CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES
|
|
|
- Marian Butler
- 5 years ago
- Views:
Transcription
1 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.
2 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 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.
3 Secion. Prmerizions of Plne Curves 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.
4 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.
5 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.
6 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 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.
7 Secion. Prmerizions of Plne Curves 65.. () () (c) 5. () () 6. () () Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.
8 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 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.
9 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 / / 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.
10 656 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.
11 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 (),, 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.
12 658 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 % $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 * 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 Check: sln heigh is 5 Ê Are is 5 5. Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.
13 Secion. Clculus Wih Prmeric Curves 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 M e 5e µ e e 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 ( compuer) Ê.9. Ne M µ dm 5 M 7 Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.
14 66 Chper Prmeric Equions nd Polr Coordines $ % M ( 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 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.
15 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 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 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.
16 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 ) (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.
17 (g) cos 7, sin 7 Ê Cresin coordines re (ß) (h) cos, sin Ê Cresin coordines re Š ß Secion. Polr Coordines (), 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 (), 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 (), 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.
18 66 Chper Prmeric Equions nd Polr Coordines Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.
19 Secion. Polr Coordines 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.
20 666 Chper Prmeric Equions nd Polr Coordines 6. r 6r sin ) Ê 6 Ê 6 Ê Ê ( ) 9, circle wih cener C(ß ) nd rdius 7. r 8 sin ) Ê r 8r sin ) Ê 8 Ê 8 Ê Ê ( ) 6, circle wih cener C(ß ) nd rdius 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 ) 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 ) ( ) ( 5) 6 Ê 5 6 Ê Ê r r cos ) r sin ) Coprigh Person Educion, Inc. Pulishing s Addison-Wesle.
21 Secion. Grphing in Polr Coordines (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.
FM Applications of Integration 1.Centroid of Area
FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is
MTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
PARABOLA. moves such that PM. = e (constant > 0) (eccentricity) then locus of P is called a conic. or conic section.
wwwskshieducioncom PARABOLA Le S be given fixed poin (focus) nd le l be given fixed line (Direcrix) Le SP nd PM be he disnce of vrible poin P o he focus nd direcrix respecively nd P SP moves such h PM
668 Chapter 11 Parametric Equatins and Polar Coordinates
668 Chapter Parametric Equatins and Polar Coordinates 5. sin ( sin Á r and sin ( sin Á r Ê not symmetric about the x-axis; sin ( sin r Ê symmetric about the y-axis; therefore not symmetric about the origin
( ) ( ) ( ) ( ) ( ) ( y )
8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll
EXERCISE - 01 CHECK YOUR GRASP
UNIT # 09 PARABOLA, ELLIPSE & HYPERBOLA PARABOLA EXERCISE - 0 CHECK YOUR GRASP. Hin : Disnce beween direcri nd focus is 5. Given (, be one end of focl chord hen oher end be, lengh of focl chord 6. Focus
Physics 2A HW #3 Solutions
Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS
CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6. VOLUMES USING CROSS-SECTIONS. A() ;, ; (digonl) ˆ Ȉ È V A() d d c d 6 (dimeter) c d c d c ˆ 6. A() ;, ; V A() d d. A() (edge) È Š È Š È ;, ; V A() d d 8
10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e
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
A 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m
PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl
660 Chapter 10 Conic Sections and Polar Coordinates
Chpter Conic Sections nd Polr Coordintes 8. ( (b (c (d (e r r Ê r ; therefore cos Ê Ê ( ß is point of intersection ˆ ˆ Ê Ê Ê ß ß ˆ ß 9. ( r cos Ê cos ; r cos Ê r Š Ê r r Ê (r (b r Ê cos Ê cos Ê, Ê ß or
698 Chapter 11 Parametric Equations and Polar Coordinates
698 Chapter Parametric Equations and Polar Coordinates 67. 68. 69. 70. 7. 7. 7. 7. Chapter Practice Eercises 699 75. (a Perihelion a ae a( e, Aphelion ea a a( e ( Planet Perihelion Aphelion Mercur 0.075
ENGR 1990 Engineering Mathematics The Integral of a Function as a Function
ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under
Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.
Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl
Chapter 2. Motion along a straight line. 9/9/2015 Physics 218
Chper Moion long srigh line 9/9/05 Physics 8 Gols for Chper How o describe srigh line moion in erms of displcemen nd erge elociy. The mening of insnneous elociy nd speed. Aerge elociy/insnneous elociy
CBSE 2014 ANNUAL EXAMINATION ALL INDIA
CBSE ANNUAL EXAMINATION ALL INDIA SET Wih Complee Eplnions M Mrks : SECTION A Q If R = {(, y) : + y = 8} is relion on N, wrie he rnge of R Sol Since + y = 8 h implies, y = (8 ) R = {(, ), (, ), (6, )}
September 20 Homework Solutions
College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum
MATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)
MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide
Available Online :
fo/u fopjr Hh# u] ugh vjehs de] foifr ns[ NsMs rqjr e/;e eu dj ';ea iq#" flg ldyi dj] lgrs foifr vusd] ^cu^ u NsMs /;s; ds] j?qcj j[s VsdAA jfpr% euo /ez iz.sr ln~xq# Jh j.nsmnlh egj STUDY PACKAGE Subjec
1. Consider a PSA initially at rest in the beginning of the left-hand end of a long ISS corridor. Assume xo = 0 on the left end of the ISS corridor.
In Eercise 1, use sndrd recngulr Cresin coordine sysem. Le ime be represened long he horizonl is. Assume ll ccelerions nd decelerions re consn. 1. Consider PSA iniilly res in he beginning of he lef-hnd
JEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse
Chapter Direct Method of Interpolation
Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o
Physics 101 Lecture 4 Motion in 2D and 3D
Phsics 11 Lecure 4 Moion in D nd 3D Dr. Ali ÖVGÜN EMU Phsics Deprmen www.ogun.com Vecor nd is componens The componens re he legs of he righ ringle whose hpoenuse is A A A A A n ( θ ) A Acos( θ) A A A nd
AP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
graph of unit step function t
.5 Piecewie coninuou forcing funcion...e.g. urning he forcing on nd off. The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion"
Three Dimensional Coordinate Geometry
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer Three Dimensionl Coordine Geomer. Coordine of Poin in Spe Z XOX, YOY nd ZOZ re he oordine-es. P,, is poin on he oordine plne nd is lled ordered riple. P,, X Y
P 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
f 5 x 3 d x x 12 x 16 x a x < 1 b x < 2 a c x e f b i c d + +
ANSWERS Chper Eercise. 0 0 c d 0 0 e f 0 [,] ],[ c ]0,] d ],0] e ],[ f ],[],[ c + c d + + c + d e - + + 0 + f + i - + ii 0 + i - + ii - + 0 {±} {±0} c Ø d {,} e {,} f {0,} - 0 0 c d 0 ],[ ],[ c ],[ + Eercise..
Section P.1 Notes Page 1 Section P.1 Precalculus and Trigonometry Review
Secion P Noe Pge Secion P Preclculu nd Trigonomer Review ALGEBRA AND PRECALCULUS Eponen Lw: Emple: 8 Emple: Emple: Emple: b b Emple: 9 EXAMPLE: Simplif: nd wrie wi poiive eponen Fir I will flip e frcion
CONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
e t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
3 Motion with constant acceleration: Linear and projectile motion
3 Moion wih consn ccelerion: Liner nd projecile moion cons, In he precedin Lecure we he considered moion wih consn ccelerion lon he is: Noe h,, cn be posiie nd neie h leds o rie of behiors. Clerl similr
Motion in a Straight Line
Moion in Srigh Line. Preei reched he mero sion nd found h he esclor ws no working. She wlked up he sionry esclor in ime. On oher dys, if she remins sionry on he moing esclor, hen he esclor kes her up in
/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2
SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the
Chapter 11. Parametric, Vector, and Polar Functions. aπ for any integer n. Section 11.1 Parametric Functions (pp ) cot
Secion. 6 Chaper Parameric, Vecor, an Polar Funcions. an sec sec + an + Secion. Parameric Funcions (pp. 9) Eploraion Invesigaing Cclois 6. csc + co co +. 7. cos cos cos [, ] b [, 8]. na for an ineger n..
