Chapter 11. Parametric, Vector, and Polar Functions. aπ for any integer n. Section 11.1 Parametric Functions (pp ) cot

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1 Secion. 6 Chaper Parameric, Vecor, an Polar Funcions. an sec sec + an + Secion. Parameric Funcions (pp. 9) Eploraion Invesigaing Cclois 6. csc + co co cos cos cos [, ] b [, 8]. na for an ineger n.. a > an cos so. 8. sin cos sin. An arch is prouce b one complee urn of he wheel. Thus, he are congruen.. The maimum value of is a an occurs when ( n+ ) a for an ineger n. 6. The funcion represene b he ccloi is perioic wih perio a, an each arch represens one perio of he graph. In each arch, he graph is concave own, has an absolue maimum of a a he mipoin, an an absolue minimum of a he wo enpoins. Quick Review.. + ( ) cos + sin + since for.. cos + sin + since for. Secion. Eercises. Yes, is a funcion of. 9 (9, 9). sin cos cos + sin + (, ). sin sin cos Coprigh 6 Pearson Eucaion, Inc.

2 6 Secion.. Yes, is a funcion of. + 7 (, ) 6. No, is no a funcion of. sin ( ) (,.9) (, ). Yes, is a funcion of. + (, ) (, ). No, is no a funcion of. + (, ) (, ). Yes, is a funcion of. (, ) 7. (a) 8. (a) 9. (a) (, ) sin an cos cos sec cos sec 8 ( an ) sin sin ( ) sin / ( ) / ( )( + ) ( ) + / / ( )( + ) (, -). (a) ( ) Coprigh 6 Pearson Eucaion, Inc.

3 Secion. 6. (a) 7. (a) ( ) 6 8 ( ). (a) +. (a). (a) ( ) + + ( + ) sec an sin sec sin cos sec sin ( ) cos sin 8. (a) (.,.) (c) We seek o minimize as a funcion of, so we compue +, which is negaive for <. an posiive for. <. There is a relaive minimum a., where (, ) (.,.). ( cos ) sin. (a) ( ) 6. (a) ( ) e e ( e ) e + e 9. (a) (, 6) (c) We seek o minimize as a funcion of, so we compue +, which is negaive for < an posiive for <. There is a relaive minimum a, where (, ) (, 6). (, ) Coprigh 6 Pearson Eucaion, Inc.

4 6 Secion. (c) We seek o maimize as a funcion of, so we compue cos, which is posiive for < an negaive for <. There is a relaive maimum a, where (, ) (, ).. (a). (a) ( ln( ), ln()) (.9,.99) (, ) (c) We seek o minimize as a funcion of, so we compue secan, which is negaive for < an posiive for <. There is a relaive minimum a, where (, ) (, ).. (a) (, ) (c) We seek o maimize as a funcion of, so we compue sin( ), which is posiive for. < an negaive for <.. There is a relaive maimum a, where (, ) (, ). (c) We seek o maimize as a funcion of, so we compue, which is posiive for all >. There is an enpoin maimum a, where (, ) (ln(), ln()).. ( + sin ) cos an ( + cos ) sin. (a) Tangen is horizonal when an. If, hen cos, an + cos. If cos, hen sin ±, an + sin or. The poins are (, ) an (, ). Tangen is verical when an. If, hen sin, an + sin. If sin, hen cos ±, an + cos or. The poins are (, ) an (, ). Coprigh 6 Pearson Eucaion, Inc.

5 . (an ) sec an (sec ) sec an. Secion ( + sin ) cos an ( + cos ) sin. (a) Tangen is horizonal when an. Since sec > for all, here are no poins where he angen line is horizonal. (a) If, hen cos an + cos. If cos, hen sin ±, an + sin or. The poins are (, ) an (, ).. Tangen is verical when an. If, hen secan, so an. If an, hen sec ±. The poins are (, ) an (, ). ( ) an ( ). (a) If, hen an ±. Using he parameric formulas for an, he poins are 6, 6 an 9, + 9. (In ecimal form, he are (.8,.79) an (.,.79).) Tangen is verical when an. If, hen sin, an + cos. If sin, hen cos ±, an + cos or. The poins are (, ) an (, ). S ( sin ) + (cos ) S S S S ( cos ) + ( sin ) S 9 S S Tangen is verical when an. Since for all, here are no poins where he angen line is verical. Coprigh 6 Pearson Eucaion, Inc.

