Review Exercises for Chapter 10

Transcription

1 60_00R.q //0 :8 M age CHATER 0 Conics, arameric Equaions, an olar Coorinaes Review Eercises for Chaper 0 See for worke-ou soluions o o-numbere eercises. In Eercises 6, mach he equaion wih he correc graph. [The graphs are labele (a), (b), (c), (), (e), an (f).] (a) (c) (e) (b) () (f) In Eercises 7, analze he equaion an skech is graph. Use a graphing uili o confirm our resuls In Eercises an, fin an equaion of he parabola.. Vere: 0, ; irecri:. Vere:, ; focus:, 0 In Eercises 5 an 6, fin an equaion of he ellipse. 5. Verices:, 0, 7, 0; foci: 0, 0,, 0 6. Cener: 0, 0; soluion poins: (, ), (, 0) 8 6 In Eercises 7 an 8, fin an equaion of he hperbola. 7. Verices: ±, 0; foci: ±6, 0 8. Foci: 0, ±8; asmpoes: ± In Eercises 9 an 0, use a graphing uili o approimae he perimeer of he ellipse A line is angen o he parabola an perpenicular o he line. Fin he equaion of he line.. A line is angen o he parabola 6 an perpenicular o he line 5. Fin he equaion of he line.. Saellie Anenna A cross secion of a large parabolic anenna is moele b he graph of 00, The receiving an ransmiing equipmen is posiione a he focus. (a) Fin he coorinaes of he focus. (b) Fin he surface area of he anenna.. Fire Truck Consier a fire ruck wih a waer ank 6 fee long whose verical cross secions are ellipses moele b he equaion (a) Fin he volume of he ank. (b) Fin he force on he en of he ank when i is full of waer. (The ensi of waer is 6. pouns per cubic foo.) (c) Fin he eph of he waer in he ank if i is full (b volume) an he ruck is on level groun. () Approimae he ank s surface area. In Eercises 5 0, skech he curve represene b he parameric equaions (inicae he orienaion of he curve), an wrie he corresponing recangular equaion b eliminaing he parameer. 5., 6., 7. 6 cos, 6 sin 8. cos, 5 sin 9. sec, an 0. 5 sin, 5 cos

2 60_00R.q //0 :9 M age 757 REVIEW EXERCISES 757 In Eercises, fin a parameric represenaion of he line or conic.. Line: passes hrough, 6 an,. Circle: cener a (5, ); raius. Ellipse: cener a, ; horizonal major ais of lengh 8 an minor ais of lengh 6. Hperbola: verices a 0, ±; foci a 0, ±5 5. Roar Engine The roar engine was evelope b Feli Wankel in he 950s. I feaures a roor, which is a moifie equilaeral riangle. The roor moves in a chamber ha, in wo imensions, is an epirochoi. Use a graphing uili o graph he chamber moele b he parameric equaions. cos 5 cos an sin 5 sin. 6. Serpenine Curve Consier he parameric equaions co an sin cos, 0 < <. (a) Use a graphing uili o graph he curve. (b) Eliminae he parameer o show ha he recangular equaion of he serpenine curve is 8. In Eercises 5 an 5, (a) use a graphing uili o graph he curve represene b he parameric equaions, (b) use a graphing uili o fin /, /, an / for an (c) use a graphing uili o graph he angen line o he curve when Arc Lengh In Eercises 5 an 5, fin he arc lengh of he curve on he given inerval. Surface Area In Eercises 55 an 56, fin he area of he surface generae b revolving he curve abou (a) he -ais an (b) he -ais Area sin 0,, cos, /6. 5. co 5. sin rsin sin 5. rcos 5. 6 cos cos 0 sin, 0 In Eercises 57 an 58, fin he area of he region. cos 6 sin 0 /6, In Eercises 7 6, (a) fin / an all poins of horizonal angenc, (b) eliminae he parameer where possible, an (c) skech he curve represene b he parameric equaions. 7., 8., 9., 57. sin 58. cos cos sin 0 0.,... cos. 6 cos 5 sin 5. cos 6. e sin In Eercises 7 50, fin all poins (if an) of horizonal an verical angenc o he curve. Use a graphing uili o confirm our resuls ,, sin, cos, cos sin 6 sin e In Eercises 59 6, plo he poin in polar coorinaes an fin he corresponing recangular coorinaes of he poin. 59., 60., 6 6.,.56 6.,.5 In Eercises 6 an 6, he recangular coorinaes of a poin are given. lo he poin an fin wo ses of polar coorinaes of he poin for 0 <. 6., 6.,

