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Guia de Estudo de C\303\241lculo \302\251 Maplesoft, a division of Waterloo Maple Inc., 2005 Cap\303\255tulo 30: Comprimento de uma Curva Plana
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field> O comprimento de uma curva (curve) no plano \303\251 tamb\303\251m conhecido como o comprimento de arco da curva (curve). Para uma curva com dobras (bends) e tor\303\247\303\265es (twists), esta quantidade geom\303\251trica unimensional pode ser dif\303\255cil de medir diretamente. Os comprimentos de segmentos retos, entretanto, s\303\243o f\303\241ceis de computar. O comprimento de um segmento de reta entre dois pontos A = ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB ) e B = ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB ) pode ser encontrado (pelo Teorema de Pit\303\241goras) como 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 onde a dist\303\242ncia horizontal entre os pontos (ou degrau (run)) \303\251 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 e a diferen\303\247a (difference) vertical entre os dois pontos (ou espelho (rise)) \303\251 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 . Para um caminho curvo, C, a id\303\251ia \303\251 dividir a curva em pequenos segmentos. Se a curva n\303\243o tem cantos ou c\303\272spides, e se as partes s\303\243o pequenas o suficiente, ent\303\243o cada um dos segmentos de curva pode ser aproximado para um segmento de reta. A soma destes segmentos de reta ser\303\241 pr\303\263xima do comprimento do caminho curvo; No limite (limit) isto nos dar\303\241 o comprimento da curva. Designando por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIkNGJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA o comprimento da curva C, estas id\303\251ias levam \303\240 defini\303\247\303\243o 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
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8"> 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). bound circle continuous curve definiteintegral derivative difference differentiable domain endpoint evaluate function graph implicit indefiniteintegral integral integrand interval limit linear mean-valuetheorem parameter piecewise radius riemannsum value volume
<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( Student[Calculus1] ):
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Procedimentos Auxiliares para Desenho Gr\303\241fico </Font></Text-field> Ptriangle := display( plot( [[1,1],[3,6]], style=point, color=blue ), plot( [[1,1],[3,6]], color=red, linestyle=DOT ), plot( [[1,1],[3,1],[3,6]], color=pink ), textplot( [0.8,0.8,"A"],align={below,left} ), textplot( [3.2,6.2,"B"],align={above,right} ), textplot( [2,0.8,"D"],align={below},font=[SYMBOL,12] ), textplot( [2.2,0.8,"x"],align={below} ), textplot( [3.2,3.5,"D"],align={right},font=[SYMBOL,12] ), textplot( [3.4,3.5,"y"],align={right} ), scaling=constrained, axes=none ): ArcLengthApprox := proc( f, domain, N ) local x, a, b, F, frames, ArcLengthApproxFrame; x,a,b := lhs(domain),op(rhs(domain)); if type(f,procedure) then F:=f else F:=unapply(f,x) end if; ArcLengthApproxFrame := proc( F, a, b, N ) local Xpt, XYpt, XY2pt, P0, P1, P2; Xpt := [ (b-a)*i/N $ i=0..N ]; XYpt := [ seq( [X,F(X)], X=Xpt ) ]; XY2pt := [ seq( op([[Xpt[i],F(Xpt[i])],[Xpt[i+1],F(Xpt[i])]]), i=1..N ), [Xpt[N+1],F(Xpt[N+1])] ]; P0 := plot( F(x), x=a..b, linestyle=DOT, legend="curve: C" ); P1 := plot( [ XYpt$2 ], style=[point,line], color=blue, legend=["points",""] ); P2 := plot( XY2pt, color=pink, legend="line segments" ); return display( P0,P1,P2, title=sprintf("Arclength Approximation\134nwith n=%a Subintervals", N) ); end proc; if type(N,posint) then return ArcLengthApproxFrame( F, a, b, N ) else frames := seq( ArcLengthApproxFrame( F, a, b, n ), n=N ); return display( [frames], insequence=true ) end if; end proc:
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Demonstra\303\247\303\265es e Casos</Font></Text-field> O comando especial Maple, ArcLengthApprox (Aproxima\303\247\303\243o do Comprimento de Arco) definido na se\303\247\303\243o Inicializa\303\247\303\243o pode apresentar uma anima\303\247\303\243o de como os pequenos segmentos se aproximam do caminho curvo \303\240 medida que o n\303\272mero de segmentos cresce. Um primeiro exemplo pode ser encontrado na Figura 30.1. ArcLengthApprox( x/2+sin(x), x=0..20, [5,10,20,40,80] ); Figura 30.1 Comprimento de Arco: Sendo medido pela linha Poligonal aproximada \303\240 curva
O processo geral parece ser muito similar ao procedimento de divis\303\243o em fatias, aproxima\303\247\303\243o, limite e avalia\303\247\303\243o (evaluate) usado em outras aplica\303\247\303\265es de integral definida (definite integral). O \303\272nico problema \303\251 que n\303\243o se parece como uma Soma de Riemann (Riemann Sum).
