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

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Cap\303\255tulo 22

Equa\303\247\303\265es Diferenciais

Equa\303\247\303\265es Diferenciais s\303\243o freq\303\274entemente encontradas em modelos matem\303\241ticos para sistemas mec\303\242nicos, popula\303\247\303\265es biol\303\263gicas, rea\303\247\303\265es qu\303\255micas e circuitos el\303\251tricos. Apesar das equa\303\247\303\265es diferenciais consideradas neste cap\303\255tulo serem relativamente simples, algumas quest\303\265es interessantes podem ser formuladas -- e repondidas. A chave para resolver uma equa\303\247\303\243o diferencial \303\251 a manipula\303\247\303\243o dela para uma forma que necessite a avalia\303\247\303\243o de uma ou mais integrais indefinidas.

Este cap\303\255tulo tem dois objetivos :

come\303\247ar a desenvolver a habilidade de escrever equa\303\247\303\265es diferenciais que descrevam os comportamentos de alguns sistemas do mundo real;

apresentar um pequeno n\303\272mero de m\303\251todos para resolver equa\303\247\303\265es diferenciais simples;

Resolver uma equa\303\247\303\243o diferencial envolve frequentemente encontrar antiderivadas, assim deve ser poss\303\255vel verificar que uma fun\303\247\303\243o, de fato, satisfaz uma dada equa\303\247\303\243o diferencial.

<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.

acelera\303\247\303\243o

limite

constante

curva

integraldefinida

vari\303\241veldependente

derivada

avaliar

par

expl\303\255cito

expoente

fam\303\255lia

primeira-ordem

fun\303\247\303\243o

solu\303\247\303\243ogeral

gr\303\241fico

impl\303\255cito

integralindefinida

vari\303\241velindependente

desigualdade

infinito

condi\303\247\303\243oinicial

integra\303\247\303\243o

intervalo

linear

m\303\241ximo

m\303\255nimo

m\303\272ltiplo

negativo

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

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

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">O que \303\251 uma Equa\303\247\303\243o Diferencial? </Text-field>

Uma equa\303\247\303\243o diferencial \303\251 uma equa\303\247\303\243o que expressa uma rela\303\247\303\243o entre uma fun\303\247\303\243o, suas derivadas e a vari\303\241vel independente para esta fun\303\247\303\243o. Resolver uma equa\303\247\303\243o diferencial \303\251 encontrar a fun\303\247\303\243o que torna a equa\303\247\303\243o verdadeira. A fun\303\247\303\243o pode ser expressa tanto explicitamente como implicitamente. Existem muitos m\303\251todos para encontrar a fun\303\247\303\243o (ou fun\303\247\303\265es) que satisfaz(em) uma dada equa\303\247\303\243o, mas por fim, a prova de que uma solu\303\247\303\243o foi encontrada consiste na demonstra\303\247\303\243o que a suposta solu\303\247\303\243o realmente "satisfaz" a equa\303\247\303\243o diferencial.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Um Recurso Notacional </Text-field>

Para simplificar a nota\303\247\303\243o na qual a equa\303\247\303\243o diferencial \303\251 exibida no Maple, temos usado a declara\303\247\303\243o de fun\303\247\303\243o do pacote PDEtools. Quando a funcionalidade instalada por esta fun\303\247\303\243o est\303\241 ativa, as equa\303\247\303\265es diferenciais devem ser introduzidas no Maple no modo padr\303\243o, a saber, com toda refer\303\252ncia \303\240 vari\303\241vel dependente explicitamente, referenciando a vari\303\241vel independente. Portanto, para entrar com a derivada 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, escreva diff(y(x),x),e para entrar com a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, escreva y(x). Maple devolver\303\241 a derivada como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeSdGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn e devolver\303\241 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 como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 22.1: Verificando Solu\303\247\303\265es </Text-field>

Determine qual das seguintes fun\303\247\303\265es:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR5MWFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR5MWJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR5MWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR5MWRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR5MWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0YlNiMsJComIiIjIiIiSSJ0R0YmRjRGNDcjLy1GKzYjRjVGLA==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0YlNiNJInRHRiY3Iy8tRitGMEYs

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0YlNiMsJComIiIjIiIiSSJ0R0YmRjRGNDcjLy1GKzYjRjVGLA==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJiomSSJ0R0YmIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHRiU2IywkKiYiIiNGLkYtRi5GLkYuNyMvLUYrNiNGLUYs

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJiwmKiYmSSJDR0YmNiMiIiJGMS1JJHNpbkc2JCUqcHJvdGVjdGVkR0YlNiMsJComIiIjRjFJInRHRiZGMUYxRjFGMSomJkYvNiNGOUYxLUkkY29zR0Y0RjZGMUYxNyMvLUYrNiNGOkYs

s\303\243o solu\303\247\303\265es da equa\303\247\303\243o diferencial:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVElb2RlMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2KEYrLUYjNihGKy1GIzYoLUkmbWZyYWNHRiQ2KC1GIzYlLUklbXN1cEdGJDYlLUY9Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZiby1JI21uR0YkNiRRIjJGJ0ZALyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lK2V4ZWN1dGFibGVHRkRGQC1GIzYnRistRiM2JkZeby1GXG82JS1GLDYlUSJ0RidGNkY5RmRvRmhvRltwRkBGK0ZbcEZALyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZbcS8lKWJldmVsbGVkR0ZELUY9Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZARkJGRUZHRklGS0ZNRk9GYW9GY28tRiM2Jy1GLDYlUSJ5RidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GT0Zhb0Zjby1JKG1mZW5jZWRHRiQ2JC1GIzYlRmNwRltwRkBGQEZbcEZARitGW3BGQC1GPTYtUSIrRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRmRyLUYjNictRmVvNiRRIjRGJ0ZARmBxRmNxRitGQEYrRkAtRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtRmVvNiRGam9GQEZbcEZARitGW3BGQEYrRltwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGW3BGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVvZGUxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLywmSSR5JydHRiYiIiIqJiIiJUYtSSJ5R0YmRi1GLSIiITcjLywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GNjYkLUYwNiNJInRHRiZGPUY9Ri0qJkYvRi1GO0YtRi1GMQ==

Para verificar se uma dada fun\303\247\303\243o satisfaz uma equa\303\247\303\243o diferencial, insira a fun\303\247\303\243o na equa\303\247\303\243o diferencial, avalie todas as derivadas e simplifique. Tudo isto \303\251 feito no Maple com a sintaxe

LyIiIUYj

Se o resultado \303\251 uma declara\303\247\303\243o verdadeira -- para todos os valores relevantes da vari\303\241vel independente -- ent\303\243o a fun\303\247\303\243o realmente satisfaz a equa\303\247\303\243o diferencial. Neste caso, a equa\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI9RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNRjFGNUYrRjU= \303\251 verdadeira para tidos os valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e assim 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 \303\251 uma solu\303\247\303\243o desta equa\303\247\303\243o diferencial.

Repetindo estes passos para a segunda fun\303\247\303\243o temos:

LywkKiYiIiQiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGLEYmRiYiIiE=

Como esta equa\303\247\303\243o \303\251 verdadeira somente quando t \303\251 um m\303\272ltiplo inteiro de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy , 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 n\303\243o \303\251 uma solu\303\247\303\243o desta equa\303\247\303\243o diferencial.

