C\303\241lculo IILi\303\247\303\243o 28: Equa\303\247\303\265es Polares e Param\303\251tricas Coordenadas PolaresO plano, bidimensional, requer dois n\303\272meros para idetificar unicamente um determinado ponto. Embora o ponto seja \303\272nico, estes dois n\303\272meros, ou coordendas, n\303\243o s\303\243o \303\272nicos, e dependem inteiramente sistema de coordenadas em quest\303\243o. Pode-se, por exemplo, selecionar um sistema de coordenadas no qual os eixos formam um \303\242ngulo igual a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjUGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiMtSSNtbkdGJDYkUSI0RidGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2. As coordenadas de um ponto em tal sistema s\303\243o diferentes quando o mesmo ponto \303\251 descrito em coordenadas cartesianas retangulares. As equa\303\247\303\265es que conectam as coordenadas de um sistema com as coordenadas de outro sistema s\303\243o chamadas equa\303\247\303\265es de transforma\303\247\303\243o. Um dos sistemas de coordenadas de maior uso \303\251 o sistema de coordenadas polares. Neste caso, para um determinado ponto, digamos (2,2), deve-se tra\303\247ar uma reta da origem a este ponto, cuja dist\303\242ncia \303\251 denominada por r, e cujo \303\242ngulo com o eixo x \303\251 denominado por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=. restart: with(plots): pointplot([[0,0],[2,2]],connect=true);

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r:=sqrt(2^2+2^2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoLCQqJCIiIyMiIiJGMEYwNyNGLg==

theta:=arctan(2/2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0aGV0YUdGKCwkSSNQaUdGJiMiIiIiIiU3I0Yu

Pode-se recuperar os valores de x e y: restart: X:=r*cos(theta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJYR0YoKiZJInJHRigiIiItSSRjb3NHRiU2I0kmdGhldGFHRihGMDcjRi4=

Y:=r*sin(theta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJZR0YoKiZJInJHRigiIiItSSRzaW5HRiU2I0kmdGhldGFHRihGMDcjRi4=

Coordenadas polares s\303\243o especialmente \303\272teis no tratamento de fun\303\247\303\265es peri\303\263dicas, curvas fechadas e se\303\247\303\265es c\303\264nicas. Uma importante quest\303\243o que surge \303\251 "Qual vari\303\241vel deve ser usada como vari\303\241vel independente?". No caso de coordenadas retantulares, convencionou-se usar x como a vari\303\241vel independente. J\303\241 no caso de coordenadas polares, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= \303\251 usada como a vari\303\241vel independente. O uso de coordenadas polares podem acarretar em simplifica\303\247\303\265es significativas nos c\303\241lculos e nas f\303\263rmulas. Para um c\303\255rculo de raio a, por exemplo, tem-se a equa\303\247\303\243o: r = a. Por\303\251m, n\303\243o existe uma fun\303\247\303\243o do Maple que converte automaticamente uma equa\303\247\303\243o para a forma polar. Para encontrar-se a equa\303\247\303\243o polar de um c\303\255rculo de raio r deve-se digitar os comandos: restart:with(plots): Eq1:=x^2+y^2=a^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTFHRigvLCYqJEkieEdGKCIiIyIiIiokSSJ5R0YoRjJGMyokSSJhR0YoRjI3I0Yu

x:=r*cos(theta): y:=r*sin(theta): simplify(Eq1);

LyokSSJyRzYiIiIjKiRJImFHRiVGJg==

r^2 = a^2;

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Existem dois m\303\251todos separados para tra\303\247ar fun\303\247\303\265es em coordenadas polares. O primeiro m\303\251todo utiliza a mesma sintaxe do comando plot para equa\303\247\303\265es param\303\251tricas, com a op\303\247\303\243o "coords=polar". plot([theta,theta, theta=0..2*Pi], coords=polar, title="Espiral de Arquimedes",scaling=constrained);

