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+ft,a:t2-4t, 0<r<5 3. r : 5sint, A :t2, -r 1 t 1 n r:3t-5, A:2t+l (a) t -1 0 t234 r -11-B-5-2r47 v -3 I 3579 (b)r:3,-5 + 3t:r15 + t: j(z+s) a :2. i@+5) + 1,soe Z" E. 7 ,: Jt,a | - | (al t 01234 r 0 1 t.4r4 1.732 2 a 0 o e (b) ^ft -2 L VU Sincel)0,r)0. U:t-t:I-12. t 01 2345 T 2 r+ \O r+Jl 3 r+J6 2.4r 2.73 3.24 a 0-3-4-305 t -7r -r/2 r/2 7r r 0 0 0 a n2 n'14 o r2ll n" s.87 2.47 9.87 271
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272 ! CHAPTERg PARAMETRICEQUATIONSANDPOLARCOORDINATES 9. (a) r: sin0, !/: cos?, 0<.01n. ,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. r:: sinl, 'Y: csct' 0 < t < +' (b) - 1. For0 < I < f,,wehave0 < :r I laruJ'y ) I. Y: cscf "inf- - , Thus, is the portion of the hyperbolay :11" with y > 1. 13.(a)t::r'.2t + 2t-Inr + 1-jl:nL:. .!t:t1 1:jlnzf1. r:Ji2cost,,g-l+2sint,rf2<t<3trf2. ByExample4withr':2,h,:3,andA:l,thernotionoftheparticle takes place on a circle centered at (3, 1) with a radius of 2. As I goes from 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 an ellipse centered at (0,0). As I goes -rr to 5r, the particle starts at the point (0, -2) and moves clockwise around the ellipse 3 times. 19. When _7, (t:,'y) - (0, -1). As I increases to 0, r decreases -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 gt increase. For I ( -1, z is positive and 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. 21. When I lncreas e increas e arflve at to 0 whi achieve corresp c 23. As in E> of [*2r 25.(a) .r = thro t h(' F para tln (a) To g (b) To g with (c) To s 15. 27. AI
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SECTION 9.1 PARAMETRIC CURVES tr 273 2'f. When f : 0 we see that r : 0 andU 0, so the curve starts at the origin. As I increases from 0 to |, the graphs show that y increases
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9.1 - 9 tr PARAMETRIC EQUATIONS POLAR AND COORDINATES 9,1...

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