Answer
1; 1.05; 1.04875
Work Step by Step
We are given:
$f(x)=\sqrt{1+x}$
and the first three Taylor polynomials for $f(x)$ centered at 0:
$p_0(x)=1$
$p_1(x)=1+\dfrac{x}{2}$
$p_2(x(=1+\dfrac{x}{2}-\dfrac{x^2}{8}$
Consider $x=0.1$.
Determine the first approximation using $p_0(x)$:
$p_0(0.1)=1$
Determine the second approximation using $p_1(x)$:
$p_1(0.1)=1+\dfrac{0.1}{2}=1+0.05=1.05$
Determine the third approximation using $p_2(x)$:
$p_2(0.1)=1+\dfrac{0.1}{2}-\dfrac{0.1^2}{8}=1+0.05-0.00125=1.04875$
