CPM Homework Help : APCALC Problem 12-56

Write the equation of the third-degree Taylor polynomial, $p_3(x)$ , about $x = 1$ for $f(x) =\sqrt { x }$ . Then use substitution to write a sixth-degree Taylor polynomial for $f ( x ) = \sqrt { x ^ { 2 } + 1 }$ .

Step 1:

$f(x)=x^{1/2},f^\prime(x)=\frac{1}{2}x^{-1/2},f^{\prime\prime}(x)=-\frac{1}{4}x^{-3/2},f^{\prime\prime\prime}(x)=\frac{3}{8}x^{-5/2}$

Step 2:

$f(1)=1,f^\prime(1)=\frac{1}{2},f^{\prime\prime}(1)=-\frac{1}{4},f^{\prime\prime\prime}(1)=\frac{3}{8}$

Step 3:

$p_3(x)=1+\frac{1}{2}(x-1)-\frac{1}{4\cdot 2!}(x-1)^2+\frac{3}{8\cdot 3!}(x-1)^3$

Step 4:

Substitute $x^{2} + 1$ for $x$ into your equation from Step 3 to obtain the sixth-degree power series.