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

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Cap\303\255tulo 19

Otimiza\303\247\303\243o

A Otimiza\303\247\303\243o se refere a encontrar a solu\303\247\303\243o "\303\263tima" ou "melhor" para um problema. Os problemas de otimiza\303\247\303\243o s\303\243o geralmente classificados como problemas com v\303\255nculo ou sem v\303\255nculo. Em um problemas de otimiza\303\247\303\243o com v\303\255nculos a meta \303\251 encontrar o maior ou o menor valor de uma fun\303\247\303\243o sujeita a um ou mais v\303\255nculos. A fun\303\247\303\243o cujo valor \303\263timo a ser encontrado \303\251 chamada de fun\303\247\303\243o objetivo. Uma rela\303\247\303\243o que conecta as vari\303\241veis que aparecem na fun\303\247\303\243o objetivo \303\251 chamada de v\303\255nculo. Para um problema de otimiza\303\247\303\243o sem v\303\255nculos, n\303\243o existem tais restri\303\247\303\265es sobre a vari\303\241vel independente.

Uma abordagem geral para um problema de otimiza\303\247\303\243o inclui os seguintes passos:

identificar a fun\303\247\303\243o objetivo e todos os v\303\255nculos;

selecionar uma vari\303\241vel para ser a vari\303\241vel independente (expressar todas as outras vari\303\241veis em termos desta vari\303\241vel);

escrever a fun\303\247\303\243o objetivo em termos da vari\303\241vel independente (e, possivelmente, par\303\242metros);

determinar o intervalo de poss\303\255veis valores para a vari\303\241vel independente;

encontrar o m\303\241ximo ou o m\303\255nimo global ou m\303\255nimo da fun\303\247\303\243o neste intervalo;

O procedimento para resolver um problema de otimiza\303\247\303\243o com uma fun\303\247\303\243o objetivo cont\303\255nua em um intervalo fechado e limitado \303\251 o procedimento para encontrar extremos globais introduzidos no Cap\303\255tulo 16 sobre An\303\241lise Global: Um m\303\241ximo global (ou um m\303\255nimo) existe e ocorre em um ponto cr\303\255tico. Quando um m\303\241ximo global de uma fun\303\247\303\243o \303\251 procurado em um intervalo que n\303\243o \303\251 limitado e fechado, ou seja, ou \303\251 aberto em pelo menos um dos limites do intervalo ou \303\251 ilimitado, a exist\303\252ncia de um m\303\241ximo ou um m\303\255nimo global n\303\243o \303\251 garantida. Nestas situa\303\247\303\265es, um pouco mais de esfor\303\247o ser\303\241 requerido para prover um argumento (matem\303\241tico) convincente de que realmente existe uma solu\303\247\303\243o \303\263tima para o problema.

Os poucos exemplos iniciais s\303\243o exemplos cl\303\241ssicos. \303\200 medida que sua compreens\303\243o do processo geral melhora, os exemplos se tornam mais complicados, frequentemente envolvendo par\303\242metros simb\303\263licos.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">\303\215ndice de Termos</Text-field>

Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio do Maple; para mais informa\303\247\303\265es v\303\241 em Dicion\303\241rio Matem\303\241tico.

\303\241rea

teoremadebolzano

c\303\255rculo

intervalo

concavidadeparacima

constante

restri\303\247\303\243o

cont\303\255nuo

pontocr\303\255tico

degenerar

derivada

diferen\303\247a

diferenci\303\241vel

dom\303\255nio

pontofinal

equilateral

extremo

pontoflutuante

fun\303\247\303\243o

minimo

gr\303\241fico

impl\303\255cito

vari\303\241vel independente

desigualdade

infinidade

integra\303\247\303\243o

intervalo

limite

m\303\241ximo

significado-teoremadovalor

pontom\303\251dio

minimo

negativo

n\303\243o-negativo

um-lado

intervalo

optimizar

optimiza\303\247\303\243o

par\303\242metro

positivo

raio

satisfazer

solu\303\247\303\243o

resolver

substitui\303\247\303\243o

semlimite

valor

vari\303\241vel

trabalho

zero

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

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<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Defini\303\247\303\265es Auxiliares</Text-field>

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<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.1</Text-field>

Mostre que entre todos os ret\303\242ngulos com um per\303\255metro fixo, o quadrado tem a \303\241rea m\303\241xima.

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

A primeira frase fornece um v\303\255nculo para este problema. Denotamso o per\303\255metro por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e o comprimento e a largura do ret\303\242ngulo por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, respectivamente. Assim, a equa\303\247\303\243o vinculada fica:

LywmKiYiIiMiIiJJIkxHNiJGJkYmKiZGJUYmSSJXR0YoRiZGJkkiUEdGKA==

Est\303\241 impl\303\255cito neste problema que os v\303\255nculos geom\303\251tricos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= n\303\243o s\303\243o negativos. Estes v\303\255nculos impl\303\255citos s\303\243o expressos no Maple como:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVElQ29uMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1JI21uR0YkNiRRIjBGJ0ZALUY9Ni1RJSZsZTtGJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYsNiVRIkxGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0Zcb0ZARitGXG9GQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZcb0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVElQ29uM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1JI21uR0YkNiRRIjBGJ0ZALUY9Ni1RJSZsZTtGJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYsNiVRIldGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0Zcb0ZARitGXG9GQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZcb0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVElQ29uMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVElQ29uM0YnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEYrRlpGQEYrRlpGQA==

NiQxIiIhSSJMRzYiMUYkSSJXR0Ym

Como a \303\241rea, designada por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251 a quantidade a ser maximizada, a fun\303\247\303\243o objetivo \303\251:

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

L0kiQUc2IiomSSJMR0YkIiIiSSJXR0YkRic=

A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEtUHJpbnRPcHRQcm9iRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2LS1GLDYlUSRtYXhGJy9GN0ZERkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRJE9iakYnRjZGOUZpbi1GLDYlUSRDb25GJ0Y2RjlGaW4tRiw2JVElQ29uMkYnRjZGOUZpbi1GLDYlUSVDb24zRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARkBGW3BGQEYrRltwRkBGK0ZbcEZA

max A = L*W s.t. 2*L+2*W = P 0 <= L 0 <= W

onde s.t. \303\251 uma abrevia\303\247\303\243o para "sujeito a" (em ingl\303\252s "subject to").

O pr\303\263ximo passo \303\251 reformular o problema com uma fun\303\247\303\243o objetivo dependente somente de uma vari\303\241vel e do v\303\255nculo expresso como um intervalo apropriado. O per\303\255metro vinculado pode ser usadopara encontrar uma solu\303\247\303\243o para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em termos da outra vari\303\241vel ( e o par\303\242metro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ). Assim o per\303\255metro vinculado \303\251 reescrito como:

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

L0kiTEc2IiwmKiYjIiIiIiIjRihJIlBHRiRGKEYoSSJXR0YkISIi

(19.1)

A fun\303\247\303\243o objetivo, expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251:

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

L0kiQUc2IiomLCYqJiMiIiIiIiNGKUkiUEdGJEYpRilJIldHRiQhIiJGKUYsRik=

A n\303\243o-negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= prov\303\252 um limite para o intervalo. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outro limite. De fato, usando a substitui\303\247\303\243o em (19.1), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito como:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjcTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRiw2JVElZXZhbEYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSVDb24yRidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRiw2JVEkU3ViRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARkBGXHBGQEYrRlxwRkBGK0ZccEZALUY9Ni1RIjpGJ0ZARkJGRUZHRklGS0ZNRk9GUUZURlxwRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

MSIiISwmKiYjIiIiIiIjRidJIlBHNiJGJ0YnSSJXR0YqISIi

ou, resolvendo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5

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

MUkiV0c2IiwkKiYjIiIiIiIjRihJIlBHRiRGKEYo

O problema de otimiza\303\247\303\243o resultante \303\251:

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

max A = (1/2*P-W)*W s.t. 0 <= W W <= 1/2*P

Como este problema est\303\241 pedindo por um m\303\241ximo global de uma fun\303\247\303\243o cont\303\255nua em um intervalo limitado e fechado, sabemos que o valor m\303\241ximo ocorrer\303\241 em um ponto cr\303\255tico da fun\303\247\303\243o objetivo no intervalo. Os limites do intervalo s\303\243o:

NiQiIiEsJComIyIiIiIiI0YnSSJQRzYiRidGJw==

Pontos estacion\303\241rios ocorrem quando a derivada da fun\303\247\303\243o objetivo \303\251 zero. A equa\303\247\303\243o resultante \303\251:

LywmKiYiIiMiIiJJIldHNiJGJiEiIiomI0YmRiVGJkkiUEdGKEYmRiYiIiE=

Um ponto estacion\303\241rio \303\251:

LCQqJiMiIiIiIiVGJUkiUEc2IkYlRiU=

(Note que este est\303\241 no intervalo; de fato, \303\251 o ponto m\303\251dio do intervalo.) E, como n\303\243o existem pontos singulares, entramos com:

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

Agora, avaliando a fun\303\247\303\243o objetivo em cada ponto cr\303\255tico tem-se:

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

NiU3JC9JIldHNiIiIiEvSSJBR0YmRic3JC9GJSwkKiYjIiIiIiIjRi9JIlBHRiZGL0YvRig3JC9GJSwkKiYjRi8iIiVGL0YxRi9GLy9GKSwkKiYjRi8iIztGLylGMUYwRi9GLw==

Desta lista fica bastante claro que a \303\241rea m\303\241xima, 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 , ocorre quando 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 , ou seja, no ponto estacion\303\241rio.