6 66 Secion. 9.. / / S (( 8sin + 8sin + 8cos ) + ( 8cos 8cos+ 8sin ) ) / / S (( 8cos ) + ( 8sin ) ) / S 8 / S S / ([ 6cos ()sin()] + [ 6sin ()cos()] ) 6 cos ()sin () + sin ()cos () 6 [cos ( ) + sin ( )][cos ( )sin ( )] 6 cos ( )sin ( ) / 6cos sin / sin / cos 6 [ ( )] / S + + ( + ) / S ( + + ) S. ( ) / S ( + ) 9 / S ( + + ) S. ( ). / (( ) + ( ) ) / S ( + ) S S Coprigh 6 Pearson Eucaion, Inc.

7 Secion. 67. / S (sec cos ) + ( sin ) / / ln an an ln cos /. (a) sin, cos, so / Lengh ( sin ) + ( cos ). cos, sin, so Lengh ( cos ) ( sin ) sin, cos, so Lengh.. ( sin ) ( cos ) which using NINT evaluaes o + 7. In he firs inegral, replace wih. Then becomes. 8. Parameerize he curve as g(),, c. The parameer is iself, so replace wih in he general formula. Then becomes. 9. a( cos ) (Noe: inegrae wih respec o from o a ; inegrae wih respec o from o.) a Area a( cos ) a( cos ) a ( cos+ cos ) a sin + + sin a Coprigh 6 Pearson Eucaion, Inc.

8 68 Secion.. a( cos ), so Volume [ a( cos )] a( cos ) a ( cos+ cos cos ) a sin + + sin sin sin a. a bsin an a b cos ( < b< a). a bsin an a b cos ( a < b< a).. S (( cos ) + ( sin ) ) S ( 9 cos+ cos + sin ) S ( cos ) S. S (( cos ) + ( sin )) S ( cos+ 9cos + 9sin ) S ( cos ) S.. False. Inee, ma no even be a funcion of. (See Eample.) 6. True. The orere pairs (, f()) an (, f()) are eacl he same. 7. B 8. C; if sin an csc, hen we can eliminae he parameer o ge. Since sin an csc are boh posiive for < <, he pah follows a porion of he curve in he firs quaran, where is ecreasing an concave up. 9. C; ln ( ) ln ( ). D e. (a) QP has lengh, so P can be obaine b saring a Q an moving sin unis righ an cos unis ownwar. (If eiher quani is negaive, he corresponing irecion is reverse.) Since Q (cos, sin ), he Coprigh 6 Pearson Eucaion, Inc.

9 Secion. 69 coorinaes of P are cos + sin an sin cos. Q sin cos P(, ) (, ) cos, sin, so Lengh (cos) + (sin). All isances are a imes as big as before. (a) a(co + sin ), a(sin cos ) Lengh a v For eercises 6, sin v cos an v sin, an for or. The maimum heigh 6 v sin v is aaine in mi-fligh a. sin / 6 To fin he pah lengh, evaluae ( vcos ) ( vsin ) + vsin vsin using NINT. To fin he maimum heigh, calculae ma ( v sin ) 6.. (a) The projecile his he groun when. ( sin 6) 7 or sin. 6 8 cos, sin ( 7sin )/ 8 Lengh ( cos ) + ( sin ) which, using NINT, evaluaes o 6.79 f The maimum heigh of he projecile occurs when, so. (a) 6.6 f 7 6 sin, 7 f 6 sin f 6. (a) 8. f Coprigh 6 Pearson Eucaion, Inc.

10 6 Secion. 6, f 6 6. (a) I is no necessar o use NINT Lengh ( ) [ 6 ] f 6. 6 f / (( sin ) cos ) S + S () ( + sin ) ( cos ) 8 / / / S (( ) + ( ) ) / S + / / / + + / ( + ). / S (( ) + ( ) ) / ( ) / ( )( ) S / / ((sec cos ) ( sin ) ) / / cos (sec ) / cosan / sin / ( cos ) S + S + Secion. Vecors in he Plane (pp. 6) Quick Review.. ( ) + ( ) 7. Solve b ; b.. Slope of PQ, so RQ an b. b. Slope of AB Slope of CD, so a an a. 6. Slope of AB Slope of CD, so b 6 8 an b v () (sin ) v () sin+ cos a () (sin+ cos) a () cos sin 8. () v () ( ) () 6 + C () 6() + C C ( ) 6( ) + 8 Coprigh 6 Pearson Eucaion, Inc.

