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Guia de Estudo de C\303\241lculo \302\251 Maplesoft, a division of Waterloo Maple Inc., 2005 Cap\303\255tulo 18: Desenhar Gr\303\241fico (Graphing)

Os dois cap\303\255tulos anteriores introduziram todas as ferramentas necess\303\241rias para fazer gr\303\241fico (graph) de quase todo tipo de fun\303\247\303\243o (function) que voc\303\252 encontrar\303\241. Enquanto o comando FunctionChart ( visualiza\303\247\303\243o informativa da fun\303\247\303\243o), no pacote Student[Calculus1] (Estudante[C\303\241culo]), pode produzir gr\303\241ficos excelentes, a habilidade de fazer gr\303\241fico de uma fun\303\247\303\243o(function) a m\303\243o \303\251 importante por muitas raz\303\265es, incluindo: algumas fun\303\247\303\265es t\303\252m propriedades sutis que s\303\243o dif\303\255ceis de serem completamente representadas graficamente (Exemplo 18.2) a maior parte dos softwares \303\251 incapaz de fazer gr\303\241fico de fun\303\247\303\265es que envolvam um par\303\242metro (parameter) (Exemplo 18.3) \303\240s vezes \303\251 necess\303\241rio construir um gr\303\241fico quando somente a primeira derivada (derivative) \303\251 conhecida (Exemplo 18.5) um computador e um software apropriado n\303\243o est\303\243o sempre dispon\303\255veis algum conhecimento sobre a curva (curve) \303\251 necess\303\241rio para usar efetivamente o software a cria\303\247\303\243o de um gr\303\241fico de uma fun\303\247\303\243o \303\251 um bom teste da compreens\303\243o de c\303\241lculo diferencial de uma pessoa Uma abordagem geral para fazer gr\303\241fico de uma fun\303\247\303\243o ser\303\241 dada e ent\303\243o demonstrada em uma cole\303\247\303\243o variada de fun\303\247\303\265es. Algumas das fun\303\247\303\265es consideradas nestes exemplos s\303\243o um pouco mais complicadas que as fun\303\247\303\265es t\303\255picas consideradas em exemplos e exerc\303\255cios. Estas fun\303\247\303\265es foram escolhidas para enfatizar as t\303\251cnicas e abordagem gerais e deixar o Maple fazer alguns dos tediosos c\303\241lculos. Em apoio a esta abordagem, esta planilha (worksheet) introduz o comando SignChart (definido na inicializa\303\247\303\243o). A aplica\303\247\303\243o geral deste comando \303\251 SignChart( f, domain, orders ); onde f \303\251 a express\303\243o contendo a fun\303\247\303\243o a ser analisada, domain \303\251 a equa\303\247\303\243o da forma var = left .. right onde var \303\251 a vari\303\241vel (variable) em f e left (esquerda) e right (direita) s\303\243o os limites (endpoints) \303\240 esquerda e \303\240 direita do intervalo (interval) em que o sinal indicativo (sign chart) \303\251 requerido, e orders \303\251 um \303\272nico inteiro ou uma lista de inteiros indicando a ordem da(s) derivada(s) para incluir no sinal indicativo (visualiza\303\247\303\243o informativa do sinal) (use 0 para f, 1 para f ', e 2 para f ''). Cada sinal indicativo \303\251 marcado e usa rosa para indicar intervalos onde a fun\303\247\303\243o (ou derivada) \303\251 positiva (positive) e verde azulado para indicar regi\303\265es onde a fun\303\247\303\243o (ou derivada) \303\251 negativa (negative). Como a \303\252nfase est\303\241 em obter a informa\303\247\303\243o necess\303\241ria para construir o gr\303\241fico de uma fun\303\247\303\243o, o maplet Analisador de Fun\303\247\303\243o (FunctionAnalyzer maplet) \303\251 recomendado acima dos maplets An\303\241lise de Curva (CurveAnalysis) e Gr\303\241fico da Derivada (DerivativePlot). Para exibir uma fun\303\247\303\243o em uma planilha do Maple, \303\251 recomendado o comando FunctionChart (Visualiza\303\247\303\243o Informativa da Fun\303\247\303\243o)no pacote Student[Calculus1] (Estudante[C\303\241lculo]).

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). asymptote concavedown concaveup concavity continuous criticalpoint curve decreasing denominator derivative discontinuity domain endpoint evaluate even explicit extremum factor family finite function minimum graph increasing infinity inflectionpoint intercept interval limit longdivision maximum minimum multiple negative numerator oblique odd one-sided optimization parameter period periodicfunction positive product quadratic range rationalfunction realline real root satisfy set sign singularpoint singularity solution solve stationarypoint subset tables unbounded value variable work zero

restart; with(plots): with(DocumentTools): with(Student[Precalculus]): with(Student[Calculus1]): interface(rtablesize=100):

Inserir o cursor em qualquer lugar no seguinte c\303\263digo Maple e pressionar a chave Enter InInterval := proc(x,S::list(numeric)) if has(x,I) then return false end if; return evalb( evalf(x)>=op(1,S) and evalf(x)<=op(2,S) ) end proc: TestIncrDecr := proc( df, pts ) local c, m, fmt1, fmt2; Digits := 4; fmt1 := "f'(%a)=%a; the function is %a on the subinterval containing %a"; fmt2 := "f'(%a)=%a; the function has a stationary point at x = %a"; for c in evalf(pts) do m := eval( df, x=c ); if m>0 then print( printf(fmt1, c, m, `increasing`, c) ); elif m<0 then print( printf(fmt1, c, m, `decreasing`, c) ); else print( printf(fmt2, c) ); end if; end do: end proc: TestConcavity := proc( d2f, pts ) local a, c, fmt1, fmt2; Digits := 4; fmt1 := "f''(%a)=%a; the function is concave %a on the subinterval containing %a"; fmt2 := "f''(%a)=%a; the function has a possible inflection point at x = %a"; for c in evalf(pts) do a := eval( d2f, x=c ); if a>0 then print( printf(fmt1, c, a, `up`, c) ); elif a<0 then print( printf(fmt1, c, a, `down`, c) ); else print( printf(fmt2, c, a, c) ); end if end do: end proc: SignPlot := proc( f, dom, desc, Y, C ) local char_fn, neg_part, pos_part, P1, P2, x, xhi, xlo; char_fn := proc( test, value ) local val; if nargs=1 then val := 1 else val := value end if; proc(f,y) local v; if nargs=1 then v := val else v := y end if; if type(f,numeric) then return `if`( test(f), v, undefined ) else return 'procname'( 'args' ) end if; end proc end proc; pos_part := char_fn( v->signum(v)= 1 ); neg_part := char_fn( v->signum(v)=-1 ); x, xlo, xhi := lhs(dom), op(evalf(rhs(dom))); P1 := plot( [pos_part(f,Y),neg_part(f,Y)], dom, color=C, thickness=3 ); P2 := plots[textplot]( [(xlo+39*xhi)/40, Y, desc], align=ABOVE ); return plots[display]( [P1, P2] ); end proc: SignChart := proc( f, dom, orders ) local F, L, ORDERS, Y; if type(orders,list) then ORDERS := seq(ord+1,ord=orders) else ORDERS := [orders+1] end if; F := [ f, diff(f,x), diff(f,x,x) ]; L := [ `y`, `y'`, `y''` ]; Y := [ -1/4, -1/2, -3/4 ]; return plots[display]( [seq( SignPlot( F[i], dom, L[i], Y[i], [pink,cyan] ), i=ORDERS )], view=[DEFAULT,-1..0], ytickmarks=0 ) end proc:

O processo de cria\303\247\303\243o de um bom gr\303\241fico de uma fun\303\247\303\243o depende da fun\303\247\303\243o espec\303\255fica a ter seu gr\303\241fico feito. Aseguinte lista de passos pode ser usada para a maior parte das fun\303\247\303\265es que voc\303\252 provavelmente deve encontrar. Alguns dos passos podem ser omitidos para alguns problemas, mas pular um passo, ou uma parte de um passo, precisa ser uma decis\303\243o consciente mais do que simplesmente uma omiss\303\243o.