Enquanto encontramos o m\303\241ximo global neste intervalo, n\303\263s n\303\243o fizemos tudo. Ainda resta verificar que o ret\303\242ngulo resultante \303\251 um quadrado. Isto ser\303\241 feito mostrando que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJMRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRIldGJ0Y0RjdGPkYrRj4= para a largura \303\263tima. Isto \303\251 feito como se segue:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

L0kiV0c2IiwkKiYjIiIiIiIlRihJIlBHRiRGKEYo

L0kiTEc2IiwkKiYjIiIiIiIlRihJIlBHRiRGKEYo

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

Note que se os v\303\255nculos geom\303\251tricos tivessem sido formulados como inequa\303\247\303\265es restritas , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 0 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 0, ent\303\243o o intervalo expresso somente em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= seria um intervalo aberto: 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIjJGJ0Y9Rj0vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkovJSliZXZlbGxlZEdRJmZhbHNlRidGK0Y9 . Neste caso o argumento de que o ponto cr\303\255tico \303\251 um m\303\241ximo global requer a considera\303\247\303\243o apropriada de um limite unilateral. Geralmente \303\251 muito mais f\303\241cil trabalhar com um intervalo limitado e fechado, aceitando objetos geom\303\251trico degenerados que correspondem aos limites (do intervalo). Algumas vezes estes limites (de intervalo) o fazem, e efetivamente, fornecem a desejada configura\303\247\303\243o \303\263tima.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.2</Text-field>

Mostre que entre todos os ret\303\242ngulos com uma \303\241rea fixa, o quadrado tem o per\303\255metro m\303\255nimo

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

Note a similaridade - e diferen\303\247as - entre este problema e aquele no Exemplo 19.1. As \303\272nicas mudan\303\247as no modo de expressar s\303\243o: i) permutar \303\241rea e per\303\255metro e ii) trocar m\303\241ximo por m\303\255nimo. A primeira diferen\303\247a \303\251 que o v\303\255nculo e as fun\303\247\303\265es objetivo s\303\243o permutadas.

A primeira frase prov\303\252 um v\303\255nculo para este problema. Denote a \303\241rea por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e denote comprimento e largura do ret\303\242ngulo por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , respectivamente. Assim, a equa\303\247\303\243o vinculada fica:

L0kiQUc2IiomSSJMR0YkIiIiSSJXR0YkRic=

Outra vez, a n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= est\303\241 impl\303\255cita, assim, escrevemos as inequa\303\247\303\265es:

NiQxIiIhSSJMRzYiMUYkSSJXR0Ym

Como o per\303\255metro, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251 a quantidade a ser minimizada, a fun\303\247\303\243o objetivo \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRPYmpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

L0kiUEc2IiwmKiYiIiMiIiJJIkxHRiRGKEYoKiZGJ0YoSSJXR0YkRihGKA==

A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEtUHJpbnRPcHRQcm9iRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2LS1GLDYlUSRtaW5GJy9GN0ZERkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRJE9iakYnRjZGOUZpbi1GLDYlUSRDb25GJ0Y2RjlGaW4tRiw2JVElQ29uMkYnRjZGOUZpbi1GLDYlUSVDb24zRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARkBGW3BGQEYrRltwRkBGK0ZbcEZA

min P = 2*L+2*W s.t. A = L*W 0 <= L 0 <= W

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser eliminado. Como no Exemplo 19.1, escolhemos manter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, para resolver a \303\241rea vinculada para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, obtendo:

L0kiTEc2IiomSSJBR0YkIiIiSSJXR0YkISIi

(19.2)

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVPYmoxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

L0kiUEc2IiwmKigiIiMiIiJJIkFHRiRGKEkiV0dGJCEiIkYoKiZGJ0YoRipGKEYo

A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= prov\303\252 um limite (de intervalo) para o dom\303\255nio da fun\303\247\303\243o 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. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outro limite. De fato, usando a substitui\303\247\303\243o em (19.2), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito como:

MSIiISomSSJBRzYiIiIiSSJXR0YmISIi

ou, resolvendo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= no Maple, encontramos:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVDb241RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

PCMyIiIhSSJXRzYi

O problema de otimiza\303\247\303\243o resultante \303\251:

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

min P = 2*A/W+2*W s.t. {0 < W}

Como este problema est\303\241 pedindo por um m\303\241ximo global de uma fun\303\247\303\243o cont\303\255nua em um intervalo aberto e sem fronteiras, temos que proceder com alguma cautela. Primeiro, os pontos estacion\303\241rios ocorrem onde:

LywmKigiIiMiIiJJIkFHNiJGJilJIldHRihGJSEiIkYrRiVGJiIiIQ==

Existem duas solu\303\247\303\265es para esta equa\303\247\303\243o:

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

NiQqJClJIkFHNiIjIiIiIiIjRigsJEYjISIi

Somente uma deles \303\251 positiva, e que \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSdTdGF0UHRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

KiQpSSJBRzYiIyIiIiIiI0Yn

A segunda derivada da fun\303\247\303\243o objetivo \303\251:

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

LCQqKCIiJSIiIkkiQUc2IkYlKUkiV0dGJyIiJCEiIkYl

que \303\251 positiva para todo 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. Portanto, a fun\303\247\303\243o objetivo \303\251 c\303\264ncava para cima para todo 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

e o \303\272nico ponto estacion\303\241rio \303\251 o m\303\255nimo global. Assim, o ret\303\242ngulo com a menor \303\241rea tem largura e comprimento:

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

L0kiV0c2IiokKUkiQUdGJCMiIiIiIiNGKQ==

L0kiTEc2IiokKUkiQUdGJCMiIiIiIiNGKQ==

com per\303\255metro:

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

L0kiUEc2IiwkKiYiIiUiIiIpSSJBR0YkI0YoIiIjRihGKA==

Em particular, observe que \303\251 um quadrado.

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

In the following execution group, add an algorithmic variable.

with(RandomTools): randomize(): n1 := Generate(integer(range=20..75)); n2 := Generate(choose([1,2])); type1 := ('maximum','minimum')[n2]; if (n2 = 1) then if ((n1 mod 2)=1) then x1 := n1-n1/2+1/2; else x1 := n1/2; end if; else x1 := 1; end if; ans1 := x1,n1-x1;

IiNs

IiIj

SShtaW5pbXVtRzYi

IiIi

NiQiIiIiI2s=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 19.1</Text-field>

In the following text region, enter the full contents of the question.

Encontre dois n\303\272meros inteiros cuja soma \303\251 `$n1`, e cujo produto \303\251 um `$type1`.

Na sua resposta, separe os dois n\303\272meros por uma v\303\255rgula.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans1;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region.

evalb({`$ANSWER`}={`$RESPONSE`});

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

Enter your hints in the following table.

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

In the following text region, enter feedback to an incorrect response.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea5`, `value`); ANSWER := ans1; if evalb({ANSWER}={eval(parse(StudentResponse))}) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>64</mn></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext></mtext><mtext> </mtext><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>64</mn></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.3</Text-field>

Um fazendeiro quer criar um campo retangular de 1 acre (43560 p\303\251s2). Este campo ser\303\241 subdividido em tr\303\252s pastos id\303\252nticos. A cerca para a parte externa do campo custa $5 por p\303\251 e a cerca interna que cria os pastos menores custa $3.50 por p\303\251. Qual \303\251 o custo m\303\255nimo para construir este campo? (Quais s\303\243o as dimens\303\265es destes campos menores e do campo total?)

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

A geometria deste problema est\303\241 resumida na Figura 19.1.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNQMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

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

Figura 19.1

O campo retangular cercado e dividido em tr\303\252s pastos id\303\252nticos

Cada lado azul \303\251 constru\303\255do com cerca de $5/p\303\251 e os dois lados vermelhos s\303\243o constru\303\255dos usando a cerca mais barata. O custo total \303\251 portanto a fun\303\247\303\243o objetivo:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRPYmpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

L0kiQ0c2IiwmKiYiI0kiIiJJInhHRiRGKEYoKiYkIiRxIiEiIkYoSSJ5R0YkRihGKA==

A equa\303\247\303\243o de v\303\255nculo:

LywkKigiIiQiIiJJInhHNiJGJkkieUdGKEYmRiYiJmdOJQ==

expressa a necessidade de que a \303\241rea cercada contenha 1 acre.

A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= est\303\241 impl\303\255cita, assim, escrevemos as inequa\303\247\303\265es:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVElQ29uMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1JI21uR0YkNiRRIjBGJ0ZALUY9Ni1RJSZsZTtGJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYsNiVRInhGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0Zcb0ZARitGXG9GQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZcb0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVElQ29uM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1JI21uR0YkNiRRIjBGJ0ZALUY9Ni1RJSZsZTtGJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYsNiVRInlGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0Zcb0ZARitGXG9GQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZcb0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVElQ29uMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVElQ29uM0YnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEYrRlpGQEYrRlpGQA==

NiQxIiIhSSJ4RzYiMUYkSSJ5R0Ym

A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEtUHJpbnRPcHRQcm9iRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2LS1GLDYlUSRtaW5GJy9GN0ZERkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRJE9iakYnRjZGOUZpbi1GLDYlUSRDb25GJ0Y2RjlGaW4tRiw2JVElQ29uMkYnRjZGOUZpbi1GLDYlUSVDb24zRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARkBGW3BGQEYrRltwRkBGK0ZbcEZA

min C = 30*x+17.0*y s.t. 3*x*y = 43560 0 <= x 0 <= y

escolhendo usar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= como a vari\303\241vel independente, resolvemos o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para obter a f\303\263rmula de substitui\303\247\303\243o:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEkU3ViRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYnLUYsNiVRKGlzb2xhdGVGJ0Y2RjktRj02LVEwJkFwcGx5RnVuY3Rpb247RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRmluLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEkQ29uRidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARlxwRkBGK0ZccEZARitGXHBGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZccEZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRTdWJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

L0kieEc2IiwkKiYiJj9YIiIiIkkieUdGJCEiIkYo

(19.3)