11 Secion isance ( ) v () v ( ) + an, 8 v + cos, 9. ( cos ) + ( sin ) cos sin Secion. Eercises. (, ) (, ),. (, ) (, ),. (, ) (, ),. cos( 8 ) sin( 8 ),. 6cos( 7 ) 6sin( 7 ) 6, 6. cos( ). 868 sin( ) ,. 9. P +, (, ) (, ) (, ),. cos( ). 6 sin( ). 6. 6,. 6. v + 8 an, 6. v ( ) + 7. an, v + an, 8. ( ) v + an, cos 8 sin, 8 cos 6 8 sin 6, 7. (a) (), ( ) 9, ( 6) 7 8. (a) ( ), ( ), + ( ) (a) + ( ), +, Coprigh 6 Pearson Eucaion, Inc.

12 6 Secion. +. (a) ( ),, 7 + ( 7) 7. (a) u (), ( ) 6, v ( ), ( ) 6, u v 6 ( 6),, 9 + ( 9). (a) u (), ( ) 6,. (a). (a) v ( ), ( ), u+ v 6+ ( ), + 6, 9 ( 6) u (), ( ), 8 v ( ), ( ), u+ v +, +, 97 + u (), ( ), 6 v ( ), ( ), 6 u + v +, + 7, 7 6 ( ) +. Iniial veloci is 7 norh of eas: cos 7, sin 7. 7,.. Win veloci is norh of eas: cos, sin. 7,. 6. A he wo vecors o ge 8., 6.. The spee is he magniue, 6. 7 mph. 6. The irecion is an norh of eas, or. 66 eas of norh. 6. cos sin The rue veloci is,, so he rue angle is cos 6. an he rue spee is ( ) + ( ). 9mph () () v a 6, 6 6, r() v (), 6, 6 r() v () sin, cos cos, sin v() a () cos, sin sin, cos r() v () e, e e e, e v() a () e e, e e + e, e Coprigh 6 Pearson Eucaion, Inc.

13 Secion r() v () cos, sin 6sin, 8cos v() a() 6sin, 8cos 8cos, sin r() v () + sin, cos + cos, + sin v() a() + cos, + sin sin, + cos r() v () sin, cos sin + cos, cos sin v() a() sin + cos, cos sin cos sin, sin cos. (a) The posiion vecor of he ball a an ime is r () (), (), where () 9 cos an () 9sin 6. v (), 9cos, 9sin (c) Fin () when (). [, ] b [, ] No, he hi oes no clear he -foo fence. Coprigh 6 Pearson Eucaion, Inc.

14 6 Secion. () See he graph in par (c). The ball his he fence afer abou. secons. (e) Evaluae v() a.. v () ( 9cos ) + ( 9sin ) v (. ) ( 9cos ) + ( 9sin. ) 86. f/sec. (a) The posiion vecor of he ball a an ime is r () (), (), where () 8 cos 7 an () 8sin 7 6. Noe ha, is on he groun a he puner s posiion, ars 9 fee from he puner s goal line. v (), 8cos 7, 8sin 7 (c) Fin when () cos cos7 The ball is over he plaer afer abou.8 secons. () Fin () when.8. (. 8) 8(. 8)sin 7 6(. 8). 8 I is unlikel ha he plaer will be able o cach he ball wihou backing up.. r() v () cos, sin sin, cos v() a() sin, cos 9cos, sin [.6,.6] b [.,.] p Coprigh 6 Pearson Eucaion, Inc.

15 Secion r() v () sin, cos cos, sin v() a () cos, sin 6sin, 9cos [.6,.6] b [.,.] p 7. (a) v () sincos,sin cos cos sin sin, cos, Spee: ( / ) [, ] b [.,.] p (c) To he righ 8. (a) v () e + e, e e e e, e + e e e e lim lim lim e e e + + (c) For an, ( e + e ) ( e e ) e + + e ( e + e ) Coprigh 6 Pearson Eucaion, Inc.

