Problem:
Find the second Taylor polynomial P_2(x) for the function e^xcos(x) about x_0=0

Use P_2(0.5) to approximate f(0.5). Find an upper bound for error |f(0.5)-P_2(0.5)| using the error formula and compare it to the actual error.

Now f(x)=P_n(x)+R_n(x)

P_2(x)=1+x

R_n(x)=\frac{f^{n+1}(E(x))}{(n+1)!}(x-x_0)^n

In mine case:

R_2(x)=\frac{-2e^{E(x)}(sin(E(x))+Cos(E(x)))}{3!}x^3

And substitute for R_2(0.5) and I need to find upper bound on |R_2(0.5)| for 0 < E(x) < 0.5

The upper bound is for E(x)=0.5 that is |R_2(0.5)| \leq 0.093222005.

But in the solution manual it is 0.0532. I rechecked 10 times and I start to believe that it's their error.

Thanks in advance.