A fun\303\247\303\243o objeto, expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVElT2JqMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1GLDYlUShpc29sYXRlRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYoRistRiM2Jy1GLDYlUSVldmFsRidGNkY5RmVuLUZcbzYkLUYjNictRiw2JVEkT2JqRidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRiw2JVEkU3ViRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARkBGZXBGQEZccC1GLDYlUSJDRidGNkY5RmVwRkBGQEZlcEZARitGZXBGQEYrRmVwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGZXBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVPYmoxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

L0kiQ0c2IiwmKiYiJytjViIiIkkieUdGJCEiIkYoKiYkIiRxIkYqRihGKUYoRig=

MSIiISwkKiYiJj9YIiIiIkkieUc2IiEiIkYn

ou, resolvendo para 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVDb241RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

PCMyIiIhSSJ5RzYi

O problema de otimiza\303\247\303\243o resultante \303\251:

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

min C = 435600/y+17.0*y s.t. {0 < y}

Uma vez mais o intervalo \303\251 aberto e ilimitado. Trabalhando como no Exemplo 19.2, a primeira derivada da fun\303\247\303\243o objetivo \303\251:

LCYqJiInK2NWIiIiKUkieUc2IiIiIyEiIkYqJCIkcSJGKkYl

Os pontos estacion\303\241rios ocorrem onde esta derivada se anula, assim temos a equa\303\247\303\243o:

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

LywmKiYiJytjViIiIilJInlHNiIiIiMhIiJGKyQiJHEiRitGJiIiIQ==

Existem duas solu\303\247\303\265es para esta equa\303\247\303\243o:

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

NiQkIitEXnQrOyEiKCQhK0RedCs7RiU=

Somente uma delas \303\251 positiva:

JCIrRF50KzshIig=

A segunda derivada da fun\303\247\303\243o objetivo \303\251:

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

LCQqJiInKzcoKSIiIilJInlHNiIiIiQhIiJGJQ==

que \303\251 positiva para todo 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. Portanto, a fun\303\247\303\243o objetivo \303\251 c\303\264ncava para cima para todo 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 e o \303\272nico ponto estacion\303\241rio \303\251 o minimizador global, como pode ser visto na Figura 19.2.

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Figura 19.2 Exemplo 19.3: Fun\303\247\303\243o de custo para a cerca

Note que a fun\303\247\303\243o objetivo \303\251 c\303\264ncava para cima e tem um m\303\255nimo global pr\303\263ximo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEkMTYwRidGPkY+RitGPg==, como o esperado.

Cada campo ter\303\241 dimens\303\265es:

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

L0kieEc2IiQiK3lCJDMyKiEiKQ==

L0kieUc2IiQiK0RedCs7ISIo

e a \303\241rea total cercada tem dimens\303\265es:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSdTdGF0UHRGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0ZZRkBGK0ZZRkA=

LywkKiYiIiQiIiJJInhHNiJGJkYmJCIrOChcN3MjISIo

L0kieUc2IiQiK0RedCs7ISIo

O custo total para construir esta cerca \303\251:

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

L0kiQ0c2IiQiK0QlKlxVYSEiJw==

(Como uma verifica\303\247\303\243o, note que a \303\241rea total fechada \303\251 43560 p\303\251s quadrados, ou 1 acre.)

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

In the following execution group, add an algorithmic variable.

with(RandomTools): randomize(): n2 := Generate(integer(range=30..70))*10; y2 := n2-2*x; area2 := y2*x; x2 := solve(diff(area2,x)=0,x); ans2 := x=x2,y=eval(y2,x=x2);

IiQrJA==

LCYiJCskIiIiSSJ4RzYiISIj

KiYsJiIkKyQiIiJJInhHNiIhIiNGJUYmRiU=

IiN2

NiQvSSJ4RzYiIiN2L0kieUdGJSIkXSI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 19.2</Text-field>

In the following text region, enter the full contents of the question.

Suponha que um fazendeiro queira construir uma cerca de modo a ter um campo retangular. Em um lado do campo, existe um rio (ou seja, nenhuma cerca \303\251 necess\303\241ria neste lado). Designemos o comprimento do lado paralelo ao rio de y, e o comprimento dos outros dois lados de x. Quais s\303\243o as dimens\303\265es do ret\303\242ngulo que maximizar\303\241 a \303\241rea do campo se o fazendeiro tem `$n2` m de cerca dispon\303\255vel?

Entre com sua resposta como x = comprimento1, y = comprimento2.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans2;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region.

evalb(evalf({`$ANSWER`}={`$RESPONSE`}));

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

Enter your hints in the following table.

The length of 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 in terms of 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 is 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 = `$y2`.

If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictRiM2Ny1GLDY1USJBRicvJSdmYW1pbHlHUSZzZXJpZkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lKnVuZGVybGluZUdGPC8lKnN1YnNjcmlwdEdGPC8lLHN1cGVyc2NyaXB0R0Y8LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRjwvJStleGVjdXRhYmxlR0Y8LyUpcmVhZG9ubHlHRjwvJSljb21wb3NlZEdGPC8lKmNvbnZlcnRlZEdGPC8lK2ltc2VsZWN0ZWRHRjwvJSxwbGFjZWhvbGRlckdGPC8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGPC8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GZ25RJ25vcm1hbEYnLyUmZmVuY2VHRjwvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGX3AtSShtZmVuY2VkR0YkNjUtRiM2NS1GLDY1USJ4RidGNEY3RjpGPUZARkJGREZGRklGTEZORlBGUkZURlZGWEZaRmZuRjRGN0Y6L0Y+RjxGQEZCRkRGRkZJRkxGTkZQRlJGVEZWRlhGWkZdb0Y0RjdGOkZqcEZARkJGREZGRklGTEZORlBGUkZURlZGWEZaRl1vRjRGN0Y6RmpwRkBGQkZERkZGSUZMRk5GUEZSRlRGVkZYRlpGXW9GK0Y0RjdGOkZqcEZARkJGREZGRklGTEZORlBGUkZURlZGWEZaRl1v is the area of the rectangle, solve 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 to find the length 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 that will maximize the area.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea6`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The length of </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>y</mi></mrow></mstyle><mtext> in terms of </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>x</mi></mrow></mstyle><mtext> is </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi fontfamily='serif'>y</mi><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi></mrow></mfenced></mrow></mrow></mstyle><mtext> = </mtext><mtext></mtext><mrow><mrow><mn>300</mn><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea6`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>If </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi fontfamily='serif'>A</mi><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi></mrow></mfenced></mrow></mrow></mstyle><mtext> is the area of the rectangle, solve </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mo fontfamily='serif' lspace='0.2222222em' rspace='0.2222222em'>&DifferentialD;</mo><mrow><mo fontfamily='serif' lspace='0.2222222em' rspace='0.2222222em'>&DifferentialD;</mo><mi fontfamily='serif'>x</mi></mrow></mfrac><mrow><mi>A</mi><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi></mrow></mfenced></mrow><mo fontfamily='serif' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn fontfamily='serif'>0</mn></mrow></mstyle><mtext> to find the length </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>x</mi></mrow></mstyle><mtext> that will maximize the area.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

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

In the following text region, enter feedback to an incorrect response.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea6`, `value`); ANSWER := ans2; if evalb(evalf({ANSWER}={eval(parse(StudentResponse))})) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>75</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>150</mn></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext></mtext><mtext> </mtext><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>75</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>150</mn></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.4</Text-field>

Um fazendeiro quer fechar um campo retangular com \303\241rea LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (acres). Este campo ser\303\241 subdividido em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pastos retangulares id\303\252nticos. A cerca externa do campo custa $LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= por p\303\251 e a cerca para criar os pastos menores custa $LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= por p\303\251. Qual \303\251 o custo m\303\255nimo para construir este campo? (Quais s\303\243o as dimens\303\265es dos campos menores e campo total?)

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

Este problema \303\251 essencialmente o mesmo que o Exemplo 19.3, exceto que as dimens\303\265es e pre\303\247os s\303\243o dados como par\303\242metros. Cada lado da cerca externa \303\251 constru\303\255do com a cerca de $LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= /p\303\251 e os LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= lados internos s\303\243o constru\303\255dos usando cerca de $LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=/p\303\251. O custo total desta cerca \303\251 a seguinte fun\303\247\303\243o objetivo:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRPYmpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

L0kiQ0c2IiwmKiZJIlBHRiQiIiIsJiooIiIjRihJIm5HRiRGKEkieEdGJEYoRigqJkYrRihJInlHRiRGKEYoRihGKCooSSJwR0YkRigsJkYsRihGKCEiIkYoRi9GKEYo

A necessidade da \303\241rea total ser LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn acres prov\303\252 a seguinte equa\303\247\303\243o de v\303\255nculo:

LyooSSJuRzYiIiIiSSJ4R0YlRiZJInlHRiVGJiwkKiYiJmdOJUYmSSJBR0YlRiZGJg==

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

NiQxIiIhSSJ4RzYiMUYkSSJ5R0Ym

A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251:

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

min C = P*(2*n*x+2*y)+p*(n-1)*y s.t. n*x*y = 43560*A 0 <= x 0 <= y

escolhendo usar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= como a vari\303\241vel independente, resolvemos para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para obter a seguinte f\303\263rmula de substitui\303\247\303\243o:

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

L0kieEc2IiwkKioiJmdOJSIiIkkiQUdGJEYoSSJuR0YkISIiSSJ5R0YkRitGKA==

(19.4)

A fun\303\247\303\243o objetivo, expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVPYmoxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

L0kiQ0c2IiwmKiZJIlBHRiQiIiIsJiooIiY/cilGKEkiQUdGJEYoSSJ5R0YkISIiRigqJiIiI0YoRi1GKEYoRihGKCooSSJwR0YkRigsJkkibkdGJEYoRihGLkYoRi1GKEYo

A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= prov\303\252 um dos limites para o dom\303\255nio da fun\303\247\303\243o 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. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outro limite. De fato, usando a substitui\303\247\303\243o em (19.4), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito como:

