Unformatted text preview: Math 128 ASSIGNMENT 7 Winter 2009 Submit all problems by 8:20 am on Wednesday, March 11th 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 7‘ > 0 and —7r < 9 3 7r for each of the following
Cartesian points: i) (1,—1), ii) (-1,\/§) iii) (—1,0) b) Convert each of the following equations to polar form, and sketch the curve in
R2. Assume r 2 0. (i) $2+y2=x (ii) $2+4y2=4 (iii) y=z (iv) a:=—1 2. a) Find the Cartesian co—ordinates of the following polar points: i) (4,75), ii) (2371:) iii) (1,—g) b) Convert each of the following equations to Cartesian form, and sketch the curve
in R2. (i) r=2 (ii) r=5csc6 (iii) r=33in6 (iv) 7‘=—2cos€ 3. Find the area of each of the following: a) the region enclosed by r = 2 — cos 0; b) the region inside r = 3sin6, and outside 7‘ = 1 + sin 6. 4. Show that the cardioid r = a(l + cos 6) can be represented by r = 2a cos2 (3), 0 g 0 S 271', and hence determine its length. 5. Find the length of the curve r = ie‘”, 0 g 9 g b, and prove that it has a ﬁnite t/i limit as b ——> oo. 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 bug’s 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 bug’s path.) (b) Find the distance travelled by a bug by the time it
meets the other bugs at the center. ...
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- Spring '10