Dom\303\255nio(Domain) Se um intervalo (interval) \303\251 dado, use-o como dom\303\255nio (domain). Ou ent\303\243o, determine o dom\303\255nio natural da fun\303\247\303\243o. Alcance (Range) Tente determinar o alcance (range) da fun\303\247\303\243o. Em particular, \303\251 o alcance limitado ou ilimitado ? (Algumas vezes isto fica mais claro depois de completar o passo 2.) Periodicidade(Periodicity) Se a fun\303\247\303\243o \303\251 peri\303\263dica, encontre seu per\303\255odo (period). Simetria (Symmetry) \303\211 a fun\303\247\303\243o par (even) (sim\303\251trica em rela\303\247\303\243o ao eixo- y)? \303\255mpar (odd) (sim\303\251trica em rela\303\247\303\243o \303\240 origem)?

Interse\303\247\303\243o com eixo (Intercept) Determine o 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. Zeros Tente localizar todas as solu\303\247\303\265es (solutions) para 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. Determine todos os 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 onde 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. Limites de Intervalo (Endpoints) Junte informa\303\247\303\265es para cada limite finito (finite) . Note: Para um limite de intervalo n\303\243o inclu\303\255do no dom\303\255nio , use um limite lateral (one-sided limit) apropriado. Visual informativo de sinal (Sign Chart) para 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 Determine intervalos do dom\303\255nio onde a fun\303\247\303\243o \303\251 positiva e onde a fun\303\247\303\243o \303\251 negativa

Horizontal Se o dom\303\255nio inclui LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NzM0O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUYsNiVRKCYjODczNDtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, compute o limite correspondente no infinito ( infinity). Se o limite \303\251 finito, ent\303\243o este valor (value) \303\251 uma ass\303\255ntota (asymptote) horizontal. E.g., se 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, ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRIkxGJ0Y0RjdGPkYrRj4= \303\251 uma ass\303\255ntota horizontal. Vertical Para qualquer n\303\272mero finito LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, se qualquer dos dois limites laterais de f em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 infinito, ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= \303\251 uma ass\303\255ntota vertical. Obl\303\255qua (Oblique) Se existem valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em que ou 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 = 0 ou 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 = 0 \303\251 verdadeiro, ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNidGKy1GIzYmLUYsNiVRImFGJ0Y0RjctRjs2LVExJkludmlzaWJsZVRpbWVzO0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0Zpbi1GLDYlUSJ4RidGNEY3Rj4tRjs2LVEiK0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0Ziby1GLDYlUSJiRidGNEY3Rj5GK0Y+RitGPg== \303\251 uma ass\303\255ntota obl\303\255qua (oblique) . Nota: Depois de computar as ass\303\255ntotas, verifique que esta informa\303\247\303\243o \303\251 consistente com o alcance (range) escontrado no Passo 1.

Pontos Cr\303\255ticos (Critical Points) Encontre todos os pontos estacion\303\241rios, pontos singulares e limites de intervalos. Visual informativo de sinal (Sign Chart) para 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 Determine intervalos no dom\303\255nio onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 crescente (increasing) e onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 decrescente (decreasing). Extremo Local (Local Extrema) Use o Teste da Primeira Derivada para classificar todo m\303\241ximo local (local maxima) e m\303\255nimo local. Certifique-se ao registrar ambos o valor extremo e o ponto no dom\303\255nio onde ele \303\251 obtido . Extremo Global (Global Extrema) Identifique o m\303\241ximo global, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e o m\303\255nimo, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. (verifique que os extremos global (extrema) s\303\243o consistentes com o alcance (range) encontrado no Passo1.) Identifique qualquer valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn no dom\303\255nio da fun\303\247\303\243o que satisfa\303\247a (satisfy) 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 ou 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 Nota: Classifica\303\247\303\243o de extremo local pode ser omitida do passo 3 se o Teste da Segunda Derivada for usado no Passo 4.

Visual informativo de sinal (Sign Chart) para 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 Use os poss\303\255veis pontos de inflex\303\243o (inflection points) para determinar intervalos no dom\303\255nio onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 c\303\264ncava para cima concave up e onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 c\303\264ncava para baixo (concave down). Extremo Local (Local Extrema) Use o Teste da Segunda Derivada para classificar todos os m\303\241ximo e m\303\255nimo local. Certifique-se ao registrar ambos o valor extremo e o ponto no dom\303\255nio onde ele \303\251 obtido . Inflection Points Identifique qualquer ponto onde a segunda derivada muda de sinal. Nota: Classifica\303\247\303\243o de extremo local pode ser omitida do Passo 4 se o Teste da Primeira Derivada for usado no Passo 3.

Assinale os pontos especiais do gr\303\241fico identificados nos Passos 1-5. Desenhe todas as ass\303\255ntotas escontradas no Passo 3. Use o visual informativo de sinal (sign charts) para "conectar os pontos." Verifique que o gr\303\241fico \303\251 consistente com toda a an\303\241lise.

Para a fun\303\247\303\243o f1 := x^6-10*x^5-15*x^4+140*x^3+160*x^2-528*x-800: F1 := unapply( f1, x ): f(x) = f1; crie um gr\303\241fico que mostre todo comportamento significante em toda a reta real (real line). A Figura 18.1 mostra as dificuldades que esta fun\303\247\303\243o apresenta para um software gr\303\241fico t\303\255pico. Determinar um janela de gr\303\241fico apropriada requer tentativas e erros, e ainda assim, a escala \303\251 dif\303\255cil de ajeitar para revelar as caracter\303\255sticas que s\303\243o importantes em diferentes escalas dentro de um mesmo gr\303\241fico. plot(f1, x=-3..4); plot(f1, x=-3.5...10.05); plot(f1, x=3..10); Figura 18.1 Exemplo 18.1: O gr\303\241fico centrado n\303\243o pode mostrar caracter\303\255sticas de ambos, pequenos e grandes valores da fun\303\247\303\243o. O lado esquerdo do gr\303\241fico mostra caracter\303\255sticas de pequena escala, enquanto o lado direito do gr\303\241fico mostra caracter\303\255sticas de grande escala . Esta fun\303\247\303\243o apresenta grande dificuldade para um programa (utility)gr\303\241fico t\303\255pico.