O mesmo processo pode ser usado para as tr\303\252s fun\303\247\303\265es remanescentes. A terceira fun\303\247\303\243o \303\251 uma solu\303\247\303\243o, como vemos em:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR5MWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0YlNiMsJComIiIjIiIiSSJ0R0YmRjRGNDcjLy1GKzYjRjVGLA==

LyIiIUYj

A quarta fun\303\247\303\243o n\303\243o \303\251 uma solu\303\247\303\243o, como podemos ver de:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR5MWRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVElZXZhbEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVElb2RlMUYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictRiw2JVEkeTFkRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARkBGYW9GQEYrRmFvRkBGK0Zhb0ZA

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJiomSSJ0R0YmIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHRiU2IywkKiYiIiNGLkYtRi5GLkYuNyMvLUYrNiNGLUYs

LywkKiYiIiUiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiNGJkkidEdGLEYmRiZGJiEiIiIiIQ==

A quinta fun\303\247\303\243o \303\251 uma solu\303\247\303\243o, como podemos ver de:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSR5MWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==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

LyIiIUYj

Estes resultados mostram que 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 e 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 s\303\243o tamb\303\251m solu\303\247\303\265es desta equa\303\247\303\243o diferencial enquanto 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 n\303\243o \303\251 uma solu\303\247\303\243o.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 22.2: Solu\303\247\303\243o Geral</Text-field>

Encontrar todas as fun\303\247\303\265es que satisfa\303\247am a equa\303\247\303\243o diferencial:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

\303\251 equivalente a encontrar todas as antiderivadas de:

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

Ly1JImZHNiI2I0kieEdGJSwmKiYiIiMiIiIpRidGKiEiIkYrIiIkRis=

Pelas Regras de Antideriva\303\247\303\243o no cap\303\255tulo anterior, a fam\303\255lia de todas as antiderivadas para esta fun\303\247\303\243o foi encontrada como sendo:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEjRjJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiw2JVEoZ2Vuc29sMkYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEYrRllGQEYrRllGQA==

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCYqJiIiIyIiIilJInhHRihGLCEiIkYtIiIkRi1GLywoKiZGLEYtRi9GMEYwKiZGMUYtRi9GLUYtSSJDR0YoRi0=

Portanto, para cada valor da constanteLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, a fun\303\247\303\243o:

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJiwoKiYiIiMiIiJJInhHRiYhIiJGMSomIiIkRi9GMEYvRi9JIkNHRiZGLzcjLy1GKzYjRjBGLA==

\303\251 uma solu\303\247\303\243o desta equa\303\247\303\243o diferencial. Esta solu\303\247\303\243o, com a constante, \303\251 chamada de a solu\303\247\303\243o geral da equa\303\247\303\243o diferencial porque esta express\303\243o cont\303\251m todas as solu\303\247\303\265es poss\303\255veis da equa\303\247\303\243o diferencial. De fato, esta express\303\243o \303\251 uma fam\303\255lia de um par\303\242metro de fun\303\247\303\265es, pois cada uma delas \303\251 uma solu\303\247\303\243o da equa\303\247\303\243o diferencial. As solu\303\247\303\265es se diferenciam uma das outras pelo valor da constante aditiva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 22.3: Solu\303\247\303\243o Particular </Text-field>

Considere a mesma equa\303\247\303\243o diferencial do Exemplo 22.2, a saber,

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

Desta vez, encontre a fun\303\247\303\243o que satisfaz a equa\303\247\303\243o diferencial acima e que passe pelo ponto 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.

Esta condi\303\247\303\243o extra, chamada de uma condi\303\247\303\243o inicial, \303\251 usada para selecionar uma \303\272nica solu\303\247\303\243o do n\303\272mero infinito de solu\303\247\303\265es dentro da solu\303\247\303\243o geral. Juntas, uma equa\303\247\303\243o diferencial e uma condi\303\247\303\243o inicial s\303\243o chamadas de um problema de valor inicial (PVI). A solu\303\247\303\243o para um problema de valor inicial \303\251 chamada de uma solu\303\247\303\243o particular.

Portanto, um outro modo para enunciar o problema para este exemplo \303\251: Encontre a solu\303\247\303\243o espec\303\255fica que satisfa\303\247a o problema de valor inicial:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVvZGUyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRpYzNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

NyQvSSJ4RzYiIiIiL0kieUdGJSIiJg==

A solu\303\247\303\243o geral \303\251 conhecida do Exemplo 22.2, e \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUShnZW5zb2wyRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA

L0kieUc2IiwoKiYiIiMiIiJJInhHRiQhIiJGKiomIiIkRihGKUYoRihJIkNHRiRGKA==

Quando a condi\303\247\303\243o inicial \303\251 substitu\303\255da nesta equa\303\247\303\243o, o resultado \303\251 a seguinte solu\303\247\303\243o para C:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LyIiJiwmIiIiRiVJIkNHNiJGJQ==

Esta equa\303\247\303\243o \303\251 facilmente resolvida, levando a:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjYzNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRiw2JVEoaXNvbGF0ZUYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNxM0YnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRIkNGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZccEZARitGXHBGQEYrRlxwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGXHBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNjM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

L0kiQ0c2IiIiJQ==

A solu\303\247\303\243o espec\303\255fica que soluciona este problema de valor inicial \303\251 portanto:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSlwYXJ0c29sM0YnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEYrRllGQEYrRllGQA==

L0kieUc2IiwoKiYiIiMiIiJJInhHRiQhIiJGKiomIiIkRihGKUYoRigiIiVGKA==

Para verificar se esta solu\303\247\303\243o realmente satisfaz o problema de valor inicial, cheque se esta solu\303\247\303\243o satisfaz a equa\303\247\303\243o diferencial via:

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

LywmKiYiIiMiIiIpSSJ4RzYiRiUhIiJGJiIiJEYmRiM=

e verifique que a condi\303\247\303\243o inicial \303\251 satisfeita via:

LyIiJkYj

Como ambos os testes retornaram uma equa\303\247\303\243o que \303\251 verdadeira, com a condi\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR1EmMC4wZW1GJy1JI21uR0YkNiRRIjBGJ0Y+Rj5GK0Y+, encontramos a solu\303\247\303\243o para o problema de valor inicial. (Tecnicamente, esta solu\303\247\303\243o faz sentido no maior intervalo contendo o ponto inicial e para o qual a equa\303\247\303\243o diferencial \303\251 satisfeita. Aqui, significa para 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.)

Como vimos no Exemplo 22.3, pelas Regras de Antiderivada no Cap\303\255tulo 21, a fam\303\255lia de todas as antiderivadas encontrada para esta fun\303\247\303\243o \303\251:

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCYqJiIiIyIiIilJInhHRihGLCEiIkYtIiIkRi1GLywoKiZGLEYtRi9GMEYwKiZGMUYtRi9GLUYtSSJDR0YoRi0=

Portanto, para cada valor de C, a fun\303\247\303\243o:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJiwoKiYiIiMiIiJJInhHRiYhIiJGMSomIiIkRi9GMEYvRi9JIkNHRiZGLzcjLy1GKzYjRjBGLA==

\303\251 uma solu\303\247\303\243o da equa\303\247\303\243o diferencial. Esta solu\303\247\303\243o, com a constante, \303\251 chamada a solu\303\247\303\243o geral da equa\303\247\303\243o diferencial.

<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.

randomize(): with(PDEtools,declare): with(RandomTools): declare(y(x), prime=x, quiet): nn := Generate(integer(range=1..4)): nn1 := Generate(integer(range=0..5)): nn2 := Generate(integer(range=-5..5)): dfn := diff(y(x),x)=sort(Generate(polynom(integer(range=-20..20),x,degree=nn)),x); rdfn := rhs(dfn): init1 := y(nn1)=nn2; ans1 := dsolve({dfn,init1});

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInhHRilGKywmRishIzoiIzsiIiI=

Ly1JInlHNiI2IyIiIiEiIg==

Ly1JInlHNiI2I0kieEdGJSwoKiRGJyIiIyMhIzpGKkYnIiM7IyEjPkYqIiIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 22.1</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.

Encontre a solu\303\247\303\243o particular de

y' = `$rdfn`

com a condi\303\247\303\243o inicial `$init1`

Entre com sua resposta na forma y = ...

(Observa\303\247\303\243o: Deve-se entrar com uma resposta antes de clicar em "Grade".)

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=ans1;

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(rhs(`$ANSWER`)-rhs(`$RESPONSE`)=0 );

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.

Find the antiderivative of `$rdfn` v(don't forget about the constant).

Using the result from Hint #1, solve for the constant by using `$init1`.