16 66 Secion. () The veloci a is,. ( ) + ( + ) e e [ 9, 9] b [ 6, 6]. The parameric equaions are an + sin( ) (a). (a). (a). (a), + cos +, 6 ( ), + sin + 6, + 8, 6+, 9 ( ) + ( + cos ) 9. cos, sin + 7, sin, cos + 7, 7+, + 7, ( cos ) + ( sin ) 8. ( + ), ( + ) +, ln( + ), ( + ) +, + ln, 7. (( ) + ) + (( + ) ). 9 e, e + +, e e, + +, +. 86, ,. 86 [, ] b [, ]. The parameric equaions are 7 sin( ) + an cos( ) +. (Noe: The paricle raverses he Figure-8 hree imes, finishing where i sare.) [6, 8] b [., 6.]. (a) v () cos, sin 6 6 v() sin, cos v() sin + cos 7 v( ). 99 a () sin, cos a( ) cos, sin a( ), 7 Coprigh 6 Pearson Eucaion, Inc.

17 Secion. 67 (c) cos 6 sin (a) v () sec,an v secan, sec v, spee: + ( ) 6 ( ) / (sec ) (an ) (c) The upper par of he righ branch: [,.7] b [,.] < / 7. (a) v (), + + v (), ( + ) ( + ) No; he -componen of veloci is zero onl if, while he -componen of veloci is zero onl if. A no ime will he veloci be,. (c) lim,, (a) v () sin,cos v () cos, sin v ( ), where an. Coprigh 6 Pearson Eucaion, Inc.

18 68 Secion. (c) cos sin () [, ] b [.,.] p 9. (a) e sin, e cos e sin + e cos, e cos e sin e cos e sin m e sin + e cos e sin ( ) + e cos ( ), e cos( ) e sin ( ) 76., 87. v () ( 76. ) + ( 87. ) 8. (c). (a) (c). (a) ( e sin+ e cos ) + ( e cos e sin ).,, v( ) ( ( )) + ( ( ) ) v( ) / (( ) + ( ) ) / ( + ) / ( + ) + ( + sin( )) 9. m( ) 6 sin ( + ) (c) Spee ( + sin ) + ( 6) 6. 7 () 8cos 6, ( + sin 6) + 7( 8)cos ,. Coprigh 6 Pearson Eucaion, Inc.

19 Secion. 69. (a) m( ) cos ( ) sin 8 Spee (c) () ( cos ) + (sin 8). 96 ( cos( )) + (sin( )). 7 sin( ), cos( ) + + (., 7. ). False; for eample, u an (u) have opposie irecions.. False; for eample,, +,,, which has a irecion angle of.. E; +, ln( + ), +,, + ( + ) 6. D; + cos( ), 7+ sin( ) +. 6, , B; 7cos 6. 7sin. cos 6. sin. 7 ( ) + ( ) B; cos sin Spee 9. The veloci vecor is has slope is ( cos ) + ( sin ). 8,, which. The acceleraion vecor ( ),,,, which has slope. Since he slopes are negaive reciprocals of each oher, he vecors are orhogonal. 6. The posiion vecor is cos, sin, which has slope an. The veloci vecor is sin, cos, which has slope. The an acceleraion vecor is cos, sin, which has slope an. The veloci slope is he negaive reciprocal of he posiion an acceleraion slopes, so veloci is orhogonal o posiion an o acceleraion. 6. (a) Since also solves ( ), he paricles collie when. Firs paricle: v ( ),, so he irecion uni vecor is,. Secon paricle: v (),, so he irecion uni vecor is,. 6. (a) Referring o he figure, look a he circular arc from he poin where o he poin m. On he one han, his arc has lengh given b r, bu i also has lengh given b v. Seing hese wo quaniies equal gives he resul. () sin v v v v, vcos an r r v v v v a() cos, sin r r r r v v v cos, sin r r r Coprigh 6 Pearson Eucaion, Inc.