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O segundo m\303\251todo utiliza uma fun\303\247\303\243o especial do pacote plots: polarplot polarplot(theta, theta=0..2*Pi, title="Espiral de Arquimedes",scaling=constrained);

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Embora haja uma grande variedade de curvas interessantes cujas equa\303\247\303\265es s\303\243o mais facilmente expressas em coordenadas polares, estudaremos um grupo de equa\303\247\303\265es bastante familiar, as se\303\247\303\265es c\303\264nicas. Estas podem ser expressas de uma maneira puramente geom\303\251trica, livre de coordenadas. Seja D uma reta chamada diretriz e seja O o ponto que chamaremos de foco. Al\303\251m disso, seja e uma n\303\272mero positivo, chamado excentricidade. Ent\303\243o, o conjunto de todos os pontos tal que 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 \303\251 uma se\303\247\303\243o c\303\264nica. restart:with(plots): P1:=implicitplot({y^2=x-2,x=1},x=-5..3,y=-1..1,axes=none,title="Defini\303\247\303\243o de C\303\264nicas Livre de Coordenadas"): P2:=pointplot([[1,.63],[2.4,.63],[2.25,0]],connect=true): P3:=textplot([[1,.7,'D'],[2.4,.7,'P'],[2.3,0,'O']],align=RIGHT): display(P1,P2,P3);

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O tipo de se\303\247\303\243o c\303\264nica varia com a magnitude de e. O Maple pode ser usado para a transforma\303\247\303\243o desta defini\303\247\303\243o em coordenadas polares. Para o c\303\241lculo em coordenadas cartesianas, considera-se d como sendo a dist\303\242ncia de P a D, r a dist\303\242ncia de P a O e, ainda, considera-se as coordenadas do ponto, em rela\303\247\303\243o a um conjunto de eixos com centro em O e com o eixo y paralelo a D, como sendo (x, y). restart:with(plots): Eq1:=e=r/(d+x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTFHRigvSSJlR0YoKiZJInJHRigiIiIsJkkiZEdGKEYySSJ4R0YoRjIhIiI3I0Yu

x:=r*cos(theta): Eq1:=Eq1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTFHRigvSSJlR0YoKiZJInJHRigiIiIsJkkiZEdGKEYyKiZGMUYyLUkkY29zR0YlNiNJJnRoZXRhR0YoRjJGMiEiIjcjRi4=

Eq2:=r=solve(Eq1,r);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTJHRigvSSJyR0YoLCQqKEkiZEdGKCIiIkkiZUdGKEYzLCYqJi1JJGNvc0dGJTYjSSZ0aGV0YUdGKEYzRjRGM0YzISIiRjNGO0Y7NyNGLg==

A fim de analisarmos a fun\303\247\303\243o do par\303\242metro e, geramos uma sequ\303\252ncia de gr\303\241ficos para diferentes valores de e. p1:=(d,e)->polarplot(e*d/(e*cos(theta)+1),theta=-Pi..Pi,view=[-15..7.5,-10..10],scaling=constrained): Seja d = 10. S:=seq(p1(10,2*k/10),k=1..13): display(S);

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Resultam assim as seguintes curvas: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GKw== uma elipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GKw== uma par\303\241bola LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRImVGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGKw== uma hip\303\251rbole Ex. 2.1 Dada uma c\303\264nica com e = 0.5 e d = 10, deseja-se encontrar sua representa\303\247\303\243o em coordenadas cartesianas. restart: with(plots): Eq1:=r=(1/2)*10/((1/2)*cos(theta)-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTFHRigvSSJyR0YoLCQqJCwmLUkkY29zR0YlNiNJJnRoZXRhR0YoIyIiIiIiIyEiIkY4RjoiIiY3I0Yu

r:=sqrt(x^2+y^2): cos(theta):=x/r: Eq2:=Eq1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTJHRigvKiQsJiokSSJ4R0YoIiIjIiIiKiRJInlHRihGM0Y0I0Y0RjMsJCokLCYqJkYyRjRGMCMhIiJGM0Y3Rj1GNEY9IiImNyNGLg==

solve(Eq2,{x^2});

Warning, solving for expressions other than names or functions is not recommended.

