<Text-field style="Heading 1" layout="Heading 1">Introdu\303\247\303\243o</Text-field>

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Guia de Estudo de C\303\241lculo

\302\251 Maplesoft, a division of Waterloo Maple Inc., 2005

Cap\303\255tulo 15 - Rela\303\247\303\243o de Propor\303\247\303\265es (Proporcionalidade)

Quando v\303\241rias quantidades est\303\243o relacionadas por uma equa\303\247\303\243o, a deriva\303\247\303\243o da equa\303\247\303\243o fornece uma rela\303\247\303\243o envolvendo as quantidades e suas varia\303\247\303\265es (rates of change). No problema t\303\255pico de (proporcionalidade?)rela\303\247\303\243o de propor\303\247\303\265es (related-rates),a rela\303\247\303\243o original e alguma combina\303\247\303\243o de quantidades e suas varia\303\247\303\265es est\303\243o condicionadas (relacionadas). O objetivo \303\251 determinar algumas das quantidades n\303\243o conhecidas ou as varia\303\247\303\265es( rates of change). As chaves para o sucesso nestes problemas s\303\243o reunir as informa\303\247\303\265es, identificar a vari\303\241vel para derivar em rela\303\247\303\243o a ela, derivar corretamente a rela\303\247\303\243o original e resolver para encontrar as quantidades desconhecidas.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Cap\303\255tulo Gloss\303\241rio</Text-field>

Os seguintes termos, neste cap\303\255tulo, tem conex\303\243o com o Dicion\303\241rio de Matem\303\241tica Maple ( Maple Math Dictionary).

area

asymptote

chainrule

constant

coordinatesystem

decreasing

derivative

differentiate

differentiation

even

explicit

floatingpoint

function

graph

increasing

independent variable

instantaneous

mean-valuetheorem

positive

proportion

radius

rateofchange

real

solution

solve

sphere

substitution

value

variable

volume

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Inicializa\303\247\303\243o</Text-field>

restart;

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 15.1</Text-field>

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Enunciado do Problema</Text-field>

O volume (volume) de um bal\303\243o est\303\241 aumentando (increasing) a uma raz\303\243o de 400 polegadas c\303\272bicas por minuto. Encontre a raz\303\243o de crescimento (rate of increase) do raio (radius) no momento quando o raio \303\251 8 polegadas.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Solu\303\247\303\243o 1</Text-field>

Denotamos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= como o volume e o raio do bal\303\243o esf\303\251rico (spherical) , respectivamente, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= o tempo (a vari\303\241vel independente (independent variable) neste problema). O problema nos diz que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZFZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= = 400 polegadas c\303\272bicas/min e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiOEYnRj5GPkYrRj4= polegadas. O objetivo \303\251 determinar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= no momento quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiOEYnRj5GPkYrRj4= polegadas e 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 polegadas/min.

A rela\303\247\303\243o entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= est\303\241 relacionada (condicionada) no problema. Contudo, como o bal\303\243o \303\251 uma esfera, n\303\263s sabemos que a rela\303\247\303\243o entre o raio e o volume \303\251 dado por

q1 := V(t) = 4/3*Pi*r(t)^3: q1;

A deriva\303\247\303\243o desta rela\303\247\303\243o com respeito a t fornece (pela Regra da Cadeia (Chain Rule))

q2 := diff( q1, t ): q2;

Resolver para a varia\303\247\303\243o 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 desconhecida, obtendo

q3 := isolate( q2, diff(r(t),t) ): q3;

Para obter 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, o valor (value) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= no momento quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiOEYnRj5GPkYrRj4=, substituir (substitute) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkmbWZyYWNHRiQ2KC1GIzYkLUYsNiVRI2RWRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y9USdub3JtYWxGJy1GIzYkLUYsNiVRI2R0RidGOUY8Rj8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRksvJSliZXZlbGxlZEdRJmZhbHNlRictSSNtb0dGJDYtUSI9RidGPy8lJmZlbmNlR0ZQLyUqc2VwYXJhdG9yR0ZQLyUpc3RyZXRjaHlHRlAvJSpzeW1tZXRyaWNHRlAvJShsYXJnZW9wR0ZQLyUubW92YWJsZWxpbWl0c0dGUC8lJ2FjY2VudEdGUC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRl9vLUkjbW5HRiQ2JFEkNDAwRidGP0Y/RitGPw==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiOEYnRj5GPkYrRj4=, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1GLDYlUSNkdEYnRjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ== 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 isto \303\251, usar os dados seguintes.

d1 := [ diff(V(t),t)=400, r(t)=8, diff(r(t),t)=`r'`(8) ]: d1;

O resultado final \303\251 ent\303\243o

q4 := eval(q3,d1): q4; cat(` `$22) = evalf(rhs(q4));

Quando o volume est\303\241 crescendo a 400 polegadas c\303\272bicas por minuto e o raio \303\251 8 polegadas, o raio est\303\241 crescendo a uma raz\303\243o de 0.497 polegadas por minuto

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Solu\303\247\303\243o 2 - (Rela\303\247\303\243o de Propor\303\247\303\265es?)Proporcionalidade (<Hyperlink linktarget="Dictionary:Definition,rateofchange" hyperlink="true">RelatedRates</Hyperlink>) Maplet</Text-field>

A solu\303\247\303\243o (solution) deste problema pode tambem ser obtida usando a rela\303\247\303\243o de propor\303\247\303\265es(RelatedRates maplet), como mostrado na Figura 15.1.

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Figura 15.1 Solu\303\247\303\243o do Exemplo 15.1 pela Calculadora Proporcionalidade e Diferencial Impl\303\255cita (Related Rates and Implicit Differentiation Calculator)

Para obter esta solu\303\247\303\243o, siga os passos abaixo.