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>Find the antiderivative of </mtext><mtext></mtext><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>15</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>16</mn></mrow></mrow><mtext> v</mtext><mtext>(don't forget about the constant).</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(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Using the result from Hint #1, solve for the constant by using </mtext><mtext></mtext><mrow><mrow><mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>1</mn></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);

<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(`TextArea0`, `value`); ANSWER := ans1; if evalb(rhs(ANSWER)-rhs(eval(parse(StudentResponse)))=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><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mfrac><mn>15</mn><mn>2</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>16</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mn>19</mn><mn>2</mn></mfrac></mrow></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><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mfrac><mn>15</mn><mn>2</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>16</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mn>19</mn><mn>2</mn></mfrac></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" layout="Heading 1" spaceabove="7">Exemplo 22.4: Solu\303\247\303\243o Gr\303\241fica </Text-field>

Considere o mesmo problema de valor inicial do Exemplo 22.3:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVvZGUyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRpYzNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

NyQvSSJ4RzYiIiIiL0kieUdGJSIiJg==

A solu\303\247\303\243o geral foi encontrada no Exemplo 22.2 e uma solu\303\247\303\243o particular foi encontrada no Exemplo 22.3. Para muitas equa\303\247\303\265es diferenciais isto n\303\243o \303\251 vi\303\241vel -- ou necess\303\241rio -- para obter uma representa\303\247\303\243o simb\303\263lica da solu\303\247\303\243o; frequentemente uma solu\303\247\303\243o gr\303\241fica pode fornecer a informa\303\247\303\243o necess\303\241ria. Como isto est\303\241 al\303\251m do alcance deste curso para aprofundar muito sobre an\303\241lise gr\303\241fica de equa\303\247\303\265es diferenciais, precisa-se estar ciente de algumas das ferramentas b\303\241sicas para preparar uma solu\303\247\303\243o gr\303\241fica de uma equa\303\247\303\243o diferencial ou de um problema de valor inicial.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">O Maplet Antiderivative</Text-field>

Porque a equa\303\247\303\243o diferencial neste exemplo \303\251 resolvida encontrando uma antiderivada, o maplet Antiderivative (O Tutor Antiderivadas) pode ser usado para obter uma solu\303\247\303\243o gr\303\241fica para o problema de valor inicial. O seu uso para resolver o PVI no Exemplo 22.4 \303\251 mostrado na Figura 22.1.

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Figura 22.1 Exemplo 22.4: O TutorAntiderivative usado para encontrar a solu\303\247\303\243o da equa\303\247\303\243o diferencial e desenhar o gr\303\241fico

Uma vez iniciado o maplet, entre com o lado direito da equa\303\247\303\243o diferencial na caixa Fun\303\247\303\243o, troque a pelo ponto inicial, 1, e b para alguma coisa suficientemente grande para ver o que acontece a uma certa dist\303\242ncia do ponto inicial. Na caixa Valor, entre com a condi\303\247\303\243o inicial como 5. Clique o bot\303\243o Exibir. V\303\241rias solu\303\247\303\265es da solu\303\247\303\243o geral est\303\243o desenhadas em verde, o gr\303\241fico da solu\303\247\303\243o particular \303\251 mostrado em azul. A curva vermelha do lado direito da equa\303\247\303\243o diferencial -- derivada da curva azul e de cada curva verde.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">O Comando DEplot </Text-field>

O comando DEplot, no pacote DEtools, \303\251 um m\303\251todo mais geral e poderoso para exibir a solu\303\247\303\243o gr\303\241fica de um problema de valor inicial. O pacote \303\251 acessado com o comando

Para a equa\303\247\303\243o diferencial:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

podemos obter a Figura 22.2, um gr\303\241fico da solu\303\247\303\243o atrav\303\251s do pontoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSI1RidGOEY4RjhGO0Y4com a sintaxe:

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Figura 22.2

Exemplo 22.4: Solu\303\247\303\243o gr\303\241fica obtida com o comando DEplot

Note que a sintaxe \303\251 um pouco complicada. A condi\303\247\303\243o inicial est\303\241 em uma lista porque \303\251 poss\303\255vel suprir muitas condi\303\247\303\265es iniciais para a mesma equa\303\247\303\243o diferencial. As setas verdes mostram a inclina\303\247\303\243o da solu\303\247\303\243o em uma grade de pontos, e coletivamente, s\303\243o chamadas um campo de dire\303\247\303\265es. A caracter\303\255stica mais importante destas setas \303\251 que as curvas solu\303\247\303\265es s\303\243o sempre tangentes a estas setas. (Para omitir as setas do gr\303\241fico, use o argumento opcional: arrows=none.)

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Equa\303\247\303\265es Separ\303\241veis </Text-field>

Somente equa\303\247\303\265es diferenciais que possam ser escritas na forma:

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

podem ser resolvidas diretamente encontrando um antiderivada. Se a equa\303\247\303\243o diferencial envolve ambos 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 e 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, ent\303\243o outra abordagem deve ser usada. Se a equa\303\247\303\243o diferencial pode ser escrita na forma:

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

dividindo a equa\303\247\303\243o por 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 (supondo 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,) \303\251 poss\303\255vel "separar vari\303\241veis," obtendo a forma:

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

Se ambos os lados desta equa\303\247\303\243o s\303\243o multiplicados pelo incremento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, o produto 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 \303\251 por defini\303\247\303\243o a diferencial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn. A equa\303\247\303\243o se torna:

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

Se as antiderivadas de ambos os lados desta equa\303\247\303\243o podem ser encontradas, isto dar\303\241:

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

uma forma impl\303\255cita da solu\303\247\303\243o da equa\303\247\303\243o diferencial.

J\303\241 que o Maple n\303\243o tem uma representa\303\247\303\243o verdadeira pra uma diferencial, se esta equa\303\247\303\243o fosse obtida no Maple ela pareceria mais como:

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 .

do que uma normalmente vista em livros textos .

Se uma condi\303\247\303\243o inicial \303\251 fornecida, ela poderia ser usada, depois da integra\303\247\303\243o, para obter uma equa\303\247\303\243o para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Supondo que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tenha sido encontrado, deveria finalmente ser poss\303\255vel encontrar 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 explicitamente, contanto que a fun\303\247\303\243o impl\303\255cita gerada pelas antiderivadas tenha uma forma suficientemente simples.

Basta da discuss\303\243o geral. Vamos ver alguns exemplos espec\303\255ficos.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 22.5: Um Primeiro Exemplo Separ\303\241vel </Text-field>

Encontre a solu\303\247\303\243o geral da equa\303\247\303\243o diferencial:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSVvZGU1RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJComKUkieEdGJiMiIiIiIiNGMUkieUdGJiEiIkY0NyMvLUklZGlmZkclKnByb3RlY3RlZEc2JC1GMzYjRi9GLywkKiZGLkYxRjtGNEY0

Esta equa\303\247\303\243o diferencial n\303\243o pode ser resolvida pela antideriva\303\247\303\243o direta, mas \303\251 separ\303\241vel. Multiplicando a equa\303\247\303\243o por 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 fornece:

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyomSSJ5R0YmIiIiSSN5J0dGJkYtLCQqJClJInhHRiYjRi0iIiNGLSEiIjcjLyomLUYsNiNGMkYtLUklZGlmZkclKnByb3RlY3RlZEc2JEY5RjJGLUYv

A multiplica\303\247\303\243o desta equa\303\247\303\243o pelo incremento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn d\303\241:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiYtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSNkeUYnRjZGOUZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGZ24tRiM2Ji1GPTYtUSomdW1pbnVzMDtGJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYjNidGKy1JJm1zcXJ0R0YkNiMtRiw2JVEieEYnRjZGOUY8LUYsNiVRI2R4RidGNkY5RkBGK0ZARitGQEYrRkA=

e a antideriva\303\247\303\243o de ambos os lados leva a:

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

Isto n\303\243o pode ser implemantado no Maple. Em vez disso, integre ambos os lados de 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 em rela\303\247\303\243o a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, obtendo:

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiQqJkkieUdGJiIiIkkjeSdHRiZGMkkieEdGJi1GLDYkLCQqJClGNCNGMiIiI0YyISIiRjQ3Iy8tRiw2JComLUYxNiNGNEYyLUklZGlmZkdGLjYkRkJGNEYyRjRGNQ==

Estas integrais s\303\243o bastante simples de se avaliar pela Regra da Pot\303\252ncia Generalizada (com 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=) e Regra da Pot\303\252ncia (com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiMUYnRj5GPi1GIzYkLUZaNiRRIjJGJ0Y+Rj4vJS5saW5ldGhpY2tuZXNzR0Zmbi8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zgby8lKWJldmVsbGVkR0ZCRitGPkYrRj4= ). O resultado, depois de inserir manualmente uma \303\272nica constante de integra\303\247\303\243o, \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLywkKiYjIiIiIiIjRi4pSSJ5R0YmRi9GLkYuLCYqJiNGLyIiJEYuKUkieEdGJiNGNUYvRi4hIiJJIkNHRiZGLjcjLywkKiZGLUYuKS1GMTYjRjdGL0YuRi5GMg==

Outra que multiplicando esta equa\303\247\303\243o por 2 para obter:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyokKUkieUdGJiIiIyIiIiwmKiYjIiIlIiIkRi8pSSJ4R0YmI0Y0Ri5GLyEiIiomRi5GL0kiQ0dGJkYvRi83Iy8qJCktRi02I0Y2Ri5GL0Yw

e trocando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRIkNGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= por uma nova constante, digamos K, para obter:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyokKUkieUdGJiIiIyIiIiwmKiYjIiIlIiIkRi8pSSJ4R0YmI0Y0Ri5GLyEiIkkiS0dGJkYvNyMvKiQpLUYtNiNGNkYuRi9GMA==

o resultado t\303\243o simples como a solu\303\247\303\243o geral pode ser. Antes de tirar a raiz quadrada necessitamos considerar que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiM2Ji1GOzYtUSIoRidGPi9GQUY2RkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjE2NjY2NjdlbUYnL0ZTRmZuLUYjNiQtRiw2JVEieEYnRjRGN0Y+LUY7Ni1RIilGJ0Y+RllGQ0ZaRkdGSUZLRk1GZW4vRlNRLDAuMTExMTExMWVtRidGPkYrRj5GK0Y+ pode ser positivo ou negativo.

