Fall 2007 - Hohnhold's Class - Quiz 3 (Version B)

Fall 2007- - √-4 = 3 2 ± 2 i(2(a Please sketch the area enclosed by the polar curve r θ = | cos θ |(3 points-2-1.5-1-0.5 0.5 1 1.5 2-1.5-1-0.5

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Quiz 3 for Math 20B. Fall Quarter 2007, UCSD Henning Hohnhold Name: Section: #1: #2: Total: (1) (a) Give the polar form of the complex number (5 + 5 i ) 3 . (2 points) The argument of 5+5 i is θ = π 4 (the easiest way to see this is to look at the point 5 - 5 i in the complex plane, but you can also use the formula for the argument) and its modulus is 5 2 + 5 2 = 5 2. Hence the polar form we are looking for is given by (5 + 5 i ) 3 = (5 2 · (cos( π/ 4) + i sin( π/ 4)) 3 = 5 3 2 3 · (cos(3 π/ 4) + i sin(3 π/ 4)) Note that the argument is only unique up to adding multiples of 2 π . Note also that computing the cos and sin we see that (5 + 5 i ) 3 = - 250 + 250 i . (b) Find all solutions of the equation 2 x 2 - 6 x + 25 2 = 0 . (2 points) Just apply the quadratic formula to see that x = 3 2 ±
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Unformatted text preview: √-4 = 3 2 ± 2 i. (2) (a) Please sketch the area enclosed by the polar curve r ( θ ) = | cos θ | . (3 points)-2-1.5-1-0.5 0.5 1 1.5 2-1.5-1-0.5 0.5 1 (b) Please compute the area of the ‘flower’ described by the polar curve r ( θ ) = 6-cos(9 θ ). (3 points)-10-7.5-5-2.5 2.5 5 7.5 10-5-2.5 2.5 5 We use the formula for the area enclosed by a polar curve: 1 2 Z 2 π (6-cos(9 θ )) 2 dθ = 1 2 Z 2 π 36-12 cos(9 θ ) + cos 2 (9 θ ) dθ = 1 2 ± 36 · 2 π-12 9 sin(9 θ ) | 2 π + 1 2 Z 2 π 1 + cos(18 θ ) dθ ² = 1 2 ± 72 π + 1 2 · 2 π + 1 36 sin(18 θ ) | 2 π ² = 73 2 π 2 Good Luck! 3...
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This note was uploaded on 04/29/2008 for the course MATH 20B taught by Professor Justin during the Fall '08 term at UCSD.

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Fall 2007- - √-4 = 3 2 ± 2 i(2(a Please sketch the area enclosed by the polar curve r θ = | cos θ |(3 points-2-1.5-1-0.5 0.5 1 1.5 2-1.5-1-0.5

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