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Caso Especial 1: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field> Se a curva \303\251 o gr\303\241fico (graph) de uma fun\303\247\303\243o diferenci\303\241vel (differentiable function) 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 no intervalo (interval) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInhGJ0Y0RjdGPkYrRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYsNiVRImJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, ent\303\243o 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 (pelo Teorema do Valor M\303\251dio (Mean Value Theorem) para derivadas (derivatives)) 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
onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtc3Vic3VwR0YkNictRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRImlGJ0Y1RjgvRjlRJ25vcm1hbEYnLUYsNiVRJGAqYEYnRjVGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGR0YrRkA= \303\251 qualquer n\303\272mero entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmRistRiM2Ji1GLDYlUSJpRidGNUY4LUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y5USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZZLUkjbW5HRiQ2JFEiMUYnRkZGRkYrRkYvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRkY= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRImlGJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA . (Note que o Teorema do Valor M\303\251dio (Mean Value Theorem)) \303\251 aplic\303\241vel somente porque supomos que a fun\303\247\303\243o (function) \303\251 diferenci\303\241vel (differentiable).) Quando isto \303\251 substitu\303\255do em uma f\303\263rmula geral de comprimento de arco e simplificada, a soma \303\251 reconhecida como uma Soma de Riemann (Riemann sum) e o comprimento de arco pode ser expresso como uma integral definida LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRIkNGJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA JSFH= 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= 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<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Caso Especial 2: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field> Similarmente, se a curva \303\251 o gr\303\241fico (graph) de uma fun\303\247\303\243o diferenci\303\241vel 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 no intervalo (interval) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInlGJ0Y0RjdGPkYrRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, ent\303\243o o TVM pode ser usado para escrever 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 quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtc3Vic3VwR0YkNictRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRImlGJ0Y1RjgvRjlRJ25vcm1hbEYnLUYsNiVRJGAqYEYnRjVGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGR0YrRkA= \303\251 um n\303\272mero entre 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRImlGJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA . Ent\303\243o, obtemos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJDRidGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy9GOVEnbm9ybWFsRic= JSFH= 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 =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KkYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYvUSRsaW1GJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEmaW5lcnRGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHUSV0cnVlRicvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xNjY2NjY3ZW1GJy1GIzYmLUYsNiVRIm5GJy8lJ2l0YWxpY0dGTi9GPFEnaXRhbGljRictRjU2LVEnJnJhcnI7RidGO0ZBRkRGRkZIRkovRk1GQ0ZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRl9vLUYsNiVRKCYjODczNDtGJ0ZmbkZobkY7LyUsYWNjZW50dW5kZXJHRkNGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC40ZW1GJy8lJmRlcHRoR0ZbcC8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUYjNiZGKy1GIzYqRistSSttdW5kZXJvdmVyR0YkNictRjU2L1EmJlN1bTtGJ0Y4RjtGPkZBRkQvRkdGTkZIL0ZLRk5GTEZPRlFGVC1GIzYmLUYsNiVRImlGJ0ZmbkZobi1GNTYtUSI9RidGO0ZBRkRGRkZIRkpGXW9GT0Zeb0Zgby1JI21uR0YkNiRRIjFGJ0Y7RjtGWS8lJ2FjY2VudEdGQ0Zkb0YrLUZnbzYmRmlvL0ZdcFEkNS4wRidGX3BGYXBGKy1JJm1zcXJ0R0YkNiMtRiM2KEYrLUYjNiZGKy1JJW1zdXBHRiQ2JS1GIzYmLUY1Ni1RIihGJ0Y7L0ZCRk5GREZecUZIRkpGXW9GTy9GUkZWRlQtRiM2KkYrLUYjNictRiw2JVEjZydGJ0ZmbkZobi1GNTYuUTAmQXBwbHlGdW5jdGlvbjtGJy8lJXNpemVHUSMxNkYnRjtGQUZERkZGSEZKRl1vRk9GUS9GVUZTLUYjNiZGXnMtRiM2JkYrLUkobXN1YnN1cEdGJDYnLUYsNiVRInhGJ0ZmbkZobi1GIzYkRmJxRjstRiw2JVEkYCpgRidGZm5GaG4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRmJ1RitGOy1GNTYtUSIpRidGO0Zhc0ZERl5xRkhGSkZdb0ZPRmJzL0ZVUSwwLjExMTExMTFlbUYnRjtGK0Y7LUY1Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y7RkFGREZGRkhGSkZdb0ZPRlFGYHQtRiw2JVEnJiM5MTY7RicvRmduRkNGO0ZqdS1JJW1zdWJHRiQ2JS1GLDYlUSJ5RidGZm5GaG5GW3VGY3VGK0Y7RmV1RjstRmlxNiRRIjJGJ0Y7RmB1RitGOy1GNTYtUSIrRidGO0ZBRkRGRkZIRkpGXW9GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZedy1GIzYmRistRmpyNiUtRiM2JkZecy1GIzYnRl12Rmp1RmF2RitGO0ZldUY7Rmd2RmB1RitGO0YrRjtGK0Y7RitGO0YrRjtGK0Y7 =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 =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