<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>

randomize(): with(PDEtools,declare): with(RandomTools): declare(y(x), prime=x, quiet): nn3 := Generate(integer(range=1..5)): nn4_ := Generate(choose([2,3,4,5,10])): nn4 := Generate(rational(range=0..1,denominator=nn4_)): nn5 := Generate(choose([nn3,nn3,-nn4,nn4])): nnn6 := Generate(integer(range=1..10)): nn6 := Generate(choose([-nnn6,nnn6])): ff_[1] := x^(nn5+1)*diff(y(x),x)*y(x)=nn6: ff_[2] := y(x)*diff(y(x),x)=x^nn5*nn6: ff_[3] := diff(y(x),x)=x^nn5/y(x): ff_[4] := diff(y(x),x)=y(x)^2/x^nn5: ff_[5] := diff(y(x),x)=x^nn5*y(x)^2: for i from 1 to 5 do ans2_[i] := eval(dsolve(ff_[i]), _C1 = C); od: nn7 := Generate(integer(range=1..5)); dfn2 := ff_[nn7]; rdfn2 := rhs(dfn2): ans2 := ans2_[nn7]; if (nops([ans2])>1) then ans2 := lhs(ans2[1])^2 = rhs(ans2[1])^2; fi;

IiIi

LyooSSJ4RzYiIyIiKiIjNS1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YlNiNGJEYkIiIiRi1GMCIiJQ==

NiQvLUkieUc2IjYjSSJ4R0YmKiQsJiokRigjIiIiIiM1IiMhKUkiQ0dGJkYtI0YtIiIjL0YkLCRGKSEiIg==

LyokLUkieUc2IjYjSSJ4R0YmIiIjLCYqJEYoIyIiIiIjNSIjISlJIkNHRiZGLQ==

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 22.2</Text-field>

Resolva a equa\303\247\303\243o diferencial y' = `$rdfn2` Se sua resposta est\303\241 na forma de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JlEieUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYlUSIyRidGOEY7RjsvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGOw== = ... ,ent\303\243o deixe nesta forma.

(Observa\303\247\303\243o: Deve-se entrar com uma resposta antes de clicar em "Grade")

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans2;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

checkans2 := proc(a,r) local check1,check2,check3; check1 := evalb(rhs(a)=rhs(r)); check2 := evalb(simplify(rhs(a)=rhs(r))); check3 := evalb(check1=true and check2=true); return check3; end: checkans2(`$ans2`,`$RESPONSE`);

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

Separate the variables.

Don't forget about the constant at the end.

Replace this text with hint 3.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Separate the variables.</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(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Don't forget about the constant at the end.</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>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea1`, `value`); ANSWER := ans2; checkans2 := proc(a,r) local check1,check2,check3; check1 := evalb(rhs(a)=rhs(r)); check2 := evalb(simplify(rhs(a)=rhs(r))); check3 := evalb(check1=true and check2=true); return check3;end: if checkans2(ans2,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><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mrow><mn>80</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mrow><mn>1</mn><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>10</mn></mrow></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></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><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mrow><mn>80</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mrow><mn>1</mn><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>10</mn></mrow></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></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" layout="Heading 1" spaceabove="7">Exemplo 22.6: Movimento de Proj\303\251til </Text-field>

Um objeto se movendo em uma trajet\303\263ria retil\303\255nea tem acelera\303\247\303\243o (em cent\303\255metros por segundo) dada por

L0kiYUc2IiwmKiYiIzUiIiIpLCZGKEYoSSJ0R0YkRigiIiYhIiJGKComIiIkRigtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSNQaUdGM0YoRitGKEYoRig=

Suponha que o objeto comece do repouso com a posi\303\247\303\243o inicial 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, onde a dist\303\242ncia \303\251 medida de uma origem fixa O. Encontre a velocidade e posi\303\247\303\243o do objeto para qualquer tempo 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.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Velocidade </Text-field>

A acelera\303\247\303\243o \303\251 uma taxa de varia\303\247\303\243o (em rela\303\247\303\243o ao tempo) da velocidade. Isto d\303\241 a equa\303\247\303\243o diferencial

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A solu\303\247\303\243o geral desta equa\303\247\303\243o diferencial \303\251 obtida diretamente pela integra\303\247\303\243o.

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiRJI3YnR0YmSSJ0R0YmLUYsNiQsJiomIiM1IiIiKSwmRjdGN0YxRjciIiYhIiJGNyomIiIkRjctSSRzaW5HRi02IyomSSNQaUdGLkY3RjFGN0Y3RjdGMTcjLy1GLDYkLUklZGlmZkdGLjYkLUkidkdGJjYjRjFGMUYxRjI=

O resultado \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kidkdGJiwoKiYjIiImIiIjIiIiKSwmRjFGMUkidEdGJkYxIiIlISIiRjYqKCIiJEYxSSNQaUclKnByb3RlY3RlZEdGNi1JJGNvc0c2JEY6RiU2IyomRjlGMUY0RjFGMUY2SSJDR0YmRjE3Iy8tRis2I0Y0Riw=

A condi\303\247\303\243o inicial 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 pode ser usada para determinar a solu\303\247\303\243o particular. A equa\303\247\303\243o a ser resolvida para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LyIiISwoIyIiJiIiIyEiIiomIiIkIiIiSSNQaUclKnByb3RlY3RlZEdGKEYoSSJDRzYiRis=

e a sua solu\303\247\303\243o d\303\241:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjcTRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRiw2JVEoaXNvbGF0ZUYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNxM0YnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRIkNGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZccEZARitGXHBGQEYrRlxwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGXHBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

L0kiQ0c2IiwmIyIiJiIiIyIiIiomIiIkRilJI1BpRyUqcHJvdGVjdGVkRyEiIkYp

Portanto, a fun\303\247\303\243o velocidade \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1GLDYlUSVldmFsRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRI3EyRidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRiw2JVEjcTRGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZccEZARitGXHBGQEYrRlxwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGXHBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJWRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kidkdGJiwqKiYjIiImIiIjIiIiKSwmRjFGMUkidEdGJkYxIiIlISIiRjYqKCIiJEYxSSNQaUclKnByb3RlY3RlZEdGNi1JJGNvc0c2JEY6RiU2IyomRjlGMUY0RjFGMUY2Ri5GMSomRjhGMUY5RjZGMTcjLy1GKzYjRjRGLA==

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

A taxa de varia\303\247\303\243o (em rela\303\247\303\243o ao tempo) da posi\303\247\303\243o \303\251 a velocidade. Isto d\303\241 a equa\303\247\303\243o diferencial:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjcydHRiYsKiomIyIiJiIiIyIiIiksJkYxRjFJInRHRiZGMSIiJSEiIkY2KigiIiRGMUkjUGlHJSpwcm90ZWN0ZWRHRjYtSSRjb3NHNiRGOkYlNiMqJkY5RjFGNEYxRjFGNkYuRjEqJkY4RjFGOUY2RjE3Iy8tSSVkaWZmR0Y6NiQtSSJzR0YmNiNGNEY0Riw=

