Stewart - Derivada Implícita

por Alberto Nascente Qui 15 Dez 2022, 08:07

Encontre as equações de ambas as retas tangentes à elipse

Stewart - Derivada Implícita Svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
<!-- Generated by CodeCogs with dvisvgm 2.13.3 -->
<svg version='1.1' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink' width='78.511895pt' height='14.322936pt' viewBox='-.239051 -.235254 78.511895 14.322936'>
<defs>
<path id='g2-43' d='M4.770112-2.761644H8.069738C8.237111-2.761644 8.452304-2.761644 8.452304-2.976837C8.452304-3.203985 8.249066-3.203985 8.069738-3.203985H4.770112V-6.503611C4.770112-6.670984 4.770112-6.886177 4.554919-6.886177C4.327771-6.886177 4.327771-6.682939 4.327771-6.503611V-3.203985H1.028144C.860772-3.203985 .645579-3.203985 .645579-2.988792C.645579-2.761644 .848817-2.761644 1.028144-2.761644H4.327771V.537983C4.327771 .705355 4.327771 .920548 4.542964 .920548C4.770112 .920548 4.770112 .71731 4.770112 .537983V-2.761644Z'/>
<path id='g2-51' d='M2.199751-4.291905C1.996513-4.27995 1.948692-4.267995 1.948692-4.160399C1.948692-4.040847 2.008468-4.040847 2.223661-4.040847H2.773599C3.789788-4.040847 4.244085-3.203985 4.244085-2.056289C4.244085-.490162 3.431133-.071731 2.84533-.071731C2.271482-.071731 1.291158-.3467 .944458-1.135741C1.327024-1.075965 1.673724-1.291158 1.673724-1.721544C1.673724-2.068244 1.422665-2.307347 1.08792-2.307347C.800996-2.307347 .490162-2.139975 .490162-1.685679C.490162-.621669 1.554172 .251059 2.881196 .251059C4.303861 .251059 5.355915-.836862 5.355915-2.044334C5.355915-3.144209 4.471233-4.004981 3.323537-4.208219C4.363636-4.507098 5.033126-5.379826 5.033126-6.312329C5.033126-7.256787 4.052802-7.950187 2.893151-7.950187C1.697634-7.950187 .812951-7.220922 .812951-6.348194C.812951-5.869988 1.183562-5.774346 1.362889-5.774346C1.613948-5.774346 1.900872-5.953674 1.900872-6.312329C1.900872-6.694894 1.613948-6.862267 1.350934-6.862267C1.279203-6.862267 1.255293-6.862267 1.219427-6.850311C1.673724-7.663263 2.797509-7.663263 2.857285-7.663263C3.251806-7.663263 4.028892-7.483935 4.028892-6.312329C4.028892-6.085181 3.993026-5.415691 3.646326-4.901619C3.287671-4.375592 2.881196-4.339726 2.558406-4.327771L2.199751-4.291905Z'/>
<path id='g2-52' d='M4.315816-7.782814C4.315816-8.009963 4.315816-8.069738 4.148443-8.069738C4.052802-8.069738 4.016936-8.069738 3.921295-7.926276L.32279-2.343213V-1.996513H3.466999V-.908593C3.466999-.466252 3.443088-.3467 2.570361-.3467H2.331258V0C2.606227-.02391 3.550685-.02391 3.88543-.02391S5.176588-.02391 5.451557 0V-.3467H5.212453C4.351681-.3467 4.315816-.466252 4.315816-.908593V-1.996513H5.523288V-2.343213H4.315816V-7.782814ZM3.526775-6.850311V-2.343213H.621669L3.526775-6.850311Z'/>
<path id='g2-54' d='M1.470486-4.160399C1.470486-7.185056 2.940971-7.663263 3.58655-7.663263C4.016936-7.663263 4.447323-7.531756 4.674471-7.173101C4.531009-7.173101 4.076712-7.173101 4.076712-6.682939C4.076712-6.419925 4.25604-6.192777 4.566874-6.192777C4.865753-6.192777 5.068991-6.372105 5.068991-6.718804C5.068991-7.340473 4.614695-7.950187 3.574595-7.950187C2.068244-7.950187 .490162-6.40797 .490162-3.777833C.490162-.490162 1.924782 .251059 2.940971 .251059C4.244085 .251059 5.355915-.884682 5.355915-2.438854C5.355915-4.028892 4.244085-5.092902 3.048568-5.092902C1.984558-5.092902 1.590037-4.172354 1.470486-3.837609V-4.160399ZM2.940971-.071731C2.187796-.071731 1.829141-.74122 1.721544-.992279C1.613948-1.303113 1.494396-1.888917 1.494396-2.725778C1.494396-3.670237 1.924782-4.853798 3.000747-4.853798C3.658281-4.853798 4.004981-4.411457 4.184309-4.004981C4.375592-3.56264 4.375592-2.964882 4.375592-2.450809C4.375592-1.841096 4.375592-1.303113 4.148443-.848817C3.849564-.274969 3.419178-.071731 2.940971-.071731Z'/>
<path id='g2-61' d='M8.069738-3.873474C8.237111-3.873474 8.452304-3.873474 8.452304-4.088667C8.452304-4.315816 8.249066-4.315816 8.069738-4.315816H1.028144C.860772-4.315816 .645579-4.315816 .645579-4.100623C.645579-3.873474 .848817-3.873474 1.028144-3.873474H8.069738ZM8.069738-1.649813C8.237111-1.649813 8.452304-1.649813 8.452304-1.865006C8.452304-2.092154 8.249066-2.092154 8.069738-2.092154H1.028144C.860772-2.092154 .645579-2.092154 .645579-1.876961C.645579-1.649813 .848817-1.649813 1.028144-1.649813H8.069738Z'/>
