CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES

CHAPTER PARAMETRIC EQUATIONS AND POLAR COORDINATES. PARAMETRIZATIONS OF PLANE CURVES., 9, _ _ Ê.,, Ê or, Ÿ. 5, 7, _ _.,, Ÿ Ÿ Ê Ê 5 Ê ( 5) Ê ˆ Ê 6 Ê ( 5) 7 Ê Ê, Ÿ Ÿ $ 5. cos, sin, Ÿ Ÿ 6. cos ( ), sin ( ), Ÿ Ÿ Ê cos sin Ê Ê cos ( ) sin ( ) Ê, Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

68 Chper Prmeric Equions nd Polr Coordines 7. cos, sin, Ÿ Ÿ 8. sin, 5 cos, Ÿ Ÿ 6 cos sin 6 sin 5 cos 6 6 6 5 6 5 Ê Ê Ê Ê 9. sin, cos, Ÿ Ÿ. sin, cos, Ÿ Ÿ Ê cos sin Ê Ê sin cos Ê 6.,, _ _.,, Ê Ê Ê Ê.,, Ÿ Ÿ.,, Ê Ê Ê, Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

Secion. Prmerizions of Plne Curves 69 5. sec, n, 6. sec, n, Ê sec n Ê Ê sec n Ê 7. cosh, sinh, _ _ 8. sinh, cosh, _ _ Ê cosh sinh Ê Ê cosh sinh Ê 5 9. () cos, sin, Ÿ Ÿ. () sin, cos, Ÿ Ÿ () cos, sin, Ÿ Ÿ () cos, sin, Ÿ Ÿ 9 (c) cos, sin, Ÿ Ÿ (c) sin, cos, Ÿ Ÿ (d) cos, sin, Ÿ Ÿ (d) cos, sin, Ÿ Ÿ. Using ß $ we cree he prmeric equions nd $, represening line which goes hrough ß $. We deermine nd so h he line goes hrough %ß when. Since % Ê &. Since $ Ê %. Therefore, one possile prmeerizion is &, $ %, Ÿ Ÿ.. Using ß$ we cree he prmeric equions nd $, represening line which goes hrough ß $. We deermine nd so h he line goes hrough $ß when. Since $ Ê %. Since $ Ê &. Therefore, one possile prmeerizion is %, $ &, Ÿ Ÿ.. The lower hlf of he prol is given for Ÿ. Susiuing for, we oin one possile prmeerizion,, Ÿ Þ. The vere of he prol is ß, so he lef hlf of he prol is given for Ÿ. Susiuing for, we oin one possile prmerizion:,, Ÿ. 5. For simplici, we ssume h nd re liner funcions of nd h he poin, srs ß $ for nd psses hrough ß. Then f, where f nd f.? Since slope? $, f $ $. Also, g, where g $ nd g.? Since slope?. g % $$ %. One possile prmeerizion is: $, $ %,. Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

