Question

25. Given the following parametric curve X(t) = -1 + 3 cos(t) y(t) = 1 + 2 sin(t) 0<t<21 a) Express the curve with an equation that relates x and y. 7C b) Find the slope of the tangent line to the curve at the point t c) State the pair(s) (x,y) where the curve has a horizontal/vertical tangent line. 27.A particle is traveling along the path such that its position at any time t is given by r(t) = (2 cos(t), 2 sin(t)). a) Find the speed at time t. b) Find the distance the point travels from 0 st 5311 c) Find the time it takes the particle to travel on a path of length 12 units.

Answer 1

25 geven geven parametric Cerover -) + 3 ше) yet) xt t) - ) + Kine) О< e < 24- x7) - -) + 3 ( 0 (е) г) x (4) +/ зеѕ (4) 2 x (4) + | cost) 3 cople) :) Х+ / 젤 3 also It a since) (e) : 8-) Я Sin(e) 11 8-) (n (0) 2. ) Ева and Изло» 24” 2. 2. osces + since) addring (+1) + (1-1) и+ 21+ / + 8 -2 +/ 2 2 с ) 9 ч 2 2 3 6 27 er +8 x +4 + - * +9 : х - 2 3 - 7" } * + * * а)

21+) = tt 3 cost b) a1000 - 3 Simle dx at 1+ 2510(e) yet) - a cosce) dy de dy dyldt clarat 2 cose - 3 sint slope ofa nty af t = 2x to - 2 copaty -3 simaty -3* बर - 2 C 3 zeso becomes O Cost = 77 nez a cost=0 를 Nemerafor of c) for horizontal fangent the slope La 2) t = (20+1), 스, af te cos I -/ y = it a sim 35 t = 2 -> -1 + 3 cos ī دام 1+2 ² 3

t= 32 x= 32 af - t3cos 35 = -1 1 BA y 2 (ا - را- اويه The poents Ore e it a sim 30 = 1-2=-1 (-1,3) of horizontal tangent for vertical tangent -3510t=0 => t = 15; おもて 2 - -> t = 0,5, 25 at tro, -1+3cos(o) y = It asim(o) / -4 x = at t = -1 + 3 cos(1) = It zsen(a) ya 2 -1 + 3 cor(21) > at t=25 y=it a gon(za) = the poents are so (2,1) (-4,1) azt of vertical tangent