Assignment
You must show work in order to receive any credit.
Suppose that you want to compute the integral a∫b f(x) dx using trapezoidal or Simpson's rule and you know that
|f''(x)| < M2 < +∞, | f(4)(x)| < M4 < +∞
for all x ∈ [a, b]. We have the following error analysis:
a∫b f(x) dx - T(h) = - [(b - a)/12] f''(ξ1)h2
a∫b f(x) dx - S(h) = - [(b - a)/180] f(4)(ξ2)h4
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 |a∫b a f(x) dx - T(h)| < ε is wanted. Justify your answer.
(ii) How many data points should you use if |a∫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) = ex^2 , a = 0, b = 1 and ε = 10-6 .
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.