Explanation Given: Consider the two Mandelbrot sequences with seeds s = a + b i and r = a − b i . The seeds s and r are complex conjugate of each other. Definition: An infinite sequence of complex numbers is said to be a Mandelbrot sequence if it starts with an arbitrary complex number s that is called the seed. Each successive term of the Mandelbrot sequence is calculated by adding the seeds to previous term’s square. Let the seed s be the initial term of the sequence ( s 0 = s ) . The next term in the sequence is obtained by squaring the preceding term and adding it to the seed, i.e., s N + 1 = ( s N ) 2 + s . Calculation: Consider the Mandelbrot sequence with seed s = a + b i . The next term in the sequence is obtained by squaring the preceding term and adding it to the seed, i.e., s N + 1 = ( s N ) 2 + s . Here, s 0 = s = a + b i , Substitute 0 for N and a + b i for s 0 in the given formula as, s 1 = ( s o ) 2 + s = ( a + b i ) 2 + ( a + b i ) = a 2 + 2 a b i − b 2 + a + b i = ( a 2 − b 2 + a ) + b ( 2 a + 1 ) i Further substitute 1 for N and ( a 2 − b 2 + a ) + b ( 2 a + 1 ) i for s 1 in the given formula as, s 2 = ( s 1 ) 2 + s = [ ( a 2 − b 2 + a ) + b ( 2 a + 1 ) i ] 2 + ( a + b i ) = ( a 2 − b 2 + a ) 2 + 2 b ( a 2 − b 2 + a ) ( 2 a + 1 ) i − b 2 ( 2 a + 1 ) 2 + a + b i = [ ( a 2 − b 2 + a ) 2 − b 2 ( 2 a + 1 ) 2 + a ] + [ 2 b ( 2 a + 1 ) ( a 2 − b 2 + a ) + b ] i Similarly, consider the Mandelbrot sequence with seed r = a − b i . The next term in the sequence is obtained by squaring the preceding term and adding it to the seed, i.e., r N + 1 = ( r N ) 2 + r

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Chapter 12, Problem 58E

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The Mandelbrot set has a reflection symmetry.

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Chapter 12, Problem 57E

Chapter 12, Problem 59E

Chapter 12 Solutions

MYLAB MATH FOR EXCURSIONS IN MATHEMATIC

Ch. 12 - Consider the construction of a Koch snowflake...Ch. 12 - Consider the construction of a Koch snowflake...Ch. 12 - Prob. 3E Ch. 12 - Prob. 4E Ch. 12 - Prob. 5E Ch. 12 - Prob. 6E Ch. 12 - Prob. 7E Ch. 12 - Prob. 8E Ch. 12 - Prob. 9E Ch. 12 - Exercises 9 through 12 refer to a variation of the...

Ch. 12 - Exercises 9 through 12 refer to a variation of the...Ch. 12 - Exercises 9 through 12 refer to a variation of the...Ch. 12 - Prob. 13E Ch. 12 - Prob. 14E Ch. 12 - Exercises 13 through 16 refer to the construction...Ch. 12 - Prob. 16E Ch. 12 - Prob. 17E Ch. 12 - Prob. 18E Ch. 12 - Prob. 19E Ch. 12 - Prob. 20E Ch. 12 - Prob. 21E Ch. 12 - Assume that the seed triangle of the Sierpinski...Ch. 12 - Prob. 23E Ch. 12 - Prob. 24E Ch. 12 - Prob. 25E Ch. 12 - Prob. 26E Ch. 12 - Prob. 27E Ch. 12 - Prob. 28E Ch. 12 - Prob. 29E Ch. 12 - Prob. 30E Ch. 12 - Prob. 31E Ch. 12 - Exercises 31 through 34 refer to a variation of...Ch. 12 - Prob. 33E Ch. 12 - Prob. 34E Ch. 12 - Prob. 35E Ch. 12 - Prob. 36E Ch. 12 - Prob. 37E Ch. 12 - Prob. 38E Ch. 12 - Exercises 35 through 40 are a review of complex...Ch. 12 - Prob. 40E Ch. 12 - Prob. 41E Ch. 12 - Prob. 42E Ch. 12 - Prob. 43E Ch. 12 - Prob. 44E Ch. 12 - Prob. 45E Ch. 12 - Prob. 46E Ch. 12 - Prob. 47E Ch. 12 - Prob. 48E Ch. 12 - Prob. 49E Ch. 12 - Exercises 49 and 50 refer to the Menger sponge, a...Ch. 12 - Prob. 51E Ch. 12 - Prob. 52E Ch. 12 - Consider the Mandelbrot sequence with seed s=1.25....Ch. 12 - Consider the Mandelbrot sequence with seed s=2. Is...Ch. 12 - Prob. 55E Ch. 12 - Prob. 56E Ch. 12 - Prob. 57E Ch. 12 - Prob. 58E Ch. 12 - Prob. 59E Ch. 12 - Prob. 60E

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The Mandelbrot Set A fractal is a geometric figure that consists of a pattern that is repeated infinitely on a smaller and smaller scale. The most famous fractal is the Mandelbrot Set, named after the Polish-born mathematician Benoit Mandelbrot (19242010). To draw the Mandelbrot Set, consider the sequence of numbers below. c,c2+c,(c2+c)2+c,[ (c2+c)2+c ]2+c,... The behavior of this sequence depends on the value of the complex number c. If the sequence is bounded (the absolute value of each number in the sequence, a+bi=a2+b2 is less than some fixed number N ), then the complex number c is in the Mandelbrot Set, and if the sequence is unbounded (the absolute value of the terms of the sequence become infinitely large), then the complex number c is not in the Mandelbrot Set. Determine whether the complex number c is in the Mandelbrot Set. (a)c=i (b)c=1+i (c)c=2 The figure below shows a graph of the Mandelbrot Set, where the horizontal and vertical axes represent the real and imaginary parts of c, respectively.
Answer the following questions. 4. Describe how the permutation of ii objects differs from the permutation of choosing r objects from a set of n objects. Include how each is calculated.

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