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Simpson's Rule I. Quiz II. Homework III. Simpson's Estimate Yesterday, we saw that the Trapezoidal and Midpoint estimates provided better accuracy than the Left and Right endpoint estimates. It turns out that a certain combination of the Trapezoid and Midpoint estimates is even better. Definition: Let f(x) be a function defined on [a,b]. Then S(n) = 1/3 T(n) + 2/3 M(n) where T(n) and M(n) are the Trapezoidal and Midpoint Estimates. Geometrically, if n is an even number then Simpson's Estimate gives the area under the parabolas defined by connecting three adjacent points. Let n be even then using the even subscripted x values for the trapezoidal estimate and the midpoint estimate, gives S(n) = 1/3[(b - a)/2 (f(x0) + 2f( f(x2) + f(x4)+ ... + f(xn-2) + f(xn)] + 2/3[f(x1) + f(x3) + f(x5) + ... + f(xn-1) = (b - a)/6 ( f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + 4f(x5) + ... + 2f(xn-2) + 4f(xn-1) + f(xn) Notice the 1 2 4 2 4 ... 2 4 2 4 1 pattern.
Exercise: Approximate int from 2 to 6 of 1 over 1 + x3 dx IV. Error in Simpson's Estimate Without proof, we state Let M = max |f''''(x)| and let ES be the error in using Simpson's estimate then |ES| < M(b - a)5/180n4 Examples will be given. |