Step 1 :Substitute \(x=0.08\) into the Taylor polynomial \(p_{2}(x)=1+\frac{x}{2}-\frac{x^{2}}{8}\) to approximate \(\sqrt{1.08}\)
Step 2 :Calculate \(p_{2}(0.08)=1+\frac{0.08}{2}-\frac{0.08^{2}}{8}=1+0.04-0.0008=1.0392\)
Step 3 :\(\boxed{\sqrt{1.08} \approx 1.0392}\) (rounded to four decimal places)
Step 4 :Calculate the exact value of \(\sqrt{1.08}\) using a calculator, which gives approximately 1.039230484
Step 5 :Compute the absolute error by subtracting the approximate value from the exact value, \(|1.039230484 - 1.0392| = 0.000030484\)
Step 6 :\(\boxed{\text{The absolute error is approximately } 3.05 \times 10^{-5}}\) (rounded to two decimal places)