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Let f be a function with:

$$f( 1 ) =\cfrac{1}{2},\ f^{ \prime }( 1 ) =-\cfrac{1}{2},\ f^{ \prime \prime }( 1 ) =1, \ f^{ \prime \prime \prime }( 1 ) =-3$$

Which of the following is the $3^{rd}$ degree Taylor polynomial for $f(x)$ centered at $x = 1$?

A

$\cfrac{1}{2}-\cfrac{ x-1}{2} +\cfrac{ ( x-1 ) ^{ 2 }}{2}-\cfrac{( x-1 )^{ 3 }}{2}$

B

$\cfrac{1}{2}-\cfrac{x-1}{2} +\cfrac{( x-1 )^2}{2}-( x-1 )^{ 3 }$

C

$\cfrac{1}{2}-\cfrac{x-1}{2} +1{ ( x-1 ) }^{ 2 }-3{ ( x-1 ) }^{ 3 }$

D

$\cfrac{1}{2}-\cfrac{1}{2}+1x^{ 2 }-3x^{ 3 }$

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