MSIiISwkKioiJmdOJSIiIkkiQUc2IkYnSSJuR0YpISIiSSJ5R0YpRitGJw==

ou, resolvendo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= no Maple, obtemos:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVDb241RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

PCMyIiIhSSJ5RzYi

O problema de otimiza\303\247\303\243o resultante \303\251:

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

min C = P*(87120*A/y+2*y)+p*(n-1)*y s.t. {0 < y}

Uma vez mais o intervalo \303\251 aberto e ilimitado. Trabalhando como no Exemplo 2, a primeira derivada da fun\303\247\303\243o objetivo \303\251:

LCYqJkkiUEc2IiIiIiwmKigiJj9yKUYmSSJBR0YlRiYpSSJ5R0YlIiIjISIiRi5GLUYmRiZGJiomSSJwR0YlRiYsJkkibkdGJUYmRiZGLkYmRiY=

Pontos estacion\303\241rios ocorrem quando:

LywmKiZJIlBHNiIiIiIsJiooIiY/cilGJ0kiQUdGJkYnKUkieUdGJiIiIyEiIkYvRi5GJ0YnRicqJkkicEdGJkYnLCZJIm5HRiZGJ0YnRi9GJ0YnIiIh

Existem duas solu\303\247\303\265es para esta equa\303\247\303\243o:

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

NiQsJCoqIiRLIiIiIiwoKiYiIiNGJkkiUEc2IkYmRiYqJkkicEdGK0YmSSJuR0YrRiZGJkYtISIiRi8pIiImI0YmRilGJikqKEYnRiZGKkYmSSJBR0YrRiZGMkYmRiYsJEYkRi8=

Somente uma delas \303\251 positiva:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSdTdGF0UHRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

LCQqKiIkSyIiIiIsKComIiIjRiVJIlBHNiJGJUYlKiZJInBHRipGJUkibkdGKkYlRiVGLCEiIkYuKSIiJiNGJUYoRiUpKihGJkYlRilGJUkiQUdGKkYlRjFGJUYl

A segunda derivada da fun\303\247\303\243o objetivo \303\251:

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

LCQqKiInU1U8IiIiSSJQRzYiRiVJIkFHRidGJSlJInlHRiciIiQhIiJGJQ==

a qual, como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= s\303\243o ambos positivos, \303\251 positiva para todo 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. Portanto, a fun\303\247\303\243o objetivo \303\251 c\303\264ncava para cima para todo 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 e o \303\272nico ponto estacion\303\241rio \303\251 o minimizador global. Por causa dos par\303\242metros envolvidos neste problema, n\303\243o \303\251 poss\303\255vel criar um gr\303\241fico; mas a forma geral ser\303\241 a mesma como a do Exemplo 19.3.

Cada campo ter\303\241 as dimens\303\265es:

L0kieEc2IiwkKi4iI20iIiJJIkFHRiRGKEkibkdGJCEiIiwoKiYiIiNGKEkiUEdGJEYoRigqJkkicEdGJEYoRipGKEYoRjFGK0YoKSIiJiNGKEYuRigpKihGLEYoRi9GKEYpRihGNEYrRig=

L0kieUc2IiwkKioiJEsiIiIiLCgqJiIiI0YoSSJQR0YkRihGKComSSJwR0YkRihJIm5HRiRGKEYoRi4hIiJGMCkiIiYjRihGK0YoKSooRilGKEYsRihJIkFHRiRGKEYzRihGKA==

e a \303\241rea total cercada tem as dimens\303\265es:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSdTdGF0UHRGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0ZZRkBGK0ZZRkA=

LyomSSJuRzYiIiIiSSJ4R0YlRiYsJCosIiNtRiZJIkFHRiVGJiwoKiYiIiNGJkkiUEdGJUYmRiYqJkkicEdGJUYmRiRGJkYmRjEhIiJGJikiIiYjRiZGLkYmKSooRixGJkYvRiZGK0YmRjVGMkYm

L0kieUc2IiwkKioiJEsiIiIiLCgqJiIiI0YoSSJQR0YkRihGKComSSJwR0YkRihJIm5HRiRGKEYoRi4hIiJGMCkiIiYjRihGK0YoKSooRilGKEYsRihJIkFHRiRGKEYzRihGKA==

O custo total para levantar toda a cerca \303\251:

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

L0kiQ0c2IiwkKi4iJGsjIiIiLCgqJiIiI0YoSSJQR0YkRihGKComSSJwR0YkRihJIm5HRiRGKEYoRi4hIiJGKEYsRihJIkFHRiRGKCkiIiYjRihGK0YoKSooRilGKEYsRihGMUYoRjRGMEYo

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.5</Text-field>

Um fio de comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (cm) \303\251 cortado em duas pe\303\247as. Uma pe\303\247a \303\251 curvada para formar um c\303\255rculo e a outra \303\251 dobrada para formar um tri\303\242ngulo equil\303\241tero. Quais s\303\243o os valores m\303\241ximo e m\303\255nimo para a soma destas duas regi\303\265es?

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

Designemos por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= o raio do c\303\255rculo e por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= o comprimento dos lados do tri\303\242ngulo. Assim, um c\303\255rculo com raio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tem per\303\255metro:

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

LyZJIlBHNiI2I0klY2lyY0dGJSwkKigiIiMiIiJJI1BpRyUqcHJvdGVjdGVkR0YrSSJyR0YlRitGKw==

e \303\241rea:

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

LyZJIkFHNiI2I0klY2lyY0dGJSomSSNQaUclKnByb3RlY3RlZEciIiIpSSJyR0YlIiIjRis=

Analogamente, um tri\303\242ngulo equil\303\241tero com lado de comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tem per\303\255metro:

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

LyZJIlBHNiI2I0kkdHJpR0YlLCQqJiIiJCIiIkkic0dGJUYrRis=

e \303\241rea:

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

LyZJIkFHNiI2I0kkdHJpR0YlLCQqKCMiIiIiIiVGKylJInNHRiUiIiNGKykiIiQjRitGL0YrRis=

A necessidade de ter um comprimento fixo para o fio \303\251 expressa pela equa\303\247\303\243o vinculada:

L0kiUEc2IiwmKigiIiMiIiJJI1BpRyUqcHJvdGVjdGVkR0YoSSJyR0YkRihGKComIiIkRihJInNHRiRGKEYo

E mais, o comprimento do raio e do lado n\303\243o podem ser negativos. (Se um dos dois for zero, ent\303\243o existe somente uma regi\303\243o.) Portanto, escrevemos as inequa\303\247\303\265es:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVDb24zRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

MSIiIUkickc2Ig==

MSIiIUkic0c2Ig==

A fun\303\247\303\243o objetivo \303\251 o total das \303\241reas delimitadas pelas duas regi\303\265es:

L0kiQUc2IiwmKiZJI1BpRyUqcHJvdGVjdGVkRyIiIilJInJHRiQiIiNGKUYpKigjRikiIiVGKSlJInNHRiRGLEYpKSIiJCNGKUYsRilGKQ==

A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251:

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

min A = Pi*r^2+1/4*s^2*3^(1/2) s.t. P = 2*Pi*r+3*s 0 <= r 0 <= s

Provavelmente \303\251 mais f\303\241cil manter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= como a vari\303\241vel deste problema, assim resolvemos para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para obter a equa\303\247\303\243o de substitui\303\247\303\243o:

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

L0kic0c2IiwmKiYjIiIiIiIkRihJIlBHRiRGKEYoKigjIiIjRilGKEkjUGlHJSpwcm90ZWN0ZWRHRihJInJHRiRGKCEiIg==

(19.5)

A fun\303\247\303\243o objetivo, expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkwtRiw2JVEnJiMyMzM7RicvRjBGPUY5LUY2Ni1RIjpGJ0Y5RjtGU0ZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GZ25GOQ==

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVPYmoxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

L0kiQUc2IiwmKiZJI1BpRyUqcHJvdGVjdGVkRyIiIilJInJHRiQiIiNGKUYpKigjRikiIiVGKSksJiomI0YpIiIkRilJIlBHRiRGKUYpKigjRixGNEYpRidGKUYrRikhIiJGLEYpKUY0I0YpRixGKUYp

A n\303\243o negatividade de 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 da fun\303\247\303\243o 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. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outro limite. De fato, usando a substitui\303\247\303\243o em (19.5), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

MSIiISwmKiYjIiIiIiIkRidJIlBHNiJGJ0YnKigjIiIjRihGJ0kjUGlHJSpwcm90ZWN0ZWRHRidJInJHRipGJyEiIg==

ou resolvendo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, obtemos:

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

MUkickc2IiwkKigjIiIiIiIjRihJIlBHRiRGKEkjUGlHJSpwcm90ZWN0ZWRHISIiRig=

O problema de otimiza\303\247\303\243o resultante \303\251:

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

min A = Pi*r^2+1/4*(1/3*P-2/3*Pi*r)^2*3^(1/2) s.t. 0 <= r r <= 1/2*P/Pi

A fun\303\247\303\243o objetivo cont\303\255nua, junto com o intervalo limitado e fechado, faz este problema como o do Exemplo 1.