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Caso Especial 3: Forma Param\303\251trica </Font></Text-field> O terceiro caso \303\251 uma generaliza\303\247\303\243o dos casos anteriores. Suponha que os pontos sobre a curva possam ser descritos em termos de um par de equa\303\247\303\265es param\303\251tricas da forma LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNictRiw2JVEiZkYnRjRGNy1GOzYuUTAmQXBwbHlGdW5jdGlvbjtGJy8lJXNpemVHUSMxNkYnRj5GQEZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0Zqbi1GIzYmLUY7Ni1RIihGJ0Y+L0ZBRjZGQy9GRkY2RkdGSUZLRk0vRlBRLDAuMTY2NjY2N2VtRicvRlNGZG8tRiM2JC1GLDYlUSJ0RidGNEY3Rj4tRjs2LVEiKUYnRj5GYW9GQ0Zib0ZHRklGS0ZNRmNvL0ZTUSwwLjExMTExMTFlbUYnRj5GK0Y+RitGPkYrRj4=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNictRiw2JVEiZ0YnRjRGNy1GOzYuUTAmQXBwbHlGdW5jdGlvbjtGJy8lJXNpemVHUSMxNkYnRj5GQEZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0Zqbi1GIzYmLUY7Ni1RIihGJ0Y+L0ZBRjZGQy9GRkY2RkdGSUZLRk0vRlBRLDAuMTY2NjY2N2VtRicvRlNGZG8tRiM2JC1GLDYlUSJ0RidGNEY3Rj4tRjs2LVEiKUYnRj5GYW9GQ0Zib0ZHRklGS0ZNRmNvL0ZTUSwwLjExMTExMTFlbUYnRj5GK0Y+RitGPkYrRj4=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInhGJ0Y0RjdGPkYrRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYsNiVRImJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= Isto significa que a curva come\303\247a no 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, passa pelos pontos 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 na ordem crescente de t de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0Yv para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e termina no ponto 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. Por exemplo, o segmento de reta de 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 para 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 pode ser dado em forma param\303\251trica como 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, 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, para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiMUYnRjVGNUYrRjU= O comando animatecurve do pacote plots (desenho de gr\303\241fico) apresenta um quadro muito bom de uma curva param\303\251trica \303\240 medida que o par\303\242metro ( parameter) aumenta atrav\303\251s de seu dom\303\255nio (domain). Aplicado \303\240 linha reta do exemplo acima, ele gera a anima\303\247\303\243o na Figura 30.2. animatecurve([1+2*t,2-2*t,t=0..1], view=[0..4,0..3]);
Figura 30.2 A anima\303\247\303\243o mostrando (decalque) tra\303\247os da curva param\303\251trica definida por 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, 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, para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiMUYnRjVGNUYrRjU=
Uma parametriza\303\247\303\243o de um c\303\255rculo (circle) \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNictRiw2JVEkY29zRicvRjVGQkY+LUY7Ni5RMCZBcHBseUZ1bmN0aW9uO0YnLyUlc2l6ZUdRIzE2RidGPkZARkNGRUZHRklGS0ZNL0ZQUSYwLjBlbUYnL0ZTRltvLUYjNiYtRjs2LVEiKEYnRj4vRkFGNkZDL0ZGRjZGR0ZJRktGTS9GUFEsMC4xNjY2NjY3ZW1GJy9GU0Zlby1GIzYkLUYsNiVRJyYjOTUyO0YnRllGPkY+LUY7Ni1RIilGJ0Y+RmJvRkNGY29GR0ZJRktGTUZkby9GU1EsMC4xMTExMTExZW1GJ0Y+RitGPkYrRj5GK0Y+ , 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 pra LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTUyO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU=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. A anima\303\247\303\243o na Figura 30.3 tra\303\247a o c\303\255rculo (circle) \303\241 medida que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy aumenta ao longo de seu dom\303\255nio (domain). animatecurve( [cos(theta),sin(theta),theta=0..2*Pi], scaling=constrained);
Figura 30.3 Anima\303\247\303\243o mostrando o tra\303\247ado de um c\303\255rculo definido parametricamente 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 , 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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTUyO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU=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
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">Notas sobre curvas param\303\251tricas </Font></Text-field> To plot a parametric curve without animation, change animatecurve to plot. There are many parametric forms for the same curve.