<Text-field style="Heading 3" layout="Heading 3">Passo 0</Text-field>

Inicie o RelatedRates maplet .

<Text-field style="Heading 3" layout="Heading 3">Passo 1</Text-field>

Entre com todas as informa\303\247\303\265es da seguinte forma:

Entre com a f\303\263rmula do volume de uma esfera, V=4/3*Pi*r^3, em Equa\303\247\303\243o (Equation) ou Campo de Rela\303\247\303\243o (Relationship field).

No campo da Vari\303\241vel(is) Dependente(s) (Dependent Variable(s)) , entre com V e r dentro dos colchetes: [ V, r ]

Tempo \303\251 a vari\303\241vel independente; no campo da Vari\303\241vel Independente (Independent Variable), entre com t.

N\303\243o existem constantes (constants) (par\303\242metros que n\303\243o mudam durante todo o problema); N\303\243o mude o campo das Constantes (Constants field): [ ]

A informa\303\247\303\243o de que o raio \303\251 8 polegadas \303\251 especificado pela mudan\303\247a de Valores do campo da Vari\303\241vel (Variables) para [ r = 8 ]

A \303\272nica taxa de varia\303\247\303\243o dada \303\251 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 inLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlRistSSNtbkdGJDYkUSIzRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjhGK0Y4 /min; no campo de Valores da Taxa de Varia\303\247\303\243o (Values of Rates of Change), entre com D(V) = 400 Note a nota\303\247\303\243o usada para indicar a taxa de varia\303\247\303\243o. Esta \303\251 uma nota\303\247\303\243o especial que \303\251 v\303\241lida somente no contexto do Raz\303\243o de Proprorcionalidade (RelatedRates maplet).

A taxa de varia\303\247\303\243o do raio \303\251 a quantidade que procuramos determinar; a Quantidade para o determinado campo deve conter D(r)

<Text-field style="Heading 3" layout="Heading 3">Passo 2</Text-field>

Preparar a f\303\263mula do volume para derivar, ativar o campo da Equa\303\247\303\243o ou Rela\303\247\303\243o(Equation /Relationship) . O campo ativo deve ser agora V = 4/3*Pi*r^3

<Text-field style="Heading 3" layout="Heading 3">Nota: " Equa\303\247\303\243o Ativa" </Text-field>

O maplet realiza opera\303\247\303\265es no ( Equa\303\247\303\243o Ativa/Rela\303\247\303\243o) Active Equation/Relationship. Para fazer "ativos" os conte\303\272dos de um campo, posicione o cursor sobre o campo e clique o bot\303\243o direito do mouse, selecione a segunda entrada: Make Active (ativar).

<Text-field style="Heading 3" layout="Heading 3">Passo 3</Text-field>

Derivar o volume com respeito ao tempo clicando o bot\303\243o Diff wrt indep var . O resultado desta deriva\303\247\303\243o (differentiation) \303\251 mostrada usando D(V) e D(r) para representar as varia\303\247\303\265es do volume e do raio, respectivamente. Concluir fazendo deste resultado a equa\303\247\303\243o ativa. A equa\303\247\303\243o ativa deve ser agora D(V) = 4*Pi*r^2*D(r)

<Text-field style="Heading 3" layout="Heading 3">Passo 4</Text-field>

Substituir os valores e propro\303\247\303\265es dados na rela\303\247\303\243o de derivada differentiated , clicar o bot\303\243o Subs variables, ativar resultado, clicar no bot\303\243o Subs rates of change, e fazer este resultado ativo. (As duas substitui\303\247\303\265es podem ser feitas em qualquer ordem.) No final do passo 4 a equa\303\247\303\243o ativa deve ser 400 = 256*Pi*D(r)

<Text-field style="Heading 3" layout="Heading 3">Passo 5</Text-field>

Resolver para (encontrar) a quantidade desconhecida clicando o bot\303\243o (Resolver para a quantidade desconhecida) Solve for unknown quantity.

(Para ver uma vers\303\243o com ponto flutuante (floating-point) de qualquer resultado, clicar o bot\303\243o direito do mouse sobre um campo, selecionar o a primeira op\303\247\303\243o do menu (Manipulate), e depois evalf.) O resultado deve ser D(r) = 25/16/Pi ou D(r) = .4973591970

Isto completa a solu\303\247\303\243o usando othe solution using the Rela\303\247\303\243o de Propor\303\247\303\265es (RelatedRates) maplet. Para este problema o maplet n\303\243o \303\251 realmente necess\303\241rio. Mas, quando existe mais vari\303\241veis, e uma rela\303\247\303\243o mais complicada, ou outros aspectos complicados, o maplet pode ser de grande ajuda para economizar tempo.

QUESTAO 15.1

Est\303\241 sendo removido ar de um bal\303\243o esf\303\251rico de forma que seu volume increaceses a uma taxa de 170 cm^3/s, quando o di\303\242metro \303\251 `$ D1 cm`? (O que \303\251 dr/dt?)

Digite Pi para representar"Pi".

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 15.2</Text-field> <Text-field style="Text" layout="Normal"></Text-field>

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Enunciado do Problema</Text-field>

Um avi\303\243o voando para o norte a (uma velocidade) 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 km por hora sobrevoa sua casa ao meio dia. Uma hora depois um avi\303\243o voando a 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 km por hora muda a trajet\303\263ria para o leste quando est\303\241 sobrevoando sua casa. (Assuma que os avi\303\265es voem nesta trajet\303\263ria para sempre.)

