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

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Guia de Estudo de C\303\241lculo

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

Cap\303\255tulo 22: Equa\303\247\303\265es Diferenciais

Equa\303\247\303\265es Diferenciais s\303\243o freq\303\274entemente escontradas 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 disto as equa\303\247\303\265es diferenciais consideradas neste cap\303\255tulo s\303\243o relativamente simples, algumas quest\303\265es interessantes podem ser formuladas -- e repondidas. A chave para resolver (solving) 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 (indefinite integrals).

Este cap\303\255tulo tem dois objetivos :

come\303\247ar a desenvolver a habilidade de escrever equa\303\247\303\265es diferenciais que descrevam o s 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 (function) , de fato, satisfaz (satisfy) uma dada equa\303\247\303\243o diferencial.

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

Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio de Matem\303\241tica Maple (Maple Math Dictionary).

acceleration

bound

constant

curve

definiteintegral

dependent variable

derivative

evaluate

even

explicit

exponent

family

first-order

function

generalsolution

graph

implicit

indefiniteintegral

independent variable

inequality

infinite

initialcondition

integration

interval

linear

maximum

minimum

multiple

negative

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

restart;

with( plots ): with(PDEtools,declare):

<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 \303\251 uma senten\303\247a aberta (uma contendo pelo menos um espa\303\247o em branco que deve ser preenchido), a verdade da qual pode ser determinada somente depois da senten\303\247a completa . A equa\303\247\303\243o alg\303\251brica t\303\255pica em uma vari\303\241vel variable tal como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 portanto uma sente\303\247a cuja verdade ou falsidade \303\251 determinada pelo n\303\272mero dado para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Os processos para resolver equa\303\247\303\265es alg\303\251bricas ocultam a natureza verdadeiro/falso porque s\303\243o m\303\251todos para encontrar todos os n\303\272meros que fazem a senten\303\247a verdadeira .

poderia ser mais agrad\303\241vel ler que uma equa\303\247\303\243o alg\303\251brica \303\251 uma rela\303\247\303\243o envolvendo algum n\303\272mero desconhecido LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e, resolver a equa\303\247\303\243o requer encontrar o n\303\272mero que satisfaz a rela\303\247\303\243o estabelecida.

Uma equa\303\247\303\243o diferencial (differential equation) \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 (independent variable) para esta fun\303\247\303\243o. Resolver uma equa\303\247\303\243o diferencial \303\251 fazer a equa\303\247\303\243o verdadeira. A fun\303\247\303\243o pode ser expressa tanto explicitamente como implicitamente (implicitly). Exatamente como a equa\303\247\303\243o alg\303\251brica, a concentra\303\247\303\243o no processo de solu\303\247\303\243o (solution) oculta a natureza fundamental da equa\303\247\303\243o diferencial como uma sente\303\247a aberta que se torna verdadeira ou falsa dependendo de qual fun\303\247\303\243o \303\251 inserida na "abertura" da equa\303\247\303\243o. Existem muitos m\303\251todos para encontrar a fun\303\247\303\243o(ou fun\303\247\303\265es) que satisfaz (satisfy) uma dada equa\303\247\303\243o, mas por fim, a prova (proof) de que uma solu\303\247\303\243o (solution) 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 deferencial \303\251 exibida no Maple, temos usado para declarar a fun\303\247\303\243o do pacote PDEtools (Ferramentas EDP). Quando a funcionalidade instalada por esta fun\303\247\303\243o est\303\241 operando, 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 (dependent variable explicitly referenciando a vari\303\241vel independente. Portanto, para entrar Ta derivada (derivative) 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, escrever diff(y(x),x),e para entrar com a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, escrever y(x). Maple devolver\303\241 will echo a derivada como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeSdGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn e devolver\303\241 will echo 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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

declare(y(t), prime=t, quiet): y1a := y(t) = sin( 2*t ): y1b := y(t) = sin( t ): y1c := y(t) = cos( 2*t ): y1d := y(t) = t*cos( 2*t ): y1e := y(t) = C[1]*rhs(y1a) + C[2]*rhs(y1c): y1a; y1b; y1c; y1d; y1e;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0YlNiMsJComIiIjIiIiSSJ0R0YmRjRGNDcjLy1GKzYjRjVGLA==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0YlNiNJInRHRiY3Iy8tRitGMEYs

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0YlNiMsJComIiIjIiIiSSJ0R0YmRjRGNDcjLy1GKzYjRjVGLA==

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJiomSSJ0R0YmIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHRiU2IywkKiYiIiNGLkYtRi5GLkYuNyMvLUYrNiNGLUYs

