Suppose that you know the following results on the errors of the trapezoidal rule and Simpson's...

 

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Suppose that you know the following results on the errors of the trapezoidal rule and Simpson's rule: f(x) dx - T(f, h) = -bªf" (1) 1² 12 f(x) dax-S(f, h)= (² <-a 180 where h is the length of the uniform subintervals, E1, E2 are two numbers in [a, b], and T(f, h), S(f, h) are the estimates to f f(x) dx calculated by the two methods. = Using these results, estimate the numbers of data points required by both methods when f(x) e (e to the power of x2), a = 0, b= 1 and = 10-6. (Hint: It should be helpful to first calculate an upper bound for f"(x) and an upper bound for f(4)(x)| over [a, b].) Suppose that you know the following results on the errors of the trapezoidal rule and Simpson's rule: f(x) dx - T(f, h) = -bªf" (1) 1² 12 f(x) dax-S(f, h)= (² <-a 180 where h is the length of the uniform subintervals, E1, E2 are two numbers in [a, b], and T(f, h), S(f, h) are the estimates to f f(x) dx calculated by the two methods. = Using these results, estimate the numbers of data points required by both methods when f(x) e (e to the power of x2), a = 0, b= 1 and = 10-6. (Hint: It should be helpful to first calculate an upper bound for f"(x) and an upper bound for f(4)(x)| over [a, b].)