20 6 Secion. v (c) From par, a() r(). So, b Newon s secon law, r he law of graviaion gives he resul. v F m r. Subsiuing for F in r () Se vt r an solve for vt. (e) Subsiue r T r GM T r r GM T r GM T r T r GM for v in v GM an solve for T. r 6. Use he hin. TRACE unil Y is approimael 8. A closer view: The screens confirm ha he ball is a a heigh of 8 fee afer abou.869 secons when i is abou fee from home plae. 6. Le u a, b be one of he vecors. I has slope b a, so he perpenicular vecor v mus have slope a Thus v kb, ka for some nonzero scalar k, an he o prouc is ( ) u v a, b, kb, ka kab + kab. 6. (a) The iagram shows, b vecor aiion, ha v + w u, so w u v. This is jus he Law of Cosines applie o he riangle, he sies of which are he magniues of he vecors.. b Coprigh 6 Pearson Eucaion, Inc.

21 Secion. 6 (c) B he HMT Rule, w u v, u v.so u + v w ( u + u) + ( v + v) ( u v) + ( u v) u + u + v + v u uv+ v + u uv + v uv + uv ( uv + uv) () From par, w u + v u v cos, so u + v w u v cos. u + v w uv + uv. Subsiuing, we ge ( uv + uv ) uv cos, From par (c), ( ) so u v uv + uv u v cos. Secion. Polar Funcions (pp. 6 7) Quick Review.. cos sin,... A r () A r () Graph / an / ( sin ) ( cos ) cos sin co 76 co( ). Coprigh 6 Pearson Eucaion, Inc.

22 6 Secion. 8. co, so or cos cos sin sin (, ) an (, ). D C B A 9. sin, s or. cos cos sin sin (, ) an (, ). ( cos ) + ( sin ). 76 Secion. Eercises (a) cos 6 sin 6,. D C (a) cos sin (, ) B A cos() sin () (, ). cos an sin an (, ) (c) cos7 sin 7 (, ) () cos sin, ( ) (c) cos sin (, ) A D C B () cos sin (, ) (a) r ( ) + an,, an, Coprigh 6 Pearson Eucaion, Inc.

23 Secion. 6 r + ( ) ± (c) an,, an, r + ± an,, an, () r + ± an, (, ) an (, ).. () ( ) r + ± an, (, ) an (, ) 6. B A C D 7. r + ( ) ± (a) ( ) 7 an, 6 6 7, an, r + ± an, + an an, an (c) r +( ) ± an,,, an, Coprigh 6 Pearson Eucaion, Inc.

24 6 Secion. 9.. Limaçon 6 6 [.,.] b [, ] q p 6. Limaçon. [, ] b [, ] q p 7. Lemniscae. Carioi [, ] b [, ] p / q p / [, ] b [, ] q p 8. Lemniscae. Carioi [.,.] b [, ] q p / [ 6, 6] b [, ] q p 9. Circle. Rose [ 6, 6] b [, ] q p [, ] b [, ] q p. Circle. Rose [.,.] b [, ] q p [.,.] b [, ] q p Coprigh 6 Pearson Eucaion, Inc.

25 Secion. 6. r csc r sin, a horizonal line. r sec r cos avericalline,. I is a parabola. [ 6, 6] b [, ] q p. I is a parabola.. + aline, (slope, -inercep ).. 6. r +, a circle (cener (, ), raius ) r sin cos rsin rcos, a line (slope, -inercep ) r sin r ( sin cos ), a hperbola [ 6, 6] b [, ] q p. I is a parabola. [ 6, 6] b [, ] q p. I is a parabola. 7. rcos rsin, he union of wo lines: ± r rcos ( + ) +, a circle ( cener (, ), raius ) r 8 rsin + 8rsin ( ) 6, a circle ( cener (, ), raius ) [ 6, 6] b [, ] q p. I is a hperbola. [ 6, 6] b [, ] q p 6. I is an ellipse.. r cos + sin r rcos + r sin + + ( ) + ( ), acircle ( cener (, ), raius ) [ 6, 6] b [, ] q p Coprigh 6 Pearson Eucaion, Inc.