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJiwmKiYmSSJDR0YmNiMiIiJGMS1JJHNpbkc2JCUqcHJvdGVjdGVkR0YlNiMsJComIiIjRjFJInRHRiZGMUYxRjFGMSomJkYvNiNGOUYxLUkkY29zR0Y0RjZGMUYxNyMvLUYrNiNGOkYs

S\303\243o solu\303\247\303\265es da equa\303\247\303\243o diferencial

ode1 := diff( y(t), t$2 ) + 4*y(t) = 0: ode1;

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 ( evaluate) todas as derivadas e simplifique. Tudo isto \303\251 feito no Maple com a sintaxe

eval( ode1, y1a );

LyIiIUYj

Se o resultado \303\251 uma declara\303\247\303\243o verdadeira -- para todos os valores (values) relevantes da independente vari\303\241vel -- 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 (values) 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 leva a

eval( ode1, y1b );

LywkKiYiIiQiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGLEYmRiYiIiE=

Como esta equa\303\247\303\243o \303\251 verdadeira somente quando t \303\251 um m\303\272ltiplo (multiple) 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.

The same process can be used for the remaining three functions. The third function is a solution, as we see from

y1c; eval( ode1, y1c );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0YlNiMsJComIiIjIiIiSSJ0R0YmRjRGNDcjLy1GKzYjRjVGLA==

LyIiIUYj

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

y1d; eval( ode1, y1d );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kieUdGJiomSSJ0R0YmIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHRiU2IywkKiYiIiNGLkYtRi5GLkYuNyMvLUYrNiNGLUYs

LywkKiYiIiUiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiNGJkkidEdGLEYmRiZGJiEiIiIiIQ==

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

y1e; eval( ode1, y1e );

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

declare(y(x),prime=x,quiet): ode2 := diff( y(x), x ) = 2/x^2 + 3: ode2;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

\303\251 eq\303\274ivalente a encontrar todas as antiderivadas de

f2 := rhs( ode2 ): f(x) = f2;

Ly1JImZHNiI2I0kieEdGJSwmKiYiIiMiIiIpRidGKiEiIkYrIiIkRis=

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

F2 := Int( f2, x ): gensol2 := value( F2 ) + C: F2 = gensol2;

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCYqJiIiIyIiIilJInhHRihGLCEiIkYtIiIkRi1GLywoKiZGLEYtRi9GMEYwKiZGMUYtRi9GLUYtSSJDR0YoRi0=

Portanto, para cada valor da constante (constant) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , a fun\303\247\303\243o

y(x) = gensol2;

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 (general solution) da 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 (one-parameter family) de fun\303\247\303\265es, 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 Espec\303\255fica </Text-field>

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

ode2;

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktRiM2Ji1GMjYtUSIoRidGNS9GOVEldHJ1ZUYnRjsvRj5GUkY/RkFGQ0ZFL0ZIUSwwLjE2NjY2NjdlbUYnL0ZLRlUtRiM2Ji1JI21uR0YkNiRRIjFGJ0Y1LUYyNi1RIixGJ0Y1RjgvRjxGUkY9Rj9GQUZDRkVGRy9GS1EsMC4zMzMzMzMzZW1GJy1GWjYkUSI1RidGNUY1LUYyNi1RIilGJ0Y1RlFGO0ZTRj9GQUZDRkVGVC9GS1EsMC4xMTExMTExZW1GJ0Y1RitGNUYrRjU=.

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 gerals . Juntas, uma equa\303\247\303\243o diferencial e uma condi\303\247\303\243o inicial (initial condition) 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 espec\303\255fica.

Portanto, um outro modo para enuciar 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

ic3 := [ x=1, y=5 ]: ode2; ic3;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

NyQvSSJ4RzYiIiIiL0kieUdGJSIiJg==

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

y = gensol2;

L0kieUc2IiwoKiYiIiMiIiJJInhHRiQhIiJGKiomIiIkRihGKUYoRihJIkNHRiRGKA==

Quando a condi\303\247\303\243o inicial \303\251 substituida nesta equa\303\247\303\243o, o resultado \303\251 a seguinte solu\303\247\303\243o para C.

q3 := eval( y=gensol2, ic3 ): q3;

LyIiJiwmIiIiRiVJIkNHNiJGJQ==

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

c3 := isolate( q3, C ): c3;

L0kiQ0c2IiIiJQ==

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

partsol3 := eval( gensol2, c3 ): y = partsol3;

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

eval( ode2, y(x)=partsol3 );

LywmKiYiIiMiIiIpSSJ4RzYiRiUhIiJGJiIiJEYmRiM=

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

eval( y=partsol3, ic3 );

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 (interval) contendo o ponto inicial e para o qual a equa\303\247\303\243o diferencial \303\251 satisfeita. Aqui, significa para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=.)