Se ambas as fun\303\247\303\265es f e g s\303\243o cont\303\255nuas (continuous) em 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 e diferenci\303\241vel em 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, ent\303\243o o Teorema do Valor M\303\251dio (para Derivadas) (Mean Value Theorem (for Derivatives)) pode ser aplicado a ambos 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 e 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 . para obter 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 onde 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtc3Vic3VwR0YkNictRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRImlGJ0Y1RjgvRjlRJ25vcm1hbEYnLUYsNiVRJGAqYEYnRjVGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGR0YrRkA= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRImlGJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZA e 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KkYrLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2L1EmJmludDtGJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEmaW5lcnRGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHRkMvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZSLUYsNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GPFEnaXRhbGljRictRiw2JVEiYkYnRlhGZW4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1JJm1zcXJ0R0YkNiMtRiM2KEYrLUYjNiZGKy1JJW1zdXBHRiQ2JS1GIzYnLUYsNiVRJGZ+J0YnRlhGZW4tRjU2LlEwJkFwcGx5RnVuY3Rpb247RicvJSVzaXplR1EjMTZGJ0Y7RkFGREZGRkhGSkZMRk5GUEZTLUYjNiYtRjU2LVEiKEYnRjsvRkJGWkZEL0ZHRlpGSEZKRkxGTi9GUVEsMC4xNjY2NjY3ZW1GJy9GVEZdcS1GIzYkLUYsNiVRInRGJ0ZYRmVuRjstRjU2LVEiKUYnRjtGanBGREZbcUZIRkpGTEZORlxxL0ZUUSwwLjExMTExMTFlbUYnRjtGK0Y7LUkjbW5HRiQ2JFEiMkYnRjsvRltvRl9vRitGOy1GNTYtUSIrRidGO0ZBRkRGRkZIRkpGTEZOL0ZRUSwwLjIyMjIyMjJlbUYnL0ZURmJyLUYjNiZGKy1GaG82JS1GIzYnLUYsNiVRI2cnRidGWEZlbkZfcEZlcEYrRjtGaXFGXXJGK0Y7RitGO0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmJzLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYsNiVRI2R0RidGWEZlbkY7RitGOw==
Note que n\303\243o existe raz\303\243o para esperar que 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 e assim a soma final n\303\243o \303\251 uma soma de Riemann. Usando t\303\251cnicas bem al\303\251m do campo deste curso pode ser demonstrado que -- no limite-- que isto n\303\243o \303\251 importante e que a integral definida (definite integral) realmente d\303\241 o comprimento de arco de uma curva param\303\251trica.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Coment\303\241rios Gerais sobre comprimento de Arco </Font></Text-field> Se uma curva tem um canto ou uma c\303\272spide, computa-se o comprimento de arco de cada parte suave e depois soma-se os resultados. Para a maioria dos problemas de comprimento de arco, as integrais definidas n\303\243o podem ser avaliadas (evaluated) em termos de fun\303\247\303\265es elementares. O maplet Comprimento de Arco (ArcLength maplet) (inserido no Arc Length Tutor) prov\303\252 uma interface conveniente para vizualizar e computar comprimento de arco quando a curva pode ser escrita como 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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em um intervalo, i.e., para o Caso especial 1.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 30.1: Curva Plana com <Equation executable="false" style="2D Math_44" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYmUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkjbW9HRiQ2LlEiPUYnLyUlYm9sZEdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHRkMvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVi1GIzYoLUYsNiZRImZGJ0Y0RjdGOi1GPjYuUTAmQXBwbHlGdW5jdGlvbjtGJy8lJXNpemVHUSMxNkYnRkRGRkZIRkpGTEZORlBGUi9GVVEmMC4wZW1GJy9GWEZfby1GIzYnLUY+Ni5RIihGJ0ZBRkQvRkdGNkZIL0ZLRjZGTEZORlBGUi9GVVEsMC4xNjY2NjY3ZW1GJy9GWEZpby1GIzYlLUYsNiZRInhGJ0Y0RjdGOi9GOEY8RjotRj42LlEiKUYnRkFGREZmb0ZIRmdvRkxGTkZQRlJGaG8vRlhRLDAuMTExMTExMWVtRidGYHBGOkYrRmBwRjpGK0ZgcEY6RitGYHBGOg==">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</Equation> </Text-field> Encontre o comprimento de arco da curva f1 := (x^4+3)/(6*x): y = f1; entre lo1 := 1/2: x = lo1; e hi1 := 3: x = hi1; Uma solu\303\247\303\243o copleta pelo maplet Comprimento de Arco (ArcLength maplet) (o inserido no Arc Length Tutor) aparece na Figura 30.4. 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Figura 30.4 Solu\303\247\303\243o do Exemplo 30.1 pelo Tutor Comprimento de Arco (Arc Length Tutor)
O gr\303\241fico de uma curva, da qual \303\251 para ser computado o comprimento de arco, aparece na Figura 30.5. plot( f1, x=lo1..hi1, view=[0..3,0..4] ); Figura 30.5 A curva do Exemplo 30.1
