Assign7 - = a (1 + cos ) can be represented by r = 2 a cos...

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Math 128 ASSIGNMENT 7 Winter 2010 Submit all problems by 8:20 am on Wednesday, March 10 th in the drop boxes across from MC4066, or in class depending on your instructor’s preference. All solutions must be clearly stated and fully justified . 1. a) Find polar co-ordinates with r > 0 and - π < θ π for each of the following Cartesian points: i) (1 , - 1) , ii) ( - 1 , 3) iii) ( - 1 , 0) b) Convert each of the following equations to polar form, and sketch the curve in R 2 . Assume r 0. (i) x 2 + y 2 = x (ii) x 2 + 4 y 2 = 4 (iii) y = x (iv) x = - 1 2. a) Find the Cartesian co-ordinates of the following polar points: i) ± - 1 , π 2 ² , ii) ³ 2 , 3 π 4 ´ , iii) ± 1 , - π 3 ² b) Convert each of the following equations to Cartesian form, and sketch the curve in R 2 . (i) r = 2 (ii) r = 5 csc θ (iii) r = 3 sin θ (iv) r = - 2 cos θ 3. Find the area of each of the following: a) the region enclosed by r = 2 - cos θ ; b) the region inside r = 3 sin θ , and outside r = 1 + sin θ . 4. Show that the cardioid r
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Unformatted text preview: = a (1 + cos ) can be represented by r = 2 a cos 2 2 , 2 , and hence determine its length. 5. Find the length of the curve r = a 2 e- , 0 b , and prove that it has a nite limit as b . 6.* Four bugs are placed at the four corners of a square with side length a . The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the center of the square along spiral paths. (a) Find the polar equation of a bugs path assuming the pole is at the center of the square. (Use the fact that the line joining one bug to the next is tangent to the bugs path.) (b) Find the distance travelled by a bug by the time it meets the other bugs at the center....
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