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de Estudo de C\303\241lculo\302\251 Maplesoft, a division of Waterloo Maple Inc., 2005Cap\303\255tulo 23: Soma de Riemann
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field>Este cap\303\255tulo coloca a base para a defini\303\247\303\243o da Integral Definida (Definite Integral). Os objetivos espec\303\255ficos deste cap\303\255tulo s\303\243o introduzir a nota\303\247\303\243o (sigma) e aprender algumas f\303\263rmulas b\303\241sicas de Somat\303\263ria (summation),usar ret\303\242ngulos para aproximar a \303\241rea (area) sob o gr\303\241fico (graph) de uma fun\303\247\303\243o positiva (positive) ,definir uma soma de Riemann (Riemann sum) de uma fun\303\247\303\243o (function) em uma divis\303\243o (partition), eassociar somas de Riemann e \303\241reas aproximadas por ret\303\242ngulos.Estes t\303\263picos n\303\243o s\303\243o diretamente relacionados \303\240s discuss\303\265es de Integrais indefinidas (indefinite integrals) ou equa\303\247\303\265es diferenciais encontradas anteriormente nesta unidade. Existe uma conex\303\243o entre os cap\303\255tulos anteriores e este material, mas que n\303\243o ser\303\243o vistos at\303\251 o cap\303\255tulo sobre Teoremas Fundamentais de C\303\241lculo mais tarde nesta unidade.O desenvolvimento de uma compreens\303\243o visual das Somas de Riemann \303\251 auxiliado pelo uso do comando ApproximateInt do pacote Student[Calculus1] e do maplet ApproximateIntegration maplet.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Cap\303\255tulo Gloss\303\241rio</Font></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).
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Inicializa\303\247\303\243o</Font></Text-field>restart;with( plots ):
with( plottools ):
with( Student[Calculus1] ):
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Procedimentos Auxiliares </Text-field>SpherePyramid := proc( N::posint )
local COLORS, layer;
layer := proc( n, N, C )
local CTR, G;
G := seq( -(n-1)+2*i, i=0..n-1 );
CTR := [seq( seq( [i,j,N+1-n], j=G ), i=G )];
return seq( sphere( c, 1, color=C ), c=CTR );
end proc:
COLORS := [ red, blue, green, cyan, pink,
magenta, navy, orange, violet, wheat ]:
display( seq( layer( i, N, COLORS[i] ), i=1..N ),
view=[-(N+2)..N+2,-(N+2)..N+2,-2..N+2],
style=patchnogrid, scaling=constrained );
end proc:RiemannSumComparison := proc( F, A, B, Part, Meth )
local MM;
MM := Matrix( nops(Part), nops(Meth), (i,j)->evalf(ApproximateInt( F, x=A..B, partition=Part[i], method=Meth[j] )) ):
< < `` | op(Meth) >,
< Vector(Part) | MM > >;
end proc:
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Nota\303\247\303\243o Sigma para Somas </Font></Text-field>Consider a pir\303\242mide constru\303\255da em 4 camadas. Cada camada \303\251 constru\303\255da de esferas id\303\252nticas. A camada inferior \303\251 um arranjo de 4 \303\227 4 de esferas, a pr\303\263xima camada \303\251 um arranjo de 3 \303\227 3 , a camada acima, um arranjo de 2 \303\227 2 esferas, e a camada superior, uma \303\272nica esfera (sphere). Percebemos claramente que esta pir\303\242mide cont\303\251m um total de 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 esferas, como tamb\303\251m pode ser determinado da Figura 23.1, que mostra a pir\303\242mide de esferas .