De acordo com a f\303\263rmula de comprimento de arco para uma curva dada por 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, o comprimento desta curva \303\251 dado pela integral definida com o integrando q1 := sqrt( diff( f1, x )^2 + 1 ): q1; Nesta forma n\303\243o existe nenhuma esperan\303\247a de encontrar uma antiderivada (antiderivative). Mas, se a express\303\243o interna for expandida e simplificada (com suposi\303\247\303\265es apropriadas para x), o resultado \303\251 ent\303\243o q2 := expand( q1 ): q3 := simplify( q2 ): q4 := simplify( q3 ) assuming x>1/2: ds1 := expand( q4 ): q1 = q2; `` = q3; `` = q4; `` = ds1; Esta fun\303\247\303\243o n\303\243o ser\303\241 dif\303\255cil de integrar! A integral definida para o comprimento desta curva \303\251 L1 := Int( ds1, x=lo1..hi1 ): L = L1; `` = value( L1 ); Como vimos na Figura 30.4, os mesmos resultados podem ser obtidos com o maplet Comprimento de Arco (ArcLength maplet ) (inserido no Arc Length Tutor). A integral pode tamb\303\251m ser obtida com o comando ArcLength do pacote Estudante[C\303\241lculo1] (Student[Calculus1]) , como mostrado abaixo. q5 := ArcLength( f1, lo1..hi1, output=integral ): L = q5; `` = value( q5 );
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Nota Final <Font opaque="true" background="[0,255,255]">Concluding Remark</Font></Text-field> Observe que este exemplo n\303\243o podia ser tratado como uma curva plana na forma 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 porque a fun\303\247\303\243o n\303\243o \303\251 1-1 no intervalo (interval) [ LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GNTYkUSIyRidGOEY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZFLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== , 3 ].
<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 := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique. with(RandomTools): with(plots): randomize(): d := Generate(integer(range=2..5)): fn1 := sort(Generate(polynom(integer(range=-5..5),x,degree=d))); low1 := Generate(integer(range=-3..0)); high1 := Generate(integer(range=1..3)); ans1 := Int(sqrt(sort(diff(fn1,x)^2+1,x,descending)),x=low1..high1); LCwqJEkieEc2IiIiJSIiJCokRiRGJyIiIyokRiRGKUYnRiQiIiIiIiZGKw== IiIh IiIi LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCwwKiRJInhHRiciIiciJFciKiRGLCIiJkYuKiRGLCIiJSIkIT0qJEYsIiIkIiMnKiokRiwiIiMiI1tGLCIjN0Y4IiIiI0Y7RjgvRiw7IiIhRjs= JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 30.1</Font></Text-field> The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question. Qual \303\251 o comprimento de arco de y = `$fn1` entre x = `$low1` e x = `$high1`? 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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 `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. JSFHevalf(value(ans1)); JSFH
<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field> To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$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( fnormal(evalf(`$ANSWER`-`$RESPONSE`), 5 ) = 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 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. The definite integral is `$ans1` = `$ANSWER`
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := evalf(value(ans1)); if evalb( fnormal(evalf(ANSWER-eval(parse(StudentResponse))), 5 ) = 0 ) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="&lt;NO RESPONSE&gt;"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect.&nbsp;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer:&nbsp;</mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>9.112150053</mn></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>The&MediumSpace;definite&MediumSpace;integral&MediumSpace;is&MediumSpace;</mtext><mtext></mtext><mrow><mrow><msubsup superscriptshift='2'><mo mathcolor='#909090' lspace='0.0em' rspace='0.0em'>&amp;int;</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msqrt><mrow><mrow><mn>144</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi>x</mi><mn>6</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>144</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi>x</mi><mn>5</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>180</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>96</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>48</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>12</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>2</mn></mrow></msqrt></mrow><mspace