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

F2 := Int( f2, x ): gensol2 := value( F2 ) + C: F2 = gensol2;

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCYqJiIiIyIiIilJInhHRihGLCEiIkYtIiIkRi1GLywoKiZGLEYtRi9GMEYwKiZGMUYtRi9GLUYtSSJDR0YoRi0=

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

y(x) = gensol2;

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});

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInhHRilGKywmRisiIzUhIiMiIiI=

Ly1JInlHNiI2IyIiJSIiJA==

Ly1JInlHNiI2I0kieEdGJSwoKiRGJyIiIyIiJkYnISIjISNwIiIi

JSFH

<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 espec\303\255fica de

y' = `$rdfn`

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

Entre com sua resposta na forma y = ...`

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

JSFH

(Sugest\303\243o#1Encontre a solu\303\247\303\243o espec\303\255fica de `$rdfn` (n\303\243o sew esque\303\247a sobre a constante)

(Sugest\303\243o#2:use o resultado da sugest\303\243o#1, Encontre a constante usando `$init1`)

JSFH

<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.

JSFHans1;

JSFH

<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 );

JSFH

<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><mrow><mn>10</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>2</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>4</mn></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>3</mn></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><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>69</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><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>69</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 22.4: Solu\303\247\303\243o Gr\303\241fica </Text-field>

Considere o mesmo problema de valor inicial do Exemplo 22.3, a saber,

ode2; ic3;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

NyQvSSJ4RzYiIiIiL0kieUdGJSIiJg==

A solu\303\247\303\243o geral foi encontrada no Exemplo 22.2 e uma solu\303\247\303\243o espec\303\255fica 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; freq\303\274entemente 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 Antiderivada (Antiderivative Maplet)</Text-field>

Porque a equa\303\247\303\243o diferencial neste exemplo \303\251 resolvida encontrando uma antiderivada, o maplet antiderivada (Antiderivative maplet )(o Antiderivative Tutor) pode ser usado para obter um gr\303\241fico solu\303\247\303\243o 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: Tutor Antiderivada (Antiderivative Tutor) usado para encontrar a solu\303\247\303\243o e fazer o gr\303\241fico

Uma vez iniciado o maplet, entre com o lado direito da equa\303\247\303\243o diferencial na caixa (espa\303\247o) Function , 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 Value (Valor) , entre com a condi\303\247\303\243o inicial como 5. Clique o bot\303\243o Display (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 (graph) da solu\303\247\303\243o espec\303\255fica \303\251 mostrado em azul. A curva (curve) 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 (Gr\303\241fico de Equa\303\247\303\265es Diferenciais), no pacote ED ferramentas (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

with( DEtools ):

Para a equa\303\247\303\243o diferncial

ode2;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjeSdHRiYsJiomIiIjIiIiKUkieEdGJkYuISIiRi8iIiRGLzcjLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YmNiNGMUYxRiw=

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

DEplot( ode2, y(x), x=0..5, [[1,5]], y=0..20, linecolor=black );

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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 (slope) da solu\303\247\303\243o em uma grade de pontos, e coletivamente, s\303\243o chamadas uma dire\303\247\303\243o de campo . A caracter\303\255stica mais importante destas setas \303\251 que as curvas solu\303\247\303\265es s\303\243o sempre tangentes (tangent) 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 (product) 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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 antiderivadas de ambos os lados desta equa\303\247\303\243o podem ser encontradas, isto dar\303\241

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

uma forma implicita da solu\303\247\303\243o da equa\303\247\303\243o diferencial.

J\303\241 que o Maple n\303\243o tem um verdadeira representa\303\247\303\243o pra uma diferencial, se esta equa\303\247\303\243o fosse obtida no Maplei 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

ode5 := diff( y(x), x ) = -sqrt(x)/y(x): ode5;

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 (separable). Multiplicando a equa\303\247\303\243o por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiM2Ji1GOzYtUSIoRidGPi9GQUY2RkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjE2NjY2NjdlbUYnL0ZTRmZuLUYjNiQtRiw2JVEieEYnRjRGN0Y+LUY7Ni1RIilGJ0Y+RllGQ0ZaRkdGSUZLRk1GZW4vRlNRLDAuMTExMTExMWVtRidGPkYrRj5GK0Y+ fornece

q1 := ode5*y(x): q1;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyomSSJ5R0YmIiIiSSN5J0dGJkYtLCQqJClJInhHRiYjRi0iIiNGLSEiIjcjLyomLUYsNiNGMkYtLUklZGlmZkclKnByb3RlY3RlZEc2JEY5RjJGLUYv