<path id='g1-50' d='M2.247572-1.625903C2.375093-1.745455 2.709838-2.008468 2.83736-2.12005C3.331507-2.574346 3.801743-3.012702 3.801743-3.737983C3.801743-4.686426 3.004732-5.300125 2.008468-5.300125C1.052055-5.300125 .422416-4.574844 .422416-3.865504C.422416-3.474969 .73325-3.419178 .844832-3.419178C1.012204-3.419178 1.259278-3.53873 1.259278-3.841594C1.259278-4.25604 .860772-4.25604 .765131-4.25604C.996264-4.837858 1.530262-5.037111 1.920797-5.037111C2.662017-5.037111 3.044583-4.407472 3.044583-3.737983C3.044583-2.909091 2.462765-2.303362 1.522291-1.338979L.518057-.302864C.422416-.215193 .422416-.199253 .422416 0H3.57061L3.801743-1.42665H3.55467C3.53076-1.267248 3.466999-.868742 3.371357-.71731C3.323537-.653549 2.717808-.653549 2.590286-.653549H1.171606L2.247572-1.625903Z'/>
<path id='g0-120' d='M5.66675-4.877709C5.284184-4.805978 5.140722-4.519054 5.140722-4.291905C5.140722-4.004981 5.36787-3.90934 5.535243-3.90934C5.893898-3.90934 6.144956-4.220174 6.144956-4.542964C6.144956-5.045081 5.571108-5.272229 5.068991-5.272229C4.339726-5.272229 3.93325-4.554919 3.825654-4.327771C3.550685-5.224408 2.809465-5.272229 2.594271-5.272229C1.374844-5.272229 .729265-3.706102 .729265-3.443088C.729265-3.395268 .777086-3.335492 .860772-3.335492C.956413-3.335492 .980324-3.407223 1.004234-3.455044C1.41071-4.782067 2.211706-5.033126 2.558406-5.033126C3.096389-5.033126 3.203985-4.531009 3.203985-4.244085C3.203985-3.981071 3.132254-3.706102 2.988792-3.132254L2.582316-1.494396C2.402989-.777086 2.056289-.119552 1.422665-.119552C1.362889-.119552 1.06401-.119552 .812951-.274969C1.243337-.358655 1.338979-.71731 1.338979-.860772C1.338979-1.099875 1.159651-1.243337 .932503-1.243337C.645579-1.243337 .334745-.992279 .334745-.609714C.334745-.107597 .896638 .119552 1.41071 .119552C1.984558 .119552 2.391034-.334745 2.642092-.824907C2.833375-.119552 3.431133 .119552 3.873474 .119552C5.092902 .119552 5.738481-1.446575 5.738481-1.709589C5.738481-1.769365 5.69066-1.817186 5.618929-1.817186C5.511333-1.817186 5.499377-1.75741 5.463512-1.661768C5.140722-.609714 4.447323-.119552 3.90934-.119552C3.490909-.119552 3.263761-.430386 3.263761-.920548C3.263761-1.183562 3.311582-1.374844 3.502864-2.163885L3.921295-3.789788C4.100623-4.507098 4.507098-5.033126 5.057036-5.033126C5.080946-5.033126 5.415691-5.033126 5.66675-4.877709Z'/>
<path id='g0-121' d='M3.144209 1.338979C2.82142 1.793275 2.355168 2.199751 1.769365 2.199751C1.625903 2.199751 1.052055 2.175841 .872727 1.625903C.908593 1.637858 .968369 1.637858 .992279 1.637858C1.350934 1.637858 1.590037 1.327024 1.590037 1.052055S1.362889 .681445 1.183562 .681445C.992279 .681445 .573848 .824907 .573848 1.41071C.573848 2.020423 1.08792 2.438854 1.769365 2.438854C2.964882 2.438854 4.172354 1.338979 4.507098 .011955L5.678705-4.65056C5.69066-4.710336 5.71457-4.782067 5.71457-4.853798C5.71457-5.033126 5.571108-5.152677 5.391781-5.152677C5.284184-5.152677 5.033126-5.104857 4.937484-4.746202L4.052802-1.231382C3.993026-1.016189 3.993026-.992279 3.897385-.860772C3.658281-.526027 3.263761-.119552 2.689913-.119552C2.020423-.119552 1.960648-.777086 1.960648-1.099875C1.960648-1.78132 2.283437-2.701868 2.606227-3.56264C2.737733-3.90934 2.809465-4.076712 2.809465-4.315816C2.809465-4.817933 2.450809-5.272229 1.865006-5.272229C.765131-5.272229 .32279-3.53873 .32279-3.443088C.32279-3.395268 .37061-3.335492 .454296-3.335492C.561893-3.335492 .573848-3.383313 .621669-3.550685C.908593-4.554919 1.362889-5.033126 1.829141-5.033126C1.936737-5.033126 2.139975-5.033126 2.139975-4.638605C2.139975-4.327771 2.008468-3.981071 1.829141-3.526775C1.243337-1.960648 1.243337-1.566127 1.243337-1.279203C1.243337-.143462 2.056289 .119552 2.654047 .119552C3.000747 .119552 3.431133 .011955 3.849564-.430386L3.861519-.418431C3.682192 .286924 3.56264 .753176 3.144209 1.338979Z'/>
</defs>
<g id='page1' transform='matrix(1.13 0 0 1.13 -63.986043 -62.969593)'>
<use x='56.413267' y='65.753425' xlink:href='#g0-120'/>
<use x='63.065354' y='60.817239' xlink:href='#g1-50'/>
<use x='70.454333' y='65.753425' xlink:href='#g2-43'/>
<use x='82.215648' y='65.753425' xlink:href='#g2-52'/>
<use x='88.068638' y='65.753425' xlink:href='#g0-121'/>
<use x='94.20529' y='60.817239' xlink:href='#g1-50'/>
<use x='102.258434' y='65.753425' xlink:href='#g2-61'/>
<use x='114.683915' y='65.753425' xlink:href='#g2-51'/>
<use x='120.536905' y='65.753425' xlink:href='#g2-54'/>
</g>
</svg>

