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Please help me with Taylor Polynomials

  1. Apr 9, 2008 #1

    I'm having trouble with this question and was wondering if someone could give me hints or suggestions on how to solve it. Any help would be greatly appreciated thankyou! :)

    Find the Taylor polynomial of degree 3 of f (x) = e^x

    about x = 0 and hence find an approximate value for e. Give an estimate for the error in the approximation.

    I know from following the Big O Notation...

    e^x = 1 + x + (x^2) / 2! + (x^3) / 3! + ... + (x^n) / n! + O (x^(n+1))

    So I'm thinking for a polynomial of degree 3 we have...

    e^x = 1 + x + (x^2) / 2! + (x^3) / 3! + O (x^4)

    And so from there i'm really not sure what comes next?? Could someone help me please??
  2. jcsd
  3. Apr 9, 2008 #2
    What is the definition of the Taylor polynomial of degree 3?
  4. Apr 9, 2008 #3
    I think the Taylor polynomial of degree 3 is the last line I have above... after substituting
    n = 3 in the Taylor formulae (next equation up).

    But I'm not sure if I'm doing it the right way?? Any help please??
  5. Apr 9, 2008 #4
    Your last line is not a polynomial (due to the big O term).

    If you omit it, your expression is indeed the Taylor polynomial of degree three.
    Last edited: Apr 9, 2008
  6. Apr 9, 2008 #5
    Oh ok, I didn't realise that...
    So from there do you know how i can find the error??
  7. Apr 9, 2008 #6
    Yes, I do.
    Are you familiar with the various forms of the remainder term in a Taylor approximation?
    If not you should look it up in your textbook or on wikipedia

  8. Apr 9, 2008 #7
    Ah yes! We use Lagrange's form for the error. But doesn't that just give me an expression for the error? And not a value for the error which the question asked?
  9. Apr 9, 2008 #8
    Yes, it gives you an expression which you have to estimate.
  10. Apr 9, 2008 #9
    ummm, I don't really understand... I've tried looking in my text book but it doesn't help much...
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