Esta fun\303\247\303\243o \303\251 definida para todos os n\303\272meros reais (real numbers). O alcance(range) \303\251 dif\303\255cil de determinar. Note os limites q1 := Limit( f1, x=infinity ): q2 := Limit( f1, x=-infinity ): q1 = value( q1 ); q2 = value( q2 ); O m\303\241xima global \303\251 +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NzM0O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Deve existir o m\303\255nimo global, mas n\303\243o temos ainda informa\303\247\303\243o suficiente nem mesmo para uma estimativa. Esta fun\303\247\303\243o n\303\243o \303\251 nem par nem \303\255mpar e definitivamente n\303\243o \303\251 peri\303\263dica (periodic). N\303\243o existem limites de intervalo para considerar. A interse\303\247\303\243o com eixo-y (intercept) \303\251 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, como vimos de int1 := [ 0, F1(0), `intercept` ]: int1; assim, um ponto no gr\303\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. Esta informa\303\247\303\243o ser\303\241 coletada para relacionar em uma tabela ( table) imediatamente antes de criar o gr\303\241fico. (Note que j\303\241 temos informa\303\247\303\243o significativa sugerindo que o alcance desta fun\303\247\303\243o ser\303\241 bem grande .) N\303\243o \303\251 poss\303\255vel encontrar as ra\303\255zes \303\240 m\303\243o. Como um polin\303\264mio de sexto grau, tem seis zeros mas n\303\243o existe nenhuma f\303\263rmula geral para estas ra\303\255zes. O comando solve (solve) do Maple prov\303\252 aproxima\303\247\303\265es para todos os seis zeros deste polin\303\264mio: q3 := solve( f1=0., x ): q3; As duas ra\303\255zes reais s\303\243o, aproximadamente, q4 := op(remove( has, [q3], I )): evalf[4]( q4 ); O ponto correspondente no gr\303\241fico s\303\243oLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRLyZ1bWludXMwOzIuNTg4RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjU2JFEiMEYnRjhGOEY4RitGOA== eLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRIzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjU2JFEiMEYnRjhGOEY4RitGOA==. A informa\303\247\303\243o necess\303\241ria para a tabela de resumo \303\251 zero1 := seq( [x,0, `zero`], x=q4 ): evalf[4]( zero1 ); Porque a fun\303\247\303\243o \303\251 cont\303\255nua (continuous) e tem somente dois zeros, o sinal da fun\303\247\303\243o pode mudar somente nestes dois pontos. O comando SignChart (sinal indicador ou Visual Informativo de Sinal) (definido na se\303\247\303\243o inicializa\303\247\303\243o) pode ser usado para criar a Figura 18.2, um visual informativo de sinal mostrando onde a fun\303\247\303\243o \303\251 positiva (rosa) e negativa (verde azulado). SignChart( f1, x=-10..15, 0 ); Figura 18.2 Exemplo 18.1: comando SignChart aplicado a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, com rosa indicando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= e azul, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= Observe que, agora sabemos, o m\303\255nimo global deve ocorrer em algum lugar entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEoJm1pbnVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRJjIuNTg4RidGL0Yv e 10.00. N\303\243o existem ass\303\255ntotas.

A primeira derivada \303\251 df1 := diff( f1, x ): `f '`(x) = df1; Os pontos estacion\303\241rios s\303\243o stat1 := op({solve( df1=0., x )}): evalf[4]( stat1 ); onde a raiz repetida LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRKyZ1bWludXMwOzJGJ0Y5Rjk= foi listada somente uma vez. N\303\243o existem limites de intervalo e nem pontos singulares; assim entramos com end1 := NULL: sing1 := NULL: Os pontos cr\303\255ticos (critical points) desta fun\303\247\303\243o s\303\243o crit1 := [ stat1, sing1, end1 ]: evalf[4]( crit1 ); Os pontos cr\303\255ticos dividem o dom\303\255nio em intervalos em que a derivada n\303\243o muda de sinal. (O visual informativo do sinal na Figura 18.3 mostra: a primeira derivada \303\251 positiva nos intervalos rosa e negativa nos intervalos azul esverdeado . Note que o terceiro argumento no comando SignChart (visual informativo de sinal) \303\251 a ordem da derivada.) SignChart( f1, x=-10..15, 1 ); Figura 18.3 Exemplo 18.1: comando SignChart aplicado a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ==, com rosa indicando 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 e azul, 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 Do visual indicativo de sinal (sign chart), e do Teste da Primeira Derivada, fica claro que a fun\303\247\303\243o tem m\303\255nimo local onde o sinal indicador muda de azul esverdeado para rosa , i.e., em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 0.8661 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 8.467, e tem m\303\241ximo local onde o sinal indicador muda de rosa para para azul, i.e., em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 3. Os pontos correspondentes no gr\303\241fico da fun\303\247\303\243o s\303\243o cp1 := seq( [ x, F1(x), `critical point` ], x=crit1 ): evalf[4]( cp1 ); O m\303\255nimo global pode agora ser determinado pela compara\303\247\303\243o dos valores da fun\303\247\303\243o dos m\303\255nimos locais; portanto, o m\303\255nimo global \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEoJm1pbnVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRJjUyNjIwRidGL0Yv e ocorre pr\303\263ximo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJjguNDY3RidGOUY5.

A segunda derivada \303\251 ddf1 := diff( f1, x,x ): `f ''`(x) = ddf1; Os poss\303\255veis pontos de inflex\303\243o s\303\243o possinfl1 := solve( ddf1=0., x ): evalf[4]( possinfl1 ); A concavidade nos cinco intervalos definidos por estes quatro pontos \303\251 mais f\303\241cil de mostrar em termos de um visual inoformativo para a segunda derivada. (Na Figura 18.4, intervalos onde a fun\303\247\303\243o \303\251 c\303\264ncava para cima s\303\243o mostrados em rosa; c\303\264ncava para baixo, em azul.) SignChart( f1, x=-10..15, 2 ); Figura 18.4 Exemplo 18.1: Comando SignChart aplicado a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ==, com rosa indicando que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1LUYsNiVRKSZlYWN1dGU7RidGL0YyRjVGOQ==c\303\264ncava para cima, e azul, c\303\264ncava para baixo Existe uma troca de sinal na segunda derivada em cada zero da segunda derivada. As coordenadas destes quatro pontos de inflex\303\243o s\303\243o ent\303\243o ip1 := seq( [ x, F1(x), `inflection point` ], x=[possinfl1] ): evalf[4]( ip1 );

