Consider ∫₁² f (x)dx, where f (x) = 3 ln x.

(a) Make a rough sketch of the graph of the fourth derivative of f (x) for 1 ≤ x ≤ 2.

(b) Find a number A such that |f″″(x)| ≤ A for all x satisfying 1 ≤ x ≤ 2.

(c) Obtain a bound on the error of using Simpson’s rule with n = 2 to approximate the definite integral.

(d) The exact value of the definite integral (to four decimal places) is 1.1589, and Simpson’s rule with n = 2 gives 1.1588. What is the error for the approximation by Simpson’s rule? Does this error satisfy the bound obtained in part (c)?

(e) Redo part (c) with the number of intervals tripled to n = 6. Is the bound on the error divided by 3?