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Taylor Series Approximation

  1. Jan 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Hi!

    I have a couple of problems on Taylor Series Approximation.

    For the following equations, write out the second-order Taylor‐series approximation.
    Let x*=1 and, for x=2, calculate the true value of the function and the approximate value given by the Taylor series approximation.

    f(x)=(x0.5-1)/0.5
    f(x)=(x-1)2


    3. The attempt at a solution

    I was able to do the approximation at x=1 for the first problem. However, for the second problem, the second derivative yields a constant meaning that I cannot plug in x=1. What should I do in this case?

    I do not understand the following direction:

    "for x=2, calculate the true value of the function and the approximate value given by the Taylor series approximation."

    Please help.

    P.S. I am sorry - I don't know how to paste formulas within this message, thus, I have to attach a Word document.
     

    Attached Files:

  2. jcsd
  3. Jan 14, 2012 #2
    have you learned about binomial series? If you have you can solve the second problem with it without even thinking much if you know about binomial series and just get the right answer that way.

    (1 + x)^p = summation from k=0 to ∞ of (p chose k)x^k

    wait nvm i don't think that will work
    oh actually ya i think you can manipulate it so that way it will

    oh my bad you want the power series not taylor series...
    so than just do from k=0 to k=2

    sorry i forgot how to do the Taylor series and just recalled the summation for (1+x)^p as a power series. You get to the point were you just do these problems so much it's just best to remeber some of the easier ones...

    I'm kinda rusty on this stuff lol

    look up power series of (1+x)^p as a power series and just do from k=0 to k=2
    you can also just put the equation in wolfram alpha to get the answer =O
    note that you still have to figure out why the answer is correct =P

    P.S. I think there's some other way to do this that I forgot about

    but if f(x)=constant
    then f(2) = constant
    it's not a problem
     
    Last edited: Jan 14, 2012
  4. Jan 14, 2012 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    Why do you say you can't plug in x = 1 in the second case? What prevents you from substitution x = 1 in the formula f''(x) = 2 (for all x)?

    RGV
     
  5. Jan 14, 2012 #4
    So you mean that given that f"(x)=2, I can say that f"(1)=2?
     
  6. Jan 14, 2012 #5
    Yes.
    You may imagine it by thinking it is 2 independently by x.
     
  7. Jan 14, 2012 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Of course. That is what we mean when we say f''(x) = 2 for all x. It holds for x = 1 or x = 943.76/sqrt(pi) or x = 10 million, or whatever.

    RGV
     
  8. Jan 14, 2012 #7
    Oh, I see. Thanks.

    Do you know how do I calculate the true value of the function and the approximate value given by the Taylor series approximation for x=2?
     
  9. Jan 14, 2012 #8

    Mark44

    Staff: Mentor

    You're given the formulas for both functions, and you presumably have your Taylor polynomials. Just evaluate these functions at the appropriate values.
     
  10. Jan 14, 2012 #9
    I got it! Thanks!
     
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