26 66 Secion. 7. I is an ellipse. [ 6, 6] b [, ] q p 8. I is a hperbola. [ 6, 6] b [, ] q p 9. r + sin (( + sin )sin ) (( + sin ) cos ) ( sin )cos cos sin sin A : A :. r cos (cos sin ) (cos cos ) cos cos sin sin sin cos cos sin A : unefine A : A : A : unefine. r sin ( sin )sin ( sin )cos ( 6sin )cos sin ( sin ) cos A (, ): A, : A (, ): A, :. r ( cos ) ( ( cos )sin ) ( ( cos ) cos ) 6cos + cos + sin ( cos ) A., : unefine A., : A (, 6 ): unefine A, :.. ( + cos ) 6 6 ( 8 8cos cos ) ( + cos + cos ) sin + sin 8 ( + sin ) ( 8 sin sin ) ( sin cos ) cos sin Coprigh 6 Pearson Eucaion, Inc.

27 (cos ) / / ( sin ) / ( cos ) / / 6 ( sin ) / cos ( ) / cos + / sin sin ( ) ( cos 8) sin 8 8 sin / / / cos ( ) ( cos ) ( 9 cos + cos ) 9 6 cos + + cos 6sin + sin /6 /6 ( sin ) /6 sin sin /6 + /6 cos sin /6 + /6 sin + cos / Secion. 67 / / ( sin ) sin / ( cos ) / [ sin ] + /6 / ( sin ) + ( ) /6 /6 / sin /6 /6 ( cos ) /6 + [ sin ] + / [ ( cos )] + ( ) / / ( cos + cos ) + / / ( 8cos + ( + cos )) + / [ 6 8sin + sin ] + 8 / ( ( cos )) / ( cos cos ) / [ 6sin / sin ] + ( 8 6cos + ( + cos )) The requese area is insie of he upper semicircle an ousie of he porion of he carioi ha is in Quarans I an II. ( ( sin )) ( sin + sin ) ( sin + cos ) + cos sin 8 Coprigh 6 Pearson Eucaion, Inc.

28 68 Secion (( cos ) ) /6 /6 /6 ( 6cos ) /6 (( 8 cos ) ) /6 ( 6 cos ) /6 [ 6 8sin ] [, ] b [, ] q p for he circle q p for he carioi / + / ( 8cos cos ) / + / [ + sin sin ] (( cos ) ( cos ) ) ( ( cos ) cos ) + 9. [, ] b [, ] q p ( sin ) sin 6. (a) ( cos 6) sin 6 6 ( sin sin ) Slope ( sin cos ) / 6sincos + cossin 6cos cos sin sin / + ln / / [ 6, 6] b [, ] q p ( ( ( sin )) ) ( + sin sin ) ( sin + cos ) cos + sin ( ) 8 (c) rcos rsin + r (cos sin ) r cos Le α an. Then he area is α. cos sin sin 6. True; polar coorinaes eermine a unique poin. Coprigh 6 Pearson Eucaion, Inc.

29 Secion False; inegraing from o raverses he curve wice, giving wice he area. The correc upper limi of inegraion is. 6. D 6. E 6. B 66. D 67. (a) [ 9, 9] b [ 6, 6] 68. (a) [ 9, 9] b [ 6, 6] (c) The graph of r is he graph of r roae counerclockwise abou he origin b angle α. 69. (a) [ 9, 9] b [ 6, 6] The graphs are ellipses. As e +, he graph approaches he circle of raius cenere a he origin. 7. (a) [ 6, ] b [, ] The graphs are ellipses ha srech ou o he righ. The ellipse wih eccenrici e is he reflecion across he -ais of he ellipse wih eccenrici e. [, ] b [, ] The graphs are hperbolas. + As e, he righ branch of he hperbola goes o infini an isappears. The lef branch approaches he parabola. Coprigh 6 Pearson Eucaion, Inc.

30 6 Secion. 7. (a) [ 9, 9] b [ 6, 6] The graphs are parabolas. + As c, he limi of he graph is he negaive -ais. 7. / / / [( ) + ( ) ] [( r cos r cos ) + ( r sin r sin ) ] [ r cos + r cos + r sin + r sin + rr cos cos + rr sin sin ] [ r + r rr cos( )] 7. (a) (c) a a( cos ) [ sin ] a a a [ ] a a a acos [ sin] ( ) ( f ( )cos f( )sin ) + ( f ( )sin + f( )cos ) ( f ( )cos ) + ( f( )sin ) + ( f ( )sin ) + ( f( )cos ) ( f( )) (cos + sin ) + ( f ( )) (cos + sin ) ( f( )) + ( f ( )) r r + ( + cos ) + ( sin ) cos + cos + cos 8sin (a) [, ] b [, ] You see boh branches of he hperbola as well as he lines ha appear o be he asmpoes. Coprigh 6 Pearson Eucaion, Inc.