N\303\243o existem pontos singulares, assim, entramos com:

Os limites (do intervalo) s\303\243o:

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

NiQiIiEsJCooIyIiIiIiI0YnSSJQRzYiRidJI1BpRyUqcHJvdGVjdGVkRyEiIkYn

NiQvSSFHNiIiIiEsJComJCIrSSVcOmYiISM1IiIiSSJQR0YlRixGLA==

Os pontos estacion\303\241rios s\303\243o as solu\303\247\303\265es da equa\303\247\303\243o:

LywmKigiIiMiIiJJI1BpRyUqcHJvdGVjdGVkR0YmSSJyRzYiRiZGJioqI0YmIiIkRiYsJiomRixGJkkiUEdGKkYmRiYqKCNGJUYtRiZGJ0YmRilGJiEiIkYmKUYtI0YmRiVGJkYnRiZGMyIiIQ==

A \303\272nica solu\303\247\303\243o desta equa\303\247\303\243o \303\251:

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

LywkKiojIiIiIiIjRiYpIiIkRiVGJkkiUEc2IkYmLCYiIipGJiomRihGJkkjUGlHJSpwcm90ZWN0ZWRHRiZGJiEiIkYmLCQqJiQiK0RfI28qZiEjNkYmRipGJkYm

Para encontrar o m\303\241ximo e m\303\255nimo global no intervalo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= 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, compare os valores da fun\303\247\303\243o objetivo em cada um dos tr\303\252s pontos cr\303\255ticos (e inclua o valor de s correspondente ). Cada valor da lista a seguir corresponde a um valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, sendo o primeiro e o terceiro os valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= no limite do dom\303\255nio, e o do meio o valor cr\303\255tico no interior do dom\303\255nio.

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

NiU3JS9JInJHNiIiIiEvSSJzR0YmLCQqJiMiIiIiIiRGLUkiUEdGJkYtRi0vSSJBR0YmLCQqKCNGLSIjT0YtKUYvIiIjRi0pRi4jRi1GN0YtRi03JS9GJSwkKihGOUYtRi9GLUkjUGlHJSpwcm90ZWN0ZWRHISIiRi0vRilGJy9GMSwkKigjRi0iIiVGLUY+RkBGNkYtRi03JS9GJSwkKipGOUYtRjhGLUYvRi0sJiIiKkYtKiZGOEYtRj5GLUYtRkBGLS9GKSwkKihGLkYtRi9GLUZLRkBGLS9GMSwkKiojRi5GRkYtRjZGLSwmRj5GLSomRi5GLUY4Ri1GLUYtKUZLRjdGQEYt

Para identificar as maiores e as menores \303\241reas, converta estes resultados para a forma de ponto flutuante, obtendo:

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

NiU3JS9JInJHNiIkIiIhRigvSSJzR0YmLCQqJiQiJUxMISIlIiIiSSJQR0YmRjBGMC9JIkFHRiYsJComJCIlNlshIiZGMClGMSIiI0YwRjA3JS9GJSwkKiYkIiUjZiJGL0YwRjFGMEYwL0YqRicvRjMsJComJCIlZXpGOEYwRjlGMEYwNyUvRiUsJComJCIlJipmRjhGMEYxRjBGMC9GKiwkKiYkIiV5P0YvRjBGMUYwRjAvRjMsJComJCIlKipIRjhGMEY5RjBGMA==

A menor valor para 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSVybWluRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA

L0kickc2IiwkKiojIiIiIiIjRigpIiIkRidGKEkiUEdGJEYoLCYiIipGKComRipGKEkjUGlHJSpwcm90ZWN0ZWRHRihGKCEiIkYo

e o maior valor para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ocorre no limite (do intervalo) \303\240 direita:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSVybWF4RidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA

L0kickc2IiwkKigjIiIiIiIjRihJIlBHRiRGKEkjUGlHJSpwcm90ZWN0ZWRHISIiRig=

Para concluir, a menor \303\241rea total para as duas regi\303\265es ocorre com um c\303\255rculo de raio:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSVybWluRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA

L0kickc2IiwkKiojIiIiIiIjRigpIiIkRidGKEkiUEdGJEYoLCYiIipGKComRipGKEkjUGlHJSpwcm90ZWN0ZWRHRihGKCEiIkYo

e um tri\303\242ngulo com lado de comprimento:

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

L0kic0c2IiwkKigiIiQiIiJJIlBHRiRGKCwmIiIqRigqJilGJyNGKCIiI0YoSSNQaUclKnByb3RlY3RlZEdGKEYoISIiRig=

A \303\241rea total para estas duas regi\303\265es \303\251:

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

L0kiQUc2IiwkKiojIiIkIiIlIiIiKUkiUEdGJCIiI0YqLCZJI1BpRyUqcHJvdGVjdGVkR0YqKiZGKEYqKUYoI0YqRi1GKkYqRiopLCYiIipGKiomRjJGKkYvRipGKkYtISIiRio=

A \303\241rea m\303\241xima total para as duas regi\303\265es ocorre com um c\303\255rculo de raio:

L0kickc2IiwkKigjIiIiIiIjRihJIlBHRiRGKEkjUGlHJSpwcm90ZWN0ZWRHISIiRig=

e um tri\303\242ngulo com lado de comprimento:

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

L0kic0c2IiIiIQ==

Neste caso o comprimento total do fio \303\251 usado para formar o c\303\255rculo e n\303\243o existe tri\303\242ngulo. A \303\241rea total \303\251 a \303\241rea do c\303\255rculo:

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

L0kiQUc2IiwkKigjIiIiIiIlRihJI1BpRyUqcHJvdGVjdGVkRyEiIilJIlBHRiQiIiNGKEYo

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.6</Text-field>

A uma dist\303\242ncia LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= de uma fonte de luz, a intensidade LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, de uma fonte de luz, pode ser modelada com:

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

onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 a intensidade da fonte de luz, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 uma constante positiva.

Supondo que duas fontes de luz, com intensidades LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB e 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, est\303\243o posicionadas distantes 10 m uma da outra, e supondo que a intensidade de ilumina\303\247\303\243o em qualquer ponto seja \303\251 a soma da ilumina\303\247\303\243o de todas as fontes relevantes de luz. (Sugest\303\243o: As respostas para as seguintes quest\303\265es podem ser encontradas considerando a ilumina\303\247\303\243o ao longo da linha que conecta as duas fontes de luz.)

Encontre a a localiza\303\247\303\243o do ponto que recebe menor ilumina\303\247\303\243o. (Note que esta localiza\303\247\303\243o ser\303\241 uma fun\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.)

Quando \303\251 o m\303\255nimo no ponto m\303\251dio entre as duas fontes de luz?

Qual valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= faz o ponto a um quarto do caminho entre as duas fontes de luz receber a menor ilumina\303\247\303\243o?

Existem alguns pontos entre as lumin\303\241rias onde o m\303\255nimo n\303\243o possa ocorrer?

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

A ilumina\303\247\303\243o de cada lumin\303\241ria \303\251:

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

LyZJIkxHNiI2IyIiIiooSSJjR0YlRicmSSJTR0YlRiZGJykmSSJkR0YlRiYiIiMhIiI=

LyZJIkxHNiI2IyIiIyoqSSJjR0YlIiIiSSJrR0YlRiomSSJTR0YlNiNGKkYqKSZJImRHRiVGJkYnISIi

assim, o total de luminosidade em um objeto destas duas fontes de luz \303\251 a fun\303\247\303\243o objetivo:

LCYqKEkiY0c2IiIiIiZJIlNHRiU2I0YmRiYpJkkiZEdGJUYpIiIjISIiRiYqKkYkRiZJImtHRiVGJkYnRiYpJkYsNiNGLUYtRi5GJg==

onde as dist\303\242ncias satisfazem os v\303\255nculos:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRDb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVDb24yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVDb24zRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

LywmJkkiZEc2IjYjIiIiRigmRiU2IyIiI0YoIiM1

MiIiISZJImRHNiI2IyIiIg==

MiIiISZJImRHNiI2IyIiIw==

(Observe que n\303\243o \303\251 permitido usar dLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2I1EhRictRiM2JC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GNTYkRj1GOEY4 nem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUklbXN1YkdGJDYlLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRidGQy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRkMvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk4vJSlzdHJldGNoeUdGTi8lKnN5bW1ldHJpY0dGTi8lKGxhcmdlb3BHRk4vJS5tb3ZhYmxlbGltaXRzR0ZOLyUnYWNjZW50R0ZOLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZ24tRkA2JEZHRkNGQ0YrRkM=.)

O problema inicial de otimiza\303\247\303\243o \303\251:

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

min c*S[1]/d[1]^2+c*k*S[1]/d[2]^2 s.t. d[1]+d[2] = 10 0 < d[1] 0 < d[2]

A elimina\303\247\303\243o de uma das dist\303\242ncias, digamos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB, \303\251 direta, e \303\251 implementada obtendo a equa\303\247\303\243o de substitui\303\247\303\243o:

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

LyZJImRHNiI2IyIiIywmIiM1IiIiJkYkNiNGKiEiIg==

A nova fun\303\247\303\243o objeto, com a vari\303\241vel dependente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB, \303\251:

LCYqKEkiY0c2IiIiIiZJIlNHRiU2I0YmRiYpJkkiZEdGJUYpIiIjISIiRiYqKkYkRiZJImtHRiVGJkYnRiYpLCYiIzVGJkYrRi5GLUYuRiY=

O v\303\255nculo de n\303\243o negatividade para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JlEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiVRIjJGJ0Y4L0Y8USdub3JtYWxGJ0ZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZE \303\251 reassegurado como:

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

MiZJImRHNiI2IyIiIiIjNQ==

O novo problema de otimiza\303\247\303\243o \303\251:

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

min c*S[1]/d[1]^2+c*k*S[1]/(10-d[1])^2 s.t. 0 < d[1] d[1] < 10

A procura por pontos estacion\303\241rios no intervalo obt\303\251m um ponto cr\303\255tico real:

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

LyZJImRHNiI2IyIiIiwoKigiIzVGJywmRidGJ0kia0dGJUYnISIiKUYsIyIiIyIiJEYnRicqKEYqRicpRiwjRidGMUYnRitGLUYtKiZGKkYnRitGLUYn

Classificar este ponto cr\303\255tico pelo Teste da Segunda Derivada requer a segunda derivada da fun\303\247\303\243o objetivo, que \303\251:

