The task is to first estimate the second degree Maclaurin series of $e^{-x^2}$ and thus estimate the integral of the function from $0$ to $0.5$. This part is no problem.
The following task is to estimate the error for this estimation. I used the regular approach with calculating the third derivative and using it with the formulae for the reminder of Taylor polynomials which apparently is wrong. However doing the same thing with $4$-th derivative works but I have no idea why.
I know that in maclaurin formulae the term including third derivative becomes $0$ however I don't know how this is connected to the error calculation.
Also I'm aware of the method with $t=-x^2$ substitution of in $e^t$ expansion, but I don't understand why the normal method does not work.
Sorry if I'm too vague.