Problem
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 $e^{-0.04}$ using $f(x)=e^{-x}$ and $p_{2}(x)=1-x+\frac{x^{2}}{2}$ a. Using the Taylor polynomial $p_{2}, e^{-0.04} \approx$ (Do not round until the final answer. Then round to four decimal places as needed.)
Solution
Step 1 ：Given the Taylor polynomial \(p_{2}(x)=1-x+\frac{x^{2}}{2}\), we need to substitute \(x=-0.04\) into this polynomial to approximate \(e^{-0.04}\).
Step 2 ：Substitute \(x=-0.04\) into the polynomial to get \(p_{2} = 1.0408\).
Step 3 ：Final Answer: The approximation of \(e^{-0.04}\) using the Taylor polynomial \(p_{2}(x)=1-x+\frac{x^{2}}{2}\) is \(\boxed{1.0408}\).