Average & instantaneous velocity and acceleration Motion with constant acceleration
Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission
ECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 10. Waveguides Part 7: Transverse Equivalent Network (TEN)
EE 537-635 Microwve Engineering Fll 7 Prof. Dvid R. Jcson Dep. of EE Noes Wveguides Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model Our gol is o come up wih rnsmission line model for
PHYSICS 1210 Exam 1 University of Wyoming 14 February points
PHYSICS 1210 Em 1 Uniersiy of Wyoming 14 Februry 2013 150 poins This es is open-noe nd closed-book. Clculors re permied bu compuers re no. No collborion, consulion, or communicion wih oher people (oher
Solutions to Problems from Chapter 2
Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5
FP3 past questions - conics
Hperolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh rcosh = ln{ + } ( ) rsinh = ln{ + + } + rtnh = ln ( < ) FP3 pst questions - conics Conics Ellipse Prol Hperol Rectngulr Hperol Stndrd
!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
Collision Detection and Bouncing
Collision Deecion nd Bouncing Collisions re Hndled in Two Prs. Deecing he collision Mike Biley mj@cs.oregonse.edu. Hndling he physics of he collision collision-ouncing.ppx If You re Lucky, You Cn Deec
Lesson-5 ELLIPSE 2 1 = 0
Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).
How to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.
How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle
AP Calculus BC - Parametric equations and vectors Chapter 9- AP Exam Problems solutions
AP Calculus BC - Parameric equaions and vecors Chaper 9- AP Exam Problems soluions. A 5 and 5. B A, 4 + 8. C A, 4 + 4 8 ; he poin a is (,). y + ( x ) x + 4 4. e + e D A, slope.5 6 e e e 5. A d hus d d
THE ESSENTIALS OF CALCULUS ANSWERS TO SELECTED EXERCISES
Assignmen - page. m.. f 7 7.. 7..8 7..77 7. 87. THE ESSENTIALS OF CALCULUS ANSWERS TO SELECTED EXERCISES m.... no collinear 8...,,.,.8 or.,..78,.7 or.7,.8., 8.87 or., 8.88.,,, 7..7 Assignmen - page 7.
5.1-The Initial-Value Problems For Ordinary Differential Equations
5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil
Version 001 test-1 swinney (57010) 1. is constant at m/s.
Version 001 es-1 swinne (57010) 1 This prin-ou should hve 20 quesions. Muliple-choice quesions m coninue on he nex column or pge find ll choices before nswering. CubeUniVec1x76 001 10.0 poins Acubeis1.4fee
Properties of Logarithms. Solving Exponential and Logarithmic Equations. Properties of Logarithms. Properties of Logarithms. ( x)
Properies of Logrihms Solving Eponenil nd Logrihmic Equions Properies of Logrihms Produc Rule ( ) log mn = log m + log n ( ) log = log + log Properies of Logrihms Quoien Rule log m = logm logn n log7 =
1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
![1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.](/thumbs/71/65799522.jpg "1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.")
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
Y 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
Thomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
Magnetostatics Bar Magnet. Magnetostatics Oersted s Experiment
Mgneosics Br Mgne As fr bck s 4500 yers go, he Chinese discovered h cerin ypes of iron ore could rc ech oher nd cerin mels. Iron filings "mp" of br mgne s field Crefully suspended slivers of his mel were
Name: Per: L o s A l t o s H i g h S c h o o l. Physics Unit 1 Workbook. 1D Kinematics. Mr. Randall Room 705
Nme: Per: L o s A l o s H i g h S c h o o l Physics Uni 1 Workbook 1D Kinemics Mr. Rndll Room 705 Adm.Rndll@ml.ne www.laphysics.com Uni 1 - Objecies Te: Physics 6 h Ediion Cunel & Johnson The objecies
, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
An object moving with speed v around a point at distance r, has an angular velocity. m/s m
Roion The mosphere roes wih he erh n moions wihin he mosphere clerly follow cure phs (cyclones, nicyclones, hurricnes, ornoes ec.) We nee o epress roion quniiely. For soli objec or ny mss h oes no isor
2D Motion WS. A horizontally launched projectile s initial vertical velocity is zero. Solve the following problems with this information.