Os pontos importantes identificados nos Passos 1-5 incluem a interse\303\247\303\243o com eixo, os zeros, os pontos cr\303\255ticos e pontos de inflex\303\243o. Estes s\303\243o colecionados na Tabela 18.1 Onde eles s\303\243o organizados de acordo com os valores crescentes de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. important1 := sort( [ int1, zero1, cp1, ip1 ], (a,b)->a[1]<b[1] ): tab1 := [ [`x`, `y`, `description`], [`-----`, `-----`, `---------------`], op(important1) ]: matrix( evalf[4]( tab1 ) ); Tabela 18.1 Os pontos importantes para desenhar o gr\303\241fico da fun\303\247\303\243o no Exemplo 18.1 Nosso esbo\303\247o da fun\303\247\303\243o come\303\247a por marcar os pontos contidos na Tabela 18.1. A Figura 18.5 \303\251 o resultado. pts1 := [seq( P[1..2], P=important1 )]: Ppts1 := plot( pts1, color=blue, style=point, symbolsize=15 ): display( Ppts1, view=[-10..15,DEFAULT] ); Figura 18.5 Marca\303\247\303\243o dos pontos contidos na Tabela 18.1 O visual informativo de sinal da fun\303\247\303\243o e de suas duas primeiras derivadas, exibido na Figura 18.6, \303\251 usado para "conectar os pontos." (Note como uma lista \303\251 usada no terceiro argumento para marcar sinal indicativo de m\303\272ltiplos (multiple) em um desenho de gr\303\241fico Maple (Maple plot).) SignChart( f1, x=-10..15, [0,1,2] ); Figura 18.6 Exemplo 18.1: Visual informativo de sinal (Sign charts) para 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, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Lembre-se de que o m\303\241ximo global da fun\303\247\303\243o \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJNRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRKCYjODczNDtGJ0Y0RjdGPkYrRj4= e o m\303\255nimo global \303\251 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; isto significa que a fun\303\247\303\243o est\303\241 crescendo muito rapidamente \303\240 medida que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkotRiw2JVEieEYnLyUnaXRhbGljR0Y/L0Y2USdpdGFsaWNGJ0YxRjVGK0Y1 cresce. Isto \303\251 visto na Figura 18.7. Pfn1 := plot(f1, x=-4..10.05, numpoints=500): display( [Ppts1, Pfn1]); Figura 18.7 Exemplo 18.1: Pontos importantes e gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= Nesta escala, \303\251 dif\303\255cil ver claramente extremos locais e pontos de inflex\303\243o . J\303\241 que todos menos tr\303\252s "pontos interessantes" est\303\243o contidos no intervalo 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, faz sentido enfocar esta regi\303\243o da fun\303\247\303\243o. Para selecionar uma janela visual, identifique a maior e a menor coordenada y de coordenadas-y ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-coordinates) que correspondem aos pontos cr\303\255ticos dentro do intervalo limitado, e os limites do intervalo limitado. Um destes limites \303\251 ent\303\243o f(-3) = F1(-3); e o outro, da Tabela 18.1, \303\251 cerca de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEoJm1pbnVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRJTExMDBGJ0YvRi8=. Nesta vista da janela, os pontos importantes vistos na Figura 18.8. display(Ppts1, view=[-3..3,-1100..400]); Figura 18.8 Exemplo 18.1: Pontos importantes para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= Figura 18.9 cont\303\251m o visual informativo de sinal correspondente \303\240 Figura 18.8. SignChart( f1, x=-3..3.1, [0,1,2] ); Figura 18.9 Exemplo 18.1: visual informativo de sinal (Sign charts) para 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Com esta informa\303\247\303\243o \303\251 uma tarefa simples obter a pequena escala e gr\303\241fico da fun\303\247\303\243o . O resultado \303\251 a Figura 18.10. display([Ppts1,Pfn1], view=[-3..3,-1100..400]); Figura 18.10 Importantes pontos em pequena escala e gr\303\241fico das fun\303\247\303\243o no Exemplo 18.1 Para concluir, \303\251 interessante comparar nossos gr\303\241ficos com aqueles produzidos pelo comando FunctionChart do Maple. A Figura 18.11 aplica o comando FunctionChart para os intervalos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRKyZ1bWludXMwOzRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSUxMC4xRidGOEY4RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGK0Y4 eLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRKyZ1bWludXMwOzNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSI0RidGOEY4RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGK0Y4. FunctionChart(f1, -4..10.1, pointoptions = [symbolsize=20],slope=[thickness(2,2), color(red,blue)], concavity=[filled(pink,cyan)], title="" ); FunctionChart(f1, -3..4, pointoptions = [symbolsize=20], slope=[thickness(2,2), color(red,blue)], concavity=[filled(pink,cyan)], title="" ); Figura 18.11 Exemplo 18.1: Comando FunctionChart aplicado aLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRKyZ1bWludXMwOzRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSUxMC4xRidGOEY4RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGK0Y4 \303\240 esquerda, eLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRKyZ1bWludXMwOzNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSI0RidGOEY4RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGK0Y4 \303\240 direita

QUESTAO 18.1 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

Para a fun\303\247\303\243o f2 := (-x^3+3*x^2-5*x+6)/(x^2-4*x+3): F2 := unapply( f2, x ): f(x) = f2; criar um gr\303\241fico de 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 que mostre todo comportamento significante em toda reta real. Figura 18.12, contendo um gr\303\241fico Maple desta fun\303\247\303\243o, gerado depois de v\303\241rias experi\303\252ncias com o tamanho da janela gr\303\241fica. A an\303\241lise a seguir revelar\303\241 ambos, os subt\303\255tulos e a validade deste gr\303\241fico. plot(f2, x=-5..10, y=-15..10, discont=true); Figura 18.12 Exemplo 18.2: Gr\303\241fico Maple de 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

O denominador (denominator) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tem zeros em q1 := solve( denom(f2)=0, x ): q1; mas, exceto por estes pontos, a fun\303\247\303\243o \303\251 definida e cont\303\255nua para todos os n\303\272meros reais. O alcance \303\251LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUTEmdW1pbnVzMDsmaW5maW47RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSgmaW5maW47RidGN0Y6RkFGQUYrRkE= por causa dos limites q2 := Limit( f2, x=infinity ): q3 := Limit( f2, x=-infinity ): q2 = value( q2 ); q3 = value( q3 ); Todo extremo interessante ser\303\241 local. Esta fun\303\247\303\243o n\303\243o \303\251 nem par nem \303\255mpar e definitivamente n\303\243o \303\251 peri\303\263dica. N\303\243o existem limites de intervalo a considerar. A interse\303\247\303\243o \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GPkYrRj4=; assim, um ponto sobre o gr\303\241fico \303\251 int2 := [ 0, F2(0), `intercept` ]: int2; Os zeros do numerador (numerator) de f n\303\243o s\303\243o \303\263bvios, mas Maple informa q4 := solve( numer(f2)=0, x ); assim o zero real de f \303\251 q5 := remove( has, [q4], I ): op(q5); O ponto correspondente no gr\303\241fico \303\251 zero2 := seq( [x,0, `zero`], x=q5 ): evalf[4]( zero2 ); O visual informativo de sinal (Sign chart) para a fun\303\247\303\243o no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRKyZ1bWludXMwOzNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSI0RidGOEY4RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGK0Y4 aparece na Figura 18.13. SignChart( f2, x=-3..4, 0 ); Figura 18.13 Exemplo 18.2: O visual informativo de sinal (Sign chart) para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRKyZ1bWludXMwOzNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSI0RidGOEY4RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGK0Y4 Rosa \342\207\222positivo, Azul \342\207\222negativo Note como a fun\303\247\303\243o muda de sinal nos buracos do dom\303\255nio (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRlA2JFEiM0YnRjlGOQ==) e no zero do numerador (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=). N\303\243o existem ass\303\255ntotas horizontais; assim entramos com horiz2 := NULL: Ass\303\255ntotas verticais podem possivelmente existir em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4=. Para testar ambos os pontos, compute os quatro limites q6 := Limit( f2, x=1, right ): q7 := Limit( f2, x=1, left ): q8 := Limit( f2, x=3, right ): q9 := Limit( f2, x=3, left ): q6 = value( q6 ); q7 = value( q7 ); q8 = value( q8 ); q9 = value( q9 ); Existem duas ass\303\255ntotas verticais, a saber, vert2 := x=1, x=3: vert2; A procura por uma ass\303\255ntota obliqua \303\251 um tanto complicada, mas leva diretamente a longa divis\303\243o.Feita no Maple, d\303\241 q10 := quo(numer(f2),denom(f2),x, 'q11'): f(x) = f2; `` = q10 + q11/denom(f2); Isto sugere que 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 \303\251 uma candidata a um ass\303\255ntota obliqua f. Para confirmar isto, compute os limites q12 := Limit( f2-q10, x=infinity ): q13 := Limit( f2-q10, x=-infinity ): q12 = value( q12 ); q13 = value( q13 ); A ass\303\255ntota obliqua para esta fun\303\247\303\243o \303\251, portanto, obliq2 := y = q10: obliq2; O conjunto (set) completo de ass\303\255ntotas para este problema \303\251 asymp2 := horiz2, vert2, obliq2: asymp2;