que passam pelo ponto (12,3)
S/ gabarito.

Derivei implicitamente, como em tese o capítulo ensina a fazer.
Porém, a reta que eu achei n eh tangente a curva, ela apenas passa pelo ponto dado...
Como pode resolver isso?

Obrigado! Very Happy

por **Elcioschin** Qui 15 Dez 2022, 11:10

Sem derivar

Equação geral das retas ---> y - 3 = m.(x - 12) ---> y = m.x + 3 - 12.m

Substitua na equação da elipse e chegue numa equação do 2º grau em x

Paras as retas serem tangentes o discriminante ∆ deverá ser nulo ---> ∆ = 0 ---> Calcule m

por Alberto Nascente Qui 15 Dez 2022, 14:29

Ah sim, eu havia feito sem derivar tbm.
A questão eh q o exercício pede para ser feito usando derivada implícita, aí só consigo resolver se for pela geo. analítica.

O senhor sabe como q eu posso fazer esses cálculos derivando implicitamente?

Obrigado! Very Happy

por **tales amaral** Qui 15 Dez 2022, 14:41

Pra mim deu a mesma coisa. O ponto não tá na curva kk.

por Alberto Nascente Sex 16 Dez 2022, 08:30

Sim, vdd Tales
Quando derivei implicitamente, achei uma reta passando por um ponto dado q n está na curva, e essa reta nem passava pela curva tbm.

Seria um erro nosso? Ou de digitação da questão?

por **petras** Sex 16 Dez 2022, 09:18

Alberto Nascente escreveu:Encontre as equações de ambas as retas tangentes à elipse que passam pelo ponto (12,3)
S/ gabarito.

Derivei implicitamente, como em tese o capítulo ensina a fazer.
Porém, a reta que eu achei n eh tangente a curva, ela apenas passa pelo ponto dado...
Como pode resolver isso?

Obrigado!

[latex]\\x^2+4y^2 = 36(I) \implies 2xdx + 8ydy = 0 \implies \frac{dx}{dy} = -\frac{x}{4y} 4y^2 = 36-x^2(II)\\ r:y - y_o = \frac{dy}{dx}(x-x_o)\\ (12,3) \in r: y-3= -\frac{x}{4y}(x-12)\implies 4y^2-12y=-x^2+12x\\ De(II): (36-x^2)-12y=-x^2+12x \therefore y = -x+3\\ Substituindo(I): x^2+4(-x+3)^2=36\implies x(x-\frac{24}{5})=0\\ \therefore x = 0\implies y = -0+3=3\\ ~ou~x = \frac{24}{5}\implies y = -\frac{24}{5}+3 = -\frac{9}{5}\\ \frac{dy}{dx} = -\frac{x}{4y} = -\frac{0}{3} = 0\\ y -y_o = \frac{dy}{dx}(x-x_o)\implies y-3 = 0(x-0) = 3 \therefore \boxed{y=3}\\ (\frac{24}{5}, -\frac{9}{5}): y + \frac{9}{5}=\frac{2}{3}(x-\frac{24}{5} \implies \boxed{y=\frac{2x}{3}-5} [/latex]

(Solução:nakagumahissao)