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Homework Help: Remainder Estimation Theorem & Maclaurin Polynomials :[

  1. Jan 11, 2007 #1
    1. The problem statement, all variables and given/known data

    Use the Remainder Estimation Theorem to find an interval containing x=0 over which f(x) can be approximated by p(x) to three decimal-place accuracy throughout the interval. Check your answer by graphing |f(x) - p(x)| over the interval you obtained.

    f(x)= sinx
    p(x)= x - (x^(3)/3!)


    2. Relevant equations

    Remainder Estimation Theorem:
    |Rn(x)| is less than or equal to (M/(n+1)!)|x-xo|^(n+1)


    3. The attempt at a solution

    i honestly have NO idea how to do this problem.. am i supposed to find the Maclaurin polynomial of sinx?

    any help would be sooo appreciated!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 11, 2007 #2

    Dick

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    p(x) IS the Maclaurin series for sin(x) up to n=4. Carefully write down the remainder term R and figure out how small x should be to guarantee R<0.001. Right?
     
  4. Jan 11, 2007 #3
    Hmm well that makes sense, but how do I find the remainder term? I dont understand that..
     
  5. Jan 11, 2007 #4

    Dick

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    You wrote down a somewhat garbled form of it in part 2). Look it up and figure out what the parts are in this particular case (sin(x) expanded around x=0).
     
  6. Jan 11, 2007 #5
    Oh.. is it Rn(x) = f(x) - pn(x) = f(x) - [sigma] (f^(k)*(xo))/k! * (x-xo)^k ?

    That's the only other equation I can find..
     
  7. Jan 11, 2007 #6

    Dick

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    Noooo. No quite. It's actually more like your form in 2. Let's use that if you can tell me what M and x0 are.
     
  8. Jan 11, 2007 #7
    I guess I would say xo is 0? And M is |F^(n+1)(x)| ?
     
  9. Jan 11, 2007 #8

    Dick

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    That's a pretty good guess. Except that M is the maximum of the expression you wrote over all of the x in the interval where we are going to use R. Is that what you meant to say? Now can you think of a good ESTIMATE (upper bound) for M?
     
  10. Jan 11, 2007 #9
    Right.. M is the upper bound.. And I'm trying to find an interval with x=0 in it, so could the upper bound be pi? or 1? Ok, I'm not sure.
     
  11. Jan 11, 2007 #10

    Dick

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    What is f^(n+1)(x) in this case? While we are at it what is n? You are getting there.
     
  12. Jan 11, 2007 #11
    Isnt it the nth+1 derivative of ...sinx?
    Hahah I dont feel like I'm getting it at all :[
     
  13. Jan 11, 2007 #12

    Dick

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    Why so glum? You are exactly right. What is n (there are two right answers!).
     
  14. Jan 11, 2007 #13
    Hmm.. I dont know isnt just infinity? Or does it get a value?
     
  15. Jan 11, 2007 #14

    Dick

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    It definitely gets a value! The series expansion may be infinite, but the remainder term estimates the error in truncating the infinite series to a finite one. What is the highest power in our FINITE series? (It's on your question page.)
     
  16. Jan 11, 2007 #15
    OH! Hmm..all that's on my question page is.. 0?
     
  17. Jan 11, 2007 #16

    Dick

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    One of these is a truncated series. Exercise: DO work out the Maclaurin expansion of sin(x). Maybe this is not what you are getting.
     
  18. Jan 11, 2007 #17

    Dick

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    Sorry. Meant to say "This is maybe what you are NOT getting."
     
  19. Jan 11, 2007 #18
    So: let xo = 0?
    f(x)= sinx, f(0)=0 po(x)=0
    f'(x)= cosx, f'(0)=1 p1(x)=x
    f''(x)= -sinx, f''(0)=0 p2(x)= x
    f'''(x)= -cosx, f'''(0)=-1 p3(x)=x - (1/3!)x^3
     
  20. Jan 11, 2007 #19

    Dick

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    Bravo! So we want the remainder term for which value of n?
     
  21. Jan 11, 2007 #20
    Hahah ummmmm... the p3?
     
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