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

LCQqLiIiJyIiIkkiY0c2IkYlJkkiU0dGJzYjRiVGJSwuIiYrKyJGJSomIiUrU0YlJkkiZEdGJ0YqRiUhIiIqJiIkKydGJSlGLyIiI0YlRiUqJiIjU0YlKUYvIiIkRiVGMSokKUYvIiIlRiVGJSomSSJrR0YnRiVGO0YlRiVGJUY7RjEpLCYiIzVGMUYvRiVGPEYxRiU=

O resultado de avalia\303\247\303\243o da segunda derivada no ponto estacion\303\241rio \303\251:

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

LCQqMiMiIiQiJStdIiIiKSwmRidGJ0kia0c2IkYnIiImRidGKkYnLDAqJClGKiIiI0YnISIiKiZGJUYnKUYqI0YsRiVGJ0YnKiYiIidGJylGKiMiIiVGJUYnRjEqJiIiKEYnRipGJ0YnKiZGNkYnKUYqI0YwRiVGJ0YxKiZGJUYnKUYqI0YnRiVGJ0YnRidGMUYnSSJjR0YrRicmSSJTR0YrNiNGJ0YnKSwoRipGMSokRj1GJ0YnKiRGQEYnRjFGOUYxKSwoRkhGJ0ZJRjFGJ0YnRjlGMUYx

Determinar que isto \303\251 positivo para todos os valores do par\303\242metro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= parece ser uma tarefa dantesca. Entretanto, um momento de an\303\241lise sugere que a substitui\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSJ6RidGNEY3LUkjbW5HRiQ2JFEiM0YnRj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPkYrRj5GK0Y+ pode simplificar a express\303\243o sem mudar a sequ\303\252ncia da an\303\241lise. Obtemos:

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

LCQqLCMiIiQiJStdIiIiJkkiU0c2IjYjRidGJ0kiY0dGKkYnSSJ6R0YqISIiKSwmRi1GJ0YnRiciIiZGJ0Yn

Desta forma da express\303\243o, fica claro que a segunda derivada no ponto cr\303\255tico \303\251 positiva para todos os valores positivos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e conclu\303\255mos que o ponto cr\303\255tico \303\251 um m\303\255nimo.

Se n\303\243o tivesse sido percebida esta substitui\303\247\303\243o simplificadora, um caminho alternativo para a desejada conclus\303\243o seria o seguinte:

Os seguintes limites mostram que a luminosidade se torna (positivamente) ilimitada em cada ponto limite de intervalo.

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

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUsJiooSSJjR0YoIiIiJkkiU0dGKDYjRi1GLSkmSSJkR0YoRjAiIiMhIiJGLSoqRixGLUkia0dGKEYtRi5GLSksJiIjNUYtRjJGNUY0RjVGLS9GMiIiIUkmcmlnaHRHRihJKWluZmluaXR5R0Ym

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUsJiooSSJjR0YoIiIiJkkiU0dGKDYjRi1GLSkmSSJkR0YoRjAiIiMhIiJGLSoqRixGLUkia0dGKEYtRi5GLSksJiIjNUYtRjJGNUY0RjVGLS9GMkY6SSVsZWZ0R0YoSSlpbmZpbml0eUdGJg==

Estes limites combinados com i) a continuidade e diferenciabilidade da fun\303\247\303\243o ilumina\303\247\303\243o no intervalo aberto 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < 10 e ii) a exist\303\252ncia de um \303\272nico ponto estacion\303\241rio em 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < 10 s\303\243o suficientes para concluir que o ponto estacion\303\241rio deve ser o minimizador global para a fun\303\247\303\243o objeto.

Usando qualquer uma das abordagens, o minimizador global para a fun\303\247\303\243o objeto \303\251:

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

LyZJImRHNiI2IyIiIiwoKigiIzVGJywmRidGJ0kia0dGJUYnISIiKUYsIyIiIyIiJEYnRicqKEYqRicpRiwjRidGMUYnRitGLUYtKiZGKkYnRitGLUYn

Para concluir, precisamos responder a quest\303\265es espec\303\255ficas.

O ponto m\303\251dio (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 ) ser\303\241 o ponto com ilumina\303\247\303\243o m\303\255nima quando

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxNUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LywoKigiIzUiIiIsJkYmRiZJImtHNiJGJiEiIilGKCMiIiMiIiRGJkYmKihGJUYmKUYoI0YmRi5GJkYnRipGKiomRiVGJkYnRipGJiIiJg==

A \303\272nica solu\303\247\303\243o para esta equa\303\247\303\243o \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSNxNkYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEYrRllGQEYrRllGQA==

L0kia0c2IiIiIg==

isto \303\251, quando as duas lumin\303\241rias tiverem a mesma intensidade. Isto faz sentido pela simetria.

Para mover o ponto de ilumina\303\247\303\243o m\303\255nima para o ponto a um quarto, 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, a segunda lumin\303\241ria teria que ser mais brilhante. Assim, esperamos encontrar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 1, mas, quanto maior ? Trabalhando como acima, escrevemos a equa\303\247\303\243o apropriada e encontramos para sua solu\303\247\303\243o:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSNxOEYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEYrRllGQEYrRllGQA==

L0kia0c2IiQiI0YiIiE=

Logo, 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 .

A quest\303\243o final est\303\241 realmente perguntando se existem solu\303\247\303\265es para a equa\303\247\303\243o

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

LywoKigiIzUiIiIsJkYmRiZJImtHNiJGJiEiIilGKCMiIiMiIiRGJkYmKihGJUYmKUYoI0YmRi5GJkYnRipGKiomRiVGJkYnRipGJkkiZEdGKQ==

para todos os valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEjMTBGJ0Y1RjVGK0Y1. Para uma primeira percep\303\247\303\243o desta quest\303\243o, examine a Figura 19.3, que cont\303\251m um gr\303\241fico de:

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

como uma fun\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

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Figura 19.3

Gr\303\241fico de 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

Parece claro que a localiza\303\247\303\243o admite valores acima de 10 \303\240 medida que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn se aproxima de 0 (da direita). De fato:

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

Ly1JJUV2YWxHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCZJImRHRig2IyIiIi9JImtHRigiIiEiIzU=

N\303\243o est\303\241 t\303\243o claro que esta fun\303\247\303\243o admite valores abaixo de zero. (Pela simetria parece que deve.) Para ver que todas as localiza\303\247\303\265es acima desta posi\303\247\303\243o da lumin\303\241ria s\303\243o localiza\303\247\303\265es para ilumina\303\247\303\243o m\303\255nima, veja no limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se aproxima do infinito.

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

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsKCooIiM1IiIiLCZGLUYtSSJrR0YoRi0hIiIpRi8jIiIjIiIkRi1GLSooRixGLSlGLyNGLUY0Ri1GLkYwRjAqJkYsRi1GLkYwRi0vRi9JKWluZmluaXR5R0YmIiIh

Agora, pela continuidade da posi\303\247\303\243o de ilumina\303\247\303\243o m\303\255nima como uma fun\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, as id\303\251ias por tr\303\241s do Teorema do Valor Intermedi\303\241rio podem ser aplicadas para concluir que a ilumina\303\247\303\243o m\303\255nima pode ocorrer em qualquer ponto entre as duas lumin\303\241rias. O m\303\255nimo ocorre em um ponto limite do intervalo somente se n\303\243o existir fonte de luz neste ponto.

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

Using the Maple-graded question type, you create a question that uses Maple functionality when creating the correct answer and determining if the student's response is correct. The Maple-graded question type has an option to plot the student's response. To include the plot option, insert the corresponding task template. For information on creating questions, making assignments, and exporting Maple T.A. course modules, see Overview of Maple T.A. Integration.

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

Algorithmic variables can be used to create a question that contains randomly generated parameters. You can refer to algorithmic variables in the Question, Answer, Hints, and Feedback input regions, as well as in other variable definitions. To reference an algorithmic variable, include "`$variable`" as text or 2-D Math, for example, x := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique.

with(RandomTools): randomize(): n3 := Generate(integer(range=2..5)): fn3 := sort(Generate(polynom(integer(range=-10..10),x,degree=n3))); x3 := Generate(integer(range=-10..10)); y3 := Generate(integer(range=-10..10)); while (y3 = eval(fn3,x=x3)) do y3 := Generate(integer(range=-10..10)) end do: dist3 := sqrt((x-x3)^2+(fn3-y3)^2); ddist3 := sort(diff(dist3,x)): s3 := [fsolve(numer(ddist3) = 0,x)]; sy3 := eval(dist3,x=s3[1]): sx3 := s3[1]: if (nops(s3)>1) then for i from 2 to nops([s3]) do temp3 := sy3; sy3 := max(sy3,eval(dist3,x=s3[i])); if (sy3 <> temp3) then sx3 := s3[i]; end if; end do; end if: ans3 := evalf(sx3,4),evalf(sy3,4);

LCgqJEkieEc2IiIiIyIiJ0YkISIqISIoIiIi

ISIj

IiIj

KiQsLCokSSJ4RzYiIiIjISNFRiUiJG0iIiMmKSIiIiokRiUiIiUiI08qJEYlIiIkISQzIiNGK0Yn

NyUkIStUXkgwcCEjNSQiK0hCWidvKEYlJCIrIkcjKT08IyEiKg==

NiQkISUwcCEiJSQiJTc4ISIk

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 19.3</Text-field>

The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question.

Considere a curva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = `$fn3`. Que ponto da curva tem a menor dist\303\242ncia do ponto (`$x3`,`$y3`)? Entre com sua resposta como (x, y). Sua resposta pode ter precis\303\243o de at\303\251 4 d\303\255gitos significativos.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=``(ans3);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$RESPONSE` is automatically initialized to the student's response. Using `$ANSWER` (the correct answer) and `$RESPONSE` (the student response), one example of evaluating to a Boolean value is, evalb(`$ANSWER`-`$RESPONSE`=0); Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region.

evalb([`$ans3`]=evalf([`RESPONSE`],4));

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints.

