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Math Help - maclaurins theorem

  1. #1
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    maclaurins theorem

    Hi,I need help! We’re doing second order approximation which is ok but to find more accurate results, we have to use MACLAURINS THEOREM, I’ve been looking at books for weeks now and can’t get a grip of it.

    The question is “using Maclaurins Theorem or otherwise, obtain the power series for

    e-kh

    We have to show how we differentiate it for f `, f `` etc..
    We then have to put it into a polynominal to recheck our results,I hope this means something to someone out there!!
    THANKS FOR YOUR TIME!!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by watford73 View Post
    Hi,I need help! We’re doing second order approximation which is ok but to find more accurate results, we have to use MACLAURINS THEOREM, I’ve been looking at books for weeks now and can’t get a grip of it.

    The question is “using Maclaurins Theorem or otherwise, obtain the power series for


    e-kh


    We have to show how we differentiate it for f `, f `` etc..
    We then have to put it into a polynominal to recheck our results,I hope this means something to someone out there!!
    THANKS FOR YOUR TIME!!
    Mclaurin series for a function f(x) is:

    f(x) = f(0) + f'(0)x + f''(0) x^2/2 + ... +f^{(n)}x^n/n!+ ..

    Now if:

     f(x)=e^{-kx}

    (I am assuming h is your variable and k a constant and using x instead of h)

    Then:

     <br />
f'(x)=-ke^{-kx}<br />
     <br />
f''(x)=-k^2 e^{-kx}<br />
    and the n-th derivative:

     <br />
f^{(n)}=(-k)^{n} e^{-kx}<br />
    so:

    e^{-kx} = 1 + (-k)x + (-k)^2 x^2/2 + ... +(-k)^nx^n/n!+ ..

    or:


    e^{-kx} = 1 + (-1)(kx) + (-1)^2 (kx)^2/2 + ... +(-1)^n(kx)^n/n!+ ..

    RonL
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  3. #3
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    maclaurins theorem

    hello there captain black,
    I feel I'm inching closer to what my tutor wants.

    The assignment is about atmospheric pressure using the equation

    p = Ae to the power of -kh

    We were doing second order approximations to find the values of p pressure
    at different heights (h) from 0m to 10000m at intervals of 1000m using our intercept (A)(109.95) and our gradient k(-.00014),the second order approximation we were using was;

    1-kh+kh2/2! but every time I try to use Maclaurins for it,I end up with similar answers or just a constant answer?

    I'm tearing my hair out!!
    p.s I think e to the power of -kh may be constant (if this is possible)

    thanks again..
    Last edited by watford73; December 4th 2007 at 07:20 AM. Reason: i made an error
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by watford73 View Post
    hello there captain black,
    I feel I'm inching closer to what my tutor wants.

    The assignment is about atmospheric pressure using the equation

    p = Ae to the power of -kh

    We were doing second order approximations to find the values of p pressure
    at different heights (h) from 0m to 10000m at intervals of 1000m using our intercept (A)(109.95) and our gradient k(-.00014),the second order approximation we were using was;

    1-kh+kh2/2! but every time I try to use Maclaurins for it,I end up with similar answers or just a constant answer?

    I'm tearing my hair out!!
    p.s I think e to the power of -kh may be constant (if this is possible)


    thanks again..
    Do you know A and k?

    Then you already have your second order approximation:

    p(h) \approx A[1-k h +(kh)^2/2]

    RonL
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  5. #5
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    maclaurins theorem

    A(intercept)=109.95
    k(gradient)=0.00014

    Is it as simple as just putting these into that equation? Is that not just the same basis second order approximation we used the first time? Surely that's not using Maclaurins Theorem?
    Last edited by watford73; December 4th 2007 at 08:00 AM.
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