31 Secion. 6 The branch on he lef correspons o cos < < cos... The branch on he righ correspons o cos < < cos... (c) Wha appear o be asmpoes are acuall lines pu in b he graphing calculaor o connec he las poin i fins on one branch of he hperbola o he firs poin i fins on he oher branch. 77. c r + ecos r+ recos c r c e( rcos ) Use rcos an r +. r + ( c e) c ce+ e + c ce+ e ( e ) + ce+ c ce Complee he square on b aing o boh sies. e ce c e c e ( e ) c + e ( e ) e ce c ( e ) + + e e ( e ) ce e + + c e c c c Le a an b. e e The las equaion becomes ( + ae) ( + ae) + or +. This is an ellipse wih cener a b a b ( ae, ). 78. The isance from he cener o one focus is a c c b ( e ) e ce ( e ) ce e This is ae, which is he isance from he cener of he ellipse in Eercise 77 o he origin. 79. The area swep ou from ime o ime is given b A K, we have r K K or. r () A r. Thus A r. Since Coprigh 6 Pearson Eucaion, Inc.

32 6 Secion. 8. From Eercise 79, r K, an since c K r, his gives. ( + ecos ) ( + ecos ) c csin ( + ecos ) c cos ( esin ) ( + ecos ) csin ( + ecos ) K csin c K sin c K K K cos cos c c r K cos cr c cos ( + ecos ) csin ( esin ) ( + ecos ) ce + c cos ( + ecos ) K ( ce + c cos ) c K ( e + cos ) c K K sin sin c cr K K K Thus he acceleraion vecor is cos, sin cos, sin which has magniue cr cr cr K cr an i poins owar he origin from he poin (, ) ( rcos, rsin ). Quick Quiz Secions... A ( ). C; ( ),. D Area + sin. 8. (a) ( ) r cos ( + sin ) cos 786. Coprigh 6 Pearson Eucaion, Inc.

33 Chaper Review 6 (c) The graph is geing closer o he origin as increases from o. () Maimize r + sin for. r + cos + cos cos r Since cos for an + > < < r cos for + < < <, here is a maimum of r when b he Firs Derivaive es. The curve is farhes from he origin when. Chaper Review Eercises (pp. 7 76). (a),, 7, ( 7) + ( ). (a), +,, sin 6 cos 6, r ( ) + ( ) 7 9. (a) 8,, ,, + ( sec ) ( an ) ( sin ) sec sin ( ) + ( ). (a), 6, ( 8). (a),,. + ( ) 7 9 sin cos,. (a) + ( ) ( + ) ( ) 6 Coprigh 6 Pearson Eucaion, Inc.

34 6 Chaper Review. (a) sec sec an Horizonal: an Also, an sec ± ± The poin are, an,. an sec Verical: sec ( impossible) There are no poins where he angens are verical. cos cos an sin an sin cos an cos (, ) an(, ). (a) ( sin ) sin sin an cos an cos sin an sin ( an, ) (, ) ( cos) 9 9cos 9cos an cos cos 9sin 9 9sin 9, 9 an, 9. (a) ( sin ) ( ) ( ) sin sin or cos, cos 9sin, 9sin (, ) an (, ) ( cos). Carioi [, ] b [, ] q p 6. Conve limaçon [.,.] b [, ] q p 7. -peale rose. (a) (cos ) sin cos cos sin cos sin a (, ) is never zero. There are no cos verical angens. [.,.] b [, ] q p 8. Verical line [, ] b [, ] p / q p / Coprigh 6 Pearson Eucaion, Inc.

