$(2)$ a$)$ Let

$f(x)={\displaystyle \underset{0}{\overset{x}{}}}\frac{cos(t^2)}{e^{3t}}dt$

i$)[5$ pts$]$ Find the second degree Maclaurin polynomial for $f.$

ii$)[5$ pts$]$ Use your answer from i$)$ to estimate

${\displaystyle \underset{-0\mathrm{.}2}{\overset{0}{}}}\frac{cos(t^2)}{e^{3t}}dt.$

b$)[6$ pts$]$ Let f be a continuous function such that:

$arccos(\sqrt[\phantom{2}]{\frac{x}{8}})arctan(\frac{x}{4})+x^2={\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{x}}{}}}f(t)dt$

Find $f(2).$