The shortest distance from a curve to a known point is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYlLUYjNjlGKy1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2NS1GIzY4LUYsNjVRInhGJy8lJ2ZhbWlseUdRJnNlcmlmRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUqdW5kZXJsaW5lR0ZHLyUqc3Vic2NyaXB0R0ZHLyUsc3VwZXJzY3JpcHRHRkcvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGRy8lK2V4ZWN1dGFibGVHRkcvJSlyZWFkb25seUdGRy8lKWNvbXBvc2VkR0ZHLyUqY29udmVydGVkR0ZHLyUraW1zZWxlY3RlZEdGRy8lLHBsYWNlaG9sZGVyR0ZHLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0ZHLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNj5RKCZtaW51cztGJ0Y/RkJGRS9GSUZHRktGTUZPRlFGVEZXRllGZW5GZ25GaW5GW29GXW9GX28vRmJvUSdub3JtYWxGJy8lJmZlbmNlR0ZHLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRkcvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRltxLUklbXN1YkdGJDYmRjwtRiw2NVEma25vd25GJ0Y/RkJGRUZIRktGTUZPRlFGVEZXRllGZW5GZ25GaW5GW29GXW9GX29GYW8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Zdb0YrRj9GQkZFRmhvRktGTUZPRlFGVEZXRllGZW5GZ25GaW5GW29GXW9GX29GaW9GP0ZCRkVGaG9GS0ZNRk9GUUZURldGWUZlbkZnbkZpbkZbb0Zdb0Zfb0Zpby1JI21uR0YkNjVRIjJGJ0Y/RkJGRUZob0ZLRk1GT0ZRRlRGV0ZZRmVuRmduRmluRltvRl1vRl9vRmlvLyUxc3VwZXJzY3JpcHRzaGlmdEdGZnEtRmVvNj5RJyZwbHVzO0YnRj9GQkZFRmhvRktGTUZPRlFGVEZXRllGZW5GZ25GaW5GW29GXW9GX29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGZ3BGaXBGXHEtRjU2JS1GODY1LUYjNjgtRiw2NVEieUYnRj9GQkZFRkhGS0ZNRk9GUUZURldGWUZlbkZnbkZpbkZbb0Zdb0Zfb0Zhby1GZW82PkZnb0Y/RkJGRUZob0ZLRk1GT0ZRRlRGV0ZZRmVuRmduRmluRltvRl1vRl9vRmlvRltwRl1wRl9wRmFwRmNwRmVwRmdwL0ZqcFEsMC4yNzc3Nzc4ZW1GJy9GXXFGXHMtRl9xNiZGZnJGYXFGZHFGXW9GK0Y/RkJGRUZob0ZLRk1GT0ZRRlRGV0ZZRmVuRmduRmluRltvRl1vRl9vRmlvRj9GQkZFRmhvRktGTUZPRlFGVEZXRllGZW5GZ25GaW5GW29GXW9GX29GaW9GZ3FGW3JGK0Y/RkJGRUZob0ZLRk1GT0ZRRlRGV0ZZRmVuRmduRmluRltvRl1vRl9vRmlvRlFGVEYrRj9GQkZFRmhvRktGTUZPRlFGVEZXRllGZW5GZ25GaW5GW29GXW9GX29GaW8=, where 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 and 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 are points on the curve.

Solve 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 to find the value of 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 that will give the shortest distance.

Replace this text with hint 3.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The shortest distance from a curve to a known point is </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi><mo fontfamily='serif' lspace='0.2222222em' rspace='0.2222222em'>−</mo><msub><mi fontfamily='serif'>x</mi><mi fontfamily='serif'>known</mi></msub></mrow></mfenced><mn fontfamily='serif'>2</mn></msup><mo fontfamily='serif' lspace='0.2222222em' rspace='0.2222222em'>+</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>y</mi><mo fontfamily='serif' lspace='0.2777778em' rspace='0.2777778em'>−</mo><msub><mi fontfamily='serif'>y</mi><mi fontfamily='serif'>known</mi></msub></mrow></mfenced><mn fontfamily='serif'>2</mn></msup></mrow></msqrt></mrow></mstyle><mtext>, where </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>x</mi></mrow></mstyle><mtext> and </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>y</mi></mrow></mstyle><mtext> are points on the curve.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Solve </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi fontfamily='serif'>x</mi></mrow></mfrac><mrow><mi>distance</mi><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi></mrow></mfenced></mrow><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>⁢</mo><mo fontfamily='serif' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>⁢</mo><mn fontfamily='serif'>0</mn></mrow></mstyle><mtext> to find the value of </mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>x</mi></mrow></mstyle><mtext> that will give the shortest distance.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

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

A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response.

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea3`, `value`); ANSWER := ``(ans3); if evalb([ans3]=evalf([`eval(parse(StudentResponse))`],4)) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>0.6905</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>1.312</mn></mrow></mfenced></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>0.6905</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>1.312</mn></mrow></mfenced></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): A := Generate(integer(range=8..20))*20; m1 := Generate(choose([1,2,3])); m2 := Generate(choose([1,2,3])); while (m1=m2) do m2 := Generate(choose([1,2,3])) end do; ansH:=sqrt( (m1+m2)/(2*m2) * A ): ansW:=sqrt( (2*m2)/(m1+m2) * A ): ansP:=simplify((ansH+m1+m2)*(ansW+2*m2)): t := Generate(choose([1,2,3])): TYPE_ := `width`,`height`,`printed area`: TYPE := TYPE_[t]; ans_:= ansW,ansH,ansP: ans := ans_[t];

IiRnIg==

IiIj

IiIi

SSZ3aWR0aEc2Ig==

LCQqJCIjOiMiIiIiIiMjIiIpIiIk

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 19.4</Text-field>

Um p\303\264ster est\303\241 para ser criado com uma `$m1` polegada de margem na parte superior, `$m2` polegada de margem nas laterais e na parte inferior, e uma \303\241rea total de `$A` in2(polegadas quadradas).

Qual \303\251 a `$TYPE` do poster com a maior \303\241rea impressa? Entre com sua resposta exata, ou com precis\303\243o de at\303\251 5 d\303\255gitos significativos.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

ans;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb( fnormal(evalf(`$ANSWER`-`$RESPONSE`), 5 ) = 0 );

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

The area can be written as P = (w-2*`$m2`)*(h-(`$m1`+`$m2`))

When h is eliminated, the problem can be reformulated as P = `$A` - 2*`$m2`*`$A`/w - (`$m1`+`$m2`)*w + (2*`$m2`)*(`$m1`+`$m2`)

Replace this text with hint 3.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext></mtext><mtext> </mtext><mtext>When h is eliminated, the problem can be reformulated as  </mtext><mtext>P = </mtext><mtext></mtext><mrow><mn>160</mn></mrow><mtext></mtext><mtext> - 2*</mtext><mtext></mtext><mrow><mn>1</mn></mrow><mtext></mtext><mtext>*</mtext><mtext></mtext><mrow><mn>160</mn></mrow><mtext></mtext><mtext>/w - (</mtext><mtext></mtext><mrow><mn>2</mn></mrow><mtext></mtext><mtext>+</mtext><mtext></mtext><mrow><mn>1</mn></mrow><mtext></mtext><mtext>)*w + (2*</mtext><mtext></mtext><mrow><mn>1</mn></mrow><mtext></mtext><mtext>)*(</mtext><mtext></mtext><mrow><mn>2</mn></mrow><mtext></mtext><mtext>+</mtext><mtext></mtext><mrow><mn>1</mn></mrow><mtext></mtext><mtext>)</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

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

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := ans; if evalb( fnormal(evalf(ANSWER-eval(parse(StudentResponse))), 5 ) = 0 ) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>8</mn><mn>3</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msqrt><mn>15</mn></msqrt></mrow></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext></mtext><mtext>&NewLine;</mtext><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>8</mn><mn>3</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msqrt><mn>15</mn></msqrt></mrow></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): r5 := -1: while (r5 <= 0) do n4 := Generate(choose([1,2])); t5 := (`cylinder`,`right circular cone`)[n4]: t6 := (`bases`,`base`)[n4]: V5 := Generate(integer(range=10..30))*10*Pi: costla := Generate(integer(range=1..8)); costb := Generate(integer(range=1..10)); while (costb = costla) do costb := Generate(integer(range=1..10)) end do; if (n4 = 1) then h50 := solve(V5 = h*Pi*r^2,h); c5 := costla*h*Pi*2*r+costb*2*Pi*r^2; else h50 := solve(V5 = (h*Pi*r^2)/3,h); c5 := costla*Pi*r*sqrt(h^2+r^2) + costb*Pi*r^2; end if; c51 := eval(c5,h=h50); dc51 := simplify(diff(c51,r)); r5 := fsolve(numer(dc51)=0,r); end do; h5 := eval(h50,r=r5); ans5 := r=evalf(r5,4),h=evalf(h5,4);

IiIi

SSljeWxpbmRlckc2Ig==

SSZiYXNlc0c2Ig==

LCRJI1BpRyUqcHJvdGVjdGVkRyIkKyM=

IiIi

IiIj

LCYqJkkickc2IiEiIkkjUGlHJSpwcm90ZWN0ZWRHIiIiIiQrJSomRidGKUYkIiIjIiIl

LCQqKEkjUGlHJSpwcm90ZWN0ZWRHIiIiLCYhI11GJiokSSJyRzYiIiIkRiZGJkYqISIjIiIp

JCIrKlxKU28kISIq

JCIrKmY3T1oiISIp

NiQvSSJyRzYiJCIlJW8kISIkL0kiaEdGJSQiJXU5ISIj

hint1 := (V=h*Pi*r^2,V=h*Pi*r^2/3)[n4]; hint2 := (LSA=h*Pi*2*r,LSA=Pi*r*sqrt(h^2+r^2))[n4];

L0kiVkc2IiooSSJoR0YkIiIiSSNQaUclKnByb3RlY3RlZEdGJ0kickdGJCIiIw==

L0kkTFNBRzYiLCQqKEkiaEdGJCIiIkkjUGlHJSpwcm90ZWN0ZWRHRihJInJHRiRGKCIiIw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 19.5</Text-field>

Considere um recipiente com a forma de um `$t5`. O custo de material para uma superf\303\255cie curva \303\251 $`$costla` por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==, e o custo de material para `$t6` \303\251 $`$costb` por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==. Se o recipiente necessita ter um volume `$V5` LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==, qual deve ser a altura e o raio (em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) do recipiente que minimizar\303\241 o custo? Entre com sua resposta como r = raio, h = altura. A resposta pode entrar com precis\303\243o at\303\251 4 d\303\255gitos significativos.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans5;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(evalf({`$ANSWER`},4)=evalf({`$RESPONSE`},4));

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

The volume of a `$t5` is `$hint1`.

