Chapter 1.2, Problem 7E

Find the second Taylor polynomial $P_2(x)$ for the function $f(x)=e^x\cos x$ about $x_0=0.$

1. Use $P_2(0.5)$ to approximate $f(0.5).$ Find an upper bound for error $\left|f(0.5)-P_2(0.5)\right|$ using     the error formula, and compare it to the actual error.
2. Find a bound for the error $\left|f(x)-P_2(x)\right|$ in using $P_2(x)$ to approximate $f(x)$ on the interval    [0, 1].
3. Approximate ${\displaystyle {\int }_0^{1}f}(x)dx$ using ${\displaystyle {\int }_0^1{P}_2}(x)dx$.
4. Find an upper bound for the error in (c) using ${\displaystyle {\int }_0^1\left|R_2(x)dx\right|}$, and compare the bound to the     actual error.