SpherePyramid( 4 );Figura 23.1 Pir\303\242mide constru\303\255da com quatro camadas de esferas
Quantas esferas existem em uma pir\303\242mide construida com 10 camadas? A Figura 23.2 mostra tal pir\303\242mide
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23.2 Pir\303\242mide constru\303\255da com 10 camadas de esferas
(O comando SpherePyramid( 10 ) (Pir\303\242mide de esferas(10)) toma pelo menos 3 minutos para executar e fornecer (a resposta), mesmo em um computador r\303\241pido com ampla mem\303\263ria. N\303\263s inserimos a figura como uma imagem. Leitores passando esta planilha em um computador mais veloz, pode gerar a figura no momento removendo o coment\303\241rio (#) e pressionando a chave Enter.)#SpherePyramid( 10 );O n\303\272mero de esfera na Figura 23.2 \303\251 bastante \303\263bvia, \303\251 1 + 4 + 9 + 16 + \342\213\257 + 81 + 100 ou, usando a nota\303\247\303\243o sigma ,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Esta express\303\243o \303\251 lida como "a somat\303\263ria de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ao quadrado com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= variando de 1 at\303\251 10." A letra Grega mai\303\272scula sigma, 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, corresponde ao nosso S, \303\251 usado para representar uma soma, e \303\251 freq\303\274entemente chamado de "o sinal de somat\303\263ria." Os n\303\272meros a serem somados s\303\243o formados de acordo a f\303\263rmula seguinte ao sinal da somat\303\263ria, aqui \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRImlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIyRicvRjtRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkFGK0ZB , onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 o \303\255ndice. Os limites superiores e inferiores do alcance (range) dos valores (values) do \303\255ndice s\303\243o os inteiros que aparecem em cima e em baixo de 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, respectivamente; neste exemplo o \303\255ndice menor \303\251 1 e o \303\255ndice maior \303\251 10.Quando existem muitos n\303\272meros a serem somados, a nota\303\247\303\243o sigma prov\303\252 um modo mais eficiente para representar a soma. Um outro benef\303\255cio da nota\303\247\303\243o sigma \303\251 visto quando um ou ambos os limites (limits) da somat\303\263ria n\303\243o s\303\243o inteiros expl\303\255citos (explicit). Por exemplo, quantas camadas deve ter uma pir\303\242mide maior constru\303\255da com no m\303\241ximo 1000 esferas? Para responder esta quest\303\243o \303\251 necess\303\241rio encontrar ao maior valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para que 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 .
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">F\303\263rmulas e Propriedade de Somat\303\263ria </Font></Text-field>Existem uma poucas somat\303\263rias que aparecer\303\243o repetidamente nos pr\303\263ximos cap\303\255tulos. Elas s\303\243o
As provas destes resultados s\303\243o baseadas tipicamente em reescrever habilmente os termos da soma ou em indu\303\247\303\243o (on induction); estes detalhes n\303\243o s\303\243o importante neste momento.Tamb\303\251m, porque a nota\303\247\303\243o sigma \303\251 outra forma de escrever uma soma, todas as propriedades das somas incluem as somas representadas pala nota\303\247\303\243o sigma. Por exemplo, as somas de combina\303\247\303\265es lineares tornam-se uma combina\303\247\303\243o linear de somas, ou seja 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 para a quest\303\243o colocada - mas n\303\243o respondida - na se\303\247\303\243o anterior, o n\303\272mero de esferas em uma pir\303\242mide dez camadas \303\251s10 := Sum( i^2, i=1..10 ):
s10 = ``(10)*``(10+1)*``(2*10+1)/6;
`` = ``(10)*``(11)*``(21)/6;
`` = 385;Deixando Maple fazer o trabalho, temos novamente s10 = value( s10 );O n\303\272mero total de esferas necess\303\241rias para construir uma pir\303\242mide com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= camadas \303\251 sn := Sum( i^2, i=1..n ):