An\303\241logo \303\240 acelera\303\247\303\243o, esta equa\303\247\303\243o diferencial pode ser resolvida diretamente pela integra\303\247\303\243o. Aqui ter\303\255amos:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiRJI3MnR0YmSSJ0R0YmLUYsNiQsKiomIyIiJiIiIyIiIiksJkY5RjlGMUY5IiIlISIiRj0qKCIiJEY5SSNQaUdGLkY9LUkkY29zR0YtNiMqJkZARjlGMUY5RjlGPUY2RjkqJkY/RjlGQEY9RjlGMTcjLy1GLDYkLUklZGlmZkdGLjYkLUkic0dGJjYjRjFGMUYxRjI=

O resultado \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxNkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kic0dGJiwsKiYjIiImIiInIiIiKSwmRjFGMUkidEdGJkYxIiIkISIiRjEqKEY1RjEpSSNQaUclKnByb3RlY3RlZEciIiNGNi1JJHNpbkc2JEY6RiU2IyomRjlGMUY0RjFGMUY2KiYjRi9GO0YxRjRGMUYxKihGNUYxRjRGMUY5RjZGMUkiS0dGJkYxNyMvLUYrNiNGNEYs

A condi\303\247\303\243o inicial 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 pode ser usada para determinar a solu\303\247\303\243o espec\303\255fica. A equa\303\247\303\243o resultante para K \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjcTdGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRiw2JVElZXZhbEYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNxNkYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUZcbzYmLUYjNilGKy1GIzYnLUYsNiVRInRGJ0Y2RjktRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtSSNtbkdGJDYkUSIwRidGQC8lK2V4ZWN1dGFibGVHRkRGQEZjby1GIzYnLUYsNiVRInNGJ0Y2RjlGYnAtRmZwNiRRIzEwRidGQEZpcEZARitGaXBGQEZALyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmlwRkBGQEZpcEZARitGaXBGQEYrRmlwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGaXBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxN0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LyIjNSwmIyIiJiIiJyIiIkkiS0c2IkYo

e sua solu\303\247\303\243o \303\251:

L0kiS0c2IiMiI2IiIic=

Portanto, a fun\303\247\303\243o posi\303\247\303\243o \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1GLDYlUSVldmFsRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRI3E2RidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRiw2JVEjcThGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZccEZARitGXHBGQEYrRlxwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGXHBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Gr\303\241ficos da Solu\303\247\303\243o</Text-field>

Os gr\303\241ficos da posi\303\247\303\243o, velocidade e acelera\303\247\303\243o em 10 segundos de movimento s\303\243o mostrados na Figura 22.3

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Figura 22.3 Exemplo 22.6: Posi\303\247\303\243o, velocidade e acelera\303\247\303\243o do proj\303\251til

Observe as oscila\303\247\303\265es em todas as tr\303\252s fun\303\247\303\265es. Para valores de t maiores a acelera\303\247\303\243o e velocidade tornam-se aproximadamente peri\303\263dicas. Enquanto que a posi\303\247\303\243o oscila e ela n\303\243o \303\251 peri\303\263dica. Porque a velocidade oscila em torno de uma constante positiva (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 ) e a posi\303\247\303\243o se estabelece para oscila\303\247\303\265es em torno de uma fun\303\247\303\243o linear.

<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(): n8 := Generate(integer(range=0..3)): n9 := Generate(integer(range=-3..5)): n10 := Generate(integer(range=-10..10)): n11 := Generate(integer(range=0..10)): n12 := Generate(choose([1,2])): n13 := n12: if (n12=1) then n13 := Generate(choose([1,2])): end if: if n8 = 0 then n9 := abs(n9): n10 = 0: end if: A1 := a = n8*t + n9 + n10 * sin(Pi*t): V1 := dsolve({diff(v(t),t)=rhs(A1),v(0)=0}): S1 := simplify(dsolve({diff(s(t),t)=rhs(V1),s(0)=n11})): fn3 := (A1,V1)[n12]; ans3 := (V1,S1)[n13]; tt3 := ('acceleration','velocity')[n12]: aa3 := ('velocity','position')[n13]: aaa3 := ('v(t)','s(t)')[n13]: un3 := (cm/s^3,cm/s^2)[n12]: hh31 := ('Acceleration','Velocity')[n12]:

L0kiYUc2IiwmSSJ0R0YkIiIkISIkIiIi

Ly1JInNHNiI2I0kidEdGJSwoKiRGJyIiJCMiIiIiIiMqJEYnRi0jISIkRi0iIidGLA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 22.3</Text-field>

Um objeto se movendo ao longo de um caminho tem `t3` (em `$un3`) dado por `$fn3` Suponha que o objeto inicie do repouso com a posi\303\247\303\243o inicial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1YkdGJDYlLUYsNiVRInNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjBGJy9GO1Enbm9ybWFsRidGQy8lL3N1YnNjcmlwdHNoaWZ0R0ZCRitGQ0YrRkM= = `$n11`, onde a dist\303\242ncia \303\251 medida de um ponto fixo O. Encontre a `$aa3` do objeto para qualquer tempo t > 0 Entre com sua resposta como `$aaa3` = ...

(Note: Deve-se entrar com uma resposta antes de clicar em "Grade".)

acceleration = acelera\303\247\303\243o

velocity = velocidade

position = posi\303\247\303\243o

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=eval(ans3,Pi=`π`);

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

evalb(rhs(`$ans3`)-rhs(`$RESPONSE`)=0);

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

Remember that acceleration is the rate of change of velocity; and velocity is the rate of change of position.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea2`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Remember that acceleration is the rate of change of velocity; and velocity is the rate of change of position.</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>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea2`, `value`); ANSWER := eval(ans3,Pi=`π`); if evalb(rhs(ans3)-rhs(eval(parse(StudentResponse)))=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><mrow><mi>s</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msup><mi>t</mi><mn>3</mn></msup></mrow></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>6</mn></mrow></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><mrow><mi>s</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msup><mi>t</mi><mn>3</mn></msup></mrow></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msup><mi>t</mi><mn>2</mn></msup></mrow></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>6</mn></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" layout="Heading 1" spaceabove="7">Exemplo 22.7: Popula\303\247\303\243o de Lobos </Text-field>

A popula\303\247\303\243o de lobos, W, em um certo estado tem crescido a uma raz\303\243o proporcional \303\240 raiz c\303\272bica do tamanho da popula\303\247\303\243o atual. A popula\303\247\303\243o era estimada de ser 1000 em 1980 e 1700 em 1990.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">(a) Escreva a equa\303\247\303\243o diferencial para a popula\303\247\303\243o de lobos. Transcreva as duas condi\303\247\303\265es em equa\303\247\303\265es matem\303\241ticas. </Text-field>

Designando por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= o tempo, em anos, deste 1980. A varia\303\247\303\243o da popula\303\247\303\243o de lobos \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZFdGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0=. Dessa forma, a equa\303\247\303\243o diferencial que governa esta popula\303\247\303\243o \303\251:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjVydHRiYqJkkia0dGJiIiIilJIldHRiYjRi4iIiRGLjcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRjA2I0kidEdGJkY7KiZGLUYuKUY5RjFGLg==

onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 a constante de proporcionalidade. A informa\303\247\303\243o sobre a popula\303\247\303\243o em 1980 e 1990 pode ser expressa como:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRlcTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRlcTJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JIldHRiY2IyIiISIlKzVGKQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JIldHRiY2IyIjNSIlKzxGKQ==

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">(b) Resolver a equa\303\247\303\243o diferencial. Usar as duas condi\303\247\303\265es para determinar os valores para a constante de proporcionalidade e a constante de integra\303\247\303\243o. </Text-field>

Esta equa\303\247\303\243o diferencial \303\251 separ\303\241vel. Em vez disso, escreva:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyomKUkiV0dGJiMiIiIiIiQhIiJJI1cnR0YmRi9JImtHRiY3Iy8qJiktRi02I0kidEdGJkYuRjEtSSVkaWZmRyUqcHJvdGVjdGVkRzYkRjhGOkYvRjM=

