You must show work in order to receive any credit. Suppose that you want to compute the integral R b a f(x) dx using trapezoidal or Simpson’s rule and you know that |f 00(x)| < M2 < +∞, f (4)(x) < +∞ for all x ∈ [a, b]. We have the following error analysis: Z b a f(x) dx − T(h) = − b − a 12 f 00(ξ1)h 2 Z b a f(x) dx − S(h) = − b − a 180 f (4)(ξ2)h 4 where h, T(h), S(h) have the same meaning as in class and ξ1, ξ2 ∈ [a, b]. Your answers to parts (i) and (ii) should be in terms of a, b, M2, M4 and the tolerance ε. (i) How many data points should you use if R b a f(x) dx − T(h) < ε is wanted. Justify your answer. (ii) How many data points should you use if R b a f(x) dx − S(h) < ε is wanted? Remember that an odd number of data points is needed for Simpson’s rule. Justify your answer. (iii) Estimate the numbers of data points required by both methods when f(x) = e x 2 , a = 0, b = 1 and ε = 10−6 .