height='0.0ex' width='0.3em' depth='0.0ex' linebreak='auto'/><mo mathcolor='#909090' lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mrow><mtext></mtext><mtext>&MediumSpace;</mtext><mtext>=</mtext><mtext>&MediumSpace;</mtext><mrow><mn>9.112150053</mn></mrow><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>9.112150053</mn></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 30.2: Curva Plana com <Equation executable="false" style="2D Math_46" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYmUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkjbW9HRiQ2LlEiPUYnLyUlYm9sZEdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHRkMvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVi1GIzYoLUYsNiZRImZGJ0Y0RjdGOi1GPjYuUTAmQXBwbHlGdW5jdGlvbjtGJy8lJXNpemVHUSMxNkYnRkRGRkZIRkpGTEZORlBGUi9GVVEmMC4wZW1GJy9GWEZfby1GIzYnLUY+Ni5RIihGJ0ZBRkQvRkdGNkZIL0ZLRjZGTEZORlBGUi9GVVEsMC4xNjY2NjY3ZW1GJy9GWEZpby1GIzYlLUYsNiZRInlGJ0Y0RjdGOi9GOEY8RjotRj42LlEiKUYnRkFGREZmb0ZIRmdvRkxGTkZQRlJGaG8vRlhRLDAuMTExMTExMWVtRidGYHBGOkYrRmBwRjpGK0ZgcEY6RitGYHBGOg==</Equation> </Text-field> Encontre o comprimento da curva F2 := x^2 = y^3: F2; entre os 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 eLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkjbW9HRiQ2LlEwJkFwcGx5RnVuY3Rpb247RicvJSVzaXplR1EjMTZGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GIzYmLUYyNi1RIihGJ0Y4L0Y8USV0cnVlRidGPi9GQUZVRkJGREZGRkgvRktRLDAuMTY2NjY2N2VtRicvRk5GWC1GIzYmLUkjbW5HRiQ2JFEiOEYnRjgtRjI2LVEiLEYnRjhGOy9GP0ZVRkBGQkZERkZGSEZKL0ZOUSwwLjMzMzMzMzNlbUYnLUZnbjYkUSI0RidGOEY4LUYyNi1RIilGJ0Y4RlRGPkZWRkJGREZGRkhGVy9GTlEsMC4xMTExMTExZW1GJ0Y4RitGOEYrRjg=. Um gr\303\241fico desta curva definida implicitamente (implicitly) aparece na Figura 30.6. implicitplot(F2, x=1..8, y=1..4, scaling=constrained, view=[0..8,0..4]); Figura 30.6 Exemplo 30.2: A curva definida implicitamente por 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
Como esta curva est\303\241 t\303\243o pr\303\263xima de ser linear, um contorno ligeiramente mais baixo (quick lower bound) para o comprimento de arco \303\251 a dist\303\242ncia entre os limites (do intervalo), a saber, d2 := sqrt( (8-1)^2 + (4-1)^2 ): L[est] = d2; `` = evalf( d2 ); Esta curva passa pelo testes da reta horizontal e reta vertical; ela pode ser escrita como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUklbXN1cEdGJDYlLUYsNiVRInhGJ0Y0RjctRiw2JVEkMi8zRidGNEY3LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRj5GK0Y+ for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVEiPEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiOEYnRjVGNUYrRjU= ou como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUklbXN1cEdGJDYlLUYsNiVRInlGJ0Y0RjctRiw2JVEkMy8yRidGNEY3LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRj5GK0Y+ for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInlGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVEiPEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiNEYnRjVGNUYrRjU= Prosseguindo da op\303\247\303\243o anterior \303\251 necess\303\241ria a regra temp := simplify(convert(isolate(F2,y),radical)) assuming x>0: f2x := rhs(temp): y = f2x; para x em lo2x := 1: hi2x := 8: lo2x .. hi2x; enquanto que, prosseguindo da op\303\247\303\243o posterior, requer a regra temp := simplify(convert(isolate(F2,x),radical)) assuming y>0: f2y := rhs(temp): x = f2y; para y em lo2y := 1: hi2y := 4: lo2y .. hi2y; Tentando usar a primeira forma implica no integrando sqrt( diff( f2x, x )^2 + 1 ); As perspectivas de encontrar uma antiderivada desta fun\303\247\303\243o s\303\243o insignificantes. E, manipula\303\247\303\265es posteriores desta express\303\243o n\303\243o melhoram a situa\303\247\303\243o. Vejamos se o integando (integrand) para a segunda forma, ou seja, ds2 := sqrt( 1 + diff( f2y, y )^2 ): ds2; \303\251 um pouco melhor (ou pior). \303\211 f\303\241cil encontrar uma antiderivada para esta fun\303\247\303\243o. Continuando a trabalhar na dire\303\247\303\243o de nosso objetivo, de encontrar o comprimento desta curva, com esta representa\303\247\303\243o leva \303\240 integral definida L2 := Int( ds2, y=lo2y..hi2y ): L = L2; `` = value( L2 ); `` = evalf( value( L2 ) ); Note que este resultado, como o esperado, \303\251 ligeiramente maior que a dist\303\242ncia entre os limites do intervalo ( endpoints). Este exemplo mostra como o comprimento de arco pode ser mais f\303\241cil de ser computado em uma forma e quase imposs\303\255vel de ser computado (explicitamente) com outra representa\303\247\303\243o.
