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Taylor Polynomial (can you help me?)

  1. Mar 22, 2007 #1
    1) Let f(x) = (x^3) [cos(x^2)].
    a) Find P_(4n+3) (x) (the 4n + 3-rd Taylor polynomial of f(x) )
    b) Find f^(n) (0) for all natural numbers n. (the n-th derivative of f evaluated at 0)


    I know the definition of Taylor polynomial but I am still unable to do this quesiton. I tried to find the first few terms but I can't see an obvious pattern. I have no problem using the definition to find the first few terms...but this is a weird question. Can someone nicely give me some hints on both parts?

    Any help is greatly apprecaited!
     
  2. jcsd
  3. Mar 22, 2007 #2

    Dick

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    Do you know a formula for the Taylor expansion of cos(x) around x=0? What does this imply for cos(x^2)?
     
  4. Mar 22, 2007 #3
    Yes, I know that formula, I guess I can replace x by x^2 and get a formula for cos(x^2) too...
     
  5. Mar 22, 2007 #4
    I also have another terrible quesiton...I am dying from it...

    2) Find the 2006-th derivative evaluated at 0 and the 2007-th derivative evluated at 0 for f(x)=tan^(-1) x (inverse tan)

    Any hints?
     
  6. Mar 22, 2007 #5

    Dick

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    I'm not very clear on what you mean f(x) to be.
     
  7. Mar 22, 2007 #6
    Find the first couple of derivates. Do you see a pattern? Can you determine a formula for the nth derivative of arctan from it? Can you prove your formula works for all n?
     
  8. Mar 22, 2007 #7
    There isn't an obvious pattern...what can I do?
     
  9. Mar 22, 2007 #8
    Look up the generalized leibniz product rule.
     
  10. Mar 22, 2007 #9

    Dick

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    You could also simply look up the series expansion of arctan.
     
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