Table 13.16 gives { f(x_n)} and {x_n} for 21 regularly spaced node points over [0, 3] for an unknown function. Use the O(h²) and O(h⁴) central-difference formula to approximate {f′(x_n)}. Plot the results and comment on your answer.

Transcribed Image Text:

Table 13.16 Numerical derivatives using the O(h²) and O(ht) central-difference formula in Example 13.23 n f(xn) D2¹f(xn) DAlf(xn) n Xn f(xn) DIf(xn) Df(xn) 1 0.00 0.00000 12 1.65 0.40693 -3.01570 -2.92107 13 1.80 -0.09825 -3.57412 -3.40255 2 0.15 0.02250 0.29960 3 0.30 0.08988 4 0.45 0.20112 0.60 0.35227 5 6 0.75 0.53330 3.85632 0.59816 0.59540 14 1.95 -0.61383-3.05820 -2.85460 0.88242 0.87465 15 2.10 0.95463-1.22368 -1.08500 1.12392 1.10728 16 2.25 -0.93933 1.56027 1.51662 1.26938 1.24004 17 2.40 -0.49964 4.14189 1.24061 1.19679 18 2.55 0.21756 8 0.94566 9 1.20 0.99146 0.30857 0.25384 20 2.85 1.35 0.96849 -0.67871 -0.71128 21 3.00 7 0.90 0.72429 4.91928 4.48258 1.05 0.89234 0.89057 19 2.70 0.84513 2.81171 2.48868 0.96417 -1.44338 10 0.41212 11 1.50 0.77807 -1.89042 -1.87186 Xn