(i) Encontre a varia\303\247\303\243o da dist\303\242ncia entre os avi\303\265es \303\240s at 3 p.m..

(ii) Encontre a varia\303\247\303\243o da dist\303\242ncia entre os avi\303\265es para qualquer tempo depois de 1 p.m.. (Porque esta resposta n\303\243o se aplica para um tempo antes de 1 p.m.?)

(iii) Descreva a varia\303\247\303\243o da dist\303\242ncia entre os avi\303\265es ao longo do tempo ap\303\263s sobrevoar a sua casa .

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Solu\303\247\303\243o</Text-field>

A rela\303\247\303\243o fundamental neste problema \303\251 a dist\303\242ncia entre os avi\303\265es a qualquer tempo depois de 1 p.m.. Assuma LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= correspondendo a 1 p.m.. Defina um sistema de coordenadas (coordinate system) no qual a sua casa est\303\241 na origem, a dire\303\247\303\243o norte \303\251 o eixo-y positivo (positive) , e a dire\303\247\303\243o leste \303\251 o eixo-x positivo. Ent\303\243o, para qualquer tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 0, a posi\303\247\303\243o dos dois avi\303\265es s\303\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 e 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, respectivamente, quando 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 e 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 s\303\243o as dist\303\242ncias dos dois avi\303\265es de sua casa. Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 a dist\303\242ncia entre os avi\303\265es, temos

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 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbXNxcnRHRiQ2Iy1GIzYoRistRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJ0Y0RjctSSNtbkdGJDYkUSIyRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y+LUY7Ni1RIitGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGZm8tRiM2JkYrLUZmbjYlLUYsNiVRInlGJ0Y0RjdGW29GX29GK0Y+RitGPkYrRj5GK0Y+

Al\303\251m disso, foram dadas as varia\303\247\303\265es 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 km/h e 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 km/h.

As quantidades LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= s\303\243o todas fun\303\247\303\265es (functions) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Na maplet Proporcionalidade (RelatedRates maplet) , entre com qualquer das rela\303\247\303\265es(equa\303\247\303\265es) acima entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e as vari\303\241veis dependentes e independentes. A rela\303\247\303\243o(equa\303\247\303\243o) diferencial das dist\303\242ncias fornece

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 or 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

(Tome um minuto para convencer a voc\303\252 mesmo que estas (rela\303\247\303\265es) s\303\243o equivalentes.)

Entrando no maplet com a informa\303\247\303\243o das varia\303\247\303\265es dadas e substituindo estes valores em qualquer das f\303\263rmulas diferenciais de dist\303\242ncia leva a

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInNGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= 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 or 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

Resolvendo a primeira delas para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= fornece um modo conveniente de resumir ambos os resultados:

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

<Text-field style="Heading 3" layout="Heading 3">(i)</Text-field>

\303\201s 3 p.m., LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GPkYrRj4= , o primeiro avi\303\243o viajou

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 = 3000 km

al\303\251m da sua casa enquanto o avi\303\243o de menor velocidade viajou somente

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 = 1600 km

(Lembre que o primeiro avi\303\243o sobrevoa sua casa uma hora antes do segundo avi\303\243o.) A separa\303\247\303\243o deles no tempo \303\251

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 = 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 = 3400 km

Substituindo estes valores na f\303\263rmula para a varia\303\247\303\243o da dist\303\242ncia temos

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSYyMTQwMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y4LUYjNiQtRjU2JFEjMTdGJ0Y4RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkUvJSliZXZlbGxlZEdRJmZhbHNlRidGK0Y4 = 1258.824 km/hr

Figura 15.2 mostra como a Calculadora Proporcionalidade e Diferencial Impl\303\255cita (Related Rates and Implicit Differentiation Calculator) pode ser usada para obter estes resultados.

