Question

(a) Find the points on the polar curve r = 2(1 – cos(0)) where the tangents are horizontal. (b) Find the points on the polar curve r = 2(1 - cos(0)) where the tangents are vertical. (c) Find the length of the curve. FIGURE 3. r = 2(1 - cos(O)).

Answer 1

-Ч Cos?o +2Co9 +2 = 0 2- 2Сод-+ 2 Co50 – 2Cos*6 -0 2C1-Ce%) + 2C+0 - 2Сого 20 o = 0,5%)- 08%) 1+0445 + (-и) Фр 099y's zt9:57 - 08ๆ OvJs , Фfot- 080)+ 0,5 da do dy «Р hip do 0Р ( 05) 's + O's - ดู 89 9y's ор 001- 08921 + 0,5 % бе Фfo) (o voyt-1)+ 04:51 = Kp (out 2 (ою)t-1) +o fo) Qччя ОР do du de 0 кү (очи 2 (оњ2-1) 1+0+0) 0м'ѕt кр Q86) (087-17 + Quis onist do с do do 6Р ОРФ) atonis TP (а - 1 до dy (QчS-) At@f%) хр rd of im.flip ФУ) = x ( 0 5 x = ор. "АР 0,5 () (o'y -) - ) ОР Р to os dill.wn [1- (o] - 9. Soluz

en VIA 2. (080 - 6080-120 26050 -20080 +010-10 2 Coso (CO80-1) + ((010-1) =0 (Cog 0-1) (200104) 20 Coso-120 20080 -- 1 Coso = -2 (801 o=2a 1 4 À =0 3. 3 when 0-0 Y: 2/1-(oko] = 2(1-1) 720 (o,o) when 0-27 13 ra --211-Costa] = 261-4-4)) = 273 8=3 when = 4k 3 z 2{1- Los 47 ] = 3 xx 23 (3,4%) HT! (0,0) (3,24) (3.47) 3 sahy for v. 2 sino Coso-sino +2 sino loro to 4. sinocoso-asino =0 asino Coso - sino=0 sino (26080-1) = 0

sino-o 26000 = 1 Cosol 0 -0, OTSA 3 3 when 0-0 V-211-6050) - 2 (1-Coso) = 2(1-1) = / (0,0) :2[l-Cos ) -211-Coon) -20 LO, ) whm 8-2, 7-247-607) -2(1-1)=2x15 when o-sty 3 0-57 211-CoAŞA) 211-5)=1 (157) V.T! (0,0) CON) (1,7) (5) © Solyn length of are of polar Carme 02 - TH)2 do 0 À 0-0 W 2 VEC1-C01037 (1820) do 0 ㅈ FOS Tu citlog?0 -26050)+4sin 20 do T 2 s 14[1+ Cog?0 +520 -26050]' do 0

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