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Guia de Estudo de C\303\241lculo \302\251 Maplesoft, a division of Waterloo Maple Inc., 2005 Cap\303\255tulo 19: Otimiza\303\247\303\243o
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field> A Otimiza\303\247\303\243o se refere a encontrar a solu\303\247\303\243o (solution) "\303\263tima" ou "melhor" para um problema. Os problemas de otimiza\303\247\303\243o (optimization) s\303\243o geralmente classificados como com v\303\255nculo ou sem v\303\255nculo. Em um problema de otimiza\303\247\303\243o com v\303\255nculo o gol \303\251 encontrar o maior ou o menor valor (value) de uma fun\303\247\303\243o (function) sujeita a um ou mais v\303\255nculos (constraints). A fun\303\247\303\243o (function) cujo valor \303\263timo (optimal value) a ser encontrado \303\251 chamada de fun\303\247\303\243o objetivo. Uma rela\303\247\303\243o que conecta as vari\303\241veis (variables) que aparecem na fun\303\247\303\243o objetivo \303\251 chamada de v\303\255nculo (constraint). Para um problema de otimiza\303\247\303\243o sem v\303\255nculos, n\303\243o existem tais restri\303\247\303\265es sobre a vari\303\241vel independente (independent variable). Uma abordagem geral para um problema de otimiza\303\247\303\243o inclui os seguintes passos. identificar a fun\303\247\303\243o objetivo e todos os v\303\255nculos selecionar uma vari\303\241vel para ser a vari\303\241vel independente (expressar todas as outras vari\303\241veis em termos desta vari\303\241vel) escrever a fun\303\247\303\243o objetivo em termos da vari\303\241vel independente (e, possivelmente, par\303\242metros) determinar o intervalo (interval) de poss\303\255veis valores para a vari\303\241vel independente encontrar o m\303\241ximo ou o m\303\255nimo gobal (global maximum ou minimum) da fun\303\247\303\243o neste intervalo O procedimento para resolver (solve) um problema de otimiza\303\247\303\243o com uma fun\303\247\303\243o objetivo cont\303\255nua (continuous) em um intervalo limitado e fechado (closed e bounded) \303\251 o procedimento para encontrar extremo (extrema) global introduzido no Cap\303\255tulo 16 sobre An\303\241lise Global: Um m\303\241ximo global (ou um m\303\255nimo) existe e ocorre em um ponto cr\303\255tico (critical point). Quando um m\303\241ximo global de uma fun\303\247\303\243o \303\251 procurado em um intervalo (interval) que n\303\243o \303\251 limitado e fechado, i.e., ou \303\251 aberto em pelo menos um dos limites do intervalo (endpoint) ou \303\251 ilimitado, a exist\303\252ncia de um m\303\241ximo ou um m\303\255nimo global n\303\243o \303\251 garantida. Nestas situa\303\247\303\265es, um pouco mais de esfor\303\247o ser\303\241 requerido para prov\303\252 um argumento (matem\303\241tico) convincente de que realmente existe uma solu\303\247\303\243o \303\263tima (optimal solution) para o problema. Os poucos exemplos iniciais s\303\243o exemplos padr\303\265es. \303\200 medida que sua compreens\303\243o do processo geral melhora, os exemplos se tornam mais complicados, freq\303\274entemente envolvendo par\303\242metros (parameters) simb\303\263licos.
<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). area bolzanostheorem circle interval concaveup constant constraint continuous criticalpoint degenerate derivative difference differentiable domain endpoint equilateral extremum floatingpoint function minimum graph implicit independent variable inequality infinity integration interval limit maximum mean-valuetheorem midpoint minimum negative non-negative one-sided interval optimal optimization parameter positive radius satisfy solution solve substitution unbounded value variable work zero
<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 ):
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Defini\303\247\303\265es Auxiliares</Font></Text-field> PrintOptProb := proc( maxmin, obj ) local con; printf( " %s %a", maxmin, obj ); if nargs>2 then con := [args[3..-1]]; printf( "\134n s.t. %a", con[1] ); seq( printf( "\134n %a", con[j]), j=2..nops(con) ); end if end proc: p1 := plot( [ [0,0],[3,0],[3,1],[0,1], [0,0] ], style=line, color=blue ): p2 := seq( plot( [ [x,0],[x,1] ], style=line, color=red ), x=1..2 ): p3 := textplot( [ seq([x-0.5,-0.1,"x"], x=1..3), [2.9,0.5,"y"] ] ): P1 := display( [p1,p2,p3], axes=none, scaling=constrained ):
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.1</Text-field> Mostre que entre todos os ret\303\242ngulos com um per\303\255metro fixo, o quadrado tem a \303\241rea m\303\241xima (maximum area).
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field> A primeira frase fornece um v\303\255nculo para este problema. Denotando o per\303\255metro (perimeter) por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e o comprimento e a largura do ret\303\242ngulo por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , respectivamente. Assim, a equa\303\247\303\243o vinculada fica The first phrase provides a constraint for this problem. Denote the perimeter by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLFswLDI1NSwyNTVdRicvJSdvcGFxdWVHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJw== and let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLFswLDI1NSwyNTVdRicvJSdvcGFxdWVHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLFswLDI1NSwyNTVdRicvJSdvcGFxdWVHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJw== denote the length and width of the rectangle, respectively. Then, the constraint equation becomes Con := 2*L + 2*W = P: Con; Est\303\241 impl\303\255cito neste problema os v\303\255nculos geom\303\251tricos de que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= n\303\243o s\303\243o negativos (negative). Estes v\303\255nculos impl\303\255citos (implicit) s\303\243o expressos no Maple como Con2 := L >= 0: Con3 := W >= 0: Con2, Con3; Como a \303\241rea, designada por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251 a quantidade a ser maximizada, a fun\303\247\303\243o objetivo \303\251 Obj := A = L*W: Obj; A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251 PrintOptProb(max,Obj,Con,Con2,Con3); onde s.t. \303\251 uma abrevia\303\247\303\243o para "sujeito a" ("subject to") . O pr\303\263ximo passo \303\251 reformular o problema com uma fun\303\247\303\243o objetivo dependente somente de uma vari\303\241vel e do v\303\255nculo expresso como um intervalo apropriado. O per\303\255metro (perimeter) vinculado pode ser usado para encontrar uma solu\303\247\303\243o para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em termos da outra vari\303\241vel ( e o par\303\242metro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ). Assim o per\303\255metro vinculado \303\251 reescrito como Sub := isolate( Con, L ): Sub; (19.1)
A fun\303\247\303\243o objetivo , expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , \303\251 Obj1 := eval( Obj, Sub ): Obj1; A n\303\243o negatividade (nonnegativity) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= prov\303\252 um limite para o intervalo. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outro limite. De fato, usando a substitui\303\247\303\243o (substitution) em (19.1), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito como q1 := eval( Con2, Sub ): q1; ou, resolvendo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, Con4 := isolate( q1, W ): Con4; O problema de otimiza\303\247\303\243o resultante \303\251 PrintOptProb( max, Obj1, Con3, Con4 ); Como este problema est\303\241 pedindo por um m\303\241ximo global de uma fun\303\247\303\243o cont\303\255nua em um intervalo limitado e fechado, sabemos que o valor m\303\241ximo ocorrer\303\241 em um ponto cr\303\255tico da fun\303\247\303\243o objetivo no intervalo. Os limites do intervalo s\303\243o EndPt := 0, P/2: EndPt; Pontos estacion\303\241rios ocorrem quando a derivada (derivative) da fun\303\247\303\243o objetivo \303\251 zero (zero). A equa\303\247\303\243o resultante \303\251 dObj1 := diff( rhs(Obj1), W ): StatPtEqn := dObj1 = 0: StatPtEqn; Um ponto estacion\303\241rio \303\251 StatPt := solve( StatPtEqn, W ): StatPt; (Note que este est\303\241 no intervalo; de fato, \303\251 o ponto m\303\251dio (midpoint) do intervalo.) E, como n\303\243o existem pontos singulares, entramos com SingPt := NULL: Agora, avaliando a fun\303\247\303\243o objetivo em cada ponto cr\303\255tico tem-se seq( ['W'=W,Obj1], W=[EndPt,StatPt,SingPt] ); Desta lista fica bastante claro que a \303\241rea m\303\241xima, 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 , ocorre quando 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 , i.e., no ponto estacion\303\241rio. Enquanto encontramos o m\303\241ximo global neste intervalo, n\303\263s n\303\243o fizemos ( oque?????). Ainda resta verificar que o ret\303\242ngulo resultante \303\251 um quadrado. Isto ser\303\241 feito mostrando que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJMRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRIldGJ0Y0RjdGPkYrRj4= para a largura \303\263tima. Isto \303\251 feito como se segue . q2 := eval( Sub, W=StatPt ): W=StatPt; q2;
