Estimate $f(b)$ using Taylor Expansion for $f'(x) = cos(x^2)$

You're just not using enough terms to get a good approximation so far away from the point that you are Taylor expanding around. Taylor expansions around $0$ are most accurate near $0$.

In the following Desmos graph the function $f(x) = 1 + int_0^x cos(s^2), text ds$ is plotted in $color{red}{text{red}}$. A $pm0.1$ tolerance region is plotted around $f$ in $color{darkgreen}{text{green}}$. The first 5 terms, i.e. the 4th order Taylor approximation, plotted in $color{blue}{text{blue}}$ leaves the green region at about $x=1.9$, which explains the bad quality of your approximation at $x=2$. If you want to approximate $f$ at $x=3$ to $0.1$ error, you will need to use at least 12 terms; this is the $color{purple}{text{purple}}$ graph. ,