65 Chper Prmeric Equions nd Polr Coordines 6. For simplici, we ssume h nd re liner funcions of nd h he poin, srs ß for nd psses hrough ß. Then f, where f nd f.???? Since slope, f. Also, g, where g nd g. Since slope. g. One possile prmeerizion is:,,. 7. Since we onl wn he op hlf of circle,, so le cos, lsin l, Ÿ Ÿ 8. Since we wn o s eween nd, le sin, hen sin 9sin, hus sin, 9sin, Ÿ _ 9. Ê Ê ; le Ê Ê. Susiuion ields Ê nd, _ _. In erms of ), prmeric equions for he circle re cos ), sin ), Ÿ ). Since ) s, he rc s s s lengh prmerizions re: cos, sin, nd Ÿ Ê Ÿ s Ÿ is he inervl for s.. Drop vericl line from he poin, o he -is, hen ) is n ngle in righ ringle, nd from rigonomer we know h n ) Ê n ). The equion of he line hrough, nd, is given. Thus n) n) n) n ) Ê nd where Ÿ).. Drop vericl line from he poin, o he -is, hen ) is n ngle in righ ringle, nd from rigonomer we know h n ) Ê n ). Since Ê Ê n ) Ê co ) Ê co ) where Ÿ. ). The equion of he circle is given. Drop vericl line from he poin, on he circle o he -is, hen ) is n ngle in righ ringle. So h we cn sr, nd roe in clockwise direcion, le cos ), sin ), Ÿ ) Ÿ.. Drop vericl line from he poin, o he -is, hen ) is n ngle in righ ringle, whose heigh is nd whose se is. B rigonomer we hve n ) Ê n ). The equion of he circle is given Ê n ) Ê sec ) n ) n ). Solving for we oin É sec) sec) n ) n ) sec ) n ) n ) n) sin ) cos ) cos) sin) cos ) cos ) cos) nd Š cos ) cos ) cos) n ) sin ) cos ) sin ) cos ). Since we onl need o go from, o,, le cos cos cos, sin cos sin cos, Ÿ Ÿn ) ) ) ) ) ) ) ) ˆ. To oin he upper limi for ), noe h nd, using n ) Ê n ) Ê ) n ˆ. 5. Eend he vericl line hrough A o he -is nd le C e he poin of inersecion. Then OC AQ nd n Ê co ; sin Ê OA ; nd (AB)(OA) (AQ) Ê AB ˆ OC n OA sin sin Ê AB ˆ ˆ sin Ê AB. Ne AB sin Ê ˆ sin sin n n n sin sin cos sin. Therefore le co nd sin,. n Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

6. Arc PF Arc AF since ech is he disnce rolled nd Arc PF Arc AF nfcp Ê Arc PF ( nfcp); ) Ê Arc AF ) Ê ) ( nfcp) Ê nfcp ); nocg ) ; nocg nocp npce nocp ˆ. Now nocp nfcp ). Thus nocg ) Ê ) ) Ê ) ) ˆ ). Secion. Prmerizions of Plne Curves 65 Then OG BG OG PE ( ) cos ) cos ( ) cos ) cos ˆ ) ( ) cos ) cos ˆ ). Also EG CG CE ( ) sin ) sin ( ) sin ) sin ˆ ) ( ) sin ) sin ˆ ). Therefore ( ) cos ) cos ˆ ) nd ( ) sin ) sin ˆ ). If, hen ˆ cos ) cos Š ) ˆ ˆ cos ) cos ) cos ) (cos ) cos ) sin ) sin )) cos ) (cos )) cos ) sin ) (sin ))( sin ) cos )) $ cos ) cos ) cos ) sin ) sin ) cos ) cos ) cos $ ) (cos )) cos ) cos $ ); ˆ ˆ Š ˆ sin ) (sin )) cos ) sin ) (cos ))( sin ) cos )) $ sin ) sin ) cos ) sin ) cos ) sin ) $ sin ) sin ) cos ) sin ) $ $ sin ) (sin )) sin ) sin ) sin ). sin ) sin ) sin ) sin ) sin ) (sin ) cos ) cos ) sin )) 7. Drw line AM in he figure nd noe h namo is righ ngle since i is n inscried ngle which spns he dimeer of circle. Then AN MN AM. Now, OA, AN AM n, nd sin. Ne MN OP Ê OP AN AM n sin Ê OP n sin sin cos $ sin cos ( sin ) sec. In ringle BPO, OP sin sin n nd OP cos sin Ê sin n nd sin. Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