NiQ8Iy8qJEkieEc2IiIiIywkKiRJInlHRidGKCEiIjwjL0YlLCpGJSMiIiIiIiVGKkYsRiYhIiYiI0RGMQ==

A primeira solu\303\247\303\243o \303\251 trivial; um par de retas. Eq3:=%[2,1];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTNHRigvKiRJInhHRigiIiMsKkYvIyIiIiIiJSokSSJ5R0YoRjEhIiJGMCEiJiIjREY0NyNGLg==

Com algumas de simplifica\303\247\303\265es alg\303\251bricas, tem-se: Expr1:=rhs(Eq3)-x^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZFeHByMUdGKCwqKiRJInhHRigiIiMjISIkIiIlKiRJInlHRihGMSEiIkYwISImIiNEIiIiNyNGLg==

with(student): Expr2:=completesquare(Expr1,[x,y]);

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Eq4:=Expr2=0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTRHRigvLCgqJCwmSSJ4R0YoIiIiIyIjNSIiJEYzIiIjIyEiJCIiJSMiJCsiRjZGMyokSSJ5R0YoRjchIiIiIiE3I0Yu

implicitplot(Eq4,x=-10..4,y=-6..6,scaling=constrained);

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O Maple possui a fun\303\247\303\243o changecoords, para convers\303\243o de fun\303\247\303\265es em diferentes sistemas de coordenadas. changecoords Lamentavelmente, se o pacote plots for carregado, a fun\303\247\303\243o changecoords n\303\243o funciona. No entanto, n\303\243o \303\251 dif\303\255cil criar uma fun\303\247\303\243o para a transforma\303\247\303\243o de express\303\265es ou equa\303\247\303\265es. restart:with(plots): P:=proc(z) local f,eq: eq:=z: subs({x=r*cos(theta),y=r*sin(theta)},eq): simplify(simplify(%,trig),symbolic); end proc: h:=1=(x^2+y^2)/sqrt(x^2-y^2);

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h[p]:=P(h);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiaEdGKDYjSSJwR0YoLyIiIiomSSJyR0YoRjIsJiokLUkkY29zR0YlNiNJJnRoZXRhR0YoIiIjRjshIiJGMiNGPEY7NyNGMQ==

O leitor deve notar que, embora sejam usadas coordenadas polares, os pontos do gr\303\241fico s\303\243o gerados em coordenadas cartesianas. Isto pode n\303\243o ser satisfat\303\263rio. O pacote plots cont\303\251m uma fun\303\247\303\243o chamada coordplot, que gera gr\303\241ficos em coordenadas polares. coordplot Por tr\303\241s da apar\303\252ncia simples de sua sintaxe, esconde-se uma fun\303\247\303\243o bastante complicada. Esta fun\303\247\303\243o pode ser usada para gerar a estrutura do sistema de coordenadas polares, que pode ser exibida juntamente com a estrutura do gr\303\241fico polar. Ex. 2.2 Representar graficamente a espiral de Arquimedes em coordenadas polares. restart: with(plots): p1:=coordplot(polar,[0..32,0..10*Pi],view=[-32..32,-32..32],labelling=false, scaling=constrained, color=[black,black], numpoints=1000): O comando acima apresenta a estrutura do gr\303\241fico, cujo eixo possui 32 unidades e cuja dist\303\242ncia angular \303\251 de 10p. O op\303\247\303\243o view indica que deseja-se visualizar toda a extens\303\243o dos eixos, em cada dire\303\247\303\243o. A op\303\247\303\243o labelling fornece uma escala para as coordenadas radial e angular. p2:=polarplot(theta,theta=0..10*Pi,thickness=2): display(p1,p2);

