Step 1 :The problem is asking for the first four nonzero terms in the MacLaurin series for the function \(f(x)=e^{-x^{2}}\).
Step 2 :The MacLaurin series for a function f(x) is the representation of the function as an infinite sum of terms that are calculated from the values of the function's derivatives at zero.
Step 3 :The MacLaurin series for the function \(e^x\) is \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...\).
Step 4 :We can derive the MacLaurin series for the function \(f(x)=e^{-x^{2}}\) by substituting \(-x^2\) for \(x\) in the series for \(e^x\).
Step 5 :The first four nonzero terms of this series are \(1 - x^{2} + \frac{x^{4}}{2}\). The term \(O(x^6)\) represents the remainder of the series and is not considered as one of the first four terms.
Step 6 :Therefore, the correct answer is \(\boxed{B. 1-x^{2}+\frac{x^{4}}{2}-\frac{x^{6}}{3 !}}\)