Agora, a integral indefinida de cada lado em rela\303\247\303\243o a t \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjcTJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNilGKy1GIzYqLUY9Ni9RKCYjODc0NztGJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRidGQC9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RidGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZeby1GIzYnLUYsNiVRJGxoc0YnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk9GXW9GX28tSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSNxMUYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmBwRkBGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0ZncC8lKmxpbmVicmVha0dRJWF1dG9GJy1GPTYvUTAmRGlmZmVyZW50aWFsRDtGJ0ZnbkZARmpuRkJGRUZHRklGS0ZNRk9GXW9GX28tRiw2JVEidEYnRjZGOUZgcEZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYjNipGWi1GIzYnLUYsNiVRJHJoc0YnRjZGOUZlb0Zob0ZgcEZARitGYnBGYHFGY3FGYHBGQEYrRmBwRkBGK0ZgcEZARitGYHBGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZgcEZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiQqJilJIldHRiYjIiIiIiIkISIiSSNXJ0dGJkY0SSJ0R0YmLUYsNiRJImtHRiZGODcjLy1GLDYkKiYpLUYyNiNGOEYzRjYtSSVkaWZmR0YuNiRGQkY4RjRGOEY5

Similar ao Exemplo 22.4, estas integrais s\303\243o bastante f\303\241ceis de avaliar pela Regra de Pot\303\252ncia Generalizada (com 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 e 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 ) e a Regra da Constante. O resultado, depois de inserir manualmente uma \303\272nica constante de integra\303\247\303\243o, \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLywkKiYjIiIkIiIjIiIiKUkiV0dGJiNGL0YuRjBGMCwmKiZJImtHRiZGMEkidEdGJkYwRjBJIkNHRiZGMDcjLywkKiZGLUYwKS1GMjYjRjdGM0YwRjBGNA==

Logo, \303\251 poss\303\255vel obter uma f\303\263rmula expl\303\255cita para a popula\303\247\303\243o de lobos:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTRHRiYvSSJXR0YmKiQpLCYqKCMiIiMiIiQiIiJJImtHRiZGNUkidEdGJkY1RjUqJkYyRjVJIkNHRiZGNUY1I0Y0RjNGNTcjLy1GLTYjRjdGLg==

N\303\243o est\303\241 claro que esta forma seja uma mais f\303\241cil de usar para avaliar as constantes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

Para determinar valores para k e C, use a informa\303\247\303\243o dos dois pontos fornecidos na descri\303\247\303\243o do problema na solu\303\247\303\243o impl\303\255cita. O sistema resultantes de duas equa\303\247\303\265es (lineares) para k e C \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjcTVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRiw2JVElZXZhbEYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNxM0YnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUZcbzYmLUYjNilGKy1GIzYnLUYsNiVRInRGJ0Y2RjktRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtSSNtbkdGJDYkUSIwRidGQC8lK2V4ZWN1dGFibGVHRkRGQEZjby1GIzYoRistRiM2Jy1GLDYlUSJXRidGNkY5RmVuLUZcbzYkLUYjNiVGX3BGaXBGQEZARmlwRkBGYnAtRmZwNiRRJTEwMDBGJ0ZARmlwRkBGK0ZpcEZARkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGaXBGQEZARmlwRkBGK0ZpcEZARitGaXBGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZpcEZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjcTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRiw2JVElZXZhbEYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNxM0YnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUZcbzYmLUYjNilGKy1GIzYnLUYsNiVRInRGJ0Y2RjktRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtSSNtbkdGJDYkUSMxMEYnRkAvJStleGVjdXRhYmxlR0ZERkBGY28tRiM2KEYrLUYjNictRiw2JVEiV0YnRjZGOUZlbi1GXG82JC1GIzYlRl9wRmlwRkBGQEZpcEZARmJwLUZmcDYkUSUxNzAwRidGQEZpcEZARitGaXBGQEZALyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmlwRkBGQEZpcEZARitGaXBGQEYrRmlwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGaXBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxNUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJy1JI21vR0YkNi1RIjtGJ0Y9LyUmZmVuY2VHRjwvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGOkY9LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxNkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LywkKiYjIiIkIiIjIiIiKSIlKzUjRidGJkYoRihJIkNHNiI=

LywkKiYjIiIkIiIjIiIiKSIlKzwjRidGJkYoRigsJiomIiM1RihJImtHNiJGKEYoSSJDR0YwRig=

<Text-field style="Heading 3" layout="Heading 3">Note: Compara\303\247\303\243o com a Solu\303\247\303\243o Expl\303\255cita </Text-field>

O sistema obtido da forma impl\303\255cita da solu\303\247\303\243o da equa\303\247\303\243o diferencial \303\251 mais simples que o sistema de equa\303\247\303\265es que seria obtido se a forma expl\303\255cita da solu\303\247\303\243o fosse usada. Este outro sistema seria:

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

LyIlKzUsJCoqIyIiIyIiKiIiIilGJyNGKUYnRikpIiIkRitGKSlJIkNHNiIjRi1GJ0YpRik=

LyIlKzwqJCksJiomIyIjPyIiJCIiIkkia0c2IkYrRisqJiMiIiNGKkYrSSJDR0YtRitGKyNGKkYwRis=

O valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser obtido da primeira equa\303\247\303\243o. E \303\251:

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

L0kiQ0c2IiIkXSI=

Inserindo este valor na segunda equa\303\247\303\243o, \303\251 simples de resolver para a constante de proporcionalidade. E \303\251

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSV2YWxrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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

L0kia0c2IiwmKiYjIiIkIiM/IiIiKSIlKzwjIiIjRihGKkYqIiM6ISIi

L0khRzYiJCIqJT4ubWohIik=

Acabamos de resolvemos o problema pelo m\303\251todo passo a passo, semelhante ao modo que se resolveria a m\303\243o. Se em vez disso estiv\303\251ssemos interessados em resolver o problema rapidamente, o comando solve do Maple teria sido usado nestas equa\303\247\303\265es. O resultado do comando solve \303\251:

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

PCQvSSJDRzYiLCQqJiMiIiQiIiMiIiIpIiUrNSNGKkYpRitGKy9JImtHRiUsJiomI0YpIiM/RispIiUrPEYuRitGKyomRjNGK0YsRishIiI=

S\303\263 porque o Maple pode ser usado para implementar um c\303\241lculo da mesma forma que um c\303\241lculo a m\303\243o \303\251 feito, isto n\303\243o deve fazer o estudante parar de pensar sobre o que a grande quest\303\243o realmente \303\251.

Em qualquer propor\303\247\303\243o, a solu\303\247\303\243o geral pode ser v\303\241lida como anteriormente, pela substitui\303\247\303\243o direta. O resultado \303\251

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

LywkKigjIiIiIiIkRiYpLCYqKCIiJ0YmSSJrRzYiRiZJInRHRi1GJkYmKiZGK0YmSSJDR0YtRiZGJiNGJiIiI0YmRixGJkYmRiM=

Que a solu\303\247\303\243o espec\303\255fica satisfaz os dois pontos dados pode ser visto de:

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

LywkKiojIiIiIiIkRiYpIiInI0YmIiIjRiYpSSJDRzYiRipGJkkia0dGLkYmRiZGIw==

LywkKigjIiIiIiIkRiYpLCYqJiIjZ0YmSSJrRzYiRiZGJiomIiInRiZJIkNHRi1GJkYmI0YmIiIjRiZGLEYmRiZGIw==

A popula\303\247\303\243o de lobos \303\251, portanto, definida pela fun\303\247\303\243o:

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kiV0dGJiokKSwmKiYkIiskSEBTQyUhIioiIiJJInRHRiZGM0YzIiQrIkYzIyIiJCIiI0YzNyMvLUYrNiNGNEYs

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">(c) Quando, caso aconte\303\247a, ser\303\241 a popula\303\247\303\243o de lobos o dobro de sua popula\303\247\303\243o em 1990?</Text-field>

A equa\303\247\303\243o que seria satisfeita quando a popula\303\247\303\243o de lobos for 3400 \303\251:

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcThHRiYvIiUrTSokKSwmKiYkIiskSEBTQyUhIioiIiJJInRHRiZGNUY1IiQrIkY1IyIiJCIiI0Y1NyNGLA==

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjcTlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRiw2JVEoaXNvbGF0ZUYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNxOEYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZccEZARitGXHBGQEYrRlxwRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGXHBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxOUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJy1JI21vR0YkNi1RIjtGJ0Y9LyUmZmVuY2VHRjwvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGOkY9LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjYGBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1GLDYlUSZldmFsZkYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2J0YrLUYjNictRiw2JVEkcmhzRidGNkY5RmVuLUZcbzYkLUYjNiUtRiw2JVEjcTlGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZccEZARitGXHBGQEZARlxwRkBGK0ZccEZARitGXHBGQEYrRlxwRkA=

L0kidEc2IiwmKiYkIitdZURjQiEjNSIiIikiJStNIyIiIyIiJEYqRiokRighIikhIiI=

L0khRzYiJCIrPSNvOShIISIp

Portanto, a popula\303\247\303\243o de lobos dobraria em tamanho em 30 anos, isto \303\251, em 2009.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">(d) De acordo com este modelo, qual \303\251 o progn\303\263stico a longo tempo para esta popula\303\247\303\243o de lobos? </Text-field>

O comportamento a um tempo longo desta solu\303\247\303\243o \303\251 governado peloexpoente nesta f\303\263rmula, porque este expoente \303\251 3/2 > 1, e 4.224 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= +100 > 0 para todo t > 0, o modelo prediz que a popula\303\247\303\243o de lobos crescer\303\241 sem limites.

