CPM Homework Help : CALC Problem 12-57

Find the third-degree Taylor polynomial, $p_3(x)$ , about $x = 1$ for $f ( x ) = \sqrt { x }$ . Then use $p_3 (x)$ to find a sixth-degree power series for $f ( x ) = \sqrt { x ^ { 2 } + 1 }$ .

Steps:

$f(x)=x^{1/2},f^\prime(x)=\frac{1}{2}x^{-1/2},f^{\prime\prime}(x)=-\frac{1}{4}x^{-3/2},f^{\prime\prime\prime}(x)=\frac{3}{8}x^{-5/2}$
Simplify.
Substitute $x^2 + 1$ for $x$ into your equation from Step 3 to obtain the sixth-degree power series.