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Taylor's series for sin(x^2)

  1. Apr 3, 2006 #1
    Find the first three terms of the Taylor's series for sin(x^2)

    The constant term is 0 as usual. The first non-zero term is (include power of x as well as its coefficient):

    according to the book

    f(x)=f(0)+f'(0)x+f''(0)x^2/2!+f'''(0)x^3/3!...

    so I think the first non zero term is

    f'(0) = cos(x^2)2x

    but substituting x with 0, the term will be zero

    and there will always be an x in the expression if I keep differentiating.

    So how do I do it?

    THANKS!
     
  2. jcsd
  3. Apr 3, 2006 #2

    HallsofIvy

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    f(x)= sin(x^2) so f'(x)= 2x cos(x^2) as you say. However, when you find the second derivative you use the product rule: f"(x)= 2 cos(x^2)- 4x^2 sin(x^2). No, there is not always an x in the expression. f"(0)= 2.

    In fact, if you know the Taylor's series for sin(x) you can get the Taylors series for sin(x^2) just by substuting x^2 for x in it.
     
  4. Apr 3, 2006 #3
    Thanx alots!
     
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