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">Observa\303\247\303\243o:</Font></Text-field> Note que se os v\303\255nculos geom\303\251tricos tivessem sido formulados como inequa\303\247\303\265es restritas , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 0 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 0, ent\303\243o o intervalo expresso somente em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= seria um intervalo aberto: 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIjJGJ0Y9Rj0vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkovJSliZXZlbGxlZEdRJmZhbHNlRidGK0Y9 . Neste caso o argumento de que o ponto cr\303\255tico \303\251 um m\303\241ximo global requer a considera\303\247\303\243o apropriada de um limite lateral (one-sided limits). Geralmente \303\251 muito mais f\303\241cil trabalhar com um intervalo limitado e fechado, aceitando objetos geom\303\251trico degenerados (degenerate) que correspondem aos limites (do intervalo). Algumas vezes estes limites (de intervalo) o fazem, e efetivamente, fornecem a desejada configura\303\247\303\243o \303\263tima
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.2</Text-field> Mostre que entre todos os ret\303\242ngulos com uma \303\241rea fixa, o quadrado tem o per\303\255metro m\303\255nimo
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field> Note a similaridade - e diferen\303\247as - entre este problema e aquele no Exemplo 19.1. As \303\272nicas mudan\303\247as no modo de expressar s\303\243o i) permutar \303\241rea e per\303\255metro e ii) trocar m\303\241ximo por m\303\255nimo. A primeira diferen\303\247a (difference) \303\251 que o v\303\255nculo e as fun\303\247\303\265es objetivo s\303\243o permutadas. A primeira frase prov\303\252 um v\303\255nculo para este problema. Denote a \303\241rea por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e denote comprimento e largura do ret\303\242ngulo por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , respectivamente. Assim, a equa\303\247\303\243o vinculada fica Con := A = L*W: Con; Outravez, a n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= est\303\241 impl\303\255cita, assim, escrevemos as inequa\303\247\303\265es Con2 := L >= 0: Con3 := W >= 0: Con2, Con3; Como o per\303\255metro, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251 a quantidade a ser minimizada, a fun\303\247\303\243o objetivo \303\251 Obj := P = 2*L+2*W: Obj; A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251 PrintOptProb(min,Obj,Con,Con2,Con3); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser eliminado. Como no Exemplo 19.1, escolhemos manter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, para resolver a \303\241rea vinculada para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, obtendo Sub := isolate( Con, L ): Sub; (19.2)
A fun\303\247\303\243o objetivo, expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251 Obj1 := isolate( eval( Obj, Sub ), P ): Obj1; A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= prov\303\252 um limite (de intervalo) para o dom\303\255nio (domain) da fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiM2Ji1GOzYtUSIoRidGPi9GQUY2RkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjE2NjY2NjdlbUYnL0ZTRmZuLUYjNiQtRiw2JVEiV0YnRjRGN0Y+LUY7Ni1RIilGJ0Y+RllGQ0ZaRkdGSUZLRk1GZW4vRlNRLDAuMTExMTExMWVtRidGPkYrRj5GK0Y+. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outro limite. De fato, usando a substitui\303\247\303\243o em (19.2), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito como q1 := eval( Con2, Sub ): q1; ou, resolvendo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= no Maple, encontramos Con5 := solve( q1, {W} ) assuming A>0: Con5; O problema de otimiza\303\247\303\243o resultante \303\251 PrintOptProb( min, Obj1, Con5 ); Como este problema est\303\241 pedindo por um m\303\241ximo global de uma fun\303\247\303\243o cont\303\255nua em um intervalo aberto e ilimitado (unbounded) , temos que proceder com alguma cautela. Primeiro, os pontos estacion\303\241rios ocorrem onde dObj1 := diff( rhs(Obj1), W ): StatPtEqn := dObj1 = 0: StatPtEqn; Existem duas solu\303\247\303\265es para esta equa\303\247\303\243o, a saber, q1 := solve( StatPtEqn, W ): q1; Somente uma deles \303\251 positiva, e que \303\251 StatPt := op(select( u->eval(u,A=1)>0, [q1] )): StatPt; A segunda derivada da fun\303\247\303\243o objetivo \303\251 d2Obj1 := diff( dObj1, W ); que \303\251 positiva (positive)para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJXRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. Portanto, a fun\303\247\303\243o objetivo \303\251 c\303\264ncava para cima (concave up) para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJXRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= e o \303\272nico ponto estacion\303\241rio \303\251 o minimizador global. Assim, o ret\303\242ngulo com a menor \303\241rea tem largura e comprimento W = StatPt; eval( Sub, W=StatPt ); com per\303\255metro eval( Obj1, W=StatPt ); Em particular, observe que \303\251 um quadrado.
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><Font readonly="true">Maple T.A. Maple-Graded Question</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field> In the following execution group, add an algorithmic variable. with(RandomTools): randomize(): n1 := Generate(integer(range=20..75)); n2 := Generate(choose([1,2])); type1 := ('maximum','minimum')[n2]; if (n2 = 1) then if ((n1 mod 2)=1) then x1 := n1-n1/2+1/2; else x1 := n1/2; end if; else x1 := 1; end if; ans1 := x1,n1-x1; IiNZ IiIi SShtYXhpbXVtRzYi NiQiI0JGIw== JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 19.1</Font></Text-field> In the following text region, enter the full contents of the question. Encontre dois n\303\272meros inteiros cuja soma \303\251 `$n1`, e cujo produto \303\251 um `$type1`. Na sua resposta, separe os dois n\303\272meros por uma v\303\255rgula. JSFH
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. JSFHans1;
<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field> Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region. evalb({`$ANSWER`}={`$RESPONSE`});
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> Enter your hints in the following table.
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> In the following text region, enter feedback to an incorrect response.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea5`, `value`); ANSWER := ans1; if evalb({ANSWER}={eval(parse(StudentResponse))}) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&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>23</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mn>23</mn></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext></mtext><mtext>&MediumSpace;</mtext><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>23</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mn>23</mn></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.3</Text-field> Um fazendeiro quer criar um campo retangular de 1 acre (43560 sq ft). Este campo ser\303\241 subdividido em tr\303\252s pastos id\303\252nticos. A cerca para a parte externa do campo custa $5 por p\303\251 e a cerca interna que cria os pastos menores custa $3.50 por p\303\251. Qual \303\251 o custo m\303\255nimo para construir este campo? (Quais s\303\243o as dimens\303\265es destes campos menores e do campo total?)