Nme D Moion WS The equions of moion h rele o projeciles were discussed in he Projecile Moion Anlsis Acii. ou found h projecile moes wih consn eloci in he horizonl direcion nd consn ccelerion in he ericl
APPLICATIONS OF DEFINITE INTEGRALS
Chpter 6 APPICATIONS OF DEFINITE INTEGRAS OVERVIEW In Chpter 5 we discovered the connection etween Riemnn sums ssocited with prtition P of the finite closed intervl [, ] nd the process of integrtion. We
(b) 10 yr. (b) 13 m. 1.6 m s, m s m s (c) 13.1 s. 32. (a) 20.0 s (b) No, the minimum distance to stop = 1.00 km. 1.
Answers o Een Numbered Problems Chper. () 7 m s, 6 m s (b) 8 5 yr 4.. m ih 6. () 5. m s (b).5 m s (c).5 m s (d) 3.33 m s (e) 8. ().3 min (b) 64 mi..3 h. ().3 s (b) 3 m 4..8 mi wes of he flgpole 6. (b)
R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
Mathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
Chapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).
![INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).](/thumbs/78/77677920.jpg "INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).")
INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely
REAL ANALYSIS I HOMEWORK 3. Chapter 1
REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs
DE55/DC55 ENGINEERING MATHEMATICS-II DEC 2013
DE55/DC55 ENGINEERING MATHEMATICS-II DEC Q. a. Evaluae LT sin x x sin 5x + sin 6x sin x Here we have, L x sin x + sin 6x sin 5x sin x L x L x x + 6x x 6x sin + cos 5x + x 5x x cos.sin sin 4x cos( x) cos
PART - III : MATHEMATICS
JEE(Advnced) 4 Finl Em/Pper-/Code-8 PART - III : SECTION : (One or More Thn One Options Correct Type) This section contins multiple choice questions. Ech question hs four choices (A), (B), (C) nd (D) out
Mathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
Answers to Algebra 2 Unit 3 Practice
Answers o Algebra 2 Uni 3 Pracice Lesson 14-1 1. a. 0, w, 40; (0, 40); {w w, 0, w, 40} 9. a. 40,000 V Volume c. (27, 37,926) d. 27 unis 2 a. h, 30 2 2r V pr 2 (30 2 2r) c. in. d. 3,141.93 in. 2 20 40 Widh
6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
Review Exercises for Chapter 3
60_00R.qd //0 :9 M age CHATER Applicaions of Differeniaion Review Eercises for Chaper. Give he definiion of a criical number, and graph a funcion f showing he differen pes of criical numbers.. Consider
NORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y
LOCUS 50 Section - 4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to
LAPLACE TRANSFORMS. 1. Basic transforms
LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming
How to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
ES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
Solutions for homework 12
y Soluions for homework Secion Nonlinear sysems: The linearizaion of a nonlinear sysem Consider he sysem y y y y y (i) Skech he nullclines Use a disincive marking for each nullcline so hey can be disinguished
236 Chapter 4 Applications of Derivatives
26 Chapter Applications of Derivatives Î$ &Î$ Î$ 5 Î$ 0 "Î$ 5( 2) $È 26. (a) g() œ ( 5) œ 5 Ê g () œ œ Ê critical points at œ 2 and œ 0 Ê g œ ± )(, increasing on ( _ß 2) and (!ß _), decreasing on ( 2 ß!)!