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BymfLax]Wy=;gewulAhEifgiy?oiyAWdItU_smsb>AD;KDSAD_GUFKwCIBaIU@?bheB@kR?aRsKI>YTVKebOFm[HD?O:\\<:::::RU_toyI:CY:MwYiH_yI:oYEOWu_CXAgaMBBevJIdD_cHIIdAC;wEKax\\cW^_SMEhbYB=OYUIb[cG`GFFoWYiwYoYRyBxgxISvsoVUQTLchecSleSegSlWS;GXacRTQCG_F<CxVKFqEfV=VSwDL;CMKDtOEG[YacXNCgGwxDehB?VGMB?UuTUUBEt:;bUCul;E:=BI;iNyI:?xEMwY[XrxQjuucyY:\\rPaunXyLdn_@T\\ivnDxgi][_pGX_NOtgpgRXs`XeRoe\\vri?ig^g]Ikhq`rf[hoqUWo@^uyX_G^j]o`owjrYj^NqlIgQqi[>kGGhUF[^gmR?]vwwWh_Kif;VmEqsTn^=H]=if<hpown<QyCXw]i[I?uwQjJymwUw;GxAeNyqJIVcXxQey<XtY\\yluLgpq]qVQPVVtTXyo=HvOYK?EYwIxi<us@QHHq<aNqLwSdtZelq=pgdvr<RcYmqAk?HUBhwVqSyXWk]obuLjxRfpUduPpMlB`o_xPNHortuU@sMivI<xkLKotNFxkSdtDILSyY:Lrivy:FNtmVIwpamy=:wc?cxMsimocayEuybSyEaceVahusbnSQYLLUAyYYSNapBmOvircHWI@WEUj_\\WxayQLLUIJtlR\\yJJ`NmUR@yRPTKhXn@iueHNUAikGrag]PyoaxcG^q\\Hc^WqyFfdNk@Psb>lsnrLQkLhwDfrEOjFqcdYwyGvQ^tu>y=YdyAZ:Y[HguyfvW@_jqr\\HaXF`OIZyAZiX[Ioyqvbgna[ishnriWjh^anOqNY`rwjK?mS?[HIn^yl:^oLYna>uUa]OFkqGaKOxhQxDqmRn]nQwfAaDIxiaeZ@x@An@vZEhxjvZ=fy<Vt;Hwxyo_x^vY^Tvkwq[TvmxAZTwika\\gycrHq_@uy`\\@YvAQi>xncHtfyjYfxXyxvY_TWpnf_S^`q@qXXceio<ow[IdXFiOY\\BVmDP^\\@ZdHZq_ao__eocmXlQQbZPmTok\\PkdN[Dbb;f<Kcb:j[hN^\\p;<SvYsy=JIvqoYwcxYMy>]xKCxikywxiihEyI:KdaOVtEWfqVCkIXSgksidmSyGXa[xAmyvOttgsIiyA_YGai`kilid[is_AtnCS_;e\\AsCmfVEfjCd:IIS;xSAbnqvy`jKyY:<w;pny=JZitiuY^ltlEJqqqH@uy=JVyY:@sKIpy=jrxY:Hli\\my=J[]py=JtAjcyQrelu?ZL@_yE[]xy;BtTty=JjQoqyi^^dliZIYnpxA^`fwihya^Ygay]ricYny:Paioy?neyAZlibY>xTxawAZnwuvQ[wxy:^vDHa<vvKYuYVyIHyTn`yvXPmsyRxUlAyY:<nHXmyOZixisQ^I@yG;:^];XhbiwIhyAayGX=jvYVyIk]xiZp[Uxy:FjQwqvIlA_y`vK:?:tRZ`UtUq]uykuY]prAqIlqkyY:`:<w=pny?Rrbwy>I\\Z<:;CQoyHAuy;RVV^AolyAZVyiBytHyajQM:u_^uI=IGyex;RtyGxerXyI:MH?_U:cyWgvyjbAuy=jmiSyTNYAqy_J>WBrxHCSNyI:M:[@<FB^ieIWFpyqsqvYIL]Xf:nmgGyTFYZZ<=>^YBJJ;ZY@^hhiWryXHpbyEIqhvesvlI>B:vK::FJryl=cCU`YmywByTHyqjM]t=j^wIKuyLI:_NlijIYNpxqvkKivytxY:::x;:pECQh@rsvwgvy:>Q@CFHHliJ[Kt;ZyE[Bt:BCaDsQrabu>gRyy:^UKxo:ZwLlwA::`U;XHJGT=rvlI>Bb::::l;phw]U:`IJJ[;xDJM:b:JA;:uAs=M:[@<FbyJJ;ZZ:FHHL]XVyKYK<Kn;xHCSNF;>T:B^WSYujy?Q@CFHHliJ[Kt;b^K[>\\b=:\\vEtMAKhSnBhFli`Mbx:FSy=LI:^VJq;lgByT:jXCsx=::_U;YPU[nJqsT::jI:jX[s:Zp:esvlI>Bb:jA;:BA=>ZU:`IJJ[;xDByG;Z<:W>:YVEyqmoy@oQ^J>WZiMLvYM:[@<FRpEvmA_vYxi?sK]aeYtELB\\H:ZC::rcKx?w;HIrytC:>N::>Bb::jyoMn@sAi@:fQLhyZt=j^;:]@IAJ<Kn;JLj>:BjjK=CvYIAUny_AP=sA<>N:r]::RZvW>Q@sKiF:::x;:pECQhjp;::<DFjQ;T<:?:tBbyZw]v[CZJ=;@nYBb:JJ:@AAcKtItELB\\H:L<`:<G:>Lry\\EVF@AAsQbjBw:yQCwYC>KA:>SZVmbKVS^Y?NY:`fy;cA:LHbu>GVXyI:A;DFLI:vYC>KA:HOlomwHxk>eUOeL`kAFuooaB?ryAZIx^\\`qv?yDydE_lGw[FGkXqtr_b<Xea`fSW^b>uYY`F>vX`tP_t]AnnxayIU:6:\"\{\}-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YGF'-%-BOUNDS_WIDTHG6#$"%+gF*-%.BOUNDS_HEIGHTG6#$"%gZF*-%)CHILDRENG6"

Figura 15.2

Solu\303\247\303\243o do Exemplo 15.2, parte (i) pela Calculadora Proporcionalidade e Diferencial Impl\303\255cita (Related Rates and Implicit Differentiation Calculator)

<Text-field style="Heading 3" layout="Heading 3">(ii)</Text-field>

Para um tempo ap\303\263s 1 p.m., LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRidGOQ==o primeiro avi\303\243o viajou

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 km

al\303\251m da sua casa enquanto o avi\303\243o que passou mais tarde viajou

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

km al\303\251m da sua casa. Portanto, a separa\303\247\303\243o entre as naves \303\251

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbXNxcnRHRiQ2Iy1GIzYoRistRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJ0Y0RjctSSNtbkdGJDYkUSIyRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y+LUY7Ni1RIitGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGZm8tRiM2JkYrLUZmbjYlLUYsNiVRInlGJ0Y0RjdGW29GX29GK0Y+RitGPkYrRj5GK0Y+ = 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

e a varia\303\247\303\243o da separa\303\247\303\243o delas \303\251

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

Para confirmar estes resultados, note que substituindo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GPkYrRj4= na primeira f\303\263mula fornece 3400, enquanto que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= na segunda f\303\263rmula poderia ser obtida pela deriva\303\247\303\243o da f\303\263mula da dist\303\242ncia expressa explicitamente ( explicitly) em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .