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field> A geometria deste problema est\303\241 resumida na Figura 19.1. P1; Ni0tJSdDVVJWRVNHNiU3JzckJCIiISEiIiQiIiEhIiI3JCQiI0khIiIkIiIhISIiNyQkIiNJISIiJCIjNSEiIjckJCIiISEiIiQiIzUhIiI3JCQiIiEhIiIkIiIhISIiLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIzUhIiItJSZTVFlMRUc2IyUlTElORUctJSdDVVJWRVNHNiU3JDckJCIjNSEiIiQiIiEhIiI3JCQiIzUhIiIkIiM1ISIiLSUmQ09MT1JHNiYlJFJHQkckIiM1ISIiJCIiISEiIiQiIiEhIiItJSZTVFlMRUc2IyUlTElORUctJSdDVVJWRVNHNiU3JDckJCIjPyEiIiQiIiEhIiI3JCQiIz8hIiIkIiM1ISIiLSUmQ09MT1JHNiYlJFJHQkckIiM1ISIiJCIiISEiIiQiIiEhIiItJSZTVFlMRUc2IyUlTElORUctJSVURVhURzYkNyQkIiImISIiJCEiIiEiIi0lKV9UWVBFU0VURzYjUSJ4NiItJSVURVhURzYkNyQkIiM6ISIiJCEiIiEiIi0lKV9UWVBFU0VURzYjUSJ4NiItJSVURVhURzYkNyQkIiNEISIiJCEiIiEiIi0lKV9UWVBFU0VURzYjUSJ4NiItJSVURVhURzYkNyQkIiNIISIiJCIiJiEiIi0lKV9UWVBFU0VURzYjUSJ5NiItJStBWEVTTEFCRUxTRzYnUSE2IlEhNiItJSVGT05URzYlJShERUZBVUxURyUoREVGQVVMVEciIzUlK0hPUklaT05UQUxHJStIT1JJWk9OVEFMRy0lKkFYRVNTVFlMRUc2IyUlTk9ORUctJShTQ0FMSU5HRzYjJSxDT05TVFJBSU5FREctJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiNxISIiLSUpQk9VTkRTX1lHNiMkIiUhRyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiUhKVEhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlUzkhIiItJSlDSElMRFJFTkc2Ig==
Figura 19.1 O campo retangular cercado e dividido em tr\303\252s pastos id\303\252nticos
Cada lado azul \303\251 constru\303\255do com cerca de $5/ft e os dois lados vermelhos s\303\243o constru\303\255dos usando a cerca mais barata. O custo total \303\251 portanto a fun\303\247\303\243o objetivo Obj := C = 5*(6*x+2*y) + 3.5*(2*y): Obj; A equa\303\247\303\243o vinculada Con := 3*x*y = 43560: Con; expressa a necessidade de que a \303\241rea cercada contenha 1 acre. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= est\303\241 impl\303\255cita, assim, escrevemos as inequa\303\247\303\265es Con2 := x >= 0: Con3 := y >= 0: Con2, Con3; A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251 PrintOptProb(min,Obj,Con,Con2,Con3); escolhendo usar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= como a independente vari\303\241vel, resolvemos o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para obter a f\303\263rmula de substitui\303\247\303\243o Sub := isolate( Con, x ): Sub; (19.3)
A fun\303\247\303\243o objeto, expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, \303\251 Obj1 := isolate( eval( Obj, Sub ), C ): Obj1; A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= prove um limite para o intervalo. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outo limite. De fato, usando a substitui\303\247\303\243o em (19.3), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito como q1 := eval( Con2, Sub ): q1; ou, resolvendo para 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 Con5 := solve( q1, {y} ): Con5; O problema de otimiza\303\247\303\243o resultante \303\251 PrintOptProb( min, Obj1, Con5 ); Uma vez mais o intervalo \303\251 aberto e ilimitado. Trabalhando como no Exemplo 19.2, a primeira derivada da fun\303\247\303\243o objetivo \303\251 dObj1 := diff( rhs(Obj1), y ): dObj1; Os pontos estacion\303\241rios ocorrem onde esta derivada se anula, assim temos a equa\303\247\303\243o StatPtEqn := dObj1 = 0: StatPtEqn; Existem duas solu\303\247\303\265es para esta equa\303\247\303\243o, a saber, q1 := solve( StatPtEqn, y ): q1; Somente uma delas \303\251 positiva, a saber, StatPt := op(select( u->u>0, [q1] )): StatPt; A segunda derivada da fun\303\247\303\243o objetivo \303\251 d2Obj1 := diff( dObj1, y ); que \303\251 positiva para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. Portanto, a fun\303\247\303\243o objetivo \303\251 c\303\264ncava para cima para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= e o \303\272nico ponto estacion\303\241rio \303\251 o minimizador global, como pode ser visto na Figura 19.2. plot( rhs(Obj1), y=0..1000, view=[DEFAULT,0..10^5]); 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 Figura 19.2 Exemplo 19.3: Fun\303\247\303\243o de custo para a cerca
Note que a fun\303\247\303\243o objetivo \303\251 c\303\264ncava para cima e tem um m\303\255nimo global pr\303\263ximo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEkMTYwRidGPkY+RitGPg==, como o esperado. Cada campo ter\303\241 dimens\303\265es eval(Sub,y=StatPt); y = StatPt; e a \303\241rea total cercada tem dimens\303\265es 3*eval(Sub,y=StatPt); y = StatPt; O custo total para construir esta cerca \303\251 eval( Obj1,y=StatPt ); (Como uma verifica\303\247\303\243o, note que a \303\241rea total fechada \303\251 43560 p\303\251s quadrados, ou 1 acre.)
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><Font readonly="true">Maple T.A. Maple-Graded Question</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field> In the following execution group, add an algorithmic variable. with(RandomTools): randomize(): n2 := Generate(integer(range=30..70))*10; y2 := n2-2*x; area2 := y2*x; x2 := solve(diff(area2,x)=0,x); ans2 := x=x2,y=eval(y2,x=x2); IiRnJg== LCYiJGcmIiIiSSJ4RzYiISIj KiYsJiIkZyYiIiJJInhHNiIhIiNGJUYmRiU= IiRTIg== NiQvSSJ4RzYiIiRTIi9JInlHRiUiJCFH JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 19.2</Font></Text-field> In the following text region, enter the full contents of the question. Suponha que um fazendeiro queira construir uma cerca de modo a ter um campo retangular. Em um lado do campo, existe um rio (i.e., nenhuma cerca \303\251 necess\303\241ria neste lado). Designemos o comprimento do lado paralelo ao rio de y, e o comprimento dos outros 2 lados de x. Quais s\303\243o as dimens\303\265es do ret\303\242ngulo que maximizar\303\241 a \303\241rea do campo se o fazendeiro tem `$n2` m de cerca dispon\303\255vel? Entre com sua resposta como x = comprimento1, y = comprimento2. JSFH
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. JSFHans2;
<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field> Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region. evalb(evalf({`$ANSWER`}={`$RESPONSE`}));
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> Enter your hints in the following table. The length of 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 in terms of 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 is 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 = `$y2`. If 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 is the area of the rectangle, solve 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 to find the length 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 that will maximize the area.
with(StringTools): with(DocumentTools): response := GetProperty(`TextArea6`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The&MediumSpace;length&MediumSpace;of&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>y</mi></mrow></mstyle><mtext>&MediumSpace;in&MediumSpace;terms&MediumSpace;of&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>x</mi></mrow></mstyle><mtext>&MediumSpace;is&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi fontfamily='serif'>y</mi><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi></mrow></mfenced></mrow></mrow></mstyle><mtext>&MediumSpace;=&MediumSpace;</mtext><mtext></mtext><mrow><mrow><mn>560</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>x</mi></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea6`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>If&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi fontfamily='serif'>A</mi><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi></mrow></mfenced></mrow></mrow></mstyle><mtext>&MediumSpace;is&MediumSpace;the&MediumSpace;area&MediumSpace;of&MediumSpace;the&MediumSpace;rectangle,&MediumSpace;solve&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mo fontfamily='serif' lspace='0.2222222em' rspace='0.2222222em'>&amp;DifferentialD;</mo><mrow><mo fontfamily='serif' lspace='0.2222222em' rspace='0.2222222em'>&amp;DifferentialD;</mo><mi fontfamily='serif'>x</mi></mrow></mfrac><mrow><mi>A</mi><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi></mrow></mfenced></mrow><mo fontfamily='serif' lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn fontfamily='serif'>0</mn></mrow></mstyle><mtext>&MediumSpace;to&MediumSpace;find&MediumSpace;the&MediumSpace;length&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>x</mi></mrow></mstyle><mtext>&MediumSpace;that&MediumSpace;will&MediumSpace;maximize&MediumSpace;the&MediumSpace;area.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> In the following text region, enter feedback to an incorrect response.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea6`, `value`); ANSWER := ans2; if evalb(evalf({ANSWER}={eval(parse(StudentResponse))})) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&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><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>140</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>280</mn></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext></mtext><mtext>&MediumSpace;</mtext><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>140</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>280</mn></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.4</Text-field> Um fazendeiro quer fechar um campo retangular com \303\241rea LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (acres). Este campo ser\303\241 subdividido em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pastos retangulares id\303\252nticos. A cerca externa do campo custa $LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= por p\303\251 e a cerca para criar os pastos menores custa $LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= por p\303\251. Qual \303\251 o custo m\303\255nimo para construir este campo? (Quais s\303\243o as dimens\303\265es dos campos menores e campo total?)