sn = factor(value( sn ));A pir\303\242mide com n camadas pode ser constru\303\255da com 1000 esferas, desde que sn := Sum( i^2, i=1..n ):
q1 := sn <= 1000.:
q1;tal queq2 := factor(value(q1)):
q2;A solu\303\247\303\243o (solution) para esta inequa\303\247\303\243o (inequality) c\303\272bica \303\251 dif\303\255cil de encontrar manualmente, mas o Maple pode completar esta tarefa com facilidade. O resultado \303\251 solve( q2, n );que \303\251 that is, s13 := eval( sn, n=13 ):
s13 = value( s13 );Depois de construir a pir\303\242mide de 13 camadas com 819 esferas ficar\303\243o 181 esferas restantes. Com as 181 esferas que restaram, uma outra pir\303\242mide pode ser constru\303\255da. O n\303\272mero encontrado de camadas nesta pir\303\242mide foi solve( value(sn)<=181., n );que s\303\243o ent\303\243o 7 camadas. O n\303\272mero de esferas que esta pir\303\242mide de 7 camadas consome \303\251s7 := eval( sn, n=7 ):
s7 = value( s7 );Esta pir\303\242mide deve ter 140 esfera em 7 camadas. Mais pir\303\242mides podem ser feitas com as 41 esferas restantes: uma pir\303\242mide de 4 camadas com 30 esferas, duas pir\303\242mides de 2 camadas com 5 esferas cada, e uma pir\303\242mide de 1 camada
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Aproxima\303\247\303\265es de \303\201rea por Ret\303\242ngulos</Font></Text-field>Um problema fundamental de matem\303\241tica anterior a descoberta de c\303\241lculo foi a computa\303\247\303\243o da \303\241rea de um regi\303\243o. A abordagem geral para tal problema \303\251 aproximar a regi\303\243o com uma cole\303\247\303\243o finita (finite) de regi\303\265es simples -- tipicamente ret\303\242ngulos-- e ent\303\243o tomar um limite. Como foi feito para ambos limites e derivadas, nossas investiga\303\247\303\265es iniciais desenvolver\303\241 a compreens\303\243o (gr\303\241fica) intuitiva desta \303\241rea. O comando ApproximateInt do pacote Student[Calculus1] ser\303\241 usado para criar ambos gr\303\241ficos e somas na nota\303\247\303\243o sigma. O maplet ApproximateIntegration maplet pode ser usado para obter resultados semelhantes sem a complica\303\247\303\243o da sintaxe do comando ApproximateInt .
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Exemplo 23.1: \303\201rea de um Trapez\303\263ide </Font></Text-field>Encontre a \303\241rea da regi\303\243o limitada pelos gr\303\241ficos de 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4= , e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. No Maple, as curvas do contorno s\303\243o dadas por F1 := 2*x + 1:
y = F1;
a := 0;
b := 4;A Figura 23.3 mostra a regi\303\243o contornada pelas curvas.
P1 := plot(F1, x=a..b, filled=true, color=pink, tickmarks=[4,[seq(k,k=1..9)]]):
display( P1 );Figura 23.3 O Trapez\303\263ide do Exemplo 23.1
Existe um n\303\272mero de maneiras para computar a \303\241rea diretamente, uma vez que se nota que a regi\303\243o \303\251 um trapez\303\263ide. Por exemplo, a m\303\251dia das bases(aqui, vertical) vezes a atura ( aqui, horizontal) d\303\241 a \303\241rea de 20. Portanto, o prop\303\263sito deste cap\303\255tulo \303\251 ilustrar a abordagem com base de c\303\241lculo usando aproxima\303\247\303\265es pelos ret\303\242ngulos. Esta aproxima\303\247\303\243o \303\251 chamada uma Soma Riemann.
<Text-field style="Heading 3" layout="Heading 3">Passo 1: 4 subintervalos</Text-field>
<Text-field style="Heading 4" layout="Heading 4"><Font encoding="UTF-8">Gr\303\241fico</Font></Text-field>Para come\303\247ar, divida a base da regi\303\243o, o intervalo( interval)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 em 4 subintervalos de comprimento 1. Em cada uma destes intervalos, costrua o ret\303\242ngulo mais alto que se ajuste completamente sob o gr\303\241fico. Figura 23.4 mostra a regi\303\243o cheia (rosa) e os quatro ret\303\242ngulos (com bordas azuis).