Estas f\303\263mulas n\303\243o s\303\243o aplic\303\241veis para tempo anterior a 1 p.m. (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < 0) porque n\303\243o conhecemos a trajet\303\263ria do segundo avi\303\243o antes 1 p.m..

Error, invalid sequence

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

<Text-field style="Heading 3" layout="Heading 3">(iii)</Text-field>

A f\303\263rmula geral para uma varia\303\247\303\243o da separa\303\247\303\243o entre os duas naves, como uma fun\303\247\303\243o de t, foi encontrada em (ii), e \303\251

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

Figura 15.3 cont\303\251m um gr\303\241fico (graph) desta varia\303\247\303\243o e mostra como a raz\303\243o varia com o tempo.

q1 := 200*(25+41*t)/sqrt( 25 +50*t+41*t^2 ): plot( q1, t=0..10, '`ds/dt`'=0..1500 );

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

Figura 15.3 A varia\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRI2RzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRI2R0RidGNUY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZDLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRisvRjlRJ25vcm1hbEYn como uma fun\303\247\303\243o do tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

Deste gr\303\241fico fica evidente que a varia\303\247\303\243o instant\303\242na da separa\303\247\303\243o se aproxima de uma constante de uma forma bem r\303\241pida. O valor exato desta assintota (asymptote) horizontal \303\251

q2 := Limit( q1, t=infinity ): q2 = value( q2 ); `` = evalf( q2 );

QUESTAO 15.2

Uma escada 10 ft repousa contra uma parede vertical. Se a ponta da escada desliza afastada da parede, a uma taxa de 2 ft/s. O quanto r\303\241pido \303\251 o topo da escada deslizar para baixo da parede, em ft/s quando a parte inferior da escada \303\251 8 ft do muro? (O que \303\251 dy/dt, Como um valor positivo?)

QUESTAO 15.3

Uma escada 10 ft repousa contra uma parede vertical. Se o topo da escada est\303\241 sendo puxado para cima Da parede, a uma taxa de 2 ft/s. Em que a taxa, em ft/s, a parte inferior da escada que se deslocam em dire\303\247\303\243o \303\240 parede, quando o topo da escada \303\251 8 ft do ch\303\243o? (O que \303\251 dy/dt?)

QUESTAO 15.4

Dois carros se aproximam de um cruzamento em diferentes dire\303\247\303\265es. Carro 1 faz uma viagem leste em 5 mph; Carro 2 foi viajar ao sul em 60 mph. A que taxa, em km / h, a dist\303\242ncia entre os carros mudam quando o carro 1 \303\251 o `$ d1\302\273 mi e o carro 2 \303\251 '$ d2' mi da interse\303\247\303\243o?

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 15.3</Text-field>

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Enunciado do Problema</Text-field>

Um tanque de \303\241gua esf\303\251rico de 20 p\303\251s de raio \303\251 esvaziado a uma raz\303\243o de 2000 p\303\251s c\303\272bicos por hora.

(i) Assumindo que o tanque estava cheio at\303\251 a metade ao meio dia , Encontre as alturas da \303\241gua (no tanque) quando 25%, 50%, 75%, e 100% dela havia sido retirada do tanque.

(ii) Qual \303\251 a raz\303\243o em que a altura da \303\241gua no tanque est\303\241 decrescendo decreasing quando metade do volume de \303\241gua original

j\303\241 tinha sido retirada?

<Text-field style="Heading 3" layout="Heading 3">Suget\303\243o: <Hyperlink linktarget="Dictionary:Definition,volume" hyperlink="true">Volume</Hyperlink> de um Segmento de uma Esfera</Text-field>

O volume de um segmento de uma esfera de raio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= cheia at\303\251 uma altura LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251

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

(15.1)

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Solu\303\247\303\243o</Text-field>

<Text-field style="Heading 3" layout="Heading 3">(i)</Text-field>

Esta parte do problema n\303\243o envolve proporcionalidade (related rates), somente \303\241gebra.

Designando meio dia como o tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. O tanque estava cheio (at\303\251 a metade) quando 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; o volume inicial no tanque \303\251

v := Pi*h^2*(r-h/3): v0 := eval( v, h=r );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjdjBHRiYsJCooIyIiIyIiJCIiIkkjUGlHJSpwcm90ZWN0ZWRHRjEpSSJyR0YmRjBGMUYxNyNGLA==

Um quarto do volume teria sido retirado do tanque quando

q1 := v = 3/4*v0: q1;

LyooSSNQaUclKnByb3RlY3RlZEciIiIpSSJoRzYiIiIjRiYsJkkickdGKUYmKiYjRiYiIiRGJkYoRiYhIiJGJiwkKigjRiZGKkYmRiRGJilGLEYvRiZGJg==

Existem tr\303\252s solu\303\247\303\265es desta equa\303\247\303\243o, e elas s\303\243o

q2 := [solve( q1, {h} )];

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTJHRiY3JTwjL0kiaEdGJiomLCgqJiMiIiIiIiNGNCksJkY1RjQqJiomRjVGNF4jRjRGNEY0KSIjOkYzRjRGNCNGNCIiJEY0RjQqJkY1RjRGNiEiIkY0RjRGNEY0SSJyR0YmRjQ8Iy9GLyomLCoqJiNGNCIiJUY0RjZGNEZAKiRGNkZARkBGNEY0KigqJkYzRjRGOkY0RjQpRj5GM0Y0LCZGMkY0Rj9GQEY0RjRGNEZBRjQ8Iy9GLyomLCpGRkZARklGQEY0RjQqKCwkRktGNEY0RkxGNEZNRjRGQEY0RkFGNDcjRiw=