65 Chper Prmeric Equions nd Polr Coordines 8. Le he -is e he line he wheel rolls long wih he -is hrough low poin of he rochoid (see he ccompning figure). w w Le ) denoe he ngle hrough which he wheel urns. Then h ) nd k. Ne inroduce -es prllel o he -es nd hving heir origin he cener C of he wheel. Then w cos nd w w w sin, where ). I follows h cos ˆ ) sin ) nd sin ˆ ) w w cos ) Ê h ) sin ) nd k cos ) re prmeric equions of he rochoid. % 7 9. D É( ) ˆ Ê D ( ) ˆ ( ) ˆ Ê D dd $ Ê Ê. The second derivive is lws posiive for Á Ê gives locl Ê minimum for D (nd hence D) which is n solue minimum since i is he onl eremum poin on he prol is (ß ). he closes dd ˆ ˆ 5 d D d D Ê Ê d D d D 9 mimum, ( ) 9 Ê relive mimum, ˆ Ê relive minimum, nd d D ˆ 5 9 5 Ê he poin ˆ ß Ê Š ß nd Š ß re he desired poins.. D Ɉ cos (sin ) Ê D ˆ cos sin Ê cos ( sin ) sin cos ( sin ) cos Ê sin or cos, or,. Now 6 cos cos 6 sin so h () relive relive minimum. Therefore oh nd give poins on he ellipse closes o. () () (c). () () (c) Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

Secion. Prmerizions of Plne Curves 65.. () () (c) 5. () () 6. () () Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

65 Chper Prmeric Equions nd Polr Coordines 7. () () (c) 8. () () (c) (d). CALCULUS WITH PARAMETRIC CURVES. Ê cos, sin / cos ; sin, cos Ê co / sin w Ê ¹ co ; ngen line is Š or ; csc Ê / csc Ê ¹ $ / sin sin 6 6 6. Ê sin ˆ ˆ sin ˆ, cos ˆ ˆ cos ˆ ; cos, sin sin Ê n Ê ¹ nˆ ˆ nˆ ; cos c 6 6 ngen line is Š or ; w sec Ê d sec cos cos$ ¹ c 6 Ê 8 Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

Secion. Clculus Wih Prmeric Curves 655. Ê sin, cos / ; cos, sin Ê sin / cos n Ê ¹ n ; ngen line is Š or ; w w d / sec d sec $ Ê / cos 8 cos Ê ¹ sin ¹ Š ˆ w d sin. Ê cos, cos ; sin, sin sin Ê Ê ; ngen line is or ; Ê Ê ¹ / / É 5. Ê, ;, Ê Ê ¹ ; ngen line is $Î d w / $Î d w / ˆ or ; Ê Ê ¹ 6. Ê sec ˆ, n ˆ ; sec n, sec sec sec n n co ¹ co ˆ ; ngen line is c d csc $ w sec n Ê Ê ( ) ( ) or ; csc Ê co Ê ¹ c / 6 6 6 / 7. Ê sec, n ; sec n, sec Ê sec csc Ê ¹ csc ; ngen line is Š or ; sec n 6 6 w w d / csc co $ d csc co Ê / sec n co Ê ¹ 6 ˆ () ˆ ( ) 8. Ê, Î Î () ; ( ), () Ê () ¹ ; ngen line is [ ( )] or ; w cî ( ) cî () d Š Ê c Š Ê ¹ cî cî $ / / c w w d / d / ¹ c $ 9. Ê 5, ;, Ê Ê ¹ ( ) ; ngen line is ( 5) or ; Ê Ê ˆ Š w Š. Ê, ;, Ê Ê ¹ ; ngen line is ( ) ( ) or ; Ê Ê ¹ / / sin sin ˆ Š cos ¹ ; ngen line is cosˆ ˆ Š. Ê sin, cos ; cos, sin Ê Ê Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