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

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

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNictSSNtb0dGJDYtUSYmaW50O0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPC8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR1EmMC4wZW1GJy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvRjhRJ2l0YWxpY0YnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmZuLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYsNiVRI2R5RidGUkZVRjctRjQ2LVEiPUYnRjdGOkY9Rj9GQUZDRkVGRy9GSlEsMC4yNzc3Nzc4ZW1GJy9GTUZmby1GIzYnLUY0Ni1RKiZ1bWludXMwO0YnRjdGOkY9Rj9GQUZDRkVGRy9GSkZORkwtRiM2KEYzLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RidGUkZVRitGVy1GLDYlUSNkeEYnRlJGVUY3LUY0Ni1RIitGJ0Y3RjpGPUY/RkFGQ0ZFRkcvRkpRLDAuMjIyMjIyMmVtRicvRk1GXXEtRiw2JVEiQ0YnRlJGVUY3RitGN0YrRjc=

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

q2 := Int( lhs(q1), x ) = Int( rhs(q1), x ): q2;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiQqJkkieUdGJiIiIkkjeSdHRiZGMkkieEdGJi1GLDYkLCQqJClGNCNGMiIiI0YyISIiRjQ3Iy8tRiw2JComLUYxNiNGNEYyLUklZGlmZkdGLjYkRkJGNEYyRjRGNQ==

Estas integrais s\303\243o bastante simples de se avaliar pela Regra de Pot\303\252ncia (Power Rule) Generalizada (com 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=) e a Regra de Pot\303\252ncia ( Power Rule) (com 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 ). O resultado, depois de inserir manualmente um \303\272nica constante de integra\303\247\303\243o, \303\251

q3 := value( lhs(q2) ) = value( rhs(q2) ) + C: q3;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLywkKiYjIiIiIiIjRi4pSSJ5R0YmRi9GLkYuLCYqJiNGLyIiJEYuKUkieEdGJiNGNUYvRi4hIiJJIkNHRiZGLjcjLywkKiZGLUYuKS1GMTYjRjdGL0YuRi5GMg==

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

q4 := 2*q3: q4;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyokKUkieUdGJiIiIyIiIiwmKiYjIiIlIiIkRi8pSSJ4R0YmI0Y0Ri5GLyEiIiomRi5GL0kiQ0dGJkYvRi83Iy8qJCktRi02I0Y2Ri5GL0Yw

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

q5 := eval( q4, C=K/2 ): q5;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyokKUkieUdGJiIiIyIiIiwmKiYjIiIlIiIkRi8pSSJ4R0YmI0Y0Ri5GLyEiIkkiS0dGJkYvNyMvKiQpLUYtNiNGNkYuRi9GMA==

o resultado t\303\243o simples como a solu\303\247\303\243o geral pode ser. Tirar a raiz quadrada (square root) necessita a considera\303\247\303\243o que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiM2Ji1GOzYtUSIoRidGPi9GQUY2RkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjE2NjY2NjdlbUYnL0ZTRmZuLUYjNiQtRiw2JVEieEYnRjRGN0Y+LUY7Ni1RIilGJ0Y+RllGQ0ZaRkdGSUZLRk1GZW4vRlNRLDAuMTExMTExMWVtRidGPkYrRj5GK0Y+ pode ser positivo (positive) ou negativo (negative).

<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;

IiIj

LyomLUkieUc2IjYjSSJ4R0YmIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JEYkRihGKSwkRigiIiM=

NiQvLUkieUc2IjYjSSJ4R0YmKiQsJiokRigiIiNGLEkiQ0dGJiIiIiNGLkYsL0YkLCRGKSEiIg==

LyokLUkieUc2IjYjSSJ4R0YmIiIjLCYqJEYoRilGKUkiQ0dGJiIiIg==

<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.

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

(Sugest\303\243o#1: separe as vari\303\241veis)

(Sugest\303\243o#2: N\303\243o esque\303\247a sobre a constante no final)

JSFH

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

JSFHans2;

JSFH

<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>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></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>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></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 (acceleration) (em cent\303\255metros por segundo) dada por

A := a = 10*(1+t)^(-5) + 3*sin(Pi*t): A;

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 (velocity) e posi\303\247\303\243o do objeto para qualquer tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=.

