a. Use the given Taylor polynomial $p_{2}$ to approximate the given quantity.
b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.
Approximate $\sqrt{1.08}$ using $f(x)=\sqrt{1+x}$ and $p_{2}(x)=1+\frac{x}{2}-\frac{x^{2}}{8}$
a. Using the Taylor polynomial $p_{2}, \sqrt{1.08} \approx$
(Do not round until the final answer. Then round to four decimal places as needed.)
b. absolute error $\approx$
(Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to two decimal places as needed.)

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)