A primeira derivada \303\251 df2 := simplify(diff( f2, x )): `f '`(x) = df2; Os pontos estacion\303\241rios s\303\243o q14 := solve( df2=0., x ): stat2 := op(remove( has, [q14], I )): evalf[4]( stat2 ); Os zeros reais da primeira derivada. Os buracos no dom\303\255nio n\303\243o s\303\243o pontos singulares nem limites de intervalos, assim entre com sing2 := NULL: end2 := NULL: Os pontos cr\303\255ticos desta fun\303\247\303\243o s\303\243o crit2 := [ stat2, sing2, end2 ]: evalf[4]( crit2 ); Os pontos cr\303\255ticos dividem o dom\303\255nio em intervalos em que a derivada n\303\243o muda de sinal. A Figura 18.14 cont\303\251m um visual informativo de sinal para a primeira derivada, que \303\251 positivo nos intervalos rosa e negativo nos intervalos azul esverdeado. SignChart( f2, x=-10..10, 1 ); Figura 18.14 Exemplo 18.2: visual informativo de sinal (Sign chart) para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== rosa \342\207\222positivo, azul \342\207\222negativo Deste visual informativo de sinal (sign chart) e do Teste da Primeira Derivada fica claro que a fun\303\247\303\243o tem um m\303\255nimo local onde o visual informativo de sinal muda de azul para rosa, i.e., em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRMCZ1bWludXMwOzAuNTMxOEYnRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJ0Y5e tem um m\303\241ximo local onde visual informativo de sinal muda de rosa para azul, i.e., em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJzUuMjE3LkYnRjlGOQ== Os pontos correspondentes no gr\303\241fico da fun\303\247\303\243o s\303\243o cp2 := seq( [ x, F2(x), `critical point` ], x=crit2 ): evalf[4]( cp2 ); O m\303\255nimo global pode agora ser determinado pela compara\303\247\303\243o dos valores da fun\303\247\303\243o para m\303\255nimo local; portanto, o m\303\255nimo global \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEoJm1pbnVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRJjUyNjIwRidGL0Yv e ocorre pr\303\263ximo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEmOC40NjdGJ0Y+Rj5GK0Y+.

A segunda derivada \303\251 ddf2 := simplify(diff( f2, x,x )): `f ''`(x) = ddf2; Os poss\303\255veis pontos de inflex\303\243o s\303\243o zeros reais da segunda derivada. O Maple encontra q15 := solve( ddf2=0., x ): possinfl2 := op(remove( has, [q15], I )): evalf[4]( possinfl2 ); A concavidade nos quatro intervalos determinados pelos tr\303\252s pontos que correspondem a um zero real da segunda derivada e a duas ass\303\255ntotas verticais \303\251 mais f\303\241cil de mostrar em termos do visual informativo de sinal (sign chart) para a segunda derivada, dado na Figura 18.15. (Intervalos onde a fun\303\247\303\243o \303\251 c\303\264ncava para cima s\303\243o mostrados em rosa; c\303\264ncava para baixo, em azul esverdeado.) SignChart( f2, x=-10..10, 2 ); Figura 18.15 Exemplo 18.2:Visual informativo de sinal (Sign chart) para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Rosa \342\207\222positivo, Azul \342\207\222negativo Existe uma mudan\303\247a de sinal da segunda derivada no zero -- e em cada buraco no dom\303\255nio. As coordenadas no ponto de inflex\303\243o no dom\303\255nio \303\251 ip2 := seq( [ x, F2(x), `inflection point` ], x=[possinfl2] ): evalf[4]( ip2 );

Os pontos importantes identificados nos Passos 1-5 incluem a intese\303\247\303\243o com eixo-y, os zeros, os pontos cr\303\255ticos e o ponto de inflex\303\243o. Estes est\303\243o colecionados na Tabela 18.2, onde est\303\243o organizados de acordo com os valores crecentes de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 important2 := sort( [ int2, zero2, cp2, ip2 ], (a,b)->a[1]<b[1] ): tab2 := [ [`x`, `y`, `description`], [`-----`, `-----`,`---------------`], op(important2) ]: Matrix( evalf[4]( tab2 ) ); Tabela 18.2 Os pontos importantes para fazer o gr\303\241fico da fun\303\247\303\243o no Exemplo 18.2 nosso esbo\303\247o da fun\303\247\303\243o come\303\247a pela marca\303\247\303\243o destes pontos na Figura 18.16 pts2 := [seq( P[1..2], P=important2 )]: Ppts2 := plot( pts2, color=red, style=point, symbolsize=15 ): display( Ppts2, view=[-1..6,DEFAULT]); Figura 18.16 Exemplo 18.2: Pontos importantes para fazer o gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ent\303\243o continua na Figura 18.17, um visual informativo de sinal para a fun\303\247\303\243o e suas duas primeiras derivadas SignChart( f2, x=-10..10, [0,1,2] ); Figura 18.17 Exemplo 18.2: Visual informativo de sinal (Sign chart) para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkxGUEYrRlAtRjY2LVEoJnByaW1lO0YnRjlGO0ZTRkBGQkZERkZGSC9GS1EsMC4xMTExMTExZW1GJ0ZURjk= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Rosa \342\207\222positivo, azul \342\207\222negativo Olhando para estes pontos e para a classifica\303\247\303\243o de cada ponto, deve parecer imposs\303\255vel "conect\303\241-los" seguindo o que sabemos sobre o sinal da fun\303\247\303\243o e suas duas primeiras derivadas. Mas, lembre-se de que existem dois pontos onde a fun\303\247\303\243o \303\251 decont\303\255nua (discontinuous). Quando as ass\303\255ntotas s\303\243o adicionadas, o quadro se torna um pouco mais claro, como visto na figura Figura 18.18. Pasym2 := implicitplot({asymp2}, x=-10..15, y=-20..10, color=sienna): display( [Ppts2,Pasym2]); Figura 18.18 Exemplo 18.2: Ass\303\255ntotas e pontos importantes para fazer gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= O comportamento da fun\303\247\303\243o em cada ass\303\255ntota, combinada com extremo local e ponto de inflex\303\243o fornecem informa\303\247\303\243o suficiente para completar o gr\303\241fico dado na Figura 18.19. Pfn2 := plot( f2, x=-10..15, y=-20..10, discont=true, numpoints=250, color=green ): display( [Ppts2, Pasym2, Pfn2]); Figura 18.19 Exemplo 18.2: Ass\303\255ntotas, pontos importantes e gr\303\241fico para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= Para concluir, \303\251 interessante comparar nosso gr\303\241fico com o da Figura 18.20, produzido pelo comando (FunctionChart) do Maple no intervaloLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRLCZ1bWludXMwOzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjU2JFEjMTVGJ0Y4RjhGOC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0YrRjg=. FunctionChart( f2, -10..15, view=[DEFAULT,-20..10], pointoptions=[symbolsize=20], slope=[thickness(2,2), color(red,blue)], concavity=[filled(pink,cyan)],title=""); Figura 18.20 Exemplo 18.2: O comando FunctionChart aplicado em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRLCZ1bWludXMwOzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjU2JFEjMTVGJ0Y4RjhGOC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0YrRjg=