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

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

declare(v(t), prime=t, quiet): ode6v := diff( v(t), t ) = rhs(A): ode6v;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kjdidHRiYsJiomIiM1IiIiKSwmRi9GL0kidEdGJkYvIiImISIiRi8qJiIiJEYvLUkkc2luRzYkJSpwcm90ZWN0ZWRHRiU2IyomSSNQaUdGOkYvRjJGL0YvRi83Iy8tSSVkaWZmR0Y6NiQtSSJ2R0YmNiNGMkYyRiw=

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

q1 := Int( lhs(ode6v), t ) = Int( rhs(ode6v), t ): q1;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiRJI3YnR0YmSSJ0R0YmLUYsNiQsJiomIiM1IiIiKSwmRjdGN0YxRjciIiYhIiJGNyomIiIkRjctSSRzaW5HRi02IyomSSNQaUdGLkY3RjFGN0Y3RjdGMTcjLy1GLDYkLUklZGlmZkdGLjYkLUkidkdGJjYjRjFGMUYxRjI=

O resultado \303\251

q2 := value( lhs(q1) ) = value( rhs(q1) ) + C: q2;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kidkdGJiwoKiYjIiImIiIjIiIiKSwmRjFGMUkidEdGJkYxIiIlISIiRjYqKCIiJEYxSSNQaUclKnByb3RlY3RlZEdGNi1JJGNvc0c2JEY6RiU2IyomRjlGMUY0RjFGMUY2SSJDR0YmRjE3Iy8tRis2I0Y0Riw=

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 a ser resolvida para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251

q3 := eval( q2, [ t=0, v=0 ] ): q3;

LyIiISwoIyIiJiIiIyEiIiomIiIkIiIiSSNQaUclKnByb3RlY3RlZEdGKEYoSSJDRzYiRis=

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

q4 := isolate( q3, C ): q4;

L0kiQ0c2IiwmIyIiJiIiIyIiIiomIiIkRilJI1BpRyUqcHJvdGVjdGVkRyEiIkYp

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

V := eval( q2, q4 ): V;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjL0kidkdGJiwqKiYjIiImIiIjIiIiKSwmRjFGMUkidEdGJkYxIiIlISIiRjYqKCIiJEYxSSNQaUclKnByb3RlY3RlZEdGNi1JJGNvc0c2JEY6RiU2IyomRjlGMUY0RjFGMUY2Ri5GMSomRjhGMUY5RjZGMTcjLy1GKzYjRjRGLA==

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

A varia\303\247\303\243o (rate of change) (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

declare(s(t), prime=t, quiet): ode6p := diff(s(t), t) = rhs(V): ode6p;

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

q5 := Int( lhs(ode6p), t ) = Int( rhs(ode6p), t ): q5;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiRJI3MnR0YmSSJ0R0YmLUYsNiQsKiomIyIiJiIiIyIiIiksJkY5RjlGMUY5IiIlISIiRj0qKCIiJEY5SSNQaUdGLkY9LUkkY29zR0YtNiMqJkZARjlGMUY5RjlGPUY2RjkqJkY/RjlGQEY9RjlGMTcjLy1GLDYkLUklZGlmZkdGLjYkLUkic0dGJjYjRjFGMUYxRjI=

O resultado \303\251

q6 := value( lhs(q5) ) = value( rhs(q5) ) + K: q6;

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

q7 := eval( q6, [ t=0, s=10 ] ): q7;

LyIjNSwmIyIiJiIiJyIiIkkiS0c2IkYo

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

q8 := isolate( q7, K ): q8;

L0kiS0c2IiMiI2IiIic=

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

S := eval( q6, q8 ): S;

<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

plot( [ rhs(A), rhs(V), rhs(S) ], t=0..10, color=[red,blue,green], legend=["acceleration","velocity","position"] );

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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\263dica (periodic). Enquanto que a posi\303\247\303\243o oscila e ela n\303\243o \303\251 peri\303\263dica (periodic). 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 (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]:

Ly1JInZHNiI2I0kidEdGJSwqKiRGJyIiIyMiIiJGKiomSSNQaUclKnByb3RlY3RlZEchIiItSSRjb3NHNiRGL0koX3N5c2xpYkdGJTYjKiZGLkYsRidGLEYsISIlRiciIiYqJEYuRjAiIiU=