Mathematical Modeling
ME pplie Engineering nlsis Chper Mhemicl Moeling Professor Ti-Rn Hsu, Ph.D. Deprmen of Mechnicl n erospce Engineering Sn Jose Se Universi Sn Jose, Cliforni, US Jnur Chper Lerning Ojecives Mhemicl moeling
Ellipse. 1. Defini t ions. FREE Download Study Package from website: 11 of 91CONIC SECTION
FREE Downlod Stud Pckge from wesite: www.tekoclsses.com. Defini t ions Ellipse It is locus of point which moves in such w tht the rtio of its distnce from fied point nd fied line (not psses through fied
on the interval (x + 1) 0! x < ", where x represents feet from the first fence post. How many square feet of fence had to be painted?
Calculus II MAT 46 Improper Inegrals A mahemaician asked a fence painer o complee he unique ask of paining one side of a fence whose face could be described by he funcion y f (x on he inerval (x + x
MAT 266 Calculus for Engineers II Notes on Chapter 6 Professor: John Quigg Semester: spring 2017
MAT 66 Clculus for Engineers II Noes on Chper 6 Professor: John Quigg Semeser: spring 7 Secion 6.: Inegrion by prs The Produc Rule is d d f()g() = f()g () + f ()g() Tking indefinie inegrls gives [f()g
ECE Microwave Engineering
EE 537-635 Microwve Engineering Adped from noes y Prof. Jeffery T. Willims Fll 8 Prof. Dvid R. Jcson Dep. of EE Noes Wveguiding Srucures Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model
Inspiration and formalism
Inspirtion n formlism Answers Skills hek P(, ) Q(, ) PQ + ( ) PQ A(, ) (, ) grient ( ) + Eerise A opposite sies of regulr hegon re equl n prllel A ED i FC n ED ii AD, DA, E, E n FC No, sies of pentgon
( ) 2. Review Exercise 2. cos θ 2 3 = = 2 tan. cos. 2 x = = x a. Since π π, = 2. sin = = 2+ = = cotx. 2 sin θ 2+
Review Eercise sin 5 cos sin an cos 5 5 an 5 9 co 0 a sinθ 6 + 4 6 + sin θ 4 6+ + 6 + 4 cos θ sin θ + 4 4 sin θ + an θ cos θ ( ) + + + + Since π π, < θ < anθ should be negaive. anθ ( + ) Pearson Educaion
Laplace Transforms. Examples. Is this equation differential? y 2 2y + 1 = 0, y 2 2y + 1 = 0, (y ) 2 2y + 1 = cos x,
Laplace Transforms Definiion. An ordinary differenial equaion is an equaion ha conains one or several derivaives of an unknown funcion which we call y and which we wan o deermine from he equaion. The equaion
Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
10.6 Parametric Equations
0_006.qd /8/05 9:05 AM Page 77 Secion 0.6 77 Parameric Equaions 0.6 Parameric Equaions Wha ou should learn Evaluae ses of parameric equaions for given values of he parameer. Skech curves ha are represened
Be able to sketch a function defined parametrically. (by hand and by calculator)
Pre Calculus Uni : Parameric and Polar Equaions (7) Te References: Pre Calculus wih Limis; Larson, Hoseler, Edwards. B he end of he uni, ou should be able o complee he problems below. The eacher ma provide
15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
A Kalman filtering simulation
A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer
Time : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A
Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new
50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS
68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3
Drill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.
Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte
C H A P T E R 5. Integrals. PDF Created with deskpdf PDF Writer - Trial :: A r e a s a n d D i s t a n c e s
License o: jsmuels@mcc.cun.eu PDF Cree wih eskpdf PDF Wrier - Tril :: hp://www.ocuesk.com C H A P T E R 5 To compue n re we pproime region recngles n le he numer of recngles ecome lrge. The precise re
( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.
Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k
AP CALCULUS BC 2016 SCORING GUIDELINES
6 SCORING GUIDELINES Quesion A ime, he posiion of a paricle moving in he xy-plane is given by he parameric funcions ( x ( ), y ( )), where = + sin ( ). The graph of y, consising of hree line segmens, is
f t f a f x dx By Lin McMullin f x dx= f b f a. 2
Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes
a. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines.
Mah A Final Eam Problems for onsideraion. Show all work for credi. Be sure o show wha you know. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A.