QUESTAO 18.2 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

Determine o valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para o qual os gr\303\241ficos da fun\303\247\303\243o f3 := 10/(x^2+6*x+c): F3 := unapply( f3, x ): f(x) = f3; experimenta uma significativa mudan\303\247a na apar\303\252ncia. Prepara um desenho contendo gr\303\241ficos representativos de cada tipo de comportamento que podem ser induzidos pela varia\303\247\303\243o em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Um modo de obter percep\303\247\303\243o imediata do comportamento de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 dar in\303\255cio ao interativo Plot Builder (construtor de Gr\303\241fico), um Assistant (assistente) que prov\303\252 uma interface para as capacidades do Maple's plotting . Os passos necess\303\241rios para criar um gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= controlado por um cursor (slider) est\303\243o resumidos nas Figuras 18.21(a) - (c). Remova o caracter comment character (#) e execute o comando para dar in\303\255cio ao Plot Builder. # plots[interactive](f3); 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Figura 18.21(a) Base (Settings) para o Plot Builder (Construtor de Gr\303\241fico) 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Figura 18.21(b) Restringir o Alcance (Restrict the Range) via o Painel de Op\303\247\303\265es(Options panel) 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R<:A:lKPqjiDUgsm:]dwqtQmA<FdJ_yY:LBB:Bygu]kYLJv[AUs;eTui_Zj]I<Y@bO:lYA:y=B:;xtjO:Z:LD::_?Jj;tIvNjrGA:y^v]]<:P;`:B:ZY:>XjbeF:<XWrAjI;lDZr=vy::>Z;LxUQAo:m@HG=ZbZv]Z]_;B::sAuPxu:Hhl;bI:I<:V?::x=JjQAR_:ZnK?tm_;ZS;BvXGvy::r:JHJSka;@?`I>RN[DM<;;:siWY>Bxj]gV:>XvKrm:g<Jxu:rY:;BjQAh<:>JL;:bf:<HJy[a<@is:ra^YVN:jDj>:;:v=Zj=@W_;Jje=y:xCJP;>YZyA:Z:FJlYutb@RpjcS:>?^Y>VfF:;:ByRqlYAjkOH:w;rK:Ze::jY:<xt:e>:^HdJYhF:FE:[qe[yA::A:l;DExFjbjv[:]DNSLBB:Bygu]:kAXf_=Zj]I<Y@bO:lYA:y=B:;xtjO>g?^dN_b>gbRfd?O[AFabqaoXdKwYa<y<uk=mYL:vYPUHdulQtkio\\UxZ\\jE<M^Ew^hqpPU[\\NPuQpMRtIOgMRFuk`PNDXoZXlyaG::6:\"\{\}LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlIVwlRiotJS5CT1VORFNfSEVJR0hURzYjJCIlU2FGKi0lKUNISUxEUkVORzYi Figura 18.21(c) O par\303\242metro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 controlado pelo cursor (slider) A chave para analisar este problema \303\251 o denominador. Completando o quadrado nesta quadr\303\241tica (quadratic) permite o denominador ser reescrito como q1 := denom( f3 ): q1 = CompleteSquare( q1, x ); Existem tr\303\252s casos a considerar, Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 9, ent\303\243o o denominador \303\251 estritamente positivo para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, i.e., o dom\303\255nio \303\251LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUTEmdW1pbnVzMDsmaW5maW47RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSgmaW5maW47RidGN0Y6RkFGQUYrRkE=. Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 9, ent\303\243o o denominador \303\251 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, i.e., o dom\303\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 Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < 9, ent\303\243o o denominador \303\251 zero para 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 , i.e., o dom\303\255nio \303\251 a uni\303\243o de tr\303\252s conjuntos diferentes. Com base nesta an\303\241lise, parece que o valor cr\303\255tico do par\303\242metro \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 9.

Os diferentes dom\303\255nios para os tr\303\252s casos j\303\241 foram notados. Eles s\303\243o para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 9, o dom\303\255nio \303\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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 9, o dom\303\255nio \303\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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < 9, o dom\303\255nio \303\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 Para conferir as ass\303\255totas horizontais, considere os limites q1 := Limit( f3, x=infinity ): q2 := Limit( f3, x=-infinity ): q1 = value( q1 ); q2 = value( q2 ); Logo, todos os tr\303\252s casos t\303\252m ass\303\255ntota horizontal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=. A procura por ass\303\255ntotas verticais produz resultados diferentes para diferentes valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 9, n\303\243o existem ass\303\255ntotas verticais para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 9, existe uma ass\303\255ntota vertical em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRKyZ1bWludXMwOzNGJ0Y5Rjk= para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < 9, existem duas ass\303\255ntotas verticais em 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 Para nenhum valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= a fun\303\247\303\243o \303\251 par ou \303\255mpar ou peri\303\263dica. Quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 0 est\303\241 no dom\303\255nio (e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=), a interse\303\247\303\243o \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEjMTBGJ0Y+Rj4tRiM2JC1GLDYlUSJjRidGNEY3Rj4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmFvLyUpYmV2ZWxsZWRHRkJGK0Y+RitGPg==.

A primeira derivada \303\251 df3 := simplify(diff( f3, x )): `f '`(x) = df3; \303\251 definida no mesmo dom\303\255nio que o da fun\303\247\303\243o original (sem considerar o valor de c). Como resultado temos que n\303\243o existem pontos singulares O zero do numerador da primeira derivada \303\251 stat3 := solve( numer(df3)=0, x ): evalf[4]( stat3 ); Note que quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiOUYnRj5GPkYrRj4=, existe um \303\272nico ponto cr\303\255tico em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTy1JI21uR0YkNiRRKyZ1bWludXMwOzNGJ0Y8RitGPA== para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 9, 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 n\303\243o est\303\241 no dom\303\255nio da fun\303\247\303\243o e assim n\303\243o existem pontos cr\303\255ticos. Um olhar mais pr\303\263ximo na primeira derivada indica que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSRmfidGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUTAmQXBwbHlGdW5jdGlvbjtGJy8lJWJvbGRHUSZmYWxzZUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR0ZALyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtRiM2Ji1GOzYtUSIoRidGQS9GREY2RkUvRkhGNkZJRktGTUZPL0ZSUSwwLjE2NjY2NjdlbUYnL0ZVRmhuLUYjNiQtRiw2JVEieEYnRjRGN0ZBLUY7Ni1RIilGJ0ZBRmVuRkVGZm5GSUZLRk1GT0Znbi9GVVEsMC4xMTExMTExZW1GJ0ZBRitGQUYrRkE= > 0 para todo 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 (e no dom\303\255nio de I) e 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 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiM0YnRjVGNUYrRjU= (e no dom\303\255nio de I). Em particular, para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiOUYnRj5GPkYrRj4=, a fun\303\247\303\243o \303\251 crescente \303\240 esquerda de 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, decrescente \303\240 direita de 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 e 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 \303\251 um m\303\241ximo local.