3 60_00R.q //0 :9 M age CHATER 0 Conics, arameric Equaions, an olar Coorinaes In Eercises 65 7, conver he polar equaion o recangular form. 65. r cos 66. r r cos 68. r cos 69. r cos 70. r sec 7. r cos sec 7. In Eercises 7 76, conver he recangular equaion o polar form. a arcan In Eercises 77 88, skech a graph of he polar equaion. 77. r r sec 80. r csc 8. r cos 8. r cos 8. r cos 8. r 85. r cos 86. r cos r sin 88. r cos In Eercises 89 9, use a graphing uili o graph he polar equaion. 89. r 90. cos In Eercises 9 an 9, (a) fin he angens a he pole, (b) fin all poins of verical an horizonal angenc, an (c) use a graphing uili o graph he polar equaion an raw a angen line o he graph for 9. r cos 9. r sin 95. Fin he angle beween he circle r sin an he limaçon r 5 sin a he poin of inersecion, True or False? There is a unique polar coorinae represenaion for each poin in he plane. Eplain. In Eercises 97 an 98, show ha he graphs of he polar equaions are orhogonal a he poins of inersecion. Use a graphing uili o confirm our resuls graphicall. 97. r cos 98. r a sin r cos /6. 7. a r cos sec 9. r sec cos r a cos arcan a r sin cos In Eercises 99 0, fin he area of he region. 99. Inerior of r cos 00. Inerior of r 5 sin 0. Inerior of r sin 0. Common inerior of r cos an r In Eercises 0 06, use a graphing uili o graph he polar equaion. Se up an inegral for fining he area of he given region an use he inegraion capabiliies of a graphing uili o approimae he inegral accurae o wo ecimal places. 0. Inerior of r sin cos 0. Inerior of r sin 05. Common inerior of r an r 8 sin 06. Region boune b he polar ais an r e for 0 In Eercises 07 an 08, fin he lengh of he curve over he given inerval. olar Equaion 07. r a cos 08. r a cos In Eercises 09 an 0, wrie an inegral ha represens he area of he surface forme b revolving he curve abou he given line. Use a graphing uili o approimae he inegral. olar Equaion 09. r cos 0 olar ais 0. r sin Inerval 0 Inerval 0 In Eercises 6, skech an ienif he graph. Use a graphing uili o confirm our resuls.. r sin. r 6. r cos. r 5. r sin 6. r cos 5 sin 8 5 cos In Eercises 7, fin a polar equaion for he line or conic wih is focus a he pole. 7. Circle 8. Line Cener: 5, Soluion poin: (0, 0) Soluion poin: (0, 0 Slope: 9. arabola 0. arabola Vere:, Vere:,. Ellipse. Hperbola Verices: 5, 0,, Verices:, 0, 7, 0 Ais of Revoluion

