Use Theorem 1.13 (Taylor) to find fourth and fifth order polynomials that are approximations to the standard normal distribution function \(\Phi(x)\). Is there a difference between the approximations? What can be said in general about two consecutive even and odd order polynomial approximations of \(\Phi(x)\) ? Prove your conjecture using the results of Theorem 1.14.

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P Theorem 1.13 (Taylor). Let f be a function that has p+1 bounded and continuous derivatives in the interval (x, x+6). Then f(x+8)= k=0 Sk f(k) (x) + Ep(x, 8), k! where 1 Ep(x, 8) = (x+8) f(p+1)(t)dt: = p! SP+1 f(p+1)() (p+1)! for some = (x, x + d).