A segunda derivada \303\251 ddf3 := simplify(diff( f3, x,x )): `f ''`(x) = ddf3; Os poss\303\255veis pontos de inflex\303\243o s\303\243o os zeros reais da segunda derivada, a saber, q3 := solve( ddf3=0, x ): q4 := [seq( x=z, z=[q3] )]: q4; Uma vez mais, isto depende do par\303\242metro da seguinte maneira. se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 9, ambos os zeros s\303\243o reais e o dom\303\255nio de f \303\251 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 se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 9, existe no m\303\241ximo um ponto de inflex\303\243o, a saber, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRKyZ1bWludXMwOzNGJ0Y5Rjk= 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, mas relembre que 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 n\303\243o est\303\241 no dom\303\255nio para este valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < 9, n\303\243o existem solu\303\247\303\265es reais para 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 = 0 mas a concavidade pode mudar nas duas ass\303\255ntotas verticais. Assim, os dois pontos de inflex\303\243o s\303\243o os seguintes: 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 Qundo f '' \303\251 avaliado (evaluated) em cada subintervalo identificado acima, encontra-se que para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= > 9, a fun\303\247\303\243o \303\251 c\303\264ncava para cima, exceto entre os dois pontos de inflex\303\243o onde a fun\303\247\303\243o \303\251 c\303\264ncava para baixo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 9, a fun\303\247\303\243o \303\251 c\303\264ncava para cima \303\240 esquerda e \303\240 direita de 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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= < 9, a fun\303\247\303\243o \303\251 c\303\264ncava para cima nos dois intervalos abertos ilimitados (unbounded) e c\303\264ncava para baixo no intervalo entre as duas singularidades (singularities)

Passos 1-5 revelaram um n\303\272mero de diferen\303\247as entre os gr\303\241ficos para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYjNigtSSNtaUdGJDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRistRkk2JVEkYD5gRidGTEZPRistSSNtbkdGJDYkUSI5RidGL0YvLUZJNiNRIUYnRi8=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiOUYnRj5GPkYrRj4=, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiOUYnRj5GPkYrRj4=. As similaridades e diferen\303\247as est\303\243o resumidas na Tabela 18.3. Categoria(Category) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiY0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSI+RicvRjVRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYmUSI5RidGLy9GNUY5RjdGPg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiY0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSI9RicvRjVRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYmUSI5RidGLy9GNUY5RjdGPg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiY0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSI8RicvRjVRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYmUSI5RidGLy9GNUY5RjdGPg== dom\303\255nio(domain) 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 horizontal asymptote(s) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= vertical asymptote(s) none LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRKyZ1bWludXMwOzNGJ0Y5Rjk= 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 interse\303\247\303\243o(intercept) 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 ponto cr\303\255tico (critical point) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFErJnVtaW51czA7M0YnRkFGQUZBLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRitGQQ== none none extremo local (local extrema) maximum none maximum ponto(s) de inflex\303\243o(inflection point(s)) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiW0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EldHJ1ZUYnLyUqc2VwYXJhdG9yR1EmZmFsc2VGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZELUYjNigtSSNtaUdGJDYjUSFGJy1GIzYnLUZKNiVRInhGJy8lJ2l0YWxpY0dGNC9GMFEnaXRhbGljRictRiw2LVEiPUYnRi8vRjNGN0Y1L0Y5RjdGOkY8Rj5GQC9GQ1EsMC4yNzc3Nzc4ZW1GJy9GRkZmbi1GIzYoLUYsNi1RKiZ1bWludXMwO0YnRi9GWUY1RlpGOkY8Rj5GQC9GQ1EmMC4wZW1GJy9GRkZeby1JI21uR0YkNiRRIjNGJ0YvLUYsNi1RIitGJ0YvRllGNUZaRjpGPEY+RkAvRkNRLDAuMjIyMjIyMmVtRicvRkZGaG8tRiM2KEZJLUkmbWZyYWNHRiQ2KC1GYW82JFEiMUYnRi9GYG8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmdwLyUpYmV2ZWxsZWRHRjctRiw2LVExJkludmlzaWJsZVRpbWVzO0YnRi9GWUY1RlpGOkY8Rj5GQEZdb0Zfby1GIzYmRkktSSZtc3FydEdGJDYjLUYjNihGam4tRmFvNiRRIzI3RidGL0Zkby1GIzYmRmBvRlxxLUZKNiVRImNGJ0ZSRlRGL0ZJRi9GSUYvRklGL0ZJRi9GSUYvLUYsNi1RIixGJ0YvRlkvRjZGNEZaRjpGPEY+RkBGXW8vRkZRLDAuMzMzMzMzM2VtRictRiM2KUZJLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRltzLyUqbGluZWJyZWFrR1EobmV3bGluZUYnRk9GVi1GIzYoRmpuRmBvLUYsNi1RKCZtaW51cztGJ0YvRllGNUZaRjpGPEY+RkBGZ29GaW9Gam9GSUYvRklGL0ZJRi8tRiw2LVEiXUYnRi9GMkY1RjhGOkY8Rj5GQEZCL0ZGUSwwLjExMTExMTFlbUYnRi8= none 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 Tabela 18.3 Exemplo 18.3: Propriedades de 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 dependente de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== A anima\303\247\303\243o na Figura 18.22 mostra como o gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= varia com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Clist := [2*($0..4),8.5,9,9.5]: COLORS := red,blue,green,sienna,pink,cyan: COLORS := [COLORS,COLORS,COLORS]: p1 := seq( plot( eval(f3,c=Clist[i]), x=-10..6, discont=true, color=COLORS[i], title=sprintf("c = %a",Clist[i]) ), i=1..nops(Clist) ): display( [p1], view=[DEFAULT,-40..30], insequence=true ); Figura 18.22 Exemplo 18.3: Varia\303\247\303\243o nos gr\303\241ficos de 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 conforme LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= varia Todos os gr\303\241ficos na Figura 18.22 aparecem simultaneamente na Figura 18.23. display([p1], view=[DEFAULT,-40..30],title=""); Figura 18.23 Exibi\303\247\303\243o simult\303\242nea de todos os gr\303\241ficos da Figura 18.22 Os valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= s\303\243o 0, 2, 4, 6, 8, 8.5, 9, 9.5 Figura 18.24 prov\303\252 um anima\303\247\303\243o muito informativa dos gr\303\241ficos FunctionChart ( Visual Informativo da Fun\303\247\303\243o). p2 := [seq( FunctionChart( factor(f3), -10..6, view=[DEFAULT,-40..30], pointoptions=[symbolsize=20], title="", slope=[thickness(2,2), color(red,blue)], concavity=[filled(pink,cyan)] ), c=Clist )]: display( p2, insequence=true ); Figura 18.24 Exemplo 18.3: Anima\303\247\303\243o dos gr\303\241ficos FunctionChart em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21uR0YkNiRRLCZ1bWludXMwOzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjU2JFEiNkYnRjhGOEY4LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRitGOA==

Criar um gr\303\241fico da fun\303\247\303\243o f4 := sin(4*x+1/2)*cos(x/2): F4 := unapply( f4, x ): f(x) = f4; mostrando todo comportamento significativo. Figura 18.25 cont\303\251m Um gr\303\241fico gerado pelo Maple da fun\303\247\303\243o (em vermelho), e os "envelopes" 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 plot([f4,cos(x/2), -cos(x/2)], x=0..4*Pi, numpoints=1000, color=[red,black,green], tickmarks=[3,2]); Figura 18.25 Examplo 18.4: Gr\303\241fico da fun\303\247\303\243o (em vermelho) e os "envelopes" 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