4 60_00R.q //0 :9 M age 759.S. roblem Solving 759.S. roblem Solving See for worke-ou soluions o o-numbere eercises.. Consier he parabola an he focal chor. (a) Skech he graph of he parabola an he focal chor. (b) Show ha he angen lines o he parabola a he enpoins of he focal chor inersec a righ angles. (c) Show ha he angen lines o he parabola a he enpoins of he focal chor inersec on he irecri of he parabola.. Consier he parabola p an one of is focal chors. (a) Show ha he angen lines o he parabola a he enpoins of he focal chor inersec a righ angles. (b) Show ha he angen lines o he parabola a he enpoins of he focal chor inersec on he irecri of he parabola.. rove Theorem 0., Reflecive roper of a arabola, as shown in he figure.. Consier he hperbola a b F wih foci F an F, as shown in he figure. Le T be he angen line a a poin M on he hperbola. Show ha incoming ras of ligh aime a one focus are reflece b a hperbolic mirror owar he oher focus. 6. Consier he region boune b he ellipse a b, wih eccenrici e ca. (a) Show ha he area of he region is ab. (b) Show ha he soli (oblae spheroi) generae b revolving he region abou he minor ais of he ellipse has a volume V b an a surface area of (c) Show ha he soli (prolae spheroi) generae b revolving he region abou he major ais of he ellipse has a volume of V ab an a surface area of 7. The curve given b he parameric equaions an is calle a srophoi. (a) Fin a recangular equaion of he srophoi. (b) Fin a polar equaion of he srophoi. (c) Skech a graph of he srophoi. () Fin he equaions of he wo angen lines a he origin. (e) Fin he poins on he graph where he angen lines are horizonal. 8. Fin a recangular equaion of he porion of he ccloi given b he parameric equaions a sin an a cos, 0, as shown in he figure. S a b e ln e e. S b ab e arcsin e. B a M F b T a F O A θ a c O aπ Figure for Figure for 5 5. Consier a circle of raius a angen o he -ais an he line a, as shown in he figure. Le A be he poin where he segmen OB inersecs he circle. The cissoi of Diocles consiss of all poins such ha O AB. (a) Fin a polar equaion of he cissoi. (b) Fin a se of parameric equaions for he cissoi ha oes no conain rigonomeric funcions. (c) Fin a recangular equaion of he cissoi. 9. Consier he cornu spiral given b cos u 0 u an (a) Use a graphing uili o graph he spiral over he inerval. (b) Show ha he cornu spiral is smmeric wih respec o he origin. (c) Fin he lengh of he cornu spiral from 0 o a. Wha is he lengh of he spiral from o? sin u 0 u.

5 60_00R.q //0 :9 M age CHATER 0 Conics, arameric Equaions, an olar Coorinaes 0. A paricle is moving along he pah escribe b he parameric equaions an sin, for <, as shown in he figure. Fin he lengh of his pah.. Le a an b be posiive consans. Fin he area of he region in he firs quaran boune b he graph of he polar equaion r. Consier he righ riangle shown in he figure. (a) Show ha he area of he riangle is (b) Show ha (c) Use par (b) o erive he formula for he erivaive of he angen funcion. α ab a sin b cos, an 0 0 sec.. (, 0) (, 0) A 0 sec. 5. An air raffic conroller spos wo planes a he same aliue fling owar each oher (see figure). Their fligh pahs are 0 an 5. One plane is 50 miles from poin wih a spee of 75 miles per hour. The oher is 90 miles from poin wih a spee of 50 miles per hour. 90 mi (a) Fin parameric equaions for he pah of each plane where is he ime in hours, wih 0 corresponing o he ime a which he air raffic conroller spos he planes. (b) Use he resul of par (a) o wrie he isance beween he planes as a funcion of. (c) Use a graphing uili o graph he funcion in par (b). When will he isance beween he planes be minimum? If he planes mus keep a separaion of a leas miles, is he requiremen me? 6. Use a graphing uili o graph he curve shown below. The curve is given b r e cos cos sin 5. Over wha inerval mus 0 50 mi 5 var o prouce he curve? Figure for Figure for. Deermine he polar equaion of he se of all poins r,, he prouc of whose isances from he poins, 0 an, 0 is equal o, as shown in he figure.. Four ogs are locae a he corners of a square wih sies of lengh. The ogs all move counerclockwise a he same spee irecl owar he ne og, as shown in he figure. Fin he polar equaion of a og s pah as i spirals owar he cener of he square. FOR FURTHER INFORMATION For more informaion on his curve, see he aricle A Su in Sep Size b Temple H. Fa in Mahemaics Magazine. To view his aricle, go o he websie 7. Use a graphing uili o graph he polar equaion r cos 5 n cos, for 0 < an for he inegers n 5o n 5. Wha values of n prouce he hear porion of he curve? Wha values of n prouce he bell porion? (This curve, creae b Michael W. Chamberlin, appeare in The College Mahemaics Journal.)