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Math Help - Taylor expansion

  1. #1
    Super Member dhiab's Avatar
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    Taylor expansion

    Give the Taylor expansion order 1 for the function :

    and dudect that graph of f accept asymtot of which they ask
    his psition and graph of f .
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  2. #2
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    Quote Originally Posted by dhiab View Post
    Give the Taylor expansion order 1 for the function :

    and dudect that graph of f accept asymtot of which they ask
    his psition and graph of f .
    f(a)=a^2\arctan\left( \frac{1}{a+1}\right)

    f'(x)=2x\arctan\left( \frac{1}{x+1}\right)+2x\left(\frac{1}{(\frac{1}{1+  x})^2+1} \right)\left( \frac{-1}{(x+1)^2}\right)=

    f'(x)=2x\arctan\left( \frac{1}{x+1}\right)-\left(\frac{2x}{(1+x)^2+1} \right)

    f'(a)=2a\arctan\left( \frac{1}{a+1}\right)-\left(\frac{2a}{(1+a)^2+1} \right)

    Since you didn't say where it was centered I centered it at "a"

    T_1(x)=f(a)+f'(a)(x-a)

    T_1(x)=a^2\arctan\left( \frac{1}{a+1}\right)+\left[2a\arctan\left( \frac{1}{a+1}\right)-\left(\frac{2a}{(1+a)^2+1} \right) \right](x-a)
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