12. Consider the following method for approximating for f (x) dx. Divide the interval [a, b]...

 

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12. Consider the following method for approximating for f (x) dx. Divide the interval [a, b] into n equal subinter- vals. On each subinterval approximate f by a quadratic function that agrees with f at both endpoints and at the midpoint of the subinterval. (a) Explain why the integral of f on the subinterval [xi, xi+1] is approximately equal to the expression h 3 2 1/ ( f (xi) + 2 (mi) + f(xi+1) 2 where m is the midpoint of the subinterval, m = (xi + xi+1)/2. (b) Show that if we add up these approximations for each subinterval, we get Simpson's rule: [* 2. M(n) +T(n) f(x)dx 3 12. Consider the following method for approximating for f (x) dx. Divide the interval [a, b] into n equal subinter- vals. On each subinterval approximate f by a quadratic function that agrees with f at both endpoints and at the midpoint of the subinterval. (a) Explain why the integral of f on the subinterval [xi, xi+1] is approximately equal to the expression h 3 2 1/ ( f (xi) + 2 (mi) + f(xi+1) 2 where m is the midpoint of the subinterval, m = (xi + xi+1)/2. (b) Show that if we add up these approximations for each subinterval, we get Simpson's rule: [* 2. M(n) +T(n) f(x)dx 3