PA4 := ApproximateInt(F1, x=a..b, partition=4, method=lower,
output=plot, boxoptions=[color=blue], functionoptions=[color=black, thickness=2, legend=""], showarea=false, title=""):
display([P1, PA4]);Figura 23.4 Quatro ret\303\242ngulos (ajustados pela esquerda) sob o gr\303\241fico 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
A soma das \303\241reas dos quatros ret\303\242ngulos \303\251 uma aproxima\303\247\303\243o da \303\241rea total do trapez\303\263ide
<Text-field style="Heading 4" layout="Heading 4"><Font encoding="UTF-8">Num\303\251rico</Font></Text-field>As alturas dos ret\303\242ngulos s\303\243o 1, 3, 5, e 7. Porque cada ret\303\242ngulo tem base 1, o total da \303\241rea dos quatros ret\303\242ngulos \303\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Como o gr\303\241fico de 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 \303\251 sempre crescente, o ret\303\242ngulo mais alto ajustado sob a curva ocorre no limite esquerdo de cada subintervalo. A soma dos quatro ret\303\242ngulos pode ser escrita na nota\303\247\303\243o sigma e avaliada (evaluated) usando o comando ApproximateInt do pacote Student[Calculus1] como se segue.SA4 := ApproximateInt( F1, x=a..b, partition=4, method=left, output=sum ):
SA4 = value( SA4 );
<Text-field style="Heading 3" layout="Heading 3">Passo 2: 8 subintervalos</Text-field>
<Text-field style="Heading 4" layout="Heading 4"><Font encoding="UTF-8">Gr\303\241fico</Font></Text-field>Dividir a base de cada um dos quatro ret\303\242ngulos e construir ret\303\242ngulos nestes 8 intervalos do mesmo modo como foi feito anteriormente. A Figura 23.5 mostra o resultado desta nova constru\303\247\303\243o.
PA8 := ApproximateInt(F1, x=a..b, partition=8, method=left,
output=plot, boxoptions=[color=green, thickness=4],
showarea=false, title="", functionoptions=[legend=""] ):
display([P1, PA4, PA8]);Figura 23.5 Quatro e oito ret\303\242ngulos (ajustados pela esquerda) sob o gr\303\241fico de 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
Note como estes oitos ret\303\242ngulos (com bordas em verde) prov\303\252 uma aproxima\303\247\303\243o melhor para a \303\241rea exata da regi\303\243o rosa.
<Text-field style="Heading 4" layout="Heading 4"><Font encoding="UTF-8">Num\303\251rico</Font></Text-field>A base de cada um dos oito ret\303\242ngulos \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GNTYkUSIyRidGOEY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZFLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== ; as alturas s\303\243o 1, 2, 3, 4, 5, 6, 7, e 8. Logo, a \303\241rea total encerrada nestes 8 ret\303\242ngulos \303\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Esta soma \303\251 tediosa para se escrever. Como o padr\303\243o \303\251 t\303\243o simples, n\303\243o \303\251 dif\303\255cil escrever a soma na nota\303\247\303\243o sigma e ent\303\243o avali\303\241-la. Os resultados s\303\243oSA8 := ApproximateInt( F1, x=a..b, partition=8, method=left, output=sum ):
SA8 = value( SA8 );
<Text-field style="Heading 4" layout="Heading 4">Note</Text-field>Note que o Maple fatorou a base comum fora de cada termo e que a soma poderia tamb\303\251m ser escrita como 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 . Note tamb\303\251m que o Maple pode formular e computar tais somas fora do comando ApproximateInt . Por exemplo, poder\303\255amos ter escrito algo como 1/2*Sum(i,i=1..8) = 1/2*sum(i,i=1..8);
<Text-field style="Heading 3" layout="Heading 3">Passo 3: 16 subintervalos</Text-field>
<Text-field style="Heading 4" layout="Heading 4"><Font encoding="UTF-8">Gr\303\241fico</Font></Text-field>Uma vez mais, dividir a base de cada ret\303\242ngulo pela metade e construir 16 ret\303\242ngulos como anteriormente. A Figure 23.6 mostra a resultado desta nova constru\303\247\303\243o
PA16 := ApproximateInt(F1, x=a..b, partition=16, method=lower,
output=plot, boxoptions=[color=yellow, thickness=7],
showarea=false, title="", functionoptions=[legend=""]):
display([P1,PA4,PA8,PA16]);Figura 23.6 Quatro, oito e dezesseis ret\303\242ngulos(ajustados pela esquerda) sob o gr\303\241fico de 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
Visualmente, os 16 ret\303\242ngulos prov\303\252em uma melhor aproxima\303\247\303\243o qoe os 8 ret\303\242ngulos
<Text-field style="Heading 4" layout="Heading 4"><Font encoding="UTF-8">Num\303\251rico</Font></Text-field>A base de cada ret\303\242ngulo \303\251 agora 1/4. O primeiro ret\303\242ngulo tem altura 1 e a altura de cada ret\303\242ngulo sucessivo aumenta de 1/2; o \303\272ltimo ret\303\242ngulo tem altura 17/2. A \303\241rea total fechada pelos 16 ret\303\242ngulos \303\251, em nota\303\247\303\243o sigma SA16 := ApproximateInt(F1, x=a..b, partition=16, method=left, output=sum):
SA16 = value( SA16 );Note que as tr\303\252s avalia\303\247\303\265es s\303\243o menores que a real \303\241rea do trapez\303\263de.
