Programming the bisection method in a spreadsheet leads to the
following results for parts a and d of this exercise. The function
whose roots we are finding is
a | b | x | f(x) |
---|---|---|---|
-1 | -2 | -1.5 | +0.812500 |
-1 | -1.5 | -1.25 | -0.902343 |
-1.25 | -1.5 | -1.375 | -0.288818 |
-1.375 | -1.5 | -1.4375 | +0.195327 |
-1.375 | -1.4375 | -1.40625 | -0.062666 |
-1.40625 | -1.4375 | -1.421875 | +0.062262 |
-1.40625 | -1.421875 | -1.4140625 | -0.001208 |
-1.4140625 | -1.421875 | -1.41796875 | +0.030274 |
-1.4140625 | -1.41796875 | -1.416015625 | +0.014470 |
-1.4140625 | -1.416015625 | -1.4150390625 | +0.006615 |
-1.4140625 | -1.4150390625 | -1.41455078125 | +0.002699 |
-1.4140625 | -1.41455078125 | -1.414306640625 | +0.000744 |
-1.4140625 | -1.414306640625 | -1.4141845703125 | -0.000231 |
-1.4141845703125 | -1.414306640625 | -1.4142456054688 | +0.000256 |
-1.4141845703125 | -1.4142456054688 | -1.4142150878906 | +0.000012 |
-1.4141845703125 | -1.4142150878906 | -1.4141998291016 | -0.000109 |
-1.4141998291016 | -1.4142150878906 | -1.4142074584961 | -0.000048 |
-1.4142074584961 | -1.4142150878906 | -1.4142112731934 | -0.000018 |
-1.4142112731934 | -1.4142150878906 | -1.414213180542 | -0.000003 |
-1.414213180542 | -1.4142150878906 | -1.4142141342163 | +0.000004 |
-1.414213180542 | -1.4142141342163 | -1.4142136573792 | +0.000000 |
the interval [-2,-1] |
a | b | x | f(x) |
---|---|---|---|
-1 | 0 | -0.5 | +1.312500 |
-1 | -0.5 | -0.75 | -0.089843 |
-0.75 | -0.5 | -0.625 | +0.578369 |
-0.75 | -0.625 | -0.6875 | +0.232681 |
-0.75 | -0.6875 | -0.71875 | +0.068085 |
-0.75 | -0.71875 | -0.734375 | -0.011768 |
-0.734375 | -0.71875 | -0.7265625 | +0.027943 |
-0.734375 | -0.7265625 | -0.73046875 | +0.008032 |
-0.734375 | -0.73046875 | -0.732421875 | -0.001881 |
-0.732421875 | -0.73046875 | -0.7314453125 | +0.003072 |
-0.732421875 | -0.7314453125 | -0.73193359375 | +0.000594 |
-0.732421875 | -0.73193359375 | -0.732177734375 | -0.000643 |
-0.732177734375 | -0.73193359375 | -0.7320556640625 | -0.000024 |
-0.7320556640625 | -0.73193359375 | -0.7319946289063 | +0.000284 |
-0.7320556640625 | -0.7319946289063 | -0.7320251464844 | +0.000130 |
-0.7320556640625 | -0.7320251464844 | -0.7320404052734 | +0.000052 |
-0.7320556640625 | -0.7320404052734 | -0.732048034668 | +0.000014 |
-0.7320556640625 | -0.732048034668 | -0.7320518493652 | -0.000005 |
-0.7320518493652 | -0.732048034668 | -0.7320499420166 | +0.000004 |
-0.7320518493652 | -0.7320499420166 | -0.7320508956909 | -0.000000 |
-0.7320508956909 | -0.7320499420166 | -0.7320504188538 | +0.000001 |
the interval [-1,0] |
Since each table starts with an interval of length 1 and has 20 more rows, the final interval has a length of 2-20 which is less than 10-6.
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