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Thread: Proof of Theorem involving Maclaurin Polynomials

  1. #1
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    Proof of Theorem involving Maclaurin Polynomials

    I'm trying to understand the proof found here:

    Calculus 9th edition, chapter 9, section 7, exercise 71
    Calc Chat Free Solutions

    The theorem is:
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    If $\displaystyle f$ is an even function, then it's nth Maclaurin polynomial contains only terms with even powers of x.

    ------------------------------------------------

    I don't understand why the coefficients of the terms are zero for the odd derivatives.
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  2. #2
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    Which part of the proof do you get hung up on?
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  3. #3
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    Quote Originally Posted by Ackbeet View Post
    Which part of the proof do you get hung up on?
    I don't understand why the coefficients of the terms are zero for the odd derivatives.
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  4. #4
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    Take a MacLaurin series: $\displaystyle f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}\,x^{2}+\dots$

    If you take the first derivative of this, you're going to make the constant term disappear, and you'll get $\displaystyle f'(0)+f''(0)x+\dots$

    If the $\displaystyle f'(0)$ is not zero, then in general, you can't have $\displaystyle f'(x)$ be an odd function (which it must be, since it is the derivative of an even function). Get the idea?
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