<Text-field style="Heading 3" layout="Heading 3">Passo 4: Tomemos isto ao limite</Text-field>
<Text-field style="Heading 4" layout="Heading 4"><Font encoding="UTF-8">Gr\303\241fico</Font></Text-field>O processo de sbdivis\303\243o iniciado dos Passos 1-3 pode continuar para sempre. Falando praticamente, entretanto, \303\251 muito dif\303\255cil ver mais que 500 ret\303\242ngulos em um \303\272nico gr\303\241fico. A anima\303\247\303\243o na Figura 23.7 mostra a aproxima\303\247\303\243o do ret\303\242ngulo menor (e correspondendo a \303\241rea para 4, 8, 16, 32, 64, 128, 256, e 512 ret\303\242ngulos. (Os ret\303\242ngulos mais baixos t\303\252m suas alturas determinadas pelo mais baixo valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== no subintervalo formando a base do ret\303\242ngulo. Como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+LUY7Ni1RIj1GJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjc3Nzc3OGVtRicvRlNGam4tSSNtbkdGJDYkUSIyRidGPi1GOzYtUTEmSW52aXNpYmxlVGltZXM7RidGPkZARkNGRUZHRklGS0ZNRmluRltvRlktRjs2LVEnJnBsdXM7RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRmdvLUZdbzYkUSIxRidGPkZgb0Y+\303\251 crescente, os ret\303\242ngulos mais baixos s\303\243o eq\303\274ivalentes aos ret\303\242ngulos ajustados pela esquerda.)
Figura 23.7 Anima\303\247\303\243o mostrando os ret\303\242ngulos mais baixos preenchendo a \303\241rea sob 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
Note como os \303\272ltimos ajustes quase cobrem completamente o trapez\303\263ide. Por causa do modo como foram selecionados os ret\303\242ngulos, todas estas aproxima\303\247\303\265es para a \303\241rea s\303\243o muito pequenas. Por estas raz\303\265es, e por outras, \303\251 tentador sugerir que a \303\241rea deste trapez\303\263ide \303\251 20 (unidades ao quadrado).