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field> Este problema \303\251 essencialmente o mesmo que o Exemplo 19.3, exceto que as dimens\303\265es e pre\303\247os s\303\243o dados como par\303\242metros. Cada lado da cerca externa \303\251 constru\303\255do com a cerca de $LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= /ft e os LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= lados internos s\303\243o constru\303\255dos usando cerca de $LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= /ft . O custo total desta cerca \303\251 a seguinte fun\303\247\303\243o objetivo. Obj := C = P * ( 2*n*x + 2*y ) + p * ( (n-1)*y ): Obj; A necessidade da \303\241rea total ser LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn acres prov\303\252 a seguinte equa\303\247\303\243o vinculada. Con := n*x*y = 43560*A: Con; A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= est\303\241 impl\303\255cita, assim n\303\263s escrevemos as inequa\303\247\303\265es Con2 := x >= 0: Con3 := y >= 0: Con2, Con3; A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251 PrintOptProb(min,Obj,Con,Con2,Con3); escolhendo usar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= como a independente vari\303\241vel, resolvemos para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para obter a seguinte f\303\263rmula de substitui\303\247\303\243o Sub := isolate( Con, x ): Sub; (19.4)
A fun\303\247\303\243o objetivo, expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , \303\251 Obj1 := isolate( eval( Obj, Sub ), C ): Obj1; A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= prov\303\252 um dos limites para o dom\303\255nio da fun\303\247\303\243o 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. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outro limite. De fato, usando a substitui\303\247\303\243o em (19.4), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito como q1 := eval( Con2, Sub ): q1; ou, resolvendo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= no Maple, obtemos Con5 := solve( q1, {y} ) assuming A>0, n::posint: Con5; O problema de otimiza\303\247\303\243o resultante \303\251 PrintOptProb( min, Obj1, Con5 ); Uma vez mais o intervalo \303\251 aberto e ilimitado. Trabalhando como no Exemplo 2, a primeira derivada da fun\303\247\303\243o objetivo \303\251 dObj1 := diff( rhs(Obj1), y ): dObj1; Pontos estacion\303\241rios ocorrem quando StatPtEqn := dObj1 = 0: StatPtEqn; Existem duas solu\303\247\303\265es para esta equa\303\247\303\243o, a saber q1 := solve( StatPtEqn, y ): q1; Somente uma delas \303\251 positiva, a saber, StatPt := op(select( u->evalf(eval(u,[p=1,P=2,n=1,A=1]))>0, [q1] )): StatPt; A segunda derivada da fun\303\247\303\243o objetiva \303\251 d2Obj1 := diff( dObj1, y ); a qual, porque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= s\303\243o ambos positivos, \303\251 positiva para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. Portanto, a fun\303\247\303\243o objetivo \303\251 c\303\264ncava para cima para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= e o \303\272nico ponto estacion\303\241rio \303\251 o minimizador global. Por causa dos par\303\242mentros envolvidos neste problema, n\303\243o \303\251 poss\303\255vel criar um gr\303\241fico; mas a forma geral ser\303\241 a mesma como a do Exemplo 19.3. Cada campo ter\303\241 as dimens\303\265es eval(Sub,y=StatPt); y = StatPt; e a \303\241rea total cercada tem as dimens\303\265es n*eval(Sub,y=StatPt); y = StatPt; O custo total para levantar toda a cerca \303\251 simplify( eval( Obj1,y=StatPt ) );
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.5</Text-field> Um fio de comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (cm) \303\251 cortado em duas pe\303\247as. Uma pe\303\247a \303\251 curvada para formar um c\303\255rculo (circle) e a outra \303\251 dobrada para formar um tri\303\242ngulo equ\303\255latero (equilateral) . Quais s\303\243o os valores m\303\241ximo e m\303\255nimo para a soma destas duas regi\303\265es?
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field> Designemos por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= a raio (radius) do c\303\255rculo e por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= o comprimento dos lados do tri\303\242ngulo. Assim, um c\303\255rculo com raio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tem per\303\255metro Pcirc := 2*Pi*r: P[circ] = Pcirc; e \303\241rea Acirc := Pi*r^2: A[circ] = Acirc; Analogamente, um tri\303\242ngulo equil\303\241tero com lado de comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tem per\303\255metro Ptri := 3*s: P[tri] = Ptri; e \303\241rea Atri := 1/2 * s * sqrt(3)/2*s: A[tri] = Atri; A necessidade de ter um comprimento fixo para o fio \303\251 expressa pela equa\303\247\303\243o vinculada Con := P = Pcirc + Ptri; E mais, o comprimento do raio e do lado n\303\243o podem ser negativos.(Se um dos dois for zero, ent\303\243o existe somente uma regi\303\243o.) Portanto, escrevemos as inequa\303\247\303\265es Con2 := r >= 0: Con3 := s >= 0: Con2; Con3; A fun\303\247\303\243o objetivo \303\251 o total das \303\241reas delimitadas pelas duas regi\303\265es. Obj := A = Acirc + Atri; A formula\303\247\303\243o inicial deste problema como um problema de otimiza\303\247\303\243o \303\251 PrintOptProb(min,Obj,Con,Con2,Con3); Provavelmente \303\251 mais f\303\241cil manter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= como a vari\303\241vel deste problema, assim resolvemos para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para obter a equa\303\247\303\243o de substitui\303\247\303\243o Sub := isolate( Con, s ): Sub; (19.5)
A fun\303\247\303\243o objetivo, expressa em termos da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkwtRiw2JVEpJmVhY3V0ZTtGJy9GMEY9RjlGOQ== Obj1 := isolate( eval( Obj, Sub ), A ): Obj1; A n\303\243o negatividade de 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 da fun\303\247\303\243o 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. A n\303\243o negatividade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= deve fornecer o outro limite. De fato, usando a substitui\303\247\303\243o em (19.5), o v\303\255nculo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser escrito q1 := eval( Con3, Sub ): q1; ou resolvendo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, obtemos Con4 := isolate( q1, r ): Con4; O problema de otimiza\303\247\303\243o resultante \303\251 PrintOptProb( min, Obj1, Con2, Con4 ); A fun\303\247\303\243o objetivo cont\303\255nua, junto com o intervalo limitado e fechado, faz este problema como o do Exemplo 1. N\303\243o existem pontos singulares, assim, entramos com SingPt := NULL: Os limites (do intervalo) s\303\243o EndPt := 0, P/(2*Pi); `` = 0,evalf(P/2/Pi); Os pontos estacion\303\241rios s\303\243o as solu\303\247\303\265es da equa\303\247\303\243o dObj1 := diff( rhs(Obj1), r ): StatPtEqn := dObj1 = 0: StatPtEqn; A \303\272nica solu\303\247\303\243o desta equa\303\247\303\243o \303\251 StatPt := solve( StatPtEqn, r ): StatPt = evalf(StatPt); Para encontrar o m\303\241ximo e m\303\255nimo global no intervalo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= 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, compare os valores da fun\303\247\303\243o objetivo em cada um dos tr\303\252s pontos cr\303\255ticos (e inclua o valor de s correspondente ). Cada valor da lista a seguir corresponde a um valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, sendo o primeiro e o terceiro os valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= no limite do dom\303\255nio, e o do meio o valor cr\303\255tico no interior do dom\303\255nio. CritPt := [ EndPt, StatPt, SingPt ]: q2 := seq( ['r'=r,simplify(Sub),simplify(Obj1)], r=CritPt ); Para identificar as maiores e as menores \303\241reas, converta estes resultados para a forma de ponto flutuante ( floating-point ), obtendo evalf[4]( q2 ); A menor valor para 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 rmin := StatPt: r = rmin; e o maior valor para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ocorre no limite (do intervalo) \303\240 direita rmax := EndPt[2]: r = rmax; Para concluir, a menor \303\241rea total para as duas regi\303\265es ocorre com um c\303\255rculo de raio r = rmin; e um tri\303\242ngulo com lado de comprimento simplify(eval( Sub, r=rmin )); A \303\241rea total para estas duas regi\303\265es \303\251 simplify(eval( Obj1, r=rmin )); A \303\241rea m\303\241xima total para as duas regi\303\265es ocorre com um c\303\255rculo de raio r = rmax; e um tri\303\242ngulo com lado de comprimento simplify(eval( Sub, r=rmax )); Neste caso o comprimento total do fio \303\251 usado para formar o c\303\255rculo e n\303\243o existe tri\303\242ngulo. A \303\241rea total \303\251 a \303\241rea do c\303\255rculo, a saber, simplify(eval( Obj1, r=rmax ));
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 19.6</Text-field> A uma dist\303\242ncia LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= de uma fonte de luz, a intensidade LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, da ilumina\303\247\303\243o (luminosidade?)pela fonte de luz, pode ser modelada com 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 onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 a intensidade da fonte de luz, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 uma constante (constant) positiva. Supondo que duas fontes de luz, com intensidades LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB e 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, est\303\243o posicionadas distantes 10 m uma da outra, e supondo que a intensidade de ilumina\303\247\303\243o em qualquer ponto seja \303\251 a soma da ilumina\303\247\303\243o de todas as fontes relevantes de luz . (Sugest\303\243o: As respostas para as seguintes quest\303\265es podem ser encontradas considerando a ilumina\303\247\303\243o ao longo da linha que conecta as duas fontes de luz.) Encontre a a localiza\303\247\303\243o do ponto que recebe menor ilumina\303\247\303\243o. (Note que esta localiza\303\247\303\243o ser\303\241 uma fun\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.) Quando \303\251 o m\303\255nimo no ponto m\303\251dio entre as duas fontes de luz? Qual valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= faz o ponto a um quarto do caminho entre as duas fontes de luz receber a menor ilumina\303\247\303\243o? Existem alguns pontos entre as lumin\303\241rias onde o m\303\255nimo n\303\243o possa ocorrer?
