Consider ∫₀² f (x)dx, where f(x) = 1/12x⁴ + 3x².

(a) Make a rough sketch of the graph of f″(x) for 0 ≤ x ≤ 2.

(b) Find a number A such that |f″(x)| ≤ A for all x satisfying 0 ≤ x ≤ 2.

(c) Obtain a bound on the error of using the midpoint rule with n = 10 to approximate the definite integral.

(d) The exact value of the definite integral (to four decimal places) is 8.5333, and the midpoint rule with n = 10 gives 8.5089. What is the error for the midpoint approximation? Does this error satisfy the bound obtained in part (c)?

(e) Redo part (c) with the number of intervals doubled to n = 20. Is the bound on the error halved? Quartered?