers, but I uploaded mines. If someone would look at it and see it its correct? IT's attached in the thumbnail View attachment 68644
     

    Attached Files:

    Last edited: Apr 14, 2014
  2. jcsd
  3. Apr 14, 2014 #2

    vela

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    Probably too many terms. The problem asked you to find the fifth-degree Taylor polynomial, right?
     
  4. Apr 14, 2014 #3
    i tried taking out one or two terms.still didnt work :/
     
  5. Apr 14, 2014 #4

    vela

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    What's the highest power of ##x## that should appear (in principle)?
     
  6. Apr 14, 2014 #5
    Shouldnt it be 20?
    because that would be where n=5
     
  7. Apr 14, 2014 #6

    vela

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    No. Suppose you didn't know about the Maclaurin series for cos x and just did the problem the hard way by calculating derivatives of f. How many derivatives would you have to take to calculate ##P_5(x)##? Surely not 20.
     
  8. Apr 14, 2014 #7
    5 derivatives
     
  9. Apr 14, 2014 #8
    but i tried taking out one term, and that didnt work.
     
  10. Apr 14, 2014 #9

    vela

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    Right. So what would be the power of ##x## in the highest-order term?
     
  11. Apr 14, 2014 #10
    would it be 5?
     
  12. Apr 14, 2014 #11

    vela

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    Exactly. The problem asked for a fifth-degree polynomial, so the highest-power should be ##x^5##, so throw out any terms with a higher power of ##x##.
     
  13. Apr 14, 2014 #12
    Oh! so, if it asks for a certain polynomial, the power can't be higher than what they're asking for? That P(5) refers to the power, and not the term?
     
  14. Apr 14, 2014 #13
    Thank you:)
     
  15. Apr 14, 2014 #14

    vela

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    Right. For ##P_5(x)##, you can have up to 6 terms, but if some vanish, you'll have fewer.
     
  16. Apr 14, 2014 #15
    If I had a function asking the same thing as above but the function was 4+15x+x^2sinx, how would I fin a taylor series for this? Would I have to expand out the x^2sinx? What would I do with the 4 and the 15x? I know if I expand out the x^2sinx, I would multiply them to each other, but where would the 4 and 15 x come into play?

    Actually, how would i make expand the x^2*sin(x)?
    I know that sinx x trend is x- x^3/3! + x^5/5!
    How do I do the x^2? Since the derivatives are 2x and 2? I plug in 0?
     
    Last edited: Apr 14, 2014
  17. Apr 14, 2014 #16
    Oh actually, that was silly. I got it.
     
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