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field> A ilumina\303\247\303\243o de cada lumin\303\241ria \303\251 L1 := c*S[1]/d[1]^2: L2 := c*(k*S[1])/d[2]^2: L[1] = L1; L[2] = L2; assim, o total de ilumina\303\247\303\243o (luminosidade) em um objeto destas duas fontes de luz \303\251 a fun\303\247\303\243o objetivo Obj := L1 + L2; onde as dist\303\242ncias satisfazem (satisfy) os v\303\255nculos Con := d[1] + d[2] = 10: Con2 := d[1] > 0: Con3 := d[2] > 0: Con; Con2; Con3; (Observe que n\303\243o \303\251 permitido usar 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 nem 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 .) O problema inicial de otimiza\303\247\303\243o \303\251 PrintOptProb( min, Obj, Con, Con2, Con3 ); A elimina\303\247\303\243o de uma das dist\303\242ncias, digamos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB, \303\251 direto, e \303\251 implementado obtendo a equa\303\247\303\243o de substitui\303\247\303\243o Sub := isolate( Con, d[2] ); A nova fun\303\247\303\243o objeto, com a vari\303\241vel dependente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB, \303\251 Obj1 := eval( Obj, Sub ); O v\303\255nculo de n\303\243o negatividade para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JlEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiVRIjJGJ0Y4L0Y8USdub3JtYWxGJ0ZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZE \303\251 reassegurado como Con4 := isolate(eval( Con3, Sub ),d[1]); (Isto devia ser \303\263bvio!) O novo problema de otimiza\303\247\303\243o \303\251 PrintOptProb( min, Obj1, Con2, Con4 ); A procura por pontos estacion\303\241rios no intervalo obt\303\251m um ponto cr\303\255tico real, a saber dObj1 := simplify( diff( Obj1, d[1] ) ): StatPtEq := dObj1 = 0: q2 := solve( StatPtEq, d[1] ): StatPt := op(remove( has, [q2], I )): d[1] = StatPt; Classificar este ponto cr\303\255tico pelo Teste da Segunda Derivada requer a segunda derivada da fun\303\247\303\243o objetivo, que \303\251 d2Obj1 := simplify(diff( Obj1, d[1]$2 )); O resultado de avalia\303\247\303\243o da segunda derivada no ponto estacion\303\241rio \303\251 Q := simplify( eval( d2Obj1, d[1]=StatPt ) ); Determinar que isto \303\251 positivo para todos os valores do par\303\242metro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= parece ser uma tarefa dantesca. Entretanto, um momento de an\303\241lise (inspe\303\247\303\243o) sugere que a substitui\303\247\303\243o 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 pode simplificar a express\303\243o sem mudar a sequ\303\252ncia da an\303\241lise. Obtemos Qz := simplify(eval(Q,k=z^3)) assuming z>0; Desta forma da express\303\243o, fica claro que a segunda derivada no ponto cr\303\255tico \303\251 positiva para todos os valores positivos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e conclu\303\255mos que o ponto cr\303\255tico \303\251 um m\303\255nimo. Se n\303\243o tivesse sido percebida esta substitui\303\247\303\243o simplificadora, um caminho alternativo para a desejada conclus\303\243o seria o seguinte: Os seguintes limites mostram que a ilumina\303\247\303\243o(luminosidade) se torna (positivamente) ilimitadas em cada ponto limite de intervalo. q3 := Limit( Obj1, d[1]=0, right ): q4 := Limit( Obj1, d[1]=10, left ): q3 = value( q3 ) assuming k>0,c>0,S[1]>0; q4 = value( q4 ) assuming k>0,c>0,S[1]>0; Estes limites combinados com i) a continuidade e diferenciabilidade (differentiability) da fun\303\247\303\243o ilumina\303\247\303\243o no intervalo aberto 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < 10 e ii) a exist\303\252ncia de um \303\272nico ponto estacion\303\241rio em 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0ZB < 10 s\303\243o suficientes para concluir que o ponto estacion\303\241rio deve ser o minimizador global para a fun\303\247\303\243o objeto. Usando qualquer uma das abordagens, o minimizador global para a fun\303\247\303\243o objeto \303\251 dmin := StatPt: d[1] = dmin; Para concluir, precisamos responder a quest\303\265es espec\303\255ficas. O ponto m\303\251dio (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 ) ser\303\241 o ponto com ilumina\303\247\303\243o m\303\255nima quando q5 := dmin = 5: q5; A \303\272nica solu\303\247\303\243o para esta equa\303\247\303\243o \303\251 q6 := solve( q5, k ): k = q6; isto \303\251, quando as duas lumin\303\241rias tiverem a mesma intensidade. Isto faz sentido pela simetria. Para mover o ponto de ilumina\303\247\303\243o m\303\255nima para o ponto a um quarto, 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, a segunda lumin\303\241ria teria que ser mais brilhante. Assim, esperamos encontrar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 1 -- mas, quanto maior ? Trabalhando como acima, escrevemos a equa\303\247\303\243o apropriada e encontramos para sua solu\303\247\303\243o q7 := dmin = 2.5: q8 := solve( q7, k ): k = q8; Logo, 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 . A quest\303\243o final est\303\241 realmente perguntando se existem solu\303\247\303\265es para a equa\303\247\303\243o q9 := dmin = d; para todos os valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkjbW9HRiQ2LVElJmxlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEjMTBGJ0Y1RjVGK0Y1. Para uma primeira percep\303\247\303\243o desta quest\303\243o, examine Figura 19.3, que cont\303\251m um gr\303\241fico (graph) de 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 como uma fun\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. plot( dmin, k=0..10^4, view=[DEFAULT,0..10] ); 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
Figura 19.3 Gr\303\241fico de 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
Parece claro que a localiza\303\247\303\243o admite valores acima de 10 \303\240 medida que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn se aproxima de 0 (da direita). De fato Eval(d[1],k=0) = eval( dmin, k=0 ); N\303\243o \303\251 t\303\243o claro que esta fun\303\247\303\243o admite valores abaixo de zero. (Pela simetria parece que devia.) Para ver que todas as localiza\303\247\303\265es acima desta posi\303\247\303\243o da lumin\303\241ria s\303\243o localiza\303\247\303\265es para ilumina\303\247\303\243o m\303\255nima, veja no limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se aproxima do infinito (infinity). q10 := Limit( dmin, k=infinity ): q10 = value( q10 ); Agora, pela continuidade da posi\303\247\303\243o de ilumina\303\247\303\243o m\303\255nima como uma fun\303\247\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, as id\303\251ias por tr\303\241s do Teorema do Ponto M\303\251dio ?? (Intermediate Value Theorem) podem ser aplicadas para concluir que a ilumina\303\247\303\243o m\303\255nima pode ocorrer em qualquer ponto entre as duas lumin\303\241rias. O m\303\255nimo ocorre em um ponto limite do intervalo somente se n\303\243o existir fonte de luz neste limite(de intervalo).
