636 CHAPTER 9 Polar Coordinates; Vectors ‘Are You Prepared?’ Answers 1. − + i 4 3 2. (a) + A B A B sin cos cos sin (b) − A B A B cos cos sin sin 3. − 3 2 ; 1 2 4. e e ; 7 12 Determine which complex numbers in part (a) are in this set by plotting them on the graph. Do the complex numbers that are not in the Mandelbrot set have any common characteristics regarding the values of a6 found in part (a)? (c) Compute = + z x y 2 2 for each of the complex numbers in part (a). Now compute a6 for each of the complex numbers in part (a). For which complex numbers is ≤ a z 6 and ≤ z 2? Conclude that the criterion for a complex number to be in the Mandelbrot set is that ≤ a z n and ≤ z 2. –2 –1 y 1 Imaginary axis x 1 Real axis 75. Challenge Problem Solve = + e 7. x yi 76. Challenge Problem Solve = + e i6 . x yi Applications and Extensions 65. Find the four complex fourth roots of unity, 1, and plot them. 66. Find the six complex sixth roots of unity, 1, and plot them. 67. Show that each complex nth root of a nonzero complex number w has the same magnitude. 68. Use the result of Problem 67 to draw the conclusion that each complex nth root lies on a circle with center at the origin. What is the radius of this circle? 69. Refer to Problem 68. Show that the complex nth roots of a nonzero complex number w are equally spaced on the circle. 70. Prove formula (6). 71. Prove = θ θ π ( ) + re re k , an integer. i i k 2 72. Euler’s Identity Show that + = π e 1 0. i 73. Prove that De Moivre’s Theorem is true for all integers n by assuming it is true for integers ≥ n 1 and then showing it is true for 0 and for negative integers. [Hint: Multiply the numerator and the denominator by the conjugate of the denominator, and use even-odd properties.] 74. Mandelbrot Sets (a) Consider the expression ( ) = + − a a z, n n 1 2 where z is some complex number (called the seed) and = a z. 0 Compute ( ) ( ) ( ) = + = + = + a aza aza azaa , , , , , 1 0 2 2 1 2 3 2 2 4 5 and a6 for the following seeds: = − z i 0.1 0.4 , 1 = + z i 0.5 0.8 , 2 = − + z i 0.9 0.7 , 3 = − + z i 1.1 0.1 , 4 = − z i 0 1.3 , 5 and = + z i 1 1 . 6 (b) The dark portion of the graph represents the set of all values = + z x yi that are in the Mandelbrot set. Retain Your Knowledge Problems 77–86 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 77. Find the area of the triangle with = = a b 8, 11, and = ° C 113 . 78. Convert ° 240 to radians. Express your answer as a multiple of π. 79. Simplify: x y 24 2 5 3 80. Determine whether = − + f x x x ( ) 5 12 4 2 has a maximum value or a minimum value, and then find the value. 81. Solve the triangle: = = = a b c 6, 8, 12 82. Write as a single logarithm: + − x y z 3 log 2 log 5 log a a a 83. Solve: + = x log 4 2 5 84. Given = − f x x x ( ) 3 4 2 and = g x x ( ) 5 ,3 find ( )( ) f g x . 85. Find an equation of the line perpendicular to the graph of = − = f x x x ( ) 2 3 5 at 6. 86. Show that − = x x 16 sec 16 4 tan . 2
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