656 Chper Prmeric Equions nd Polr Coordines ˆ c ( cos )(cos ) (sin )(sin ) d / c cos ( cos ) cos / cos w w Ê ; Ê ( cos ) Ê ¹ cos sin. Ê cos, sin ; sin, cos Ê co d csc $ d sin Ê ¹ co ; ngen line is ; csc Ê csc Ê ¹ w w Ê Ê ¹. Ê, ;, Ê Ê ¹ 9; ngen line is 9 ; 8 e e e e w e e e e e ¹ e 8. Ê e, e ; e, e Ê Ê ¹ ; ngen line is ; Ê Ê 5. 9 Ê Ê Ê ; $ 6 / Š ( ) 6 / c Š ( ) $ Ê 6 6 Ê ; hus ; Ê () 9 Ê 8 9 Ê Ê ; Ê () $ $ 6 Ê 6 Ê 8 Ê ; herefore ¹ 6. É5 Ê ˆ 5 ˆ Î ; ( ) Ê ( ) Î Î É5 c c Ê Ê ; hus c É5c ˆ É& Š & 9 ; Ê É5 ; Ê Ê herefore, ¹ É5 $Î Î 7. Ê Ê ˆ Î Ê ; Ê ˆ Î ( ) ˆ Î Ê Š c Š Š ; hus Š ( ) cc Œ / ( ) $Î / ; ˆ Î ; Š Î Ê Ê Ê Ê Ê cc() Œ Ê ( ) () () Ê ; herefore ¹ 6 () Œ () Î Î cos sin sin cos ˆ c cos sin 8. sin Ê sin cos Ê (sin ) cos Ê ; sin Ê sin cos ; hus ; Ê sin sin cos 8 c Š cos sin Ê ; herefore ¹ Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

Secion. Clculus Wih Prmeric Curves 657 ¹ 9., Ê, 6 Ê 6 Ê Ê. ln, e Ê ˆ Ê Ê, e e ; e e e e ; ln ¹ Ê Ê Ê Ê ˆ cos ˆ. A cos cos cos cos cos cos cos cos sin sin. A e u Ê du ; dv e Ê v e e º e u Ê du ; dv e Ê v e e º e º e e e e º e e e e e e e cos. A sin sin sin cos sin 6 6 7 8 9 9. (),, Ÿ Ÿ Ê A (),, Ÿ Ÿ Ê A e 5. sin nd cos Ê Êˆ Š É sin cos cos Ê Lengh cos cos sin Ɉ ( cos ) É cos cos sin (since sin on [ ß ]); [u cos Ê du sin ; Ê u, cos Î Î Ê u ] Ä u du u 6. nd Ê Êˆ Š É () 9% 9 Š since on ß Ê Lengh ; u Ê du ; Ê u, Ê u Ä u du u (8 ) 7 Î $Î % 7. nd ( ) Ê Êˆ Š É k k since Ÿ Ÿ Î Ê Lengh 8 % Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

658 Chper Prmeric Equions nd Polr Coordines Î 8. nd Ê Êˆ Š É k k since Ÿ Ÿ Ê Lengh ( ) 9. 8 cos nd 8 sin Ê Êˆ Š É8 cos 8 sin 6 cos 6 sin Î Î k8k 8 since Ÿ Ÿ Ê Lengh 8 c d. sec n sec cos sec cos nd ˆ sin Ê Êˆ Š sec n Ésec cos sin sec n kn k n since Ÿ Ÿ Î Î sin Î$ cos c k kd Ê Lengh n ln cos ln ln ln. sin nd cos Ê Êˆ Š É sin cos Ê Are ds sin c cos d [ ] 8 Î Î. nd Ê Êˆ Š É Ê Are ds ˆ É $Î ; cu Ê du ; Ê u, $Î % 8 9 9 $Î Ê u Ä u du u Noe: ˆ É is n improper inegrl u lim f eiss nd is equl o, where Ä f ˆ É $Î. Thus he disconinui is removle: define F f for nd F Ê 8 9 F.. nd Ê Êˆ Š Ê Š É Ê Are ds c Š É ; u Ê du Š ; Ê u, Ê u 9 Ä u du u $Î * 7 9 5 Î Î. From Eercise, ʈ Š n Ê Are ds cos n sin Î$ c cos d ( ) 5. nd Ê Êˆ Š 5 Ê Are ds 5 5 5. Check: sln heigh is 5 Ê Are is 5 5. Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

