1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    Bravo! So we want the remainder term for which value of n?
     
  21. Jan 11, 2007 #20
    Hahah ummmmm... the p3?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Remainder Estimation Theorem & Maclaurin Polynomials :[
Loading...