<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(plots): randomize(): n1 := 2: n2 := Generate(integer(range=3..8)): n3 := Generate(integer(range=2..20)): n4 := Generate(integer(range=2..20)): while (n3=n4) do n4 := Generate(integer(range=2..20)) end do: fn2 := n3*x^n1=n4*y^n2; fn2y := simplify(convert(isolate(fn2,x),radical)) assuming y > 0: l2 := Generate(integer(range=1..5)); h2 := Generate(integer(range=2..10)); while (h2 <= l2) do h2 := Generate(integer(range=2..10)) end do; xl2 := floor(evalf(abs(solve(eval(fn2, y = l2), x)[1]))); xh2 := ceil(evalf(abs(solve(eval(fn2, y = h2), x)[1]))); dfn2y := sqrt(simplify(1 + diff(rhs(fn2y),y)^2)): ans2 := Int(dfn2y,y=l2..h2); evalf(value(ans2)); #implicitplot(fn2,x=xl2..xh2,y=l2..h2); LywkKiRJInhHNiIiIiMiIiUsJCokSSJ5R0YmIiIkIiIn IiIj IiIp IiIk IiNH LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJCokLCYiIzsiIiJJInlHRiciI2EjRi0iIiMjRi0iIiUvRi47RjEiIik= JCIrJW8kWytEISIp JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 30.2</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 o comprimento de arco de `$fn2` entre y = `$l2` e y = `$h2`? 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JSFH
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. JSFHevalf(value(ans2)); 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`-`$RESPONSE`=0); Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region. evalb(evalf(`$ANSWER`-`$RESPONSE`,5)=0); JSFH
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints. Replace this text with hint 1. Replace this text with hint 2. Replace this text with hint 3.
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response. The definite integral is `$ans2` = `$ANSWER`
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea1`, `value`); ANSWER := evalf(value(ans2)); if evalb(evalf(ANSWER-eval(parse(StudentResponse)),5)=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="&lt;NO RESPONSE&gt;"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect.&nbsp;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer:&nbsp;</mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>25.00483684</mn></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>The&MediumSpace;definite&MediumSpace;integral&MediumSpace;is&MediumSpace;</mtext><mtext></mtext><mrow><mrow><msubsup superscriptshift='2'><mo mathcolor='#909090' lspace='0.0em' rspace='0.0em'>&amp;int;</mo><mn>2</mn><mn>8</mn></msubsup><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><msqrt><mrow><mn>16</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>54</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi></mrow></mrow></msqrt></mrow></mrow><mspace height='0.0ex' width='0.3em' depth='0.0ex' linebreak='auto'/><mo mathcolor='#909090' lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mrow><mtext></mtext><mtext>&MediumSpace;</mtext><mtext>=</mtext><mtext>&MediumSpace;</mtext><mrow><mn>25.00483684</mn></mrow><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>25.00483684</mn></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Exemplo 30.3: Curva Plana em Forma Param\303\251trica </Font></Text-field> Considere a curva param\303\251trica dada por f3 := sqrt(5)*sin(2*t)-2: g3 := sqrt(5)*cos(2*t)-sqrt(3): lo3 := 0: hi3 := Pi/2: x = f3; y = g3; t = lo3..hi3; De acordo com a an\303\241lise geral anterior, neste cap\303\255tulo, o integrando para a integral (integral) do comprimento de arco ser\303\241 q1 := sqrt( diff(f3,t)^2 + diff(g3,t)^2 ): ds3 := simplify( q1 ): q1 = ds3; Isto n\303\243o poderia ser mais simples para integrar. A integral definida (trivial)resultante para o comprimento de arco desta curva \303\251 L3 := Int( ds3, t=0..Pi/2 ): L = L3; `` = value( L3 ); Qual \303\251 esta curva e por que o problema se simplifica tanto ? Para responder a primeira quest\303\243o relembre a discuss\303\243o do comando animatecurve do pacote plots . Seu uso, aqui, fornece a anima\303\247\303\243o na Figura 30.7. animatecurve([f3,g3,t=lo3..hi3], view=[-3..1,-5..1], scaling=constrained); Figura 30.7 Tra\303\247os animados da curva parametricamente definida no Exemplo 30.3
Aha! Isto parece ser um semic\303\255rculo. O centro \303\251 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 e o raio (radius) \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUkjbW5HRiQ2JFEiNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjY=. Da geometria, a circunfer\303\252ncia de um semic\303\255rculo com raio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUkjbW5HRiQ2JFEiNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjY= \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjVGJ0Y3RitGN0YrRjc=, exatamente o valor (value) encontrado atrav\303\251s do c\303\241lculo.
