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Math Help - maclaurin series help

  1. #1
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    maclaurin series help

    find the maclaurin series for f(x) = coshx

    the answer is = 1 + x^2 / 2! + x^4 / 4! + x^6 / 6! + ...

    my question: How is the 6th derivative positive? i believe its a -coshx?

    also, whats the difference between coshx and just cosx?

    this particular problem was unique to me. thanks
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  2. #2
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    Quote Originally Posted by rcmango View Post
    find the maclaurin series for f(x) = coshx

    the answer is = 1 + x^2 / 2! + x^4 / 4! + x^6 / 6! + ...

    my question: How is the 6th derivative positive? i believe its a -coshx?

    also, whats the difference between coshx and just cosx?

    this particular problem was unique to me. thanks
    Third time lucky.

    cosh(x) = [e^x+e^(-x)]/2

    sinh(x) = [e^x-e^(-x)]/2

    d/dx cosh(x) = sinh(x)

    d/dx sinh(x) = cosh(x)

    sinh(0) = 0

    cosh(0) = 1.

    From which the result follows from the definition of a Maclaurin series

    (or even more easily by observing from the definition of cosh its power
    series representation about 0 contains the even power terms of the
    series for the exponential function, and the odd power terms all have
    0 for their coefficient, and so don't appear in the expansion)

    RonL
    Last edited by CaptainBlack; February 18th 2007 at 10:33 PM.
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  3. #3
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    Quote Originally Posted by rcmango View Post
    find the maclaurin series for f(x) = coshx

    the answer is = 1 + x^2 / 2! + x^4 / 4! + x^6 / 6! + ...

    my question: How is the 6th derivative positive? i believe its a -coshx?

    also, whats the difference between coshx and just cosx?

    this particular problem was unique to me. thanks

    The hyperbolic cosine and sine are not sine and cosine functions at all, however they are called like that because they are very similar related through the derivatives. And furthermore the analogue of a sine and cosine on a hyperbolic geometry are that of a sinh and cosh x.
    Many important identities can be remembered through them as thinking about them as sine and cosine. For example, the infinite series you gave seems like the sin x series however this one has positive ters and is therefore a hyperbolic sine.
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