Ly1JInNHNiI2I0kidEdGJSwkKiYsLComRiciIiRJI1BpRyUqcHJvdGVjdGVkRyIiIyIiIi1JJHNpbkc2JEYuSShfc3lzbGliR0YlNiMqJkYtRjBGJ0YwISNDKiZGJ0YvRi1GLyIjOkY2IiNDKiRGLUYvIiNbRjBGLSEiIyNGMCIiJw==

JSFH

<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 `$tt3` (em `$un3`) dado por `$fn3` Suponha que o objeto inicie do repouso com a posi\303\247\303\243o inicial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1YkdGJDYlLUYsNiVRInNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjBGJy9GO1Enbm9ybWFsRidGQy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0YrRkM= = `$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

(Sugest\303\243o#1:Lembre -se que acelera\303\247\303\243o \303\251 a varia\303\247\303\243o da velocidade e velocidade \303\251 a varia\303\247\303\243o da posi\303\247\303\243o)

JSFH

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

JSFHeval(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><mfrac><mn>1</mn><mn>6</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfrac><mrow><mrow><mrow><msup><mi>t</mi><mn>3</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>&\134#x3c0;</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>24</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mi>&\134#x3c0;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>t</mi></mrow></mrow></mfenced></mrow></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>15</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>t</mi><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>&\134#x3c0;</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>24</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>&\134#x3c0;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>t</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>48</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>&\134#x3c0;</mi><mn>2</mn></msup></mrow></mrow></mrow><mrow><msup><mi>&\134#x3c0;</mi><mn>2</mn></msup></mrow></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><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><mfrac><mn>1</mn><mn>6</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfrac><mrow><mrow><mrow><msup><mi>t</mi><mn>3</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>&\134#x3c0;</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>24</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mi>&\134#x3c0;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>t</mi></mrow></mrow></mfenced></mrow></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>15</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>t</mi><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>&\134#x3c0;</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>24</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>&\134#x3c0;</mi><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>t</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>48</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>&\134#x3c0;</mi><mn>2</mn></msup></mrow></mrow></mrow><mrow><msup><mi>&\134#x3c0;</mi><mn>2</mn></msup></mrow></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 22.7: 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 (proportional) a raiz (root) c\303\272bica do tamanho da corrente popula\303\247\303\243o. A popula\303\247ao 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. Transforme 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, o equa\303\247\303\243o diferencial que governa esta popula\303\247\303\243o \303\251

declare(W(t), prime=t, quiet): ode7 := diff(W(t), t ) = k * W(t)^(1/3): ode7;

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

eq1 := W(0) = 1000: eq2 := W(10) = 1700: eq1; eq2;

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 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 (separable). Em vista disso, escreva

q1 := ode7/W(t)^(1/3): q1;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyomKUkiV0dGJiMiIiIiIiQhIiJJI1cnR0YmRi9JImtHRiY3Iy8qJiktRi02I0kidEdGJkYuRjEtSSVkaWZmRyUqcHJvdGVjdGVkRzYkRjhGOkYvRjM=

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

q2 := Int( lhs(q1), t ) = Int( rhs(q1), t ): q2;

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 de Constante. O resultado, depois de inserir manualmente uma \303\272nica constante de integra\303\247\303\243o, \303\251

q3 := value( lhs(q2) ) = value( rhs(q2) ) + C: q3;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLywkKiYjIiIkIiIjIiIiKUkiV0dGJiNGL0YuRjBGMCwmKiZJImtHRiZGMEkidEdGJkYwRjBJIkNHRiZGMDcjLywkKiZGLUYwKS1GMjYjRjdGM0YwRjBGNA==

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

q4 := isolate( q3, W(t) );

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

q5 := eval( q3, [t=0,W(t)=1000]): q6 := eval( q3, [t=10,W(t)=1700]): q5; q6;

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

eval( q4, [t=0,W(t)=1000]); eval( q4, [t=10,W(t)=1700]);

LyIlKzUsJCoqIyIiIyIiKiIiIilGJyNGKUYnRikpIiIkRitGKSlJIkNHNiIjRi1GJ0YpRik=

LyIlKzwqJCksJiomIyIjPyIiJCIiIkkia0c2IkYrRisqJiMiIiNGKkYrSSJDR0YtRitGKyNGKkYwRis=

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

valC := isolate( simplify( q5 ), C ): valC;

L0kiQ0c2IiIkXSI=

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

valk := isolate( eval( q6, valC ), k ): valk; `` = evalf(rhs(valk));

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 estivesse interessado 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:

solve({q5,q6},{C,k});

PCQvSSJDRzYiLCQqJiMiIiQiIiMiIiIpIiUrNSNGKkYpRitGKy9JImtHRiUsJiomI0YpIiM/RispIiUrPEYuRitGKyomRjNGK0YsRishIiI=