Secion. Clculus Wih Prmeric Curves 659 6. h nd r Ê Êˆ Š h r Ê Are ds rh r r h r r h r r h r. Check: sln heigh is h r Ê Are is r h r. 7. Le he densi e $. Then cos sin Ê cos, nd sin cos Ê sin Ê dm ds ʈ Š ( cos ) ( sin ) kk since Ÿ Ÿ. The curves mss is Î Î Î Î M dm. Also M µ dm sin cos sin cos 8 Î Î csin cos d c sin sin cos d, where we inegred prs. Therefore, M Š. Ne, M µ dm cos sin cos sin M Î Î Î Š 8 Î Î M ˆ M Š 8 ccos sin d c cos cos sin d, gin inegring prs. Hence. Therefore ß ˆ ß. 8. Le he densi e $. Then e cos Ê e cos e sin, nd e sin Ê e sin e cos Ê dm ds ʈ Š Ée cos e sin e sin e cos e e. The curves mss is M dm e e. Also M µ dm e sin Š e Š e e e 5 5 e 5 5 5 M e 5e µ e e 5 5 5 M Š e 5 5 e e e M 5 e. Therefore 5 e 5 e. e Š M e sin ( sin cos ) Š Ê. Ne M dm e cos Š e e cos cos sin Š Ê ß ß 9. Le he densi e $. Then cos Ê sin, nd sin Ê cos Ê dm ds ʈ Š É sin cos cos. The curves mss is M dm cos cos É cos ˆ cos ˆ cos ˆ ˆ since Ÿ Ÿ Ê Ÿ Ÿ sin ˆ. Also M µ dm sin ˆ cos cos ˆ sin cos ˆ ˆ ˆ ˆ ˆ 6 M ˆ 6 M µ sin ˆ ˆ ˆ ˆ M ˆ M. Therefore ˆ. cos sin cos cos Ê. Ne M dm cos cos cos cos sin Ê ß ß $. Le he densi e $. Then Ê, nd Ê Ê dm ds ʈ Š É () kk since Ÿ Ÿ. The curves mss is M dm $Î 7. Also M µ dm Š 9 $ 87 M 7. 7. ( compuer) Ê.9. Ne M µ dm 5 M 7 Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

66 Chper Prmeric Equions nd Polr Coordines $ % M 6.89 6.89 ( compuer) Ê.5. Therefore, ß.5ß.9. M 7. () sin nd cos Ê Êˆ Š É sin cos Î Î Ê Lengh cd () cos nd sin Ê Êˆ Š Écos sin Î Î Î Ê Lengh c d Î w d d d c c c Î Î Î Î 9 8 Î 8 Î 56 7 7 7 Î Î Î Î É É É Î lim Ä Î Î Î Î Î Î Î Î lim ˆ ˆ Î lim lim Ä ˆ Ä Š ˆ Ä. () g hs he prmerizion g nd for c Ÿ Ÿ d Ê g nd ; hen Lengh ÊŠ Š Ê Š [gw ] Î Î (), Ÿ Ÿ Ê Ê L É ˆ Î 9 9 É ˆ (c), Ÿ Ÿ Ê Ê L d) d) cos ) sin ) cos ) sin ) cos ) sin ) cos ) sin ) cos ) cos ) sin ) sin ) cos ) sin ) sin ) cos ) sin ). sin ) cos ), sin ) sin ) Ê cos ) sin ) sin ), cos ) sin ) cos ) sin ) Ê sin cos ) cos sin sinˆ ˆ cosˆ ) / cosˆ ˆ sinˆ ˆ ˆ sinˆ ˆ cosˆ º cosˆ ˆ sinˆ ) / () sincos, sin sin ; º () ˆ sinˆ cosˆ, ˆ sinˆ sinˆ ; º (c) ˆ sin cos ˆ, ˆ sin sin ˆ ; Š sin d d cos., cos, Ÿ Ÿ Ê, sin Ê sin Ê Š cos Ê cos. The mimum nd minimum slope will occur poins h mimize/minimize, in oher words, poins where Ê cos Ê or Ê ± ± Î Î () he mimum slope is º sinˆ, which occurs, cosˆ Î () he minimum slope is º sinˆ, which occurs, cosˆ Î d / cos cos cos / cos cos cos 5 7 sin ˆ Ê Š ß is he poin where he ngen line is horizonl. A he origin: nd 5. cos nd cos Ê ; hen Ê Ê cos Ê cos Ê,,,. In he s qudrn: Ê sin nd Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