No Maple, o limite relevante:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRxMTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLUkmTGltaXRHNiQlKnByb3RlY3RlZEdGJTYkL0kiV0dGJiokKSwmKiYkIiskSEBTQyUhIioiIiJJInRHRiZGOEY4IiQrIkY4IyIiJCIiI0Y4L0Y5SSlpbmZpbml0eUdGLTcjLUYrNiQvLUYwNiNGOUYxRj4=

avalia para:

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJmxpbWl0RzYkJSpwcm90ZWN0ZWRHRiU2JEkiV0dGJi9JInRHRiZJKWluZmluaXR5R0YuJCIiIkYzNyMvLUYsNiQtRjA2I0YyRjFGNA==

<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>

randomize(): with(PDEtools,declare): with(RandomTools): declare(P(t), prime=t, quiet): year1 := Generate(integer(range=185..199))*10; year2 := Generate(integer(range=195..200))*10; while (year1 >= year2) do year2 := Generate(integer(range=195..200))*10; end do: pop1 := Generate(integer(range=5..10))*10; pop2 := Generate(integer(range=100..500))*10; while (pop1 >= pop2) do pop2 := Generate(integer(range=70..300))*10; end do: n14 := Generate(integer(range=2..20)): r1 := diff(P(t),t)=k*P(t)^(1/n14); r2 := P(t)^(1/n14): d1 := isolate(dsolve(r1),P(t)): e1 := isolate(eval(d1,{P(t)=pop1,t=0}),_C1): e2 := isolate(eval(d1,{P(t)=pop2,t=year2-year1,e1}),k): d2 := P(t) = evalf(eval(rhs(d1),{e1,e2}),6); n15 := Generate(choose([1,2,3])): growth := ('double','triple','quadruple')[n15]: e3 := evalf(isolate(eval(d2,P(t)=(n15+1)*pop2),t)); year3 := floor(year2 + rhs(e3)); if (evalf(Im(rhs(e3))) <> 0) then year3 := 'na' end if: ans4 := d2,year3;

IiUhKT4=

IiUrPw==

IiMhKg==

IiVnVg==

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJQRzYiNiNJInRHRilGKyomSSJrR0YpIiIiRicjRi4iIzc=

Ly1JIlBHNiI2I0kidEdGJSokLCZGJyQiJ1VgNSEiJCQiJ3EmPSchIiUiIiIjIiM3IiM2

L0kidEc2IiQiK3BcbnhzISIp

IiVzPw==

NiQvLUkiUEc2IjYjSSJ0R0YmKiQsJkYoJCInVWA1ISIkJCIncSY9JyEiJSIiIiMiIzciIzYiJXM/

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 22.4</Text-field>

A popula\303\247\303\243o de coelhos em uma certa \303\241rea tem crescido a uma taxa proporcional a `$r2`, onde P \303\251 a popula\303\247\303\243o atual. A popula\303\247\303\243o estava estimada a ser `$pop1` em `$year1` e `$pop2` em `$year2`. Qual \303\251 a equa\303\247\303\243o que representa esta popula\303\247\303\243o de coelhos ? Quando (em que ano), caso aconte\303\247a, a popula\303\247\303\243o de coelhos ser\303\241 o `$growth` de sua popula\303\247\303\243o em `$year2`? Entre com sua resposta como P = ..., year Se a popula\303\247\303\243o n\303\243o for o `$growth`, entre com sua resposta como na. (Observa\303\247\303\243o: Deve-se entrar com uma resposta antes de clicar em "Grade".)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnZG91YmxlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUYsNiVRJmRvYnJvRidGL0YyRjk=

triple = triplo quadruple = qu\303\241druplo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans4;

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

evalb((rhs([`$ANSWER`][1]),[`$ANSWER`][2])=(rhs([`$RESPONSE`][1]),[`$RESPONSE`][2]));

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

Your differential equation should look something like `$r1`

When solving the differential equation, remember that the year `$year1` is at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRIjBGJ0ZARkBGK0ZARitGQA== here.

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></mtext><mtext> </mtext><mtext>Your differential equation should look something like </mtext><mtext></mtext><mrow><mrow><mi>P'</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>P</mi><mrow><mn>1</mn><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>12</mn></mrow></msup></mrow></mrow></mrow><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(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>When solving the differential equation, remember that the year </mtext><mtext></mtext><mrow><mn>1980</mn></mrow><mtext></mtext><mtext> is at </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><mrow><mi mathbackground='#000000'>t</mi><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&equals;</mo><mn mathbackground='#000000'>0</mn></mrow></mrow></mrow></mstyle><mtext> here.</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>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea3`, `value`); ANSWER := ans4; if evalb((rhs([ANSWER][1]),[ANSWER][2])=(rhs([eval(parse(StudentResponse))][1]),[eval(parse(StudentResponse))][2])) 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><mrow><mi>P</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><mn>105.342</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>t</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>61.8570</mn></mrow></mfenced><mrow><mn>12</mn><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>11</mn></mrow></msup></mrow></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2072</mn></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><mrow><mi>P</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><mn>105.342</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>t</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>61.8570</mn></mrow></mfenced><mrow><mn>12</mn><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>11</mn></mrow></msup></mrow></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2072</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 22.8: Velocidade de Escape </Text-field>

A atra\303\247\303\243o gravitacional LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= exercida pela Terra sobre um objeto com massa LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= localizado a uma diat\303\242ncia LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= do centro da terra \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSZGZ3JhdkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

L0kiRkc2IiwkKihJIm1HRiQiIiJJImdHRiRGKClJInNHRiQiIiMhIiJGLQ==

onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 a acelera\303\247\303\243o da gravidade na superf\303\255cie da Terra. Suponha que a velocidade inicial \303\251 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 e despreze a resist\303\252ncia do ar. Para pequenos valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHRkBGK0ZB um objeto lan\303\247ado da superf\303\255cie da Terra retornar\303\241 \303\240 Terra; para grandes valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHRkBGK0ZB o objeto n\303\243o cair\303\241 de volta na Terra. A velocidade de escape \303\251 o valor m\303\255nimo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHRkBGK0ZB para o qual o objeto n\303\243o retorna \303\240 Terra. Encontre a velocidade de escape como uma fun\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Passo 1: Modelando</Text-field>

A Segunda Lei do Movimento de Newton enuncia que:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNOMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

L0kiRkc2IiomSSJtR0YkIiIiSSJhR0YkRic=

onde F \303\251 a magnitude da for\303\247a gravitacional entre a Terra e o objeto lan\303\247ado e a \303\251 a acelera\303\247\303\243o do objeto. Isto d\303\241:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTFHRiYvLCQqKEkibUdGJiIiIkkiZ0dGJkYwKUkic0dGJiIiIyEiIkY1KiZGL0YwSSJhR0YmRjA3I0Ys

A acelera\303\247\303\243o \303\251 a varia\303\247\303\243o da velocidade em rela\303\247\303\243o ao tempo e a velocidade \303\251 a varia\303\247\303\243o da posi\303\247\303\243o em rela\303\247\303\243o ao tempo:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJWRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kiYUdGJkkjdidHRiY3Iy9GKy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ2R0YmNiNJInRHRiZGNg==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kidkdGJkkjcydHRiY3Iy8tRis2I0kidEdGJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJzR0YmRjBGMQ==

Isto leva \303\240 equa\303\247\303\243o diferencial:

q2 := subs( A,V, q1 );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTJHRiYvLCQqKEkibUdGJiIiIkkiZ0dGJkYwKUkic0dGJiIiIyEiIkY1KiZGL0YwSSRzJydHRiZGMDcjL0YtKiZGL0YwLUklZGlmZkclKnByb3RlY3RlZEc2JC1GPDYkLUYzNiNJInRHRiZGQ0ZDRjA=

Note que a equa\303\247\303\243o diferencial \303\251 de segunda ordem (ou seja, ela envolve a segunda derivada da fun\303\247\303\243o desconhecida. Para reduzir esta equa\303\247\303\243o a uma equa\303\247\303\243o diferencial de primeira ordem observe que:

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHNGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHNGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

A acelera\303\247\303\243o pode portanto ser expressa como:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkkb2RlR0YmLywkKihJIm1HRiYiIiJJImdHRiZGMClJInNHRiYiIiMhIiJGNSooRi9GMEkjdidHRiZGMEkidkdGJkYwNyMvRi0qKEYvRjAtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUY4NiNGM0YzRjBGQEYw

Isto \303\251 melhor. A equa\303\247\303\243o diferencial \303\251 de primeira ordem para 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. Note que a vari\303\241vel independente \303\251 s e 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 \303\251 a fun\303\247\303\243o desconhecida.