S\303\263 porque \303\241s vezes 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, n\303\243o deve fazer o estudante parar de pensar sobre o que a big picture???? 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 ( substitution) direta. O resultado \303\251

q7 := combine(simplify(eval( ode7, q4 ), assume=positive)): q7;

LywkKigjIiIiIiIkRiYpLCYqKCIiJ0YmSSJrRzYiRiZJInRHRi1GJkYmKiZGK0YmSSJDR0YtRiZGJiNGJiIiI0YmRixGJkYmRiM=

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

combine(eval( q7, t=0 )); combine(eval( q7, t=10));

LywkKiojIiIiIiIkRiYpIiInI0YmIiIjRiYpSSJDRzYiRipGJkkia0dGLkYmRiZGIw==

LywkKigjIiIiIiIkRiYpLCYqJiIjZ0YmSSJrRzYiRiZGJiomIiInRiZJIkNHRi1GJkYmI0YmIiIjRiZGLEYmRiZGIw==

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

psol7 := eval( q4, [valC,evalf(valk)] ): psol7;

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 lobo 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

q8 := eval( psol7, W(t)=3400 );

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcThHRiYvIiUrTSokKSwmKiYkIiskSEBTQyUhIioiIiJJInRHRiZGNUY1IiQrIkY1IyIiJCIiI0Y1NyNGLA==

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

q9 := isolate( q8, t ): q9; `` = evalf(rhs(q9));

L0kidEc2IiwmKiYkIitdZURjQiEjNSIiIikiJStNIyIiIyIiJEYqRiokRighIikhIiI=

L0khRzYiJCIrPSNvOShIISIp

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

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

O comportamento a longo termo desta solu\303\247\303\243o \303\251 governado pelo expoente (exponent) 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 (bound).

No Maple, o limite relevante

q10 := Limit( psol7, t=infinity ): q10;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLUkmTGltaXRHNiQlKnByb3RlY3RlZEdGJTYkL0kiV0dGJiokKSwmKiYkIiskSEBTQyUhIioiIiJJInRHRiZGOEY4IiQrIkY4IyIiJCIiI0Y4L0Y5SSlpbmZpbml0eUdGLTcjLUYrNiQvLUYwNiNGOUYxRj4=

avalia para

value( q10 );

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=

IiVdPg==

IiNn

IiVxOw==

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJQRzYiNiNJInRHRilGKyomSSJrR0YpIiIiRicjRi4iIiU=

Ly1JIlBHNiI2I0kidEdGJSokLCZGJyQiJ1MpPiIhIiUkIicjZTojRiwiIiIjIiIlIiIk

L0kidEc2IiQiK3QrQydbJCEiKQ==

IiVNPw==

NiQvLUkiUEc2IjYjSSJ0R0YmKiQsJkYoJCInUyk+IiEiJSQiJyNlOiNGLSIiIiMiIiUiIiQiJU0/

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

(Sugest\303\243o#1:A sua equa\303\247\303\243o diferencial deve parecer com algo como P' = kLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictRiM2JEYrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRitGK0YrRjE= `$r2`

(Sugest\303\243o #2:Quando estiver resolvendo a equa\303\247\303\243o diferencial, lembre-se que o ano `$year1` \303\251 em t = 0)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnZG91YmxlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUYsNiVRJmRvYnJvRidGL0YyRjk=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==triple = triplo quadruple = qu\303\241druplo

JSFH

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

JSFHans4;

<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>4</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>11.9840</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>t</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>21.5582</mn></mrow></mfenced><mrow><mn>4</mn><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>3</mn></mrow></msup></mrow></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2034</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>11.9840</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>t</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>21.5582</mn></mrow></mfenced><mrow><mn>4</mn><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>3</mn></mrow></msup></mrow></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2034</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

Fgrav := F = -m*g/s^2: Fgrav;

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB um objeto lan\303\247ado da superf\303\255cie da Terra retornar\303\241 \303\240 Terra ; para grandes valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB o objeto n\303\243o cair\303\241 de volta na Terra. A velocidade de escape \303\251 o valor m\303\255nimo (minimum) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB 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: Esquematizando (Modelando)</Text-field>

A Segunda Lei do Movimento de Newton enuncia que

N2 := F = m*a: N2;

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

q1 := eval( N2, Fgrav );

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.