Secion. Clculus Wih Prmeric Curves 66 Ê sin Ê or nd sin Ê,,, ; hus nd give he ngen lines he origin. Tngens origin: ¹ Ê nd ¹ Ê / cos (cos cos sin sin ) / cos cos 6. cos nd cos Ê c cos (cos ) sin cos sin d ( cos ) cos sin ( cos ) cos cos cos cos ; hen ( cos ) cos Ê cos Ê cos or cos : cos Ê, nd 5 7 cos Ê cos Ê 6, 6, 6, 6. In he s qudrn: 6 Ê sin ˆ 6 nd sin ˆ Ê Š 6 ß is he poin where he grph hs horizonl ngen. A he origin: Ê Ê 5 Ê nd sin nd sin,,, nd,,,,, nd give cos cos he ngen lines he origin. Tngens he origin: ¹ Ê, nd ¹ cos () cos () Ê É cos sin cos cos sin É ˆ ˆ ˆ 7. () sin, cos, Ÿ Ÿ Ê cos, sin Ê Lengh cos sin sin cos cos cos 8 cos É cos sin cos cos cos sin Î Î ˆ ˆ ˆ ˆ Î ˆ () Ê sin, cos, Ÿ Ÿ Ê cos, sin Ê Surfce re cos cos cos cos sin 8 sin u Ê du Ê du; Ê u, Ê u 6 sin u du 6 sin u sinudu 6 cos u sinudu 6 sinudu 6 cos usinudu 6 6 6 6 6 cos u cos u ˆ 6 ˆ 6 8. sin, cos, Ÿ Ÿ ; Volume cos cos cos cos cos cos ˆ cos ˆ cos cos 5 5 5 sin sin sin 5 5 ˆ cos cos sin cos ˆ cos cos sin cos 7-5. Emple CAS commnds: Mple: wih( plos ); wih( suden ); := -> ^/; := -> ^/; := ; := ; N := [,, 8 ]; for n in N do Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

66 Chper Prmeric Equions nd Polr Coordines := [seq( +i*(-)/n, i=..n )]; ps := [seq([(),()],=)]; L := simplif(dd( suden[disnce](ps[i+],ps[i]), i=..n )); T := sprinf(7() (Secion.)\nn=%d L=%8.5f\n, n, L ); P[n] := plo( [[(),(),=..],ps], ile=t ): end do: displ( [seq(p[n],n=n)], insequence=rue ); ds := ->sqr( simplif(d()()^ + D()()^) ): L := In( ds(), =.. ): L = evlf(l); () () (c). POLAR COORDINATES., e;, g; c, h; d, f., f;, h; c, g; d, e. () ˆ n nd ˆ ß ß (n ), n n ineger () (ß n ) nd ( ß (n ) ), n n ineger (c) ˆ n nd ˆ ß ß (n ), n n ineger (d) (ß (n ) ) nd ( ß n ), n n ineger. () ˆ n nd ˆ 5 ß n ß, n n ineger () ˆ n nd ˆ 5 ß n ß, n n ineger (c) ˆ n nd ˆ ß n ß, n n ineger (d) ˆ n nd ˆ ß ß n, n n ineger 5. () r cos ) cos, r sin ) sin Ê Cresin coordines re ( $ß) () r cos ) cos, r sin ) sin Ê Cresin coordines re ( $ß) (c) r cos ) cos, r sin ) sin Ê Cresin coordines re Š ß 7 7 (d) r cos ) cos, r sin ) sin Ê Cresin coordines re Š ß (e) r cos ) cos, r sin ) sin Ê Cresin coordines re (ß) (f) r cos ) cos, r sin ) sin Ê Cresin coordines re Š ß (g) r cos ) cos, r sin ) sin Ê Cresin coordines re ( ß) (h) r cos ) cos ˆ, r sin ) sin ˆ Ê Cresin coordines re Š ß 6. () cos, sin Ê Cresin coordines re (ß) () cos, sin Ê Cresin coordines re (ß) (c) cos, sin Ê Cresin coordines re (ß) (d) cos ˆ, sin ˆ Ê Cresin coordines re ( ß ) 5 5 6 6 (e) cos, sin Ê Cresin coordines re Š ß (f) 5 cos ˆ n, 5 sin ˆ n Ê Cresin coordines re ( $ß) Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