Todas as tr\303\252s solu\303\247\303\265es s\303\243o valores reais, como podemos ver de

q3 := map( fnormal@evalf@evalc, q2 );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTNHRiY3JTwjL0kiaEdGJiwkKiYkIitIei41RyEiKiIiIkkickdGJkY1RjU8Iy9GLywkKiYkIit1XyR5VCchIzVGNUY2RjUhIiI8Iy9GLywkKiYkIisnKWZYPCQpRj1GNUY2RjVGNTcjRiw=

e somente uma delas tem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRImhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkjbW9HRiQ2LVEiPEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYyNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y1RjhGO0Y9Rj9GQUZDRkUvRkhRJjAuMGVtRicvRktGUC1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjVGK0Y1. Esta solu\303\247\303\243o \303\251

q4 := select(x->is(rhs(x[1])/r>0) and is(rhs(x[1])/r<1), q3)[1][1]: q4;

L0kiaEc2IiwkKiYkIisnKWZYPCQpISM1IiIiSSJyR0YkRipGKg==

Com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEjMjBGJ0Y+Rj5GK0Y+ , isto se traduz em

eval( q4, r=20 );

L0kiaEc2IiQiKyg+IlxqOyEiKQ==

Repetindo estes c\303\241lculos para encontrar a altura quando metade do volume inicial j\303\241 tenha sido retirado requer resolver a equa\303\247\303\243o

q5 := v = 1/2*v0: q5;

LyooSSNQaUclKnByb3RlY3RlZEciIiIpSSJoRzYiIiIjRiYsJkkickdGKUYmKiYjRiYiIiRGJkYoRiYhIiJGJiwkKihGLkYmRiRGJilGLEYvRiZGJg==

Existem tr\303\252s solu\303\247\303\265es para esta equa\303\247\303\243o, a saber,

q6 := [solve( q5, {h} )];

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTZHRiY3JTwjL0kiaEdGJiomLCgqJiMiIiIiIiNGNCksJiIiJUY0KiYqJkY4RjReI0Y0RjRGNCkiIiRGM0Y0RjQjRjRGPUY0RjQqJkY1RjRGNiEiIkY0RjRGNEY0SSJyR0YmRjQ8Iy9GLyomLCoqJiNGNEY4RjRGNkY0RkAqJEY2RkBGQEY0RjQqKComRjNGNEY7RjRGNEY8RjQsJkYyRjRGP0ZARjRGNEY0RkFGNDwjL0YvKiYsKkZGRkBGSEZARjRGNCooLCRGSkY0RjRGPEY0RktGNEZARjRGQUY0NyNGLA==

Todas as tr\303\252s solu\303\247\303\265es s\303\243o valores reais, como se v\303\252 em

q7 := map( fnormal@evalf@evalc, q6 );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTdHRiY3JTwjL0kiaEdGJiwkKiYkIitVX1F6RyEiKiIiIkkickdGJkY1RjU8Iy9GLywkKiYkIitrKSkpM0smISM1RjVGNkY1ISIiPCMvRi8sJComJCIrWWsuRmxGPUY1RjZGNUY1NyNGLA==

q8 := select(x->is(rhs(x[1])/r>0) and is(rhs(x[1])/r<1), q7)[1][1]: q8;

L0kiaEc2IiwkKiYkIitZay5GbCEjNSIiIkkickdGJEYqRio=

eval( q8, r=20 );

L0kiaEc2IiQiKypHMmFJIiEiKQ==

Repetindo estes c\303\241lculos para encontrar a altura quando o tanque contem um quarto do seu volume inicial de \303\241gua requer resolver a equa\303\247\303\243o

q9 := v = 1/4*v0: q9;

LyooSSNQaUclKnByb3RlY3RlZEciIiIpSSJoRzYiIiIjRiYsJkkickdGKUYmKiYjRiYiIiRGJkYoRiYhIiJGJiwkKigjRiYiIidGJkYkRiYpRixGL0YmRiY=

Existem tr\303\252s solu\303\247\303\265es para esta equa\303\247\303\243o, a saber,

q10 := [solve( q9, {h} )];

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkkcTEwR0YmNyU8Iy9JImhHRiYqJiwoKiYjIiIiIiIjRjQpLCYiIidGNComKiZGNUY0XiNGNEY0RjQpIiIoRjNGNEY0I0Y0IiIkRjRGNComRjVGNEY2ISIiRjRGNEY0RjRJInJHRiZGNDwjL0YvKiYsKiomI0Y0IiIlRjRGNkY0RkEqJEY2RkFGQUY0RjQqKComRjNGNEY7RjRGNClGP0YzRjQsJkYyRjRGQEZBRjRGNEY0RkJGNDwjL0YvKiYsKkZHRkFGSkZBRjRGNCooLCRGTEY0RjRGTUY0Rk5GNEZBRjRGQkY0NyNGLA==

Todas as tr\303\252s solu\303\247\303\265es s\303\243o valores reais, como se v\303\252 em

q11 := map( fnormal@evalf@evalc, q10 );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkkcTExR0YmNyU8Iy9JImhHRiYsJComJCIrXj1DVUghIioiIiJJInJHRiZGNUY1PCMvRi8sJComJCIrSDpuVlEhIzVGNUY2RjUhIiI8Iy9GLywkKiYkIis8SURAV0Y9RjVGNkY1RjU3I0Ys

q12 := select(x->is(rhs(x[1])/r>0) and is(rhs(x[1])/r<1), q11)[1][1]: q12;

L0kiaEc2IiwkKiYkIis8SURAVyEjNSIiIkkickdGJEYqRio=

eval( q12, r=20 );

