| 4. (a) The first few iterates are |
| z0 = 0 |
| z1 = 0.5 |
| z2 = 0.75 |
| z3 = 1.0625 |
| z4 = 1.62891 |
| z5 = 3.15334 |
| Because z5 is farther than 2 from the origin,
|
| (b) The first few iterates are |
| z0 = 0 |
| z1 = i/4 |
| z2 = -.0625 + .25i |
| z3 = -.0586 + .2188i |
| z4 = -.0444 + .2244i |
| z5 = -.0484 + .2300i |
| z6 = -.0506 + .2277i |
| The iterates appear to be converging, so we deduce c = i/4 does belong to the Mandelbrot set. |
| (c) The first few iterates are |
| z0 = 0 |
| z1 = -5/4 |
| z2 = .3125 |
| z3 = -1.1523 |
| z4 = .0779 |
| z5 = -1.244 |
| z6 = .2974 |
| Note the values appear to converge to a 2-cycle, hence do not run away to infinity. We deduce c = -5/4 does belong to the Mandelbrot set. |
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