A := a = diff( v(t), t ): V := v(t) = diff( s(t), t ): A; V;

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 (second order) (i.e., 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 (first-order) observe que

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHNGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHRGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEjZHZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYnLUYjNiQtRiw2JVEjZHNGJ0Y3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

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

declare(v(s), prime=s, quiet); ode := eval(q2, diff(s(t),t,t)=diff(v(s),s)*v(s));

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 (esquema), o dado inicial no tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= \303\251

ic := [ s=R, v(R)=v[0] ]: ic;

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

q3 := ode / m: q3;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLywkKiZJImdHRiYiIiIpSSJzR0YmIiIjISIiRjIqJkkjdidHRiZGLkkidkdGJkYuNyMvRisqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRjU2I0YwRjBGLkY9Ri4=

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

q4 := Int( lhs(q3), s ) = Int( rhs(q3), s ): q4;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLy1JJEludEc2JCUqcHJvdGVjdGVkR0YlNiQsJComSSJnR0YmIiIiKUkic0dGJiIiIyEiIkY3RjUtRiw2JComSSN2J0dGJkYzSSJ2R0YmRjNGNTcjL0YrLUYsNiQqJi1JJWRpZmZHRi42JC1GPDYjRjVGNUYzRkVGM0Y1

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

gsol8 := value( lhs(q4) ) = value( rhs(q4) ) + C: gsol8;

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.

q5 := subs(ic,subs(ic, gsol8));

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkjcTVHRiYvKiZJImdHRiYiIiJJIlJHRiYhIiIsJiomI0YvIiIjRi8pJkkidkdGJjYjIiIhRjVGL0YvSSJDR0YmRi83I0Ys

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

q6 := isolate( q5, C ): q6;

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,

q7 := eval( gsol8, q6 ): q7;

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjLyomSSJnR0YmIiIiSSJzR0YmISIiLCgqJiNGLSIiI0YtKUkidkdGJkYzRi1GLSomRixGLUkiUkdGJkYvRi0qJkYyRi0pJkY1NiMiIiFGM0YtRi83Iy9GKywoKiZGMkYtKS1GNTYjRi5GM0YtRi1GNkYtRjhGLw==

Resolvendo para 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 d\303\241

psol8 := isolate( q7, v(s)^2 ): psol8;

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 (maximum). 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

q8 := eval( psol8, v(s)=0 ): q8;

LyIiISwoKigiIiMiIiJJImdHNiJGJ0kic0dGKSEiIkYnKihGJkYnRihGJ0kiUkdGKUYrRisqJCkmSSJ2R0YpNiNGI0YmRidGJw==

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

smax := isolate( q8, s ): smax;

L0kic0c2IiwkKioiIiMiIiJJIlJHRiRGKEkiZ0dGJEYoLCYqJkYnRihGKkYoISIiKiYpJkkidkdGJDYjIiIhRidGKEYpRihGKEYtRi0=

Como s \303\251 a dist\303\242ncia do centro da Terra, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. Portanto, esta f\303\263rmula faz sentido somente quando a seguinte inequa\303\247\303\243o (inequality)\303\251 mantida (\303\251 verdade).

q9 := rhs(smax) > 0: q9;

MiIiISwkKioiIiMiIiJJIlJHNiJGJ0kiZ0dGKUYnLCYqJkYmRidGKkYnISIiKiYpJkkidkdGKTYjRiNGJkYnRihGJ0YnRi1GLQ==

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

temp := is(numer(rhs(smax))<0) assuming R>0,g>0: if temp then q9 := denom(rhs(smax))<0 else q9 := denom(rhs(smax))>0;fi: q9;

MiwmKiYiIiMiIiJJImdHNiJGJiEiIiomKSZJInZHRig2IyIiIUYlRiZJIlJHRihGJkYmRi8=

Isolando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkobXN1YnN1cEdGJDYnLUYsNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjBGJy9GO1Enbm9ybWFsRidGQy1GQDYkUSIyRidGQy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGSkYrRkNGK0ZD tem-se a rela\303\247\303\243o

q10 := isolate( q9, v[0]^2 ) assuming v0>0, R>0, g>0: q10;

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">Cap\303\255tulo Sum\303\241rio </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 deferencial 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 encontrado usando a informa\303\247\303\243o adicional para identificar um \303\272nico membro na fam\303\255lia solu\303\247\303\243o geral 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">Qual \303\251 o pr\303\263ximo?</Text-field>

O pr\303\263ximo cap\303\255tulo sobre Soma de Riemann (Riemann Sums) \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 (Definite Integral). As duas se\303\247\303\265es finais desta Unidade enfatiza uma rela\303\247\303\243o \303\255ntima entre integrais definida e indefinida .

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