<Text-field style="Heading 4" layout="Heading 4"><Font encoding="UTF-8">Anal\303\255tico</Font></Text-field>O passo final neste processo \303\251 executar os c\303\241culos simb\303\263licos que mostrar\303\243o a \303\241rea exata deste trapez\303\263ide. Para isto \303\251 necess\303\241rio trabalhar com uma divis\303\243o geral com n subintervalos. Quando o limite esquerdo de cada intervalo ( interval) \303\251 usado para formar os ret\303\242ngulos, a \303\241rea total com n subintervalos \303\251SAn := ApproximateInt( F1, x=a..b, partition=n, method=left, output=sum ):
SAn = value( SAn );Como comentado anteriormente, cada uma destas aproxima\303\247\303\265es de \303\241rea \303\251 t\303\243o pequena que converge para a verdadeira \303\241rea do trapez\303\263ide, como vemos nos c\303\241lculos seguintes.SA := Limit( SAn, n=infinity ):
Area[trapezoid] = SA;
`` = Limit( value( SAn ), n=infinity );
`` = value( SA );Note que se os limites (de intervalo) \303\241 direita de cada subintervalo forem usados para determinar a altura do ret\303\242ngulo em cada subintervalo, ent\303\243o cada ret\303\242ngulo se estende acima da \303\241rea do trapez\303\263ide e as aproxima\303\247\303\265es para a \303\241rea s\303\243o todas um pouco maiores. Mas, no limite , o valor deve se aproximar do valor verdadeiro da \303\241rea, como se v\303\252 nos c\303\241lculos seguintes.SAn2 := ApproximateInt( F1, x=a..b, partition=n, method=right, output=sum ):
SA2 := Limit( SAn2, n=infinity ):
Area[trapezoid] = SA2;
`` = Limit( value( SAn2 ), n=infinity );
`` = Limit( simplify(value( SAn2 )), n=infinity );
`` = value( SA2 );A \303\241rea deste trapez\303\263ide \303\251, de fato, 20.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Somas de Riemann </Text-field>Cada uma das somas encontradas na discuss\303\243o de Aproxima\303\247\303\243o de \303\201rea por Ret\303\242ngulos \303\251 um tipo especial de soma chamada a soma de Riemann. A import\303\242ncia desta observa\303\247\303\243o \303\251 que limites apropriados de soma de Riemann podem ser relacionados a integrais e antiderivadas. Mas, primeiro, vamos aprender um pouco mais sobre somas de Riemann.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Defini\303\247\303\243o (Soma de Riemann)</Font></Text-field>Let f be a function defined on a 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be a partition of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RIltGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJXRydWVGJy8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGSi1GIzYmLUYsNiVRImFGJy8lJ2l0YWxpY0dGOi9GNlEnaXRhbGljRictRjI2LVEiLEYnRjUvRjlGPS9GPEY6L0Y/Rj1GQEZCRkRGRi9GSVEmMC4wZW1GJy9GTFEsMC4zMzMzMzMzZW1GJy1GLDYlUSJiRidGUkZURjUtRjI2LVEiXUYnRjVGOEY7Rj5GQEZCRkRGRkZIL0ZMUSwwLjExMTExMTFlbUYnRjVGK0Y1 into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= subintervals with endpointsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < ... < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmRistRiM2Ji1GLDYlUSJuRidGNUY4LUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y5USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZZLUkjbW5HRiQ2JFEiMUYnRkZGRkYrRkYvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRkY= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIm5GJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=A Riemann sum of f on the partition LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is any sum of the 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where, for each 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,
the width of subinterval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is 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 is a point in subinterval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, i.e., 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUklbXN1YkdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRictRiM2JC1GLDYlUSJpRidGUkZVRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRjVGK0Y1.Observe que os pontos 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 podem ser selecionados por diferentes m\303\251todos em diferentes subintervalos . Mais tipicamente, entretanto, estes pontos s\303\243o selecionados como o limite direito ou o limite esquerdo do subintervalo ou o local de maior ou menor valor de f no subintervalo. (Quando o maior valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em cada intervalo \303\251 consistentemente selecionado, a soma de Riemann \303\251 chamada uma soma superior. Quando o menor valor \303\251 selecionado, a soma Riemann resultante \303\251 chamada uma soma inferior.)Para enfatizar este aspecto de somas de Riemann, retorne para (ou reative) o maplet ApproximateIntegration maplet. Aplicado \303\240 fun\303\247\303\243o 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 no intervaloLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSIyRidGOEY4RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGK0Y4, isto resulta em uma imagem como aquela na Figura 23.8.
Figura 23.8 O maplet Approximate Integration aplicado \303\240 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 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSIyRidGOEY4RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGK0Y4 usando a op\303\247\303\243o ramd\303\264mica
Na Figura 23.8, o n\303\272mero de ret\303\242ngulos foi programado para 10, e a op\303\247\303\243o rand\303\264mica foi selecionada. Isto faz com que a altura de cada ret\303\242ngulo seja o valor da fun\303\247\303\243o em um ponto rand\303\264mico no subintervalo que forma a base do ret\303\242ngulo. O gr\303\241fico aparece quando o bot\303\243o Display \303\251 clicado. Depois que o gr\303\241fico aparece, clique novamenteo bot\303\243o Display; repita isto v\303\241rias vezes mais, registre a cada vez o valor aproximado da \303\241rea sob esta curva (curve). Clique o bot\303\243o Compare para obter a demonstra\303\247\303\243o na Figura 23.9.
