Tenth Homework Set Answers

2. (a) The first ten iterates are

z0 = 0

z1 = -0.5

z2 = -0.25

z3 = -0.4375

z4 = -0.3086

z5 = -0.4048

z6 = -0.3361

z7 = -0.3870

z8 = -0.3502

z9 = -0.3773

z10 = -0.3576

Because z10 remains closer than 2 from the origin, c = -0.5 appears to belong to the Mandelbrot set.

(b) The first ten iterates are

z0 = 0

z1 = -i

z2 = -1 + i

z3 = i

z4 = -1 - i

z5 = i

z6 = -1 - i

z7 = i

z8 = -1 - i

z9 = i

z10 = -1 - i

Note the values eventually cycle between -1 - i and i, (we can see this because z4 = z2) hence do not run away to infinity, so c = -i does belong to the Mandelbrot set.

(c) The first ten iterates are

z0 = 0

z1 = -1+i

z2 = -1-i

z3 = -1+3i

z4 = -9 - 5i

z5 = 55 + 91i

z6 = -5257 + 10011i

Not much point in continuing, is there?

Note z3 is farther than 2 from the origin, so we know the zi do run away to infinity and consequently c = -1+i does not belong to the Mandelbrot set.

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