10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e

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1 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have wo branches, and boh branches approach he verical asmpoe a as approaches a from he lef or righ. a=_ a=_ a=_.5 a=_. a= a=.5 a= a= FIGURE 7 Members of he famil =a+cos, =a an +sin, all graphed in he viewing recangle _, b _, When a, boh branches are smooh; bu when a reaches, he righ branch acquires a sharp poin, called a cusp. For a beween and he cusp urns ino a loop, which becomes larger as a approaches. When a, boh branches come ogeher and form a circle (see Eample ). For a beween and, he lef branch has a loop, which shrinks o become a cusp when a. For a, he branches become smooh again, and as a increases furher, he become less curved. Noice ha he curves wih a posiive are reflecions abou he -ais of he corresponding curves wih a negaive. These curves are called conchoids of Nicomedes afer he ancien Greek scholar Nicomedes. He called hem conchoids because he shape of heir ouer branches resembles ha of a conch shell or mussel shell. M. EXERCISES Skech he curve b using he parameric equaions o plo poins. Indicae wih an arrow he direcion in which he curve is raced as increases.. s,,. cos, cos,. 5 sin,,. e, e, 5 (a) Skech he curve b using he parameric equaions o plo poins. Indicae wih an arrow he direcion in which he curve is raced as increases. Eliminae he parameer o find a Caresian equaion of he curve. 5. 5, 6., 5, 5 7., 5, 8., 9. s,., 8 (a) Eliminae he parameer o find a Caresian equaion of he curve. Skech he curve and indicae wih an arrow he direcion in which he curve is raced as he parameer increases.. sin, cos,. cos, 5 sin,. sin, csc,. e, 5. e, e 6. ln, s, 7. sinh, cosh

2 SECTIN. CURVES DEFINED BY PARAMETRIC EQUATINS cosh, 5 sinh 9 Describe he moion of a paricle wih posiion, as varies in he given inerval. 9. cos, sin,. sin, cos,. 5 sin, cos,. sin, cos, Use he graphs of f and o skech he parameric curve f,. Indicae wih arrows he direcion in which he curve is raced as increases. 5. _ 6.. Suppose a curve is given b he parameric equaions f,, where he range of f is, and he range of is,. Wha can ou sa abou he curve?. Mach he graphs of he parameric equaions f and in (a) (d) wih he parameric curves labeled I IV. Give reasons for our choices. (a) I 7. II 8. Mach he parameric equaions wih he graphs labeled I-VI. Give reasons for our choices. (Do no use a graphing device.) (a),, s (c) sin, sin sin (d) cos 5, sin (e) sin, cos sin cos (f), I II III (c) III IV V VI (d) IV ; 9. Graph he curve 5. ;. Graph he curves 5 and and find heir poins of inersecion correc o one decimal place.

3 68 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES. (a) Show ha he parameric equaions where, describe he line segmen ha joins he poins P, and P,. Find parameric equaions o represen he line segmen from, 7 o,. ;. Use a graphing device and he resul of Eercise (a) o draw he riangle wih verices A,, B,, and C, 5.. Find parameric equaions for he pah of a paricle ha moves along he circle in he manner described. (a) nce around clockwise, saring a, Three imes around counerclockwise, saring a, (c) Halfwa around counerclockwise, saring a, ;. (a) Find parameric equaions for he ellipse a b. [Hin: Modif he equaions of he circle in Eample.] Use hese parameric equaions o graph he ellipse when a and b,,, and 8. (c) How does he shape of he ellipse change as b varies? ; 5 6 Use a graphing calculaor or compuer o reproduce he picure If a and b are fied numbers, find parameric equaions for he curve ha consiss of all possible posiions of he poin P in he figure, using he angle as he parameer. Then eliminae he parameer and idenif he curve. a b P. If a and b are fied numbers, find parameric equaions for he curve ha consiss of all possible posiions of he poin P in he figure, using he angle as he parameer. The line segmen AB is angen o he larger circle. A a b P B Compare he curves represened b he parameric equaions. How do he differ? 7. (a), 6, (c) e, e 8. (a), cos, (c) e, e 9. Derive Equaions for he case.. Le P be a poin a a disance d from he cener of a circle of radius r. The curve raced ou b P as he circle rolls along a sraigh line is called a rochoid. (Think of he moion of a poin on a spoke of a biccle wheel.) The ccloid is he special case of a rochoid wih d r. Using he same parameer as for he ccloid and, assuming he line is he -ais and when P is a one of is lowes poins, show ha parameric equaions of he rochoid are r d sin r d cos Skech he rochoid for he cases d r and d r. sec. A curve, called a wich of Maria Agnesi, consiss of all possible posiions of he poin P in he figure. Show ha parameric equaions for his curve can be wrien as a co a sin Skech he curve. =a a. (a) Find parameric equaions for he se of all poins P as shown in he figure such ha P AB. (This curve is called he cissoid of Diocles afer he Greek scholar Diocles, who inroduced he cissoid as a graphical mehod for consrucing he edge of a cube whose volume is wice ha of a given cube.) A C P