The lateral surface area of a `$t5` is `$hint2`.

Using the known volume, treat the height as a function with respect to the radius. Find the value of the radius that will minimize cost.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The volume of a </mtext><mtext></mtext><mrow><mi>cylinder</mi></mrow><mtext></mtext><mtext> is </mtext><mtext></mtext><mrow><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The lateral surface area of a </mtext><mtext></mtext><mrow><mi>cylinder</mi></mrow><mtext></mtext><mtext> is </mtext><mtext></mtext><mrow><mrow><mi>LSA</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>r</mi></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Using the known volume, treat the height as a function with respect to the radius.  Find the value of the radius that will minimize cost.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

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

The cost of material can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= = `$c5`.

Substitute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = `$h50` to get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= = `$c51`.

Differentiate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and solve 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 to find the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that will minimize cost.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea4`, `value`); ANSWER := ans5; if evalb(evalf({ANSWER},4)=evalf({eval(parse(StudentResponse))},4)) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>3.684</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>14.74</mn></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>The cost of material can be written as </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi mathbackground='#000000'>C</mi></mrow></mrow></mstyle><mtext> = </mtext><mtext></mtext><mrow><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>r</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext><mtext> </mtext><mtext>Substitute </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>h</mi></mrow></mstyle><mtext> = </mtext><mtext></mtext><mrow><mfrac><mrow><mn>200</mn></mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mrow><mtext></mtext><mtext> to get </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi mathbackground='#000000'>C</mi></mrow></mrow></mstyle><mtext> = </mtext><mtext></mtext><mrow><mrow><mfrac><mrow><mn>400</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext><mtext>Differentiate </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>C</mi></mrow></mstyle><mtext> with respect to </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>r</mi></mrow></mstyle><mtext>, and solve </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac background='#000000'><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&DifferentialD;</mo><mrow><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&DifferentialD;</mo><mi mathbackground='#000000'>r</mi></mrow></mfrac><mi mathbackground='#000000'>C</mi><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&equals;</mo><mn mathbackground='#000000'>0</mn></mrow></mstyle><mtext> to find the value of </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>r</mi></mrow></mstyle><mtext> that will minimize cost.</mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>3.684</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>14.74</mn></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): n5 := Generate(choose([1,2])); t7 := (`cylinder`,`right circular cone`)[n5]: t8 := (`bases`,`base`)[n5]: j := 0: h6 := 0: unassign('temp6'); while (j = 0 or (evalf(Im(h6)) <> 0)) do costla2 := Generate(integer(range=1..8)); costb2 := Generate(integer(range=1..10)); while (costb2 = costla2) do costb2 := Generate(integer(range=1..10)) end do; cost := Generate(integer(range=10..30))*10*Pi; if (n5 = 1) then h60 := solve(cost=costla2*h*Pi*2*r+costb2*2*Pi*r^2,h); V6 := eval(h*Pi*r^2,h=h60); else h60 := sqrt(simplify(rhs(isolate(cost=costla2*Pi*r*sqrt(h^2+r^2) + costb2*Pi*r^2,h^2)))); V6 := eval(Pi*h*r^2/3,h=h60); end if; dV6 := simplify(diff(V6,r)); s6 := [fsolve(numer(dV6)=0,r)]; for i from 2 to nops(s6) do if (s6[i] > 0) then j := j+1; temp6[j] := s6[i]; end if; end do; if (j > 1) then temp7 := temp6[1]: temp8 := eval(V6,r=temp7): for k from 2 to j do temp9 := temp8; temp8 := max(temp8,eval(V6,r=temp6[j])); if temp8 <> temp9 then temp7 := temp6[j]; end if; end do; r6 := temp7; else r6 := temp6[1]; end if; h6 := eval(h60,r=r6); end do; ans6 := r=evalf(r6,4),h=evalf(h6,4);

IiIi

IiIl

IiIm

LCRJI1BpRyUqcHJvdGVjdGVkRyIkIUc=

LCYqJkkjUGlHJSpwcm90ZWN0ZWRHIiIiSSJyRzYiIiIjIyEjOiIiJUYkIiNO

NyQkIStqLzBiSSEiKiQiK2ovMGJJRiU=

JCIrZ2hpUHchIio=

NiQvSSJyRzYiJCIlYkkhIiQvSSJoR0YlJCIlUXdGKA==

hint3 := (V=h*Pi*r^2,V=h*Pi*r^2/3)[n5]; hint4 := (LSA=h*Pi*2*r,LSA=Pi*r*sqrt(h^2+r^2))[n5];

L0kiVkc2IiooSSJoR0YkIiIiSSNQaUclKnByb3RlY3RlZEdGJ0kickdGJCIiIw==

L0kkTFNBRzYiLCQqKEkiaEdGJCIiIkkjUGlHJSpwcm90ZWN0ZWRHRihJInJHRiRGKCIiIw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 19.6</Text-field>

Considere um recipiente com a forma de um `$t7`. O custo de material para uma superf\303\255cie curva \303\251 $`$costla2` por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==, e o custo de material para `$t8` \303\251 $`$costb2` por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==. Qual deve ser a altura e o raio (em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) do recipiente para ter um volume m\303\241ximo para um custo total de $`$cost`? Entre com sua resposta como r = raio, h = altura. A reposta pode entrar com precis\303\243o at\303\251 4 d\303\255gitos significativos.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans6;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(evalf({`$ANSWER`},4)=evalf({`$RESPONSE`},4));

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

The volume of a `$t7` is `$hint3`.

The lateral surface area of a `$t5` is `$hint2`.

Using the known cost, treat the height as a function with respect to the radius. Find the value of the radius that will maximize the volume.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea7`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The volume of a </mtext><mtext></mtext><mrow><mi>cylinder</mi></mrow><mtext></mtext><mtext> is </mtext><mtext></mtext><mrow><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea7`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The lateral surface area of a </mtext><mtext></mtext><mrow><mi>cylinder</mi></mrow><mtext></mtext><mtext> is </mtext><mtext></mtext><mrow><mrow><mi>LSA</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>r</mi></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea7`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Using the known cost, treat the height as a function with respect to the radius.  Find the value of the radius that will maximize the volume.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

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

The volume of the container can be written as `$hint3`.

Substitute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = `$h60` to get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = `$V6`.

Differentiate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and solve 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 to find the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that will maximize the volume.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea7`, `value`); ANSWER := ans6; if evalb(evalf({ANSWER},4)=evalf({eval(parse(StudentResponse))},4)) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>3.055</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>7.638</mn></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>The volume of the container can be written as </mtext><mtext></mtext><mrow><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext><mtext> </mtext><mtext>Substitute </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>h</mi></mrow></mstyle><mtext> = </mtext><mtext></mtext><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfrac><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>28</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mrow><mtext></mtext><mtext> to get </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>V</mi></mrow></mstyle><mtext> = </mtext><mtext></mtext><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>28</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>r</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi mathvariant='normal'>&pi;</mi></mrow></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext><mtext>Differentiate </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>V</mi></mrow></mstyle><mtext> with respect to </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>r</mi></mrow></mstyle><mtext>, and solve </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac background='#000000'><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&DifferentialD;</mo><mrow><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&DifferentialD;</mo><mi mathbackground='#000000'>r</mi></mrow></mfrac><mi mathbackground='#000000'>V</mi><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&equals;</mo><mn mathbackground='#000000'>0</mn></mrow></mstyle><mtext> to find the value of </mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>r</mi></mrow></mstyle><mtext> that will maximize the volume.</mtext><mtext>&NewLine;</mtext><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>3.055</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>7.638</mn></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

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

A solu\303\247\303\243o de um problema de otimiza\303\247\303\243o envolve tipicamente tr\303\252s passos principais:

formula\303\247\303\243o do problema;

an\303\241lise que leva \303\240 solu\303\247\303\243o;

interpreta\303\247\303\243o da solu\303\247\303\243o;

O passo da formula\303\247\303\243o pode ser com frequ\303\252ncia o mais dif\303\255cil. A chave aqui \303\251 identificar uma vari\303\241vel apropriada (somente uma). Se isto \303\251 feito corretamente, ent\303\243o deve ser poss\303\255vel expressar a fun\303\247\303\243o objeto em termos desta vari\303\241vel e identificar um intervalo onde o m\303\241ximo e/ou m\303\255nimo \303\251 requerido.

O passo da solu\303\247\303\243o \303\251 geralmente bastante simples. \303\211 importante determinar se o intervalo \303\251 limitado e fechado. O caso em que \303\251 ou aberto em pelo menos um dos limites (do intervalo) ou \303\251 ilimitado requer um pouco mais de esfor\303\247o para verificar que um ponto cr\303\255tico \303\251 o desejado valor extremo.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Pr\303\263ximo Cap\303\255tulo</Text-field>

O cap\303\255tulo Teorema do Valor M\303\251dio \303\251 o \303\272ltimo cap\303\255tulo desta parte. Este ser\303\241 uma boa conclus\303\243o para nossa discuss\303\243o de derivadas \303\251 ser\303\241 uma ponte para integra\303\247\303\243o.

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