Chapter 1.1, Problem 12ES

Repeat Exercise 11 using x₀ = π/6.

11. Find the second Taylor polynomial P₂(x) for the function f(x) = e^x cos x about x₀ = 0.

1. a. Use P₂(0.5) to approximate f(0.5). Find an upper bound for error |f(0.5) − P₂(0.5)| using the error formula and compare it to the actual error.
2. b. Find a bound for the error |f(x) − P₂(x)| in using P₂(x) to approximate f(x) on the interval [0, 1].
3. c. Approximate ${\displaystyle \underset{0}{\overset{1}{\int }}f(x)dx}$ using ${\displaystyle \underset{0}{\overset{1}{\int }}{P}_2(x)dx}.$
4. d. Find an upper bound for the error in (c) using ${\displaystyle \underset{0}{\overset{1}{\int }}\left|R_2(x)dx\right|}$ and compare the bound to the actual error.