4 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES abou he -ais. Therefore, from Formula 7, we ge S r sin sr sin r cos d r sin sr sin cos d r sin r d r sin d r cos ] r M. EXERCISES Find dd.. sin,., s e 9. cos,. cos, sin sin 6 Find an equaion of he angen o he curve a he poin corresponding o he given value of he parameer.., ;., ; 5. e s, ln ; 6. cos sin, sin cos ; 7 8 Find an equaion of he angen o he curve a he given poin b wo mehods: (a) wihou eliminaing he parameer and b firs eliminaing he parameer. 7. ln, ;, 8. an, sec ; (, s) ; 9 Find an equaion of he angen(s) o he curve a he given poin. Then graph he curve and he angen(s) sin, ;. cos cos, sin sin ; 6 Find dd and d d. For which values of is he curve concave upward?.,.,. e, e. ln, ln 5. sin, cos, 6. cos, cos, 7 Find he poins on he curve where he angen is horizonal or verical. If ou have a graphing device, graph he curve o check our work. 7.,, 8.,, ;. Use a graph o esimae he coordinaes of he righmos poin on he curve 6, e. Then use calculus o find he eac coordinaes. ;. Use a graph o esimae he coordinaes of he lowes poin and he lefmos poin on he curve,. Then find he eac coordinaes. ; Graph he curve in a viewing recangle ha displas all he imporan aspecs of he curve..,. 8, 5. Show ha he curve cos, sin cos has wo angens a, and find heir equaions. Skech he curve. ; 6. Graph he curve cos cos, sin sin o discover where i crosses iself. Then find equaions of boh angens a ha poin. 7. (a) Find he slope of he angen line o he rochoid r d sin, r d cos in erms of. (See Eercise in Secion..) Show ha if d r, hen he rochoid does no have a verical angen. 8. (a) Find he slope of he angen o he asroid, a sin a cos in erms of. (Asroids are eplored in he Laboraor Projec on page 69.) A wha poins is he angen horizonal or verical? (c) A wha poins does he angen have slope or? 9. A wha poins on he curve, does he angen line have slope?. Find equaions of he angens o he curve, ha pass hrough he poin,.. Use he parameric equaions of an ellipse, a cos, b sin,, o find he area ha i encloses.

5 SECTIN. CALCULUS WITH PARAMETRIC CURVES 67. Find he area enclosed b he curve, s and he -ais.. Find he area enclosed b he -ais and he curve e,.. Find he area of he region enclosed b he asroid a cos, a sin. (Asroids are eplored in he Laboraor Projec on page 69.) a _a a 9. Use Simpson s Rule wih n 6 o esimae he lengh of he curve e, e, In Eercise in Secion. ou were asked o derive he parameric equaions a co, a sin for he curve called he wich of Maria Agnesi. Use Simpson s Rule wih n o esimae he lengh of he arc of his curve given b. 5 5 Find he disance raveled b a paricle wih posiion, as varies in he given ime inerval. Compare wih he lengh of he curve. 5. sin, cos, 5. cos, cos, 5. Find he area under one arch of he rochoid of Eercise in Secion. for he case d r. 6. Le be he region enclosed b he loop of he curve in Eample. (a) Find he area of. If is roaed abou he -ais, find he volume of he resuling solid. (c) Find he cenroid of. 7 Se up an inegral ha represens he lengh of he curve. Then use our calculaor o find he lengh correc o four decimal places. 7.,, 8. e,, 9. cos, sin,. ln, s, Find he eac lengh of he curve..,,. e e, 5,., ln,. cos cos, sin sin, ; 5 7 Graph he curve and find is lengh. 5. e cos, e sin, 6. cos ln(an ), sin, 7. e, e, _a Find he lengh of he loop of he curve,. CAS CAS 5. Show ha he oal lengh of he ellipse a sin, b cos, a b, is L a where e is he eccenrici of he ellipse (e ca, where c sa b ). 5. Find he oal lengh of he asroid a cos, a sin, where a. 55. (a) Graph he epirochoid wih equaions cos cos sin sin Wha parameer inerval gives he complee curve? Use our CAS o find he approimae lengh of his curve. 56. A curve called Cornu s spiral is defined b he parameric equaions C cosu du S sinu du where C and S are he Fresnel funcions ha were inroduced in Chaper 5. (a) Graph his curve. Wha happens as l and as l? Find he lengh of Cornu s spiral from he origin o he poin wih parameer value Se up an inegral ha represens he area of he surface obained b roaing he given curve abou he -ais. Then use our calculaor o find he surface area correc o four decimal places. 57. e, e, s e sin 58. sin, sin, d