Figura 23.9 Outras somas de Riemann conseq\303\274entes da Figura 23.8
Como comparar as aproxima\303\247\303\265es de outras somas de Riemann com aquelas encontradas com os pontos randomicamente selecionados em cada subintervalo? Repita esta experi\303\252ncia com o dobro de divis\303\265es, digamos 20; n\303\243o se esque\303\247a de usar o bot\303\243o Compare para obter uma lista \303\272nica com as v\303\241rias somas de Reimann. Repita este teste uma vez mais, com o Number of Partitions(n\303\272mero de divis\303\265es) programado (set) para 40. Sua tabela (table) de valores deve ficar parecida com a tabela que vem seguir; o procedimento RiemannSumComparison (definido na se\303\247\303\243o inicializa\303\247\303\243o) pode ajudar quando comparar uma lista de somas de Riemann porque isto gera tabelas da seguite forma.RiemannSumComparison( x^2-x+2, 0, 2, [ 10, 20, 40 ], [lower,random,random,upper] );Sinta se livre para experimentar este comando para diferentes fun\303\247\303\265es, intervalos, n\303\272mero de divis\303\265es e tipos de m\303\251todos no comando anterior.Deve-se ser capaz de fazer v\303\241rias observa\303\247\303\265es sobre os dados apresentados nesta matrix.todas as avalia\303\247\303\265es ficam entre os valores registrados para as somas superior e inferior, para cada tamanho da divis\303\243o,a aproxima\303\247\303\243o com pontos selecionados randomicamente muda a cada vez que uma nova avalia\303\247\303\243o \303\251 requerida, (mas cada aproxima\303\247\303\243o fica entre as somas de Riemann superior e inferior ),conforme o n\303\272mero de divis\303\265es aumenta, as somas de Riemann superiores e inferiores tendem a ficar pr\303\263ximas.De fato, no limite quando a largura das divis\303\265es decresce tendendo a zero, as somas de Riemann superiores e inferiores convergem (converge) para o mesmo valor. Como todas as outras somas de Riemann ficam dentro do intervalo criado pelas somas de Riemann superior e inferior, elas, tamb\303\251m, devem convergir para o mesmo limite. Este limite \303\251 a integral definida de f em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RIltGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJXRydWVGJy8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGSi1GIzYmLUkjbW5HRiQ2JFEiMEYnRjUtRjI2LVEiLEYnRjUvRjlGPS9GPEY6L0Y/Rj1GQEZCRkRGRi9GSVEmMC4wZW1GJy9GTFEsMC4zMzMzMzMzZW1GJy1GUDYkUSIyRidGNUY1LUYyNi1RIl1GJ0Y1RjhGO0Y+RkBGQkZERkZGSC9GTFEsMC4xMTExMTExZW1GJ0Y1RitGNQ==. (Uma defini\303\247\303\243o exata de integral definida ser\303\241 dada no Cap\303\255tulo 24.)
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><Font readonly="true">Maple T.A. Maple-Graded Question</Font></Text-field>Using the Maple-graded question type, you create a question that uses Maple functionality when creating the correct answer and determining if the student's response is correct. The Maple-graded question type has an option to plot the student's response. To include the plot option, insert the corresponding task template. For information on creating questions, making assignments, and exporting Maple T.A. course modules, see Overview of Maple T.A. Integration.
<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>Algorithmic variables can be used to create a question that contains randomly generated parameters. You can refer to algorithmic variables in the Question, Answer, Hints, and Feedback input regions, as well as in other variable definitions. To reference an algorithmic variable, include "`$variable`" as text or 2-D Math, for example, x := `$myvariable`. Adding algorithmic variables to the question is optional.
