9.1 - 9 tr PARAMETRIC EQUATIONS POLAR AND COORDINATES 9,1 Parametric Gurves 1.r:l ft,a:t2-4t t T 0<r<5 2345 01 r 2 r\\O 2.4r r Jl 2.73 3 r J6 3.24 a 0

9.1 - 9 tr PARAMETRIC EQUATIONS POLAR AND COORDINATES 9,1...

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9 tr PARAMETRIC EQUATIONS AND POLAR COORDINATES 9,1 Parametric Gurves 1 . r : l + f t , a : t 2 - 4 t , 0<r<5 3. r : 5sint, A :t2, -r 1 t 1 n r : 3 t - 5 , A : 2 t + l (a) t - 1 0 t 2 3 4 r -11-B-5-2r47 v -3 - 1 I 3 5 7 9 (b)r:3,-5 + 3t:r15 + t: j(z+s) a :2. [email protected]+5) + 1,soe : Z" + E. 7 ,: Jt,a : | - | (al t 0 1 2 3 4 r 0 1 t.4r4 1.732 2 a 0 - 1 o e (b) - ^ft r -2 L - V U S i n c e l ) 0 , r ) 0 . U : t - t : I - 1 2 . t 0 1 2 3 4 5 T r 2 r+ \O r+Jl 3 r+J6 2.4r 2.73 3.24 a 0 - 3 - 4 - 3 0 5 t -7r -r/2 0 r/2 7r r 0 0 0 a n2 n'14 o r2ll n" s.87 2.47 2.47 9.87 271
272 ! CHAPTERg PARAMETRICEQUATIONSANDPOLARCOORDINATES 9 . ( a )r : s i n 0 , ! / : c o s ? , 0 < . 0 1 n . ,t" + u2 : sin2 0 + cos2 I : 1. Since 0 I 0 I n,wehave sind ) 0, so:r ) 0. Thus, the curve isthe right half of the circle 12 i a2 : I. 11.(a) r:: sinl, 'Y: csct' 0 < t < +' (b) I - 1. For0 < I < f,,wehave0 < :r I laruJ'y ) I. Y: cscf : "inf- - , Thus, the curve is the portion of the hyperbolay :11" with y > 1. 13.(a)t::r'.2t + 2t-Inr + 1-jl:nL:. . ! t : t 1 1 : j l n z f 1 . r : J i 2 c o s t , , g - l + 2 s i n t , r f 2 < t < 3 t r f 2 . B y E x a m p l e 4 w i t h r ' : 2 , h , : 3 , a n d A : l , t h e r n o t i o n o f t h e p a r t i c l e takes place on a circle centered at (3, 1) with a radiusof 2. As I goesfrom t to * , the particle starts at the point (3, 3) and moves counterclockwise to (3, -1) [one-half of a circle]. 17. :r::5sint, s:2cosl + sint: f, rust: 1. sin2r+cos2r: 1 + (i)'* G)' -l. Themotionofthe particle takes place on anellipse centered at (0,0). As I goes from -rr to 5r, theparticle starts at thepoint (0, -2) and moves clockwise around the ellipse 3 times. 19. Whent : _7, (t:,'y) - (0, -1). As I increases to 0, r decreases to -1 and g increases to 0. As t increases from 0 to 1, r increases to 0 and y increases to 1. As I increases beyond 1, both r and gtincrease. For I ( -1, z is positiveand decreasing and gr is negative and increasing. We could achieve greater accuracy by estimating r- and g-values for selected values oft from the given graphs and plotting the corresponding points. (b) 21. When I lncreas increas arflve at to 0 whi achieve corresp 23. As in E> of [*2r (b) 25.

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