<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 := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique. with(RandomTools): randomize(): n3 := Generate(integer(range=2..5)): fn3 := sort(Generate(polynom(integer(range=-10..10),x,degree=n3))); x3 := Generate(integer(range=-10..10)); y3 := Generate(integer(range=-10..10)); while (y3 = eval(fn3,x=x3)) do y3 := Generate(integer(range=-10..10)) end do: dist3 := sqrt((x-x3)^2+(fn3-y3)^2); ddist3 := sort(diff(dist3,x)): s3 := [fsolve(numer(ddist3) = 0,x)]; sy3 := eval(dist3,x=s3[1]): sx3 := s3[1]: if (nops(s3)>1) then for i from 2 to nops([s3]) do temp3 := sy3; sy3 := max(sy3,eval(dist3,x=s3[i])); if (sy3 <> temp3) then sx3 := s3[i]; end if; end do; end if: ans3 := evalf(sx3,4),evalf(sy3,4); LCwqJEkieEc2IiIiJSEjNSokRiQiIiQiIikqJEYkIiIjRiZGJCIiIiEiIkYt IiIj ISIp KiQsNCokSSJ4RzYiIiIjIiNlRiUiIzUiI2AiIiIqJEYlIiIkIiQ/IiokRiUiIiUhJDMiKiRGJSIiKSIkKyIqJEYlIiIoISRnIiokRiUiIichIzsqJEYlIiImIiNXI0YrRic= NyUkIStHYWZgIikhIzUkIitLZScqPicpRiUkIisiKnl4JlIiISIq NiQkISVhIikhIiUkIiU8RyEiJA== JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 19.3</Font></Text-field> The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question. Considere a curva LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = `$fn3`. Que ponto da curva tem a menor dist\303\242ncia do ponto (`$x3`,`$y3`)? Entre com sua resposta como (x, y). Sua resposta pode ter precis\303\243o de at\303\251 4 d\303\255gitos significativos. 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 `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. JSFH``(ans3); JSFH
<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field> To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$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([`$ans3`]=evalf([`RESPONSE`],4)); 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. The shortest distance from a curve to a known point is 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, where 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 and 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 are points on the curve. Solve 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 to find the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbWlHRiQ2NVEieEYnLyUnZmFtaWx5R1Emc2VyaWZGJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjcvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMkY1L0Y5RjdGO0Y9Rj9GQUZERkdGSUZLRk1GT0ZRRlNGVS9GWFEnbm9ybWFsRic= that will give the shortest distance. Replace this text with hint 3.
with(StringTools): with(DocumentTools): response := GetProperty(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The&MediumSpace;shortest&MediumSpace;distance&MediumSpace;from&MediumSpace;a&MediumSpace;curve&MediumSpace;to&MediumSpace;a&MediumSpace;known&MediumSpace;point&MediumSpace;is&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi><mo fontfamily='serif' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi fontfamily='serif'>x</mi><mi fontfamily='serif'>known</mi></msub></mrow></mfenced><mn fontfamily='serif'>2</mn></msup><mo fontfamily='serif' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>y</mi><mo fontfamily='serif' lspace='0.2777778em' rspace='0.2777778em'>&minus;</mo><msub><mi fontfamily='serif'>y</mi><mi fontfamily='serif'>known</mi></msub></mrow></mfenced><mn fontfamily='serif'>2</mn></msup></mrow></msqrt></mrow></mstyle><mtext>,&MediumSpace;where&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>x</mi></mrow></mstyle><mtext>&MediumSpace;and&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>y</mi></mrow></mstyle><mtext>&MediumSpace;are&MediumSpace;points&MediumSpace;on&MediumSpace;the&MediumSpace;curve.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Solve&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&amp;DifferentialD;</mo><mrow><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&amp;DifferentialD;</mo><mi fontfamily='serif'>x</mi></mrow></mfrac><mrow><mi>distance</mi><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi fontfamily='serif'>x</mi></mrow></mfenced></mrow><mo fontfamily='serif' lspace='0.2777778em' rspace='0.2777778em'>&InvisibleTimes;</mo><mo fontfamily='serif' lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mo fontfamily='serif' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mn fontfamily='serif'>0</mn></mrow></mstyle><mtext>&MediumSpace;to&MediumSpace;find&MediumSpace;the&MediumSpace;value&MediumSpace;of&MediumSpace;</mtext><mstyle fontfamily='serif' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi fontfamily='serif'>x</mi></mrow></mstyle><mtext>&MediumSpace;that&MediumSpace;will&MediumSpace;give&MediumSpace;the&MediumSpace;shortest&MediumSpace;distance.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response. Replace this text with feedback.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea3`, `value`); ANSWER := ``(ans3); if evalb([ans3]=evalf([`eval(parse(StudentResponse))`],4)) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="&lt;NO RESPONSE&gt;"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect.&nbsp;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer:&nbsp;</mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.8154</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mn>2.817</mn></mrow></mfenced></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.8154</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mn>2.817</mn></mrow></mfenced></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><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 := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique. with(RandomTools): randomize(): A := Generate(integer(range=8..20))*20; m1 := Generate(choose([1,2,3])); m2 := Generate(choose([1,2,3])); while (m1=m2) do m2 := Generate(choose([1,2,3])) end do; ansH:=sqrt( (m1+m2)/(2*m2) * A ): ansW:=sqrt( (2*m2)/(m1+m2) * A ): ansP:=simplify((ansH+m1+m2)*(ansW+2*m2)): t := Generate(choose([1,2,3])): TYPE_ := `width`,`height`,`printed area`: TYPE := TYPE_[t]; ans_:= ansW,ansH,ansP: ans := ans_[t]; IiRnIw== IiIj IiIi SS1wcmludGVkfmFyZWFHNiI= LCQqJCwmKiQiJCFSIyIiIiIiI0YoIiIkRihGKSNGKUYq JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 19.4</Font></Text-field> The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question. Um poster est\303\241 para ser criado com uma `$m1` polegada de margem na parte superior, `$m2` polegadas de margem nas laterais e na parte inferior, e uma \303\241rea total de `$A` in2(polegadas quadradas). Qual \303\251 a `$TYPE` do poster com a maior \303\241rea impressa? Entre com sua resposta exata, ou com precis\303\243o de at\303\251 5 d\303\255gitos significativos. JSFH
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. ans; 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`-`$RESPONSE`), 5 ) = 0 );
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints. The area can be written as P = (w-2*`$m2`)*(h-(`$m1`+`$m2`)) When h is eliminated, the problem can be reformulated as P = `$A` - 2*`$m2`*`$A`/w - (`$m1`+`$m2`)*w + (2*`$m2`)*(`$m1`+`$m2`) Replace this text with hint 3.