(g) cos 7, sin 7 Ê Cresin coordines re (ß) (h) cos, sin Ê Cresin coordines re Š ß Secion. Polr Coordines 66 7. (), Ê r, sin ) nd cos ) Ê ) Ê Polr coordines re Š, (), Ê r É, sin ) nd cos ) Ê ) Ê Polr coordines re, 6 6 (c) Š, r ÊŠ Ê, sin nd cos Ê Ê Polr coordines re ˆ ) ) ), (d), Ê r É 5, sin ) nd cos ) Ê ) rcnˆ Ê Polr coordines re ˆ 5, rcnˆ 5 5 8. (), Êr É, sin ) nd cos ) Ê ) ÊPolr coordines re Š, (), Ê r, sin nd cos Ê Ê Polr coordines re ˆ ) ) ), 6 6 (c) Š, Ê r ÊŠ 5, sin nd cos Ê Ê Polr coordines re ˆ 5 ) ) ), (d) 5, Êr É5, sin ) nd cos ) 5 Ê) rcnˆ ÊPolr coordines re ˆ, rcnˆ 5 5 9. (), Êr, sin ) nd cos ) Ê ) 5 ÊPolr coordines re Š 5, (), Êr É, sin ) nd cos ) Ê) ÊPolr coordines re, (c) Š, r Ê Š 5 Ê, sin ) nd cos ) Ê) ÊPolr coordines re ˆ, 5 (d), Êr É 5, sin ) nd cos ) Ê) rcnˆ ÊPolr coordines re ˆ 5, rcnˆ 5 5. (), Êr É, sin ) nd cos ) Ê) ÊPolr coordines re, (), Ê r, sin ) nd cos ) Ê ) or ) Ê Polr coordines re, or, (c), Êr É, sin nd cos Ê ÊPolr coordines re ˆ ) ) ), 6 6 (d) Š, r ÊŠ ˆ 7 5 Ê, sin ) nd cos ) Ê) or ) ÊPolr coordines re ˆ 7, or ˆ 5, 6 6 Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

66 Chper Prmeric Equions nd Polr Coordines.... 5. 6. 7. 8. 9...... 5. Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

Secion. Polr Coordines 665 6. 7. r cos ) Ê, vericl line hrough (ß) 8. r sin ) Ê, horizonl line hrough (ß ) 9. r sin ) Ê, he -is. r cos ) Ê, he -is sin ). r csc ) Ê r Ê r sin ) Ê, horizonl line hrough (ß) cos ). r sec ) Ê r Ê r cos ) Ê, vericl line hrough ( ß). r cos ) r sin ) Ê, line wih slope m nd inercep. r sin ) r cos ) Ê, line wih slope m nd inercep 5. r Ê, circle wih cener C (ß) nd rdius 6. r r sin ) Ê Ê Ê ( ), circle wih cener C (ß) nd rdius 5 7. r sin ) cos ) Ê r sin ) r cos ) 5 Ê 5, line wih slope m nd inercep 5 8. r sin ) Ê r sin ) cos ) Ê (r sin ))(r cos )) Ê, hperol wih focl is 9. r co ) csc ) ˆ cos ) ˆ Ê r sin ) cos ) Ê r sin ) r cos ) Ê, prol wih vere (ß) which opens o he righ sin ) sin ). r n ) sec ) Ê r ˆ sin ) cos ) Ê r cos ) sin ) Ê r cos ) r sin ) Ê, prol wih vere (ß) which opens upwrd r cos ) r cos ). r (csc )) e Ê r sin ) e Ê e, grph of he nurl eponenil funcion. r sin ) ln r ln cos ) ln (r cos )) Ê ln, grph of he nurl logrihm funcion. r r cos ) sin ) Ê Ê Ê ( ) Ê, wo prllel srigh lines of slope nd -inerceps. cos ) sin ) Ê r cos ) r sin ) Ê Ê kk k k Ê, wo perpendiculr lines hrough he origin wih slopes nd, respecivel. 5. r r cos ) Ê Ê Ê Ê ( ), circle wih cener C( ß) nd rdius Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