In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique.with(RandomTools):
with( Student[Calculus1] ):
with(plots):
randomize():
nn := Generate(integer(range=1..5));
meth := ['left','lower','midpoint','right','upper'][nn]:
subint := Generate(integer(range=4..20)):
nn1 := Generate(integer(range=2..6)):
nn2 := Generate(integer(range=-5..6)):
nn3 := Generate(integer(range=-3..8)):
while (nn3 <= nn2) do nn3 := Generate(integer(range=-3..8)) end do:
ff1 := Generate(choose([sin(x),cos(x),sin(2*x),cos(2*x),sin(x)+cos(x),sin(x)-cos(x)])):
ff2 := sort(Generate(polynom(integer(range=-20..20),x,degree=nn1))):
fn1 := Generate(choose([ff1,ff2]));
ans1 := evalf(RiemannSum(fn1,x=nn2..nn3,method=meth));
JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 23.1</Font></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.Qual \303\251 a soma de Riemann `$meth` com `$subint` subintervalos para y = `$fn1` no intervalo [`$nn2`,`$nn3`]?upper = superiorlower = 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JSFH
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`.
Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed.ans1;
<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`-`$RESPONSE`=0);
Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region.evalb(evalf(`$ANSWER`-`$RESPONSE`,8)=0);
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional.
Enter your hints in the following table. You can provide up to three hints.
Replace this text with hint 1.Replace this text with hint 2.Replace this text with hint 3.
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional.
In the following text region, enter feedback to an incorrect response.Replace this text with feedback.with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := ans1; if evalb(evalf(ANSWER-eval(parse(StudentResponse)),8)=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!
</mtext><mtext>
</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>
</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>
</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>0.6540275563</mn></mrow></mrow></mstyle></math><mtext>
</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>
</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>0.6540275563</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"><Font encoding="UTF-8">Cap\303\255tulo Sum\303\241rio</Font></Text-field>Este cap\303\255tulo introduziu v\303\241rias novas id\303\251ias: nota\303\247\303\243o de somat\303\263ria, aproxima\303\247\303\243o de \303\241reas por ret\303\243ngulos e somas de Riemann e como estes t\303\263picos est\303\243o relacionados . A nota\303\247\303\243o de Somat\303\263ria prov\303\252 uma nota\303\247\303\243o abreviada para escrever a soma de uma (finita ou infinita ) seq\303\274encia de n\303\272meros. \303\211 importante que se sinta confort\303\241vel com a nota\303\247\303\243o e seja capaz de identificar e avaliar algumas somas simples. Somos tamb\303\251m capazes de reconhecer uma classe especial de somas: As somas de Riemann. As somas de Riemann s\303\243o importantes na computa\303\247\303\243o de \303\241rea de uma regi\303\243o com um contorno curvo.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Qual \303\251 o Pr\303\263ximo?</Font></Text-field>Os problemas associados com este cap\303\255tulo s\303\243o mais conceituais que na maioria dos outros cap\303\255tulos. Uma compreens\303\243o s\303\263lida de conceitos fundamentais ser\303\241 cr\303\255tico para a compreens\303\243o do restante (remainder) deste. N\303\243o se deixe enganar por um n\303\272mero relativamente pequeno de problemas que pedem para computar alguma coisa.Os primeiros tr\303\252s cap\303\255tulos desta unidade introduziu algumas id\303\251ias diferentes: Antiderivadas, Equa\303\247\303\265es Diferenciais e Somas de Riemann. O pr\303\263ximo cap\303\255tulo completa o desenvolvimento da Integral Definida. Esta \303\251 uma das mais importantes id\303\251ias deste curso. O significado conceitual da integral definida ser\303\241 esclarecido no Cap\303\255tulo 25 sobre Teoremas Fundamentais de C\303\241lculo. Os resultados neste cap\303\255tulo mostram uma rela\303\247\303\243o pr\303\263xima entre integrais definidas e indefinidas e prov\303\252 m\303\251todos para avalia\303\247\303\243o eficiente de integrais definidas.Chapter Navigation<< Prev12345678910111213141516171819202122 23 2425262728293031Next >>
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