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Equa\303\247\303\265es Param\303\251tricasO que \303\251 uma curva? Uma curva \303\251 a imagem de um segmento da reta real sob um mapeamento, ,f: t -> 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, onde as coordenadas x e y s\303\243o fun\303\247\303\265es de uma \303\272nica vari\303\241vel independente. Pela escolha de t como um ponto da reta real, tem-se um mapeamento da reta real em uma curva no plano. Na verdade, existem apenas duas formas dististas para a escolha de t; como um ponto da reta real ou como um ponto no c\303\255rculo unit\303\241rio. Obviamente, \303\251 poss\303\255vel considerar a reta real como um n\303\272mero infinito de c\303\255rculos estendidos, ou ainda, pode-se considear o c\303\255rculo como a reta real espiralando-se um n\303\272mero infinito de vezes. Os pontos na reta real s\303\243o geralmente designados por x. Portanto, a curva \303\251 o conjunto de pontos (x, f(x)) que formam o gr\303\241fico de f(x).A representa\303\247\303\243o de um mapeamento t -> 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 \303\251 uma representa\303\247\303\243o param\303\251trica com par\303\242metro t. Muitas curvas s\303\243o mais facilmente expressas nesta forma. Um exemplo cl\303\241ssico \303\251 a cicl\303\263ide; a curva determinada por um ponto na circunfer\303\252ncia de um c\303\255rculo que gira sem deslizar ao longo da reta. Ex. 2.3 Tra\303\247ar o gr\303\241fico da cicl\303\263ide. X:=t-sin(t):Y:=1-cos(t): Note que s\303\243o usadas as letra mai\303\272sculas X e Y. Lembre-se que o Maple \303\251 sens\303\255vel \303\240 varia\303\247\303\243o entre letras mai\303\272sculas e min\303\272sculas. A sintaxe do comando plot para equa\303\247\303\265es param\303\251tricas requer aten\303\247\303\243o especial. As fun\303\247\303\265es f(t), y(t) e o intervalo devem estar entre colchetes, formando assim uma lista. Se o intervalo for inserido fora dos colchetes, o Maple tra\303\247a o gr\303\241fico de duas curvas separadas. plot([X,Y,t=0..13], scaling=constrained);

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Tem-se aqui uma curva que n\303\243o \303\251 fun\303\247\303\243o. (P\303\264r que?) Qual a forma desta curva em coordenadas Cartesianas? O Maple possui uma fun\303\247\303\243o chamada eliminate, que apresenta a fun\303\247\303\243o equivalente no sistema de coordenadas padr\303\243o. eliminate Eqn:=eliminate({x=X,y=Y},t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcW5HRig3JDwjL0kidEdGKComLChJInhHRigiIzVJInlHRigiIiIhIzVGN0Y3RjQhIiI8Iyw0KiRGNCIiJCIkLCIqJkY0IiIjRjZGNyIjPyomRjRGN0Y2RkBGNyokRjRGQCEkKyQqJkY0RjdGNkY3ISNTKiRGNkZARjlGNCIkKyRGNkZBISQrIkY3NyNGLg==

A sa\303\255da deste comando \303\251 uma lista contendo dois conjuntos. No segundo conjunto, tem-se uma express\303\243o que, quando igualada a 0, \303\251 uma equa\303\247\303\243o impl\303\255cita em x e y. Estamos particularmente interessados no termo isolado do segundo conjunto, e n\303\243o propriamente no conjunto. No comando a seguir, Eqn[2,1] ou Eqn[2][1] selecionam o primeiro (e \303\272nico) membro do segundo conjunto. Introduz-se tamb\303\251m a fun\303\247\303\243o collect , que \303\251 bastante conveniente neste caso. collect Eqn:=collect(Eqn[2,1],x,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcW5HRigsKiokSSJ4R0YoIiIkIiQsIiomLCZJInlHRigiIz8hJCskIiIiRjhGMCIiI0Y4KigsJkY1RjghIzVGOEY4LCZGNUY4ISNJRjhGOEYwRjhGOCokRjtGOSEiIjcjRi4=