6 APPENDIX I ANSWERS T DD-NUMBERED EXERCISES A (a) Populaion sabilizes a 5. (i) W, R : Zero populaions (ii) W, R 5: In he absence of wolves, he rabbi populaion is alwas 5. (iii) W 6, R : Boh populaions are sable. (c) The populaions sabilize a rabbis and 6 wolves. (d) Species R Species CHAPTER 9 REVIEW N PAGE 65 True-False Quiz. True. False 5. True 7. True Eercises. (a) c ; 6,, (iv) (iii) (ii) (i) = = = =, 5 W R = W (a) P ; 56 ln e. 7. (a) L L L 5 e. L Le k 9. 5 das. k ln h h RV C. (a) Sabilizes a, (i), : Zero populaions (ii),, : In he absence of birds, he insec populaion is alwas,. (iii) 5,, 75: Boh populaions are sable. (c) The populaions sabilize a 5, insecs and 75 birds. (d) (insecs) 5, 5, 5, 5, 5, 5. (a) k cosh k a k or k cosh k k cosh kb h PRBLEMS PLUS N PAGE 68. f e C 9. f L L ln n (c) No L L. (a) 9.8 h ; 68 f h (c) 5. h. 6 5 CHAPTER EXERCISES. N PAGE 66.. =5 {+œ 5, 5} insecs birds (birds),9, f k sinh kb = (a)..8 = (, ) = (, ) 5 = (, ) 5 _ (c) and ; here is a local maimum or minimum 5. ( C)e sin 7. sln C 9. r 5e.. C ln 5. (a) (_5, ) = (_8, _) =_ (_, ) = (, 5) =

7 A APPENDIX I ANSWERS T DD-NUMBERED EXERCISES 7. (a) 5, (_, 5) = 9. (a), (, ) = 5, = (, ) = (, _) = (7, ) =_ (, _) =. 5, 7 8,. (a) cos, sin, cos, sin, 6 (c) cos, sin, 7. The curve is generaed in (a). In, onl he porion wih is generaed, and in (c) we ge onl he porion wih.. a cos, b sin ; a b, ellipse. a 5. (a) Two poins of inersecion. (a),. (a), (, ) (, _) 5. (a) ln 7. (a), (, ) ne collision poin a, when (c) There are sill wo inersecion poins, bu no collision poin. 7. For c, here is a cusp; for c, here is a loop whose size increases as c increases. _ Moves counerclockwise along he circle from, o,. Moves imes clockwise around he ellipse 5, saring and ending a,. I is conained in he recangle described b and (, ) = = _ 9. As n increases, he number of oscillaions increases; a and b deermine he widh and heigh. EXERCISES. N PAGE 66. cos sin. 5. e _ (_, ) = (, _) =_ = _,, e, e e, an, sec, 6, 6 7. Horizonal a, verical a, 9. Horizonal a (s, ) (four poins), verical a,..6, ; (5 6 65, e 6 5 ) _

8 APPENDIX I ANSWERS T DD-NUMBERED EXERCISES A , (c) 8.5 _, 7. (a) d sin r d cos 9. 7, 9 9 ),,. ab. e 5. r d 7. s d s s sin cos d.67. s ln( s) s ln( s) 5. s e 8 ( 6. (a) (c),,, 5 (, ),, _ (, s) _ 7. e e (a) s, s, (s, s) 5. (a) (i) (s, 7) (ii) (s, ) (i), (ii), r= _, r= =_ = 6 r=. = r= r= e se d (7s 6) 6. 5a (99s6 ) EXERCISES. N PAGE 67. (a) 5,, _ _. s 5. Circle, cener, radius 7. Circle, cener (, ), radius 9. Horizonal line, uni above he -ais. r sec. r co csc 5. r c cos 7. (a) = 5, =_ 6, 7,,, 5,,

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