35 Chaper Review 6 9. Lemniscae. Circle [.,.] b [, ] q p /. ( ( sin )sin ) ( sin )cos, ( ( sin )cos ) sin (sin ) cos, ± 98.. [.,.] b [, ] q p ( cos ) ( cos ) sin cos cos cos sin sin sin cos cos sin.. The ips of he peals are a he poins where r sin, which are he poins where,,, an 7. The slope is ( )(sin sin ) ( )(sin cos ) cos sin + sin cos. cos cos + sin sin. ( + cos ) ( + cos ) sin cos cos cos sin sin + cos sin cos cos sin sin. 6. cos sin ( ) sin sin cos cos + + cos, ±., ±. 79 cos cos cos sin ( ) sin cos sin +,. 67,. / (, ) ( ) (, ) (, ) 7 ( ) m Tangen line, + +, 6. The carioi crosses he -ais a he poins where an. The slope is ( )(( + sin )sin ) ( )(( + sin ) cos ) cossin + ( + sin ) cos. coscos + ( + sin ) sin Coprigh 6 Pearson Eucaion, Inc.

36 66 Chaper Review (, ) m Tangen line (, ) (, ). + 6 r cos + r sin 6, or 6 r cos + sin 7., a line r rcos +, a circle cener,, raius r sin r r cos cos cos rsin, a parabola. r cos cos sin sin cos sin or, a line r + r sin r sin + r r sin r sin.. ( + ) + ( ) 6 ( rcos + ) + ( rsin ) 6 ( cos ) (sin cos ( sin 9)) sin (sin cos ) 6 6. ( ) 7. / / ( + cos ) ( ) / + cos cos / + sin + sin + 6 sin + + sin + / Coprigh 6 Pearson Eucaion, Inc.

37 Chaper Review ( ( + sin )) ( sin ) ( + sin + sin ) sin + + ( sin cos ) ( cos ) cos sin sin (a) v () cos, sin v() sin, cos a() ( sin ), cos a() cos, sin cos, sin, ( ) ( ) Spee +. (a) v () sec, an v ( ) sec an, sec a( ) sec an, sec a ( ) (sec an + sec ), sec an sec( ) an( ), sec ( ), Spee + Coprigh 6 Pearson Eucaion, Inc.

38 68 Chaper Review. v (), + + v(), ( + ) ( + ) / / spee + / / ( ) ( ) + + spee + The maimum value of is, when +.. r( ) e cos, e sin, wih slope.. e sin an. e cos v( ) e cos e sin, e sin + e cos a( ) e sin, e cos, wih slope e cos. e sin Since he slopes are an negaive reciprocals, he angle is alwas 9. r() sin, cos r() cos,sin cos( ), sin( ), r() cos,sin + + C + C + C C + C C r (), + ( + ) + C + + C + C C + + C, C r() an, an, r() an +, +. v(), v(), + C () + C C So v(), r(), (), C C C r + + r(), 6. v (), 7. (a) v() +, r() ( + 6), ( + ) C C () C, C () C, C + 6+ C + + C 6 C C C C r(), () + () +, () + () +, r() + 6, + + ( ) sin cos cos ( ) sin sin ( ) cos ( ) + 7 Coprigh 6 Pearson Eucaion, Inc.

39 sin ( ) (c) 8. (a) cos 6 6 cos sin 6 6 ( ) ( ) ( ) cos an sin + 9 e sin + e cos e cos e sin cos+ sin cos sin (c) 9. (a) Disance v () e [ e ] e ( ) 6 () v,,, 96 v ( ) 8,, an 96 v ( ) 8 + Chaper Review 69 6 ( ) Disance ( + ) (c) +, so 6 +. e (sin + cos ), e (cos sin ) e (sin + sin cos + cos ) e ( + sin cos ) e (cos cos sin + sin ) e ( cos sin ) e e v() e v() Coprigh 6 Pearson Eucaion, Inc.

40 66 Chaper Review. egrees eas of norh is (9 ) egrees norh of eas. A he vecors: cos, sin + cos( ), sin( ) 9 cos, 8 sin 8. 96, Spee mph Direcion an 8. 8 eas of norh (a). (a) a v + + C () C + cos( + ) Posiion ( +,cos( + )) A his poin +, so. Spee ( ) + ( ( sin( + )). va(), anv B(), + ( ) 6. 6 (c) Seing A B, we fin ha. Plugging ino A an B, we fin ha boh values are he same (). Thus, he paricles collie when. (Noe: If ou graph boh pahs, he will cross a (, ). However, he paricles are here a ifferen imes.). (a) Area + sin The polar equaion is equivalen o r+ rsin. Thus, r rsin r ( rsin ) + ( ) (c) Area, 8 which, inee, is. Coprigh 6 Pearson Eucaion, Inc.