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Math Help - Taylor's inequality problem

  1. #1
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    Taylor's inequality problem

    Use the Taylor polynomial of degree up to 3 for y = cos x along with Taylor's inequality to estimate (integral from 0 to 1) of cos(x^2) and give a bound on the error that your estimate makes.

    I'm not quite sure how to do this.
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  2. #2
    Super Member Anonymous1's Avatar
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    \int_0^1 \cos(x^2)\approx\int_0^1 (1-\frac{x^4}{2}) = \int_0^1 1-\int_0^1 \frac{x^4}{2}

    Also, you need to find the remainder term to find the bound on error. You know... R_n(x)= \frac{f^{n+1}(x)}{n!}
    Last edited by Anonymous1; March 28th 2010 at 10:00 PM.
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  3. #3
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    Ok thank you, I don't know why the integral threw me off...seems straight forward enough .
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  4. #4
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    Incase anyone else is interested in this answer in the future, the x^8 term should not be included (since it is of degree 3 and the x^4 term isn't used). Thanks again for the reply though.
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  5. #5
    Super Member Anonymous1's Avatar
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    Yes, this is correct. Note that if you want equality to cos(x^2), use the \infty ^{th} order expansion, which is what I implied with +...

    I've edited the post to show we are finding an approximation.
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