eq:=solve(Eqn,y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNlcUdGKDYkKiYsKiokSSJ4R0YoIiIjISM1KiQsJiokRjIiIiUhIiIqJEYyIiIkIiIiI0Y8RjNGPEYyIiM/RjRGPEY8LCZGMkY8RjlGPEY5LCQqJiwqRjEiIzVGNUY8RjIhIz9GQ0Y8RjxGP0Y5Rjk3JEYvRkA=

with(plots): plot([eq[1],eq[2]],x=0..1,color=[red,blue]);

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O comando eliminate pode ser melhor explorado com fun\303\247\303\265es polinomiais. Em alguns casos ainda \303\251 necess\303\241rio fazer os c\303\241lculos "\303\240 moda antiga"; resolver uma equa\303\247\303\243o para t e substituir na outra. Uma fun\303\247\303\243o expressa na forma "usual" como y = f(x) pode ser expressa paramentricamente como f: t -> 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 . Para diferenciar uma fun\303\247\303\243o paramentricamente definida, na forma 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 , deve-se calcular 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 . Ex. 2.5 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 , 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 . Encontrar dy/dx . restart: Y:=t^3: X:=t^2: dY:=diff(Y,t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkWUdGKCwkKiRJInRHRigiIiMiIiQ3I0Yu

dX:=diff(X,t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkWEdGKCwkSSJ0R0YoIiIjNyNGLg==

`dy/dx`:=dY/dX;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkeS9keEdGKCwkSSJ0R0YoIyIiJCIiIzcjRi4=

Curvas paramentricamente definidas podem ser enganosas. Coisas estranhas podem acontecer. Ex. 2.6 Tra\303\247ar o gr\303\241fico da espiral 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 e da par\303\241bola 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 no mesmo conjunto de eixos. Resolver o sistema de equa\303\247\303\265es. restart: with(plots): polarplot({[t,t^2,t=0..10], [1/(-sin(t)+1),t,t=0..10]}, title="Par\303\241bola e Espiral", view = [-10..10, -10..10]);

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Embora possa parecer que existem muitas solu\303\247\303\265es para este conjunto de equa\303\247\303\265es, isto n\303\243o \303\251 verdade! Para que uma solu\303\247\303\243o exista, r e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= devem ser iguais nas duas equa\303\247\303\265es. Para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=, isto acontece quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiMtSSVtc3VwR0YkNiVGMS1JI21uR0YkNiRRIjJGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRis= , ou seja, apenas em 0 e 1. Por\303\251m, para nenhum destes valores de t, o valor r de uma equa\303\247\303\243o ser\303\241 igual ao de outra. Portanto, n\303\243o existe um valor de t onde estes gr\303\241ficos interceptam-se. Aplica\303\247\303\265es 1. a) Escrever a equa\303\247\303\243o polar para uma elipse. b) Tra\303\247ar o gr\303\241fico desta equa\303\247\303\243o na forma polar. c) Converter a equa\303\247\303\243o para coordenadas cartesianas. d) Tra\303\247ar o gr\303\241fico da equa\303\247\303\243o em coordenadas cartesianas. 2. Repetir o problema #1 para o caso de uma hip\303\251rbole e uma par\303\241bola. 3. Seja 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, 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 . a) Tra\303\247ar o gr\303\241fico da curva param\303\251trica.b) Encontrar uma equa\303\247\303\243o que relaciona x e y.c) Tra\303\247ar o gr\303\241fico da equa\303\247\303\243o que relaciona x e y. 4. Utilizar o Maple para mostrar que, caso a diretriz seja paralela ao eixo x, a forma polar de uma c\303\264nica \303\251 dada por: 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 or 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