Observe that the Error Bound for T_N (which has 12 in the denominator) is twice as large as the Error Bound for MN (which has 24 in the denominator). Compute the actual error in T_N for ∫^π₀ sin x dx for N = 4, 8, 16, 32, and 64 and compare it with the calculations of Exercise 56. Does the actual error in T_N seem to be roughly twice as large as the error in M_N in this case?

Data From Exercise 56

The Error Bound for M_N is proportional to 1/N², so the Error Bound decreases by 1/4 if N is increased to 2N. Compute the actual error in MN for ∫^π₀ sin x dx for N = 4, 8, 16, 32, and 64. Does the actual error seem to decrease by 1/4 as N is doubled?