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Math Help - taylor poly help

  1. #1
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    taylor poly help

    Find taylor poly of f(x)=ln(1+x) of order 2 about x=0

    done !

    found it P2(x)= x

    when they say order that means terms right! i,e p2(x)=f(0)+f'(0)x

    now the next part which is abit tricky

    Find the remainder R2(x)at the point x=1

    and use it to prove 13/24< ln2 <5/6

    Thanks
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Not quite - Order refers to the degree of the Taylor Polynomial.

    You want the quadratic p2(x) = x-x^2/2

    R2(x) = f ''' (t) x^3/3! 0 < t < 1

    f ''' = 2/(1+x^3) 0n [0,1] f ''' (x) < 2 anf f'''(x) > 1/4

    R2(1) = f ''' (t)/3! < 2/6 =1/3

    p2(1) = 1-1/2 =1/2

    ln(2) < 1/2 + 1/3 = 5/6

    f ''' (t) > 1/(4*3!) = 1/24

    1/2 +1/24 < ln(2)

    13/24 < ln(2)
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