Esta fun\303\247\303\243o \303\251 definida para todos os n\303\272meros reais. Como cada termo no produto (product) tem o 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, o produto deles ter\303\241 valores em 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. Se o alcance (range) \303\251 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 ou um suconjunto (subset) de 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 isto n\303\243o \303\251 importante neste momento. Desde que 'f'(-x)-'f'(x) = simplify( F4(-x)-F4(x) ); 'f'(-x)+'f'(x) = simplify( F4(-x)+F4(x) ); esta fun\303\247\303\243o n\303\243o \303\251 nem par nem \303\255mpar. 'f'(x+4*Pi)-'f'(x) = simplify( F4(x+4*Pi)-F4(x) ); mas \303\251 per\303\255odica com per\303\255odo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU=. Como a fun\303\247\303\243o \303\251 peri\303\263dica, \303\251 suficiente esbo\303\247ar seu gr\303\241fico em um per\303\255odo. Se escolhemos trabalhar no intervalo [ 0, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= ], ent\303\243o os limites do intervalo s\303\243o end4 := 0, 4*Pi: end4; A interse\303\247\303\243o (com eixo) \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=, assim um ponto no gr\303\241fico \303\251 int4 := [ 0, F4(0), `intercept` ]: int4; A fun\303\247\303\243o ser\303\241 zero sempre que qualquer um dos fatores (factor) for zero. A solu\303\247\303\243o para 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 satisfaz q1 := 4*x + 1/2 = i * Pi: q1; isto \303\251, q2 := isolate( q1, x ): q2; para um inteiro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Os zeros em 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 s\303\243o q3 := seq( rhs(q2), i=1..16 ): q3; Da mesma forma, os zeros de 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 s\303\243o q4 := x/2 = Pi/2 + i*Pi: q4; onte LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 um inteiro. Os zeros em 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 s\303\243o q5 := seq( solve(q4,x), i=0..1 ): q5; Os primeiros pontos no gr\303\241fico s\303\243o os 18 zeros do produto em 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. zero4 := seq( [x,0, `zero`], x=[q3,q5] ): evalf[4]( zero4 ); o visual informativo de sinal para a fun\303\247\303\243o em 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 aparece na Figura 18.26 SignChart( f4, x=0..4*Pi, 0 ); Figura 18.26 Exemplo 18.4: Visual informativo de sinal para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= Rosa \342\207\222positivo, azul \342\207\222negativo Note que alguns dos zeros adjacentes est\303\243o t\303\243o pr\303\263ximos que n\303\243o s\303\243o vis\303\255veis neste visual informativo de sinal. A continuidade da fun\303\247\303\243o em 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 significa que esta n\303\243o pode ter ass\303\255ntotas verticais. Da mesma forma , como o dom\303\255nio \303\251 limitado, n\303\243o existem ass\303\255ntotas horizontais e obliquas.

A primeira derivada \303\251 df4 := simplify(diff( f4, x )): `f '`(x) = df4; Os pontos estacion\303\241rios s\303\243o os zeros da primeira derivada, a saber, q6 := solve( df4=0., x ): evalf[4]( q6 ); Note que estes zeros est\303\243o no 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 Pela periodicidade, os zeros em 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 podem ser obtidos adicionando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= a cada um dos zeros acima. Portanto, temos q7 := seq(evalf(z+4*Pi),z=[q6]): stat4 := op(select( InInterval, [q6,q7], [0,evalf(4*Pi)] )): evalf[4]( stat4 ); N\303\243o existem pontos singulares e j\303\241 identificamos os limites do intervalo. Sendo assim, entramos com sing4 := NULL: end4; Os pontos cr\303\255ticos da fun\303\247\303\243o s\303\243o crit4 := [ stat4, sing4, end4 ]: evalf[4]( crit4 ); Os pontos cr\303\255ticos dividem o dom\303\255nio em intervalos em que a derivada n\303\243o muda de sinal . Figura 18.27 prov\303\252 um visual informativo de sinal para a primeira derivada, que \303\251 positivo nos intervalos rosa e negativo nos intervalos azul esverdeado. SignChart( f4, x=0..4*Pi, 1 ); Figura 18.27 Exemplo 18.4: Visual informativo de sinal para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Rosa \342\207\222positivo, Azul \342\207\222negativo Do visual informativo de sinal na Figura 18.26 e do Teste da Primeira Derivada, est\303\241 claro que a fun\303\247\303\243o tem um m\303\255nimo local em cada um dos nove pontos onde o visual de sinal muda de azul para rosa e tem um m\303\241ximo local em cada um dos nove pontos onde o visual de sinal muda de rosa para azul. (Somente nos dois limites n\303\243o h\303\241 extremo local.) Os pontos correspondentes no gr\303\241fico da fun\303\247\303\243o s\303\243o cp4 := seq( [ x, F4(x), `critical point` ], x=crit4 ): evalf[4]( cp4 ); Estes pontos mostram que o alcance (range) \303\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

A segunda derivada \303\251 ddf4 := simplify(diff( f4, x,x )): `f ''`(x) = ddf4; Os poss\303\255veis pontos de inflex\303\243o s\303\243o os zeros reais da segunda derivada, a saber, q8 := solve( ddf4=0., x ): evalf[4]( q8 ); Deslocando estas solu\303\247\303\265es de 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 seguinte lista dos 18 poss\303\255veis pontos de inflex\303\243o. q9 := seq(evalf(z+4*Pi),z=[q8]): possinfl4 := op(select( InInterval, [q8,q9], [0,evalf(4*Pi)] )): evalf[4]( possinfl4 ); A concavidade nos intervalos, definida por estes 18 pontos, \303\251 mais f\303\241cil de mostrar em termos do visual informativo de sinal (sign chart) para a segunda derivada, encontrado na Figura 18.28. Os intervalos onde a fun\303\247\303\243o \303\251 c\303\264ncava para cima s\303\243o mostrados em rosa; e onde \303\251 c\303\264ncava para baixo, em azul. SignChart( f4, x=0..4*Pi, 2 ); Figura 18.28 Exemplo 18.4: Visual informativo de sinal para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Rosa \342\207\222positivo, Azul \342\207\222negativo Existe uma troca de sinal na segunda derivada nos pontos onde 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. As coordenadas dos pontos de inflex\303\243o no dom\303\255nio s\303\243o ip4 := seq( [ x, F4(x), `inflection point` ], x=[possinfl4] ): evalf[4]( ip4 );

Os pontos importantes identificados nos Passos 1-5 incluem a interse\303\247\303\243o (com eixo), os zeros, os pontos cr\303\255ticos e pontos de inflex\303\243o. Estes est\303\243o colecionados na Tabela 18.4, arranjados de acordo com os valores crescente de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. important4 := sort( evalf([ int4, zero4, cp4, ip4 ]), (a,b)->a[1]<b[1] ): tab4 := [ [`x`, `y`, `description`], [`-----`, `-----`,`---------------`], op(important4) ]: Matrix( evalf[4]( tab4 ) ); Tabela 18.4 Exemplo 18.4: Interse\303\247\303\243o, zeros, pontos cr\303\255ticos e pontos de inflex\303\243o Esta \303\251 uma longa lista, mas cada ponto da Tabela 18.4 \303\251 importante. Quando estes pontos s\303\243o desenhados, Figura 18.29 prov. pts4 := [seq( P[1..2], P=important4 )]: Ppts4 := plot( pts4, color=red, style=point, symbol=circle,symbolsize=10 ): display( Ppts4 ); Figura 18.29 Exemplo 18.4: Pontos da Tabela 18.4 O correspondente visual informