Let f(x) be a smooth function f(x) defined on [a, b] and the integral of f(x)...

   

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Let f(x) be a smooth function f(x) defined on [a, b] and the integral of f(x) on [a, b] is denoted by las[f]. The goal is to approximate lab] by the midpoint, trapezoidal, and Simpson's quadrature rules, and their corresponding order of error terms are respectively shown as follows. f (a+b) (b-a) with E=0((b- a)) Midpoint rule: Masif = (a+b) (6-0 f(a)+f(b) (b-a) with E = O((b-a)) 2 Trapezoidal rule: Tas[] = Simpson's rule: Sa,b] = (f(a)+41 (a+b). +(b) (b-a) with E O((b-a)) 180 (d) The error term of Sas[f] can be simplified to (a)(4)() for some in [a, b]. If we consider the composite Simpson's method with h = (a 2n-interval partition of [a, b]) to evaluate f(x)dx. show that the corresponding error of the approximation is for some & in [a, b]. (5%) (e) Let f(r) = and find the smallest n to accurately approximate f(x)d up to 0.5 x 104, by using the composite Simpson's method. Hint: log1020.3010, log10 3 0.4771, and log105 (10%) -0.6990. Let f(x) be a smooth function f(x) defined on [a, b] and the integral of f(x) on [a, b] is denoted by las[f]. The goal is to approximate lab] by the midpoint, trapezoidal, and Simpson's quadrature rules, and their corresponding order of error terms are respectively shown as follows. f (a+b) (b-a) with E=0((b- a)) Midpoint rule: Masif = (a+b) (6-0 f(a)+f(b) (b-a) with E = O((b-a)) 2 Trapezoidal rule: Tas[] = Simpson's rule: Sa,b] = (f(a)+41 (a+b). +(b) (b-a) with E O((b-a)) 180 (d) The error term of Sas[f] can be simplified to (a)(4)() for some in [a, b]. If we consider the composite Simpson's method with h = (a 2n-interval partition of [a, b]) to evaluate f(x)dx. show that the corresponding error of the approximation is for some & in [a, b]. (5%) (e) Let f(r) = and find the smallest n to accurately approximate f(x)d up to 0.5 x 104, by using the composite Simpson's method. Hint: log1020.3010, log10 3 0.4771, and log105 (10%) -0.6990.