L0kiaEc2IiQiK01nXVUpKSEiKg==

<Text-field style="Heading 3" layout="Heading 3">(ii)</Text-field>

A rela\303\247\303\243o fundamental para este problema \303\251 a f\303\263mula para um segmento de uma esfera cheia at\303\251 um altura LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e tendo raio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= dada em (15.1) onde r \303\251 uma constante (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEjMjBGJ0Y+Rj5GK0Y+ ft) e V e h s\303\243o fun\303\247\303\265es da vari\303\241vel independente, t. Escrevendo (15.1) de forma a mostrar sua depend\303\252ncia de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= explicitamente, temos

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

derivando em rela\303\247\303\243o a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ent\303\243o d\303\241

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRImhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIyRicvRjtRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkFGK0ZB LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZGhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= = 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZGhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0=

Este resultado pode ser obtido no Maple, se primeiro, for colocado a f\303\263rmula do volume como

VOL := V(t) = Pi*h(t)^2*(r-h(t)/3);

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkkVk9MR0YmLy1JIlZHRiY2I0kidEdGJiooSSNQaUclKnByb3RlY3RlZEciIiIpLUkiaEdGJkYvIiIjRjQsJkkickdGJkY0KiYjRjQiIiRGNEY2RjQhIiJGNDcjRiw=

e ent\303\243o derivar. O resultado do Maple \303\251

DIFF_VOL := simplify(diff(VOL,t));

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkpRElGRl9WT0xHRiYvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIlZHRiY2I0kidEdGJkY0KipJI1BpR0YvIiIiLUkiaEdGJkYzRjctRi42JEY4RjRGNywmKiYiIiNGN0kickdGJkY3RjdGOCEiIkY3NyNGLA==

Os valores conhecidos s\303\243o

data := [diff(V(t),t)=-2000, r=20, eval(q8,r=20)];

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkklZGF0YUdGJjclLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJWR0YmNiNJInRHRiZGNSElKz8vSSJyR0YmIiM/L0kiaEdGJiQiKypHMmFJIiEiKTcjRiw=

Inserindo estes valores na equa\303\247\303\243o obtida pela deriva\303\247\303\243o da f\303\263rmula do volume, obtemos a equa\303\247\303\243o

EQ := subs(diff(h(t),t)=`h'`, data, DIFF_VOL);

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjRVFHRiYvISUrPyoqSSNQaUclKnByb3RlY3RlZEciIiItJCIrKkcyYUkiISIpNiNJInRHRiZGMUkjaCdHRiZGMSwmIiNTRjFGMiEiIkYxNyNGLA==

Resolvendo esta equa\303\247\303\243o para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjaCdGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn d\303\241

evalf(isolate(EQ, `h'`));

L0kjaCdHNiIkIStOSyUpND0hIio=

Assim, a altura da \303\241guas decresce a uma raz\303\243o de 1.81 ft/hr.

Figura 15.4 mostra esta solu\303\247\303\243o computada na Calculadora Proporcionalidade e Diferencial Impl\303\255cita.

so the water height is decreasing at a rate of 1.81 ft/hr.

Figure 15.4 shows this solution computed in the Related Rates and Implicit Differentiation Calculator.

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Figura 15.4

Solu\303\247\303\243o do Exemplo 15.3 (ii) pela Calculadora Proporcionalidade e Diferencial Impl\303\255cita (Related Rates and Implicit Differentiation Calculator)

QUESTAO 15.5

Um tanque de \303\241gua tem a forma de um cone circular \303\240 direita, com a ponta apontando para baixo. O tanque tem 13 cm de altura eo raio (no topo) \303\251 10 cm cm. Se a \303\241gua \303\251 drenada do tanque a uma taxa de 13 cm^2, encontrar o ritmo, em cm/s, em que o n\303\255vel da \303\241gua est\303\241 a mudar quando a \303\241gua \303\251 `$ h1\302\273 cm de profundidade.

Digite Pi para representar "Pi".

QUESTAO 15.6

Um tanque de \303\241gua tem a forma de um cilindro circular direito. O tanque tem 6 cm cm de altura, eo raio \303\251 de 19 cm. Se a \303\241gua est\303\241 sendo bombeada para o tanque a uma taxa de 16cm^3/s,encontrar o ritmo, em cm/s, em que o n\303\255vel da \303\241gua est\303\241 a aumentar quando a \303\241gua \303\251 `$ h2\302\273 cm de profundidade.

Digite Pi para representar "Pi".

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 15.4</Text-field>

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Enunciado do Problema</Text-field>

Uma bola de neve (esf\303\251rica) derrete uma raz\303\243o proprocional (proportional) \303\241 \303\241rea (area) de sua superf\303\255cie.

(i) Mostre que seu raio encolhe a uma raz\303\243o constantee.

(ii) Se, durante a primeira hora, metade da bola de neve derreteu-se , quanto tempo vai levar para derreter-se completamente?

