30. Using the definite integrals over bounded intervals in exercises 26

(b) and 26(c), and 27

(a) and 27

(b) (use n ¼ 10 and r ¼ 0:10 in exercise 27):

(a) Implement both the trapezoidal rule and Simpson’s rule for several values of n and compare the associated errors. (Hint: Try n ¼ 5; 10; 25, and 100, say.)

(b) For each, evaluate the error as n increases significantly, to see if the respective orders of convergence, O 1 n2



and O 1 n4



, are apparent. (Hint: If T n ¼ jI  I Tj for Dx ¼ ba n , the error T n ¼ O 1 n2



means that n2T n aCT for some constant CT as n ! y, and similarly for S n ¼ jI  I Sj, that n4S n aCS as n ! y. Attempt to verify that the CT and CS values obtained are no bigger than the values predicted in theory using the maxima of the derivatives of the given functions.)