666 Chper Prmeric Equions nd Polr Coordines 6. r 6r sin ) Ê 6 Ê 6 Ê 6 9 9 Ê ( ) 9, circle wih cener C(ß ) nd rdius 7. r 8 sin ) Ê r 8r sin ) Ê 8 Ê 8 Ê 8 6 6 Ê ( ) 6, circle wih cener C(ß ) nd rdius 9 9 9 8. r cos ) Ê r r cos ) Ê Ê Ê Ê ˆ, circle wih cener C ˆ ß nd rdius 9. r cos ) sin ) Ê r r cos ) r sin ) Ê Ê Ê ( ) ( ), circle wih cener C(ß) nd rdius 5. r cos ) sin ) Ê r r cos ) r sin ) Ê Ê 5 Ê ( ) ˆ 5, circle wih cener C ˆ ß nd rdius 6 6 6, line wih slope m nd inercep 5. r sin ˆ r ˆ sin cos cos sin ) Ê ) ) Ê r sin ) r cos ) Ê Ê 5. r sin ˆ 5 r ˆ sin cos cos sin ) Ê ) ) 5 Ê r cos ) r sin ) 5 Ê 5 Ê, line wih slope m nd inercep 5. 7 Ê r cos ) 7 5. Ê r sin ) 55. Ê r cos ) r sin ) Ê ) 56. Ê r cos ) r sin ) 57. Ê r Ê r or r 58. Ê r cos ) r sin ) Ê r cos ) sin ) Ê r cos ) 9 59. Ê 9 6 Ê r cos ) 9r sin ) 6 6. Ê (r cos ))(r sin )) Ê r cos ) sin ) Ê r cos ) sin ) Ê r sin ) 6. Ê r sin ) r cos ) Ê r sin ) cos ) 6. Ê Ê r r sin ) cos ) Ê r ( sin ) cos )) 6. ( ) Ê Ê Ê r r sin ) Ê r sin ) 6. ( 5) 5 Ê 5 5 Ê Ê r r cos ) Ê r cos ) 65. ( ) ( ) Ê 6 9 Ê 6 6 Ê r 6r cos ) r sin ) 6 66. ( ) ( 5) 6 Ê 5 6 Ê Ê r r cos ) r sin ) Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.

Secion. Grphing in Polr Coordines 667 67. (ß )) where ) is n ngle cos ) sin ) 68. () Ê r cos ) Ê r Ê r sec ) () Ê r sin ) Ê r Ê r csc ). GRAPHING IN POLAR COORDINATES. cos ( ) ) cos ) r Ê smmeric ou he -is; cos ( ) ) Á r nd cos ( ) ) cos ) Á r Ê no smmeric ou he -is; herefore no smmeric ou he origin. cos ( ) ) cos ) r Ê smmeric ou he -is; cos ( ) ) Á r nd cos ( ) ) cos ) Á r Ê no smmeric ou he -is; herefore no smmeric ou he origin. sin ( ) ) sin ) Á r nd sin ( ) ) sin ) Á r Ê no smmeric ou he -is; sin ( ) ) sin ) r Ê smmeric ou he -is; herefore no smmeric ou he origin. sin ( ) ) sin ) Á r nd sin ( ) ) sin ) Á r Ê no smmeric ou he -is; sin ( ) ) sin ) r Ê smmeric ou he -is; herefore no smmeric ou he origin Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.