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Solu\303\247\303\243o</Text-field>

<Text-field style="Heading 3" layout="Heading 3">(i)</Text-field>

O volume, V, e a \303\241rea da superf\303\255cie, S, para uma esfera de raio r s\303\243o, respectivamente,

v := 4/3*Pi*r^3: s := 4*Pi*r^2: V = v; S = s;

Como o raio muda com o tempo, o volume e a \303\241rea da superf\303\255cie ser\303\243o uma fun\303\247\303\243o do tempo. Portanto estas rela\303\247\303\265es podem ser escritas como

q1 := eval( V=v, [V=V(t),r=r(t)] ): q2 := eval( S=s, [S=S(t),r=r(t)] ): q1; q2;

A varia\303\247\303\243o do volume \303\251

q3 := diff( q1, t ): q3;

Note a presen\303\247a da \303\241rea da superf\303\255cie, 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 , nesta f\303\263rmula. Isto nos permite reescrever a varia\303\247\303\243o do volume como

q4 := lhs(q3) = S(t)*diff( r(t),t ): q4;

Se o volume muda a uma raz\303\243o proporcional a \303\241rea da superf\303\255cie, ent\303\243o 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, para uma determinada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, assim a equa\303\247\303\243o acima fica

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0=

da qual segue

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

Portanto, o raio muda a um raz\303\243o constante

<Text-field style="Heading 3" layout="Heading 3">(ii)</Text-field>

Sabemos que o raio decresce a uma raz\303\243o constante, porem a uma raz\303\243o ainda desconhecida. Se chamarmos esta raz\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , ent\303\243o 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, o raio a qualquer tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ser\303\241 o raio inicial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB menos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInRGJ0Y0RjdGPkYrRj4=. Assim, o raio varia da forma

q1 := r = r0 - k*t: q1;

A bola de neve desaparece quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=, isto \303\251, no tempo

q2 := isolate( eval(q1,r=0), t );

A informa\303\247\303\243o de que metade da bola de neve derrete na primeira hora \303\251 usada para determinar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjcjBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEia0YnRjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ== . Primeiro, Se metade da bola derrete na primeira hora, ent\303\243o

q3 := eval( v, eval(q1,t=1) ) = 1/2*(4/3*Pi*r0^3);

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTNHRiYvLCQqKCMiIiUiIiQiIiJJI1BpRyUqcHJvdGVjdGVkR0YyKSwmSSNyMEdGJkYySSJrR0YmISIiRjFGMkYyLCQqKCMiIiNGMUYyRjNGMilGN0YxRjJGMjcjRiw=

onde o lado esquerdo da equa\303\247\303\243o \303\251 o volume no tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= e o lado direito \303\251 a metade do volume inicial. Esta equa\303\247\303\243o pode ser resolvida para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcjBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, fornecendo as solu\303\247\303\265es

q4 := solve( q3, {r0} );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTRHRiY2JTwjL0kjcjBHRiYqJiwoKiQpIiIjI0Y0IiIkIiIiRjcqJClGNCNGN0Y2RjdGN0Y0RjdGN0kia0dGJkY3PCMvRi8qJiwqKiYjRjdGNEY3RjNGNyEiIiomRkFGN0Y5RjdGQkY0RjcqKComRkFGN14jRjdGN0Y3KUY2RkFGNywmRjJGN0Y4RkJGN0Y3RjdGO0Y3PCMvRi8qJiwqRkBGQkZDRkJGNEY3KigsJEZFRjdGN0ZHRjdGSEY3RkJGN0Y7Rjc3JUYtRjxGSQ==

Somente as solu\303\247\303\265es reais (real-valued) interessam, assim todas as solu\303\247\303\265es contendo I devem ser removidas. A \303\272nica solu\303\247\303\243o real \303\251

q5 := op(remove( has, [q4], I ));

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTVHRiY8Iy9JI3IwR0YmKiYsKCokKSIiIyNGMyIiJCIiIkY2KiQpRjMjRjZGNUY2RjZGM0Y2RjZJImtHRiZGNjcjRiw=

Quando esta \303\251 substituida na f\303\263mula para o tempo em que a bola de neve desaparece, a saber, em 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, n\303\263s encontramos que a bola desaparece quando

q6 := eval( q2, q5 ): q6; `` = evalf(rhs(q6));

L0kidEc2IiwoKiQpIiIjI0YoIiIkIiIiRisqJClGKCNGK0YqRitGK0YoRis=

L0khRzYiJCIrLUBLWlshIio=

Em cerca de 4 horas, 50 minutos, a bola de neve desapareceu completamente.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Cap\303\255tulo Resumo (Sum\303\241rio)</Text-field>

Problemas de rela\303\247\303\243o de propor\303\247\303\265es (proporcionalidade) podem ser dif\303\255ceis mesmo se eles envolvam em geral somente \303\241lgebra, algumas vezes geometria, e as regras de deriva\303\247\303\243o - particularmente a regra da cadeia (chain rule). A parte mais dif\303\255cil do problema pode ser identificar as quantidades vari\303\241veis e as rela\303\247\303\265es fundamentais entre as vari\303\241veis. Os erros mais comuns s\303\243o

n\303\243o lembrar que uma quantidade \303\251 na verdade uma fun\303\247\303\243o do tempo,

n\303\243o aplicar a regra da cadeia propriamente, e

substituir os valores num\303\251ricos antes de completar toda a deriva\303\247\303\243o.

Esteja alerta quanto a estas potenciais causas de erros ( pitfalls)!

Os quatros exemplos neste cap\303\255tulo fornecem uma boa estrutura para resolver problemas de proporcionalidade (related rate) com ou sem o RelatedRates maplet .

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Qual \303\251 o pr\303\263ximo?</Text-field>

Cap\303\255tulos 16-19 prov\303\252m uma completa discuss\303\243o do uso de c\303\241lculo para localizar as caracter\303\255sticas do gr\303\241fico de uma fun\303\247\303\243o. O cap\303\255tulo do Teorema do Valor M\303\251dio ( Mean Value Theorem) (Cap\303\255tulo 20) \303\251 um resultado \303\263bvio mas \303\251 um importante resultado te\303\263rico que, como o Teorema do Valor da M\303\251dia???(Intermediate Value Theorem), \303\251 tamb\303\251m usado em uma variedade de aplica\303\247\303\265es.

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