Para completar o Passo 1, encontrando o valor inicial que comp\303\265em o modelo, o dado inicial no tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNpY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjNyQvSSJzR0YmSSJSR0YmLy1JInZHRiY2I0YtJkYwNiMiIiFGKQ==

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Passo 2: Solu\303\247\303\243o para Problema de Valor Inicial </Text-field>

A equa\303\247\303\243o diferencial no problema de valor inicial encontrado no Passo 1 \303\251 separ\303\241vel. Portanto escreva:

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLywkKiZJImdHRiYiIiIpSSJzR0YmIiIjISIiRjIqJkkjdidHRiZGLkkidkdGJkYuNyMvRisqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRjU2I0YwRjBGLkY9Ri4=

A integral indefinida desta equa\303\247\303\243o com respeito a s \303\251:

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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiQsJComSSJnR0YmIiIiKUkic0dGJiIiIyEiIkY3RjUtRiw2JComSSN2J0dGJkYzSSJ2R0YmRjNGNTcjL0YrLUYsNiQqJi1JJWRpZmZHRi42JC1GPDYjRjVGNUYzRkVGM0Y1

As antiderivadas s\303\243o facilmente encontradas usando as Regras de Antideriva\303\247\303\243o no Cap\303\255tulo 21. O resultado, depois de adicionar uma constante, \303\251:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSZnc29sOEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyomSSJnR0YmIiIiSSJzR0YmISIiLCYqJiNGLSIiI0YtKUkidkdGJkYzRi1GLUkiQ0dGJkYtNyMvRissJiomRjJGLSktRjU2I0YuRjNGLUYtRjZGLQ==

Para determinar o valor da constante que seja consistente com a condi\303\247\303\243o inicial, a seguinte equa\303\247\303\243o deve ser resolvida:

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTVHRiYvKiZJImdHRiYiIiJJIlJHRiYhIiIsJiomI0YvIiIjRi8pJkkidkdGJjYjIiIhRjVGL0YvSSJDR0YmRi83I0Ys

A constante C \303\251 ent\303\243o:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxNkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

L0kiQ0c2IiwmKiZJImdHRiQiIiJJIlJHRiQhIiJGKComI0YoIiIjRigpJkkidkdGJDYjIiIhRi1GKEYq

Substitua este resultado na solu\303\247\303\243o geral para obter a solu\303\247\303\243o espec\303\255fica na forma impl\303\255cita, a saber:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxN0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyomSSJnR0YmIiIiSSJzR0YmISIiLCgqJiNGLSIiI0YtKUkidkdGJkYzRi1GLSomRixGLUkiUkdGJkYvRi0qJkYyRi0pJkY1NiMiIiFGM0YtRi83Iy9GKywoKiZGMkYtKS1GNTYjRi5GM0YtRi1GNkYtRjhGLw==

Resolvendo para 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 d\303\241:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSZwc29sOEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyokKUkidkdGJiIiIyIiIiwoKihGLkYvSSJnR0YmRi9JInNHRiYhIiJGLyooRi5GL0YyRi9JIlJHRiZGNEY0KiQpJkYtNiMiIiFGLkYvRi83Iy8qJCktRi02I0YzRi5GL0Yw

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Passo 3: An\303\241lise da Solu\303\247\303\243o</Text-field>

Se o objeto retorna \303\240 Terra , a velocidade ser\303\241 zero quando o objeto alcan\303\247a a altura m\303\241xima. Se o objeto n\303\243o retorna \303\240 Terra a velocidade nunca \303\251 zero.

Da equa\303\247\303\243o:

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

v\303\252-se que a velocidade \303\251 zero quando:

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

LyIiISwoKigiIiMiIiJJImdHNiJGJ0kic0dGKSEiIkYnKihGJkYnRihGJ0kiUkdGKUYrRisqJCkmSSJ2R0YpNiNGI0YmRidGJw==

Esta equa\303\247\303\243o d\303\241, para o m\303\241ximo de altura do objeto, a quantidade:

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

L0kic0c2IiwkKioiIiMiIiJJIlJHRiRGKEkiZ0dGJEYoLCYqJkYnRihGKkYoISIiKiYpJkkidkdGJDYjIiIhRidGKEYpRihGKEYtRi0=

Como s \303\251 a dist\303\242ncia do centro da Terra, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtSSNtbkdGJDYkUSIwRidGOUY5. Portanto, esta f\303\263rmula faz sentido somente quando a seguinte inequa\303\247\303\243o \303\251 verdadeira:

MiIiISwkKioiIiMiIiJJIlJHNiJGJ0kiZ0dGKUYnLCYqJkYmRidGKkYnISIiKiYpJkkidkdGKTYjRiNGJkYnRihGJ0YnRi1GLQ==

Em termos da velocidade inicial, que podemos supor como positiva, isto requer que se mantenha a seguinte inequa\303\247\303\243o :

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNxOUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPQ==

MiwmKiYiIiMiIiJJImdHNiJGJiEiIiomKSZJInZHRig2IyIiIUYlRiZJIlJHRihGJkYmRi8=

Isolando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkobXN1YnN1cEdGJDYnLUYsNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjBGJy9GO1Enbm9ybWFsRidGQy1GQDYkUSIyRidGQy8lMXN1cGVyc2NyaXB0c2hpZnRHRkIvJS9zdWJzY3JpcHRzaGlmdEdGQkYrRkNGK0ZD tem-se a rela\303\247\303\243o :

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

MiokKSZJInZHNiI2IyIiISIiIyIiIiwkKihGKkYrSSJnR0YnRitJIlJHRichIiJGKw==

Consequentemente, se a velocidade inicial satisfaz 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 ent\303\243o n\303\243o existe altura m\303\241xima. Logo, a velocidade de escape \303\251 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 .

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

Este cap\303\255tulo \303\251 uma introdu\303\247\303\243o \303\240 equa\303\247\303\265es diferenciais e o seu uso como modelos para problemas f\303\255sicos. As equa\303\247\303\265es diferenciais usadas neste cap\303\255tulo puderam ser resolvidas pela integra\303\247\303\243o direta uma vez que as vari\303\241veis independentes e dependentes s\303\243o separadas. A solu\303\247\303\243o geral para uma equa\303\247\303\243o diferencial corresponde a uma fam\303\255lia de antiderivadas. Uma solu\303\247\303\243o espec\303\255fica para uma problema de valor inicial \303\251 encontrada usando a informa\303\247\303\243o adicional para identificar um \303\272nico membro na fam\303\255lia de solu\303\247\303\265es que satisfa\303\247a a equa\303\247\303\243o diferencial e a condi\303\247\303\243o inicial. Todas as solu\303\247\303\265es para a equa\303\247\303\243o diferencial podem ser checadas pela substitui\303\247\303\243o da solu\303\247\303\243o proposta de volta na equa\303\247\303\243o diferencial .

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

O pr\303\263ximo cap\303\255tulo sobre Somas de Riemann \303\251 o primeiro de dois cap\303\255tulos dedicados \303\240 introdu\303\247\303\243o de um segundo tipo de integral: a Integral Definida. As duas se\303\247\303\265es finais desta Unidade enfatizam uma rela\303\247\303\243o \303\255ntima entre as integrais definidas e indefinidas.

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