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Math Help - Fundamental theorm of calculus

  1. #1
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    Fundamental theorm of calculus

    Hi can someone please help me the following question:

    Fundamental theorm of calculus-untitled.jpg


    for the first part is the answer just (cos(x))^1/2 by the Fundamental theorm of calculus part 1

    for the second part i know how to calucate taylor polynoimals but this i looks abit tricky can some please help me.


    Thanks
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  2. #2
    MHF Contributor chiph588@'s Avatar
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     T_{2,0}(x) = f(0)+f'(0)x+\frac{f''(0)}{2}x^2

    It's easy to see that  f(0)=0 .

     f'(0)=\sqrt{\cos(0)} = 1

     f''(x) = \left(\sqrt{\cos(x)}\right)' = -\frac{\sin(x)}{2\sqrt{\cos(x)}} , so  f''(0) = 0

    Thus  T_{2,0}(x) = x .
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