<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(plots): randomize(): r1 := Generate(choose([-20..-1,1..20])): n5 := Generate(integer(range=1..20)): n6 := Generate(integer(range=r1)): n7 := Generate(integer(range=1..20)): n8 := Generate(integer(range=1..20)): n9 := Generate(integer(range=r1)): while (n6=n9) do n9 := Generate(integer(range=r1)) end do: n10 := Generate(integer(range=1..20)): n11 := Generate(integer(range=2..6)): n12 := Generate(integer(range=2..6)): fn3_[1] := sqrt(n5)*sin(n6*t)-sqrt(n7): gn3_[1] := sqrt(n8)*cos(n9*t)-sqrt(n10): fn3_[2] := sort(Generate(polynom(integer(range=r1),t,degree=n11)),x,descending): gn3_[2] := sort(Generate(polynom(integer(range=r1),t,degree=n12)),x,descending): n13 := Generate(choose([1,2])): fn3 := sort(fn3_[n13]); gn3 := sort(gn3_[n13]); l3 := Generate(integer(range=1..3)): h3 := Generate(integer(range=3..8)): while (h3 <= l3) do h3 := Generate(integer(range=3..8)) end do: #plot([fn3,gn3,t=l3..h3]); with(Student[Calculus1]): ans3 := sort(ArcLength( [fn3,gn3], l3..h3, output=integral )); LCYqJiIiJCMiIiIiIiMtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YtIiM7RiYhIiIqJEYnRiUhIiQ= LCYqJiIiKCMiIiIiIiMtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YtIiM2RiZGJiokIiM+RiUhIiI= LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCwmKiQtSSRjb3NHRiQ2IywkSSJ0R0YnIiM7IiIjIiRvKCokLUkkc2luR0YkNiMsJEYwIiM2RjIiJFopIyIiIkYyL0YwO0YyIiIl JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 30.3</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. Considere a curva param\303\251trica dada por x = `$fn3` y = `$gn3` Qual \303\251 o comprimento de arco desta curva de t = `$l3` a t = `$h3`? 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 JSFH
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. JSFHevalf(value(ans3));
<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`-`$RESPONSE`,6)=0);
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints. Replace this text with hint 1. Replace this text with hint 2. Replace this text with hint 3.
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response. The definite integral is `$ans3` = `$ANSWER`
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea2`, `value`); ANSWER := evalf(value(ans3)); if evalb(evalf(ANSWER-eval(parse(StudentResponse)),6)=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="&lt;NO RESPONSE&gt;"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect.&nbsp;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer:&nbsp;</mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>54.35173944</mn></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>The&MediumSpace;definite&MediumSpace;integral&MediumSpace;is&MediumSpace;</mtext><mtext></mtext><mrow><mrow><msubsup superscriptshift='2'><mo mathcolor='#909090' lspace='0.0em' rspace='0.0em'>&amp;int;</mo><mn>2</mn><mn>4</mn></msubsup><mrow><msqrt><mrow><mrow><mn>768</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mn>16</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi></mrow></mrow></mfenced></mrow><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>847</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mn>11</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>t</mi></mrow></mrow></mfenced></mrow><mn>2</mn></msup></mrow></mrow></msqrt></mrow><mspace height='0.0ex' width='0.3em' depth='0.0ex' linebreak='auto'/><mo mathcolor='#909090' lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mrow><mtext></mtext><mtext>&MediumSpace;=&MediumSpace;</mtext><mrow><mn>54.35173944</mn></mrow><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>54.35173944</mn></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Sum\303\241rio</Font></Text-field> O procedimento de dividir em fatias, aproximar, limitar e avaliar foi usado, uma vez mais, para desenvolver integrais definidas para tr\303\252s classes gerais de curvas. O TVM representa um papel fundamental na identifica\303\247\303\243o da aproxima\303\247\303\243o de soma como uma soma de Riemann. Sem esta parte chave do conhecimento n\303\243o haveria um m\303\251todo geral para expressar o comprimento de arco como integral definida. O comando ArcLengthApprox (Aproxima\303\247\303\243o do Comprimento de Arco) pode preparar um anima\303\247\303\243o mostrando como um caminho curvo pode ser aproximado por uma fun\303\247\303\243o de aproxima\303\247\303\243o linear por partes (piecewise linear ). O maplet ArcLength maplet (inserido no Arc Length Tutor) pode ser usado para checar se a integral definida foi corretamente formulada e avaliada (exatamente e numericamente).
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Qual \303\251 o Pr\303\263ximo?</Font></Text-field> A \303\252nfase desta aplica\303\247\303\243o est\303\241 baseada na integral definida. A avalia\303\247\303\243o da integral para encontrar uma express\303\243o num\303\251rica (ou simb\303\263lica) \303\251 um assunto importante, mas secund\303\241rio. O cap\303\255tulo final nesta unidade introduz trabalho mec\303\242nico (mechanical work) como uma aplica\303\247\303\243o de integrais definidas.
Chapter Navigation << Prev 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Next >> Table of Contents \302\251 Maplesoft, a division of Waterloo Maple Inc., 2005. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation. For more information on Maplesoft products and services, visit www.maplesoft.com