with(StringTools): with(DocumentTools): response := GetProperty(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The&MediumSpace;area&MediumSpace;can&MediumSpace;be&MediumSpace;written&MediumSpace;as&MediumSpace; </mtext><mtext>P&MediumSpace;=&MediumSpace;(w-2*</mtext><mtext></mtext><mrow><mn>1</mn></mrow><mtext></mtext><mtext>)*(h-(</mtext><mtext></mtext><mrow><mn>2</mn></mrow><mtext></mtext><mtext>+</mtext><mtext>`$m2`</mtext><mtext>))</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext></mtext><mtext>&MediumSpace;</mtext><mtext>When&MediumSpace;h&MediumSpace;is&MediumSpace;eliminated,&MediumSpace;the&MediumSpace;problem&MediumSpace;can&MediumSpace;be&MediumSpace;reformulated&MediumSpace;as&MediumSpace; </mtext><mtext>P&MediumSpace;=&MediumSpace;</mtext><mtext></mtext><mrow><mn>260</mn></mrow><mtext></mtext><mtext>&MediumSpace;-&MediumSpace;2*</mtext><mtext></mtext><mrow><mn>1</mn></mrow><mtext></mtext><mtext>*</mtext><mtext></mtext><mrow><mn>260</mn></mrow><mtext></mtext><mtext>/w&MediumSpace;-&MediumSpace;(</mtext><mtext></mtext><mrow><mn>2</mn></mrow><mtext></mtext><mtext>+</mtext><mtext>`$m2`</mtext><mtext>)*w&MediumSpace;+&MediumSpace;(2*</mtext><mtext>`$m2`</mtext><mtext>)*(</mtext><mtext>`$m1`</mtext><mtext>+</mtext><mtext>`$m2`</mtext><mtext>)</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := ans; if evalb( fnormal(evalf(ANSWER-eval(parse(StudentResponse))), 5 ) = 0 ) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="&lt;NO RESPONSE&gt;"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect.&nbsp;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer:&nbsp;</mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mn>390</mn></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mrow></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext></mtext><mtext>&NewLine;</mtext><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mn>390</mn></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mrow></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><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 := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique. with(RandomTools): randomize(): r5 := -1: while (r5 <= 0) do n4 := Generate(choose([1,2])); t5 := (`cylinder`,`right circular cone`)[n4]: t6 := (`bases`,`base`)[n4]: V5 := Generate(integer(range=10..30))*10*Pi: costla := Generate(integer(range=1..8)); costb := Generate(integer(range=1..10)); while (costb = costla) do costb := Generate(integer(range=1..10)) end do; if (n4 = 1) then h50 := solve(V5 = h*Pi*r^2,h); c5 := costla*h*Pi*2*r+costb*2*Pi*r^2; else h50 := solve(V5 = (h*Pi*r^2)/3,h); c5 := costla*Pi*r*sqrt(h^2+r^2) + costb*Pi*r^2; end if; c51 := eval(c5,h=h50); dc51 := simplify(diff(c51,r)); r5 := fsolve(numer(dc51)=0,r); end do; h5 := eval(h50,r=r5); ans5 := r=evalf(r5,4),h=evalf(h5,4); IiIi SSljeWxpbmRlckc2Ig== SSZiYXNlc0c2Ig== LCRJI1BpRyUqcHJvdGVjdGVkRyIkcSI= IiIm IiIo LCYqJkkickc2IiEiIkkjUGlHJSpwcm90ZWN0ZWRHIiIiIiUrPComRidGKUYkIiIjIiM5 LCQqKEkjUGlHJSpwcm90ZWN0ZWRHIiIiLCYhJEQlRiYqJEkickc2IiIiJCIiKEYmRiohIiMiIiU= JCIrKGVULiRSISIq JCIrV2NcKzYhIik= NiQvSSJyRzYiJCIlSVIhIiQvSSJoR0YlJCIlKzYhIiM= hint1 := (V=h*Pi*r^2,V=h*Pi*r^2/3)[n4]; hint2 := (LSA=h*Pi*2*r,LSA=Pi*r*sqrt(h^2+r^2))[n4]; L0kiVkc2IiooSSJoR0YkIiIiSSNQaUclKnByb3RlY3RlZEdGJ0kickdGJCIiIw== L0kkTFNBRzYiLCQqKEkiaEdGJCIiIkkjUGlHJSpwcm90ZWN0ZWRHRihJInJHRiRGKCIiIw== JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 19.5</Font></Text-field> The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question. Considere um recipiente com a forma de um `$t5`. O custo de material para uma superf\303\255cie curva \303\251 $`$costla` por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==, e o custo de material para `$t6` \303\251 $`$costb` por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==. Se o recipiente necessita ter um volume `$V5` LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==, qual deve ser a altura e o raio (em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) do recipiente que minimizar\303\241 o custo? Entre com sua resposta como r = raio, h = altura. A resposta pode entrar com precis\303\243o at\303\251 4 d\303\255gitos significativo. JSFH
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. JSFHans5;
<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`},4)=evalf({`$RESPONSE`},4));
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints. The volume of a `$t5` is `$hint1`. The lateral surface area of a `$t5` is `$hint2`. Using the known volume, treat the height as a function with respect to the radius. Find the value of the radius that will minimize cost.
with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The&MediumSpace;volume&MediumSpace;of&MediumSpace;a&MediumSpace;</mtext><mtext></mtext><mrow><mi>cylinder</mi></mrow><mtext></mtext><mtext>&MediumSpace;is&MediumSpace;</mtext><mtext>`$hint1`</mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The&MediumSpace;lateral&MediumSpace;surface&MediumSpace;area&MediumSpace;of&MediumSpace;a&MediumSpace;</mtext><mtext></mtext><mrow><mi>cylinder</mi></mrow><mtext></mtext><mtext>&MediumSpace;is&MediumSpace;</mtext><mtext>`$hint2`</mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea4`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Using&MediumSpace;the&MediumSpace;known&MediumSpace;volume,&MediumSpace;treat&MediumSpace;the&MediumSpace;height&MediumSpace;as&MediumSpace;a&MediumSpace;function&MediumSpace;with&MediumSpace;respect&MediumSpace;to&MediumSpace;the&MediumSpace;radius.&MediumSpace;&MediumSpace;Find&MediumSpace;the&MediumSpace;value&MediumSpace;of&MediumSpace;the&MediumSpace;radius&MediumSpace;that&MediumSpace;will&MediumSpace;minimize&MediumSpace;cost.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> 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 cost of material can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= = `$c5`. Substitute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = `$h50` to get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= = `$c51`. Differentiate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and solve 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 to find the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that will minimize cost.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea4`, `value`); ANSWER := ans5; if evalb(evalf({ANSWER},4)=evalf({eval(parse(StudentResponse))},4)) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&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><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>3.930</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>11.00</mn></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>The&MediumSpace;cost&MediumSpace;of&MediumSpace;material&MediumSpace;can&MediumSpace;be&MediumSpace;written&MediumSpace;as&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi mathbackground='#000000'>C</mi></mrow></mrow></mstyle><mtext>&MediumSpace;=&MediumSpace;</mtext><mtext></mtext><mrow><mrow><mrow><mn>10</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>&amp;pi;</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>r</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>14</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>&amp;pi;</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext><mtext> </mtext><mtext>Substitute&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>h</mi></mrow></mstyle><mtext>&MediumSpace;=&MediumSpace;</mtext><mtext></mtext><mrow><mfrac><mrow><mn>170</mn></mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mrow><mtext></mtext><mtext>&MediumSpace;to&MediumSpace;get&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi mathbackground='#000000'>C</mi></mrow></mrow></mstyle><mtext>&MediumSpace;=&MediumSpace;</mtext><mtext>`$c51`</mtext><mtext>.</mtext><mtext>&NewLine;</mtext><mtext>Differentiate&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>C</mi></mrow></mstyle><mtext>&MediumSpace;with&MediumSpace;respect&MediumSpace;to&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>r</mi></mrow></mstyle><mtext>,&MediumSpace;and&MediumSpace;solve&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac background='#000000'><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&amp;DifferentialD;</mo><mrow><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&amp;DifferentialD;</mo><mi mathbackground='#000000'>r</mi></mrow></mfrac><mi mathbackground='#000000'>C</mi><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&amp;equals;</mo><mn mathbackground='#000000'>0</mn></mrow></mstyle><mtext>&MediumSpace;to&MediumSpace;find&MediumSpace;the&MediumSpace;value&MediumSpace;of&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>r</mi></mrow></mstyle><mtext>&MediumSpace;that&MediumSpace;will&MediumSpace;minimize&MediumSpace;cost.</mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>3.930</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>11.00</mn></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><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): randomize(): n5 := Generate(choose([1,2])); t7 := (`cylinder`,`right circular cone`)[n5]: t8 := (`bases`,`base`)[n5]: j := 0: h6 := 0: unassign('temp6'); while (j = 0 or (evalf(Im(h6)) <> 0)) do costla2 := Generate(integer(range=1..8)); costb2 := Generate(integer(range=1..10)); while (costb2 = costla2) do costb2 := Generate(integer(range=1..10)) end do; cost := Generate(integer(range=10..30))*10*Pi; if (n5 = 1) then h60 := solve(cost=costla2*h*Pi*2*r+costb2*2*Pi*r^2,h); V6 := eval(h*Pi*r^2,h=h60); else h60 := sqrt(simplify(rhs(isolate(cost=costla2*Pi*r*sqrt(h^2+r^2) + costb2*Pi*r^2,h^2)))); V6 := eval(Pi*h*r^2/3,h=h60); end if; dV6 := simplify(diff(V6,r)); s6 := [fsolve(numer(dV6)=0,r)]; for i from 2 to nops(s6) do if (s6[i] > 0) then j := j+1; temp6[j] := s6[i]; end if; end do; if (j > 1) then temp7 := temp6[1]: temp8 := eval(V6,r=temp7): for k from 2 to j do temp9 := temp8; temp8 := max(temp8,eval(V6,r=temp6[j])); if temp8 <> temp9 then temp7 := temp6[j]; end if; end do; r6 := temp7; else r6 := temp6[1]; end if; h6 := eval(h60,r=r6); end do; ans6 := r=evalf(r6,4),h=evalf(h6,4); IiIi IiIn IiM1 LCRJI1BpRyUqcHJvdGVjdGVkRyIkcSI= LCYqJkkjUGlHJSpwcm90ZWN0ZWRHIiIiSSJyRzYiIiIjISImRiQjIiMmKSIiJw== NyQkIStCM0QkbyIhIiokIitCM0QkbyJGJQ== JCIrdmckM2gmISIq NiQvSSJyRzYiJCIlJG8iISIkL0kiaEdGJSQiJTZjRig= hint3 := (V=h*Pi*r^2,V=h*Pi*r^2/3)[n5]; hint4 := (LSA=h*Pi*2*r,LSA=Pi*r*sqrt(h^2+r^2))[n5]; L0kiVkc2IiooSSJoR0YkIiIiSSNQaUclKnByb3RlY3RlZEdGJ0kickdGJCIiIw== L0kkTFNBRzYiLCQqKEkiaEdGJCIiIkkjUGlHJSpwcm90ZWN0ZWRHRihJInJHRiRGKCIiIw== JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 19.6</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 um recipiente com a forma de um `$t7`. O custo de material para uma superf\303\255cie curva \303\251 $`$costla2` por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==, e o custo de material para `$t8` \303\251 $`$costb2` por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==. Qual deve ser a altura e o raio (em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) do recipiente para ter um volume m\303\241ximo para um custo total de $`$cost`? Entre com sua resposta como r = raio, h = altura. A reposta pode entrar com precis\303\243o at\303\251 4 d\303\255gitos significativo. 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 `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. JSFHans6;
<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field> To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$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({`$ANSWER`},4)=evalf({`$RESPONSE`},4));
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints. The volume of a `$t7` is `$hint3`. The lateral surface area of a `$t5` is `$hint2`. Using the known cost, treat the height as a function with respect to the radius. Find the value of the radius that will maximize the volume.
with(StringTools): with(DocumentTools): response := GetProperty(`TextArea7`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The&MediumSpace;volume&MediumSpace;of&MediumSpace;a&MediumSpace;</mtext><mtext></mtext><mrow><mi>cylinder</mi></mrow><mtext></mtext><mtext>&MediumSpace;is&MediumSpace;</mtext><mtext>`$hint3`</mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea7`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>The&MediumSpace;lateral&MediumSpace;surface&MediumSpace;area&MediumSpace;of&MediumSpace;a&MediumSpace;</mtext><mtext></mtext><mrow><mi>cylinder</mi></mrow><mtext></mtext><mtext>&MediumSpace;is&MediumSpace;</mtext><mtext>`$hint2`</mtext><mtext>.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea7`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Using&MediumSpace;the&MediumSpace;known&MediumSpace;cost,&MediumSpace;treat&MediumSpace;the&MediumSpace;height&MediumSpace;as&MediumSpace;a&MediumSpace;function&MediumSpace;with&MediumSpace;respect&MediumSpace;to&MediumSpace;the&MediumSpace;radius.&MediumSpace;&MediumSpace;Find&MediumSpace;the&MediumSpace;value&MediumSpace;of&MediumSpace;the&MediumSpace;radius&MediumSpace;that&MediumSpace;will&MediumSpace;maximize&MediumSpace;the&MediumSpace;volume.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response. The volume of the container can be written as `$hint3`. Substitute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = `$h60` to get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = `$V6`. Differentiate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and solve 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 to find the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that will maximize the volume.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea7`, `value`); ANSWER := ans6; if evalb(evalf({ANSWER},4)=evalf({eval(parse(StudentResponse))},4)) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&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><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>1.683</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>5.611</mn></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>The&MediumSpace;volume&MediumSpace;of&MediumSpace;the&MediumSpace;container&MediumSpace;can&MediumSpace;be&MediumSpace;written&MediumSpace;as&MediumSpace;</mtext><mtext></mtext><mrow><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>&amp;pi;</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mtext></mtext><mtext>.</mtext><mtext>&NewLine;</mtext><mtext> </mtext><mtext>Substitute&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>h</mi></mrow></mstyle><mtext>&MediumSpace;=&MediumSpace;</mtext><mtext></mtext><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mn>5</mn><mn>6</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>17</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mrow></mrow><mtext></mtext><mtext>&MediumSpace;to&MediumSpace;get&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>V</mi></mrow></mstyle><mtext>&MediumSpace;=&MediumSpace;</mtext><mtext>`$V6`</mtext><mtext>.</mtext><mtext>&NewLine;</mtext><mtext>Differentiate&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>V</mi></mrow></mstyle><mtext>&MediumSpace;with&MediumSpace;respect&MediumSpace;to&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>r</mi></mrow></mstyle><mtext>,&MediumSpace;and&MediumSpace;solve&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac background='#000000'><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&amp;DifferentialD;</mo><mrow><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&amp;DifferentialD;</mo><mi mathbackground='#000000'>r</mi></mrow></mfrac><mi mathbackground='#000000'>V</mi><mo mathbackground='#000000' lspace='0.2777778em' rspace='0.2777778em' background='#000000'>&amp;equals;</mo><mn mathbackground='#000000'>0</mn></mrow></mstyle><mtext>&MediumSpace;to&MediumSpace;find&MediumSpace;the&MediumSpace;value&MediumSpace;of&MediumSpace;</mtext><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathbackground='#000000'>r</mi></mrow></mstyle><mtext>&MediumSpace;that&MediumSpace;will&MediumSpace;maximize&MediumSpace;the&MediumSpace;volume.</mtext><mtext>&NewLine;</mtext><mtext></mtext><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>1.683</mn></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>5.611</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> A solu\303\247\303\243o de um problema de otimiza\303\247\303\243o envolve tipicamente tr\303\252s passos principais: formula\303\247\303\243o do problema an\303\241lise que leva \303\240 solu\303\247\303\243o interpreta\303\247\303\243o da solu\303\247\303\243o O passo da formula\303\247\303\243o pode ser com freq\303\274encia o mais dif\303\255cil. A chave aqui \303\251 identificar uma vari\303\241vel apropriada (somente uma). Se isto \303\251 feito corretamente, ent\303\243o deve ser poss\303\255vel expressar a fun\303\247\303\243o objeto em termos desta vari\303\241vel e identificar um intervalo onde o m\303\241ximo e/ou m\303\255nimo \303\251 requerido. O passo da solu\303\247\303\243o \303\251 geralmente bastante simples. \303\211 importante determinar se o intervalo \303\251 limitado e fechado. O caso em que \303\251 ou aberto em pelo menos um dos limites (do intervalo) ou \303\251 ilimitado requer um pouco mais de esfor\303\247o para verificar que um ponto cr\303\255tico \303\251 o desejado valor extremo.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Qual \303\251 o pr\303\263ximo</Font></Text-field> O cap\303\255tulo Teorema do Valor M\303\251dio (Mean Value Theorem) \303\251 o \303\272ltimo cap\303\255tulo nesta unidade. Este ser\303\241 uma boa conclus\303\243o para nossa discuss\303\243o de derivadas \303\251 ser\303\241 um trampolim para integra\303\247\303\243o (integration) na Unidade 4.
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