Answer
a) 1.0246875;
b) Absolute error: $7.6\cdot 10^{-6}$
Work Step by Step
We are given the function $f$ and the Taylor polynomial $p_2$:
$f(x)=\sqrt{1+x}$
$p_2(x)=1+\dfrac{x}{2}-\dfrac{x^2}{8}$
a) Let $x=0.05$
Compute $p_2(x)$ for $x=0.05$:
$p_2(0.05)=1+\dfrac{0.05}{2}-\dfrac{0.05^2}{8}=1.0246875$
b) Use a computer to determine the exact value of $\sqrt{1.05}$:
$\sqrt{1.05}=1.0246951
Determine the absolute error:
$|\text{exact value}-\text{approximating value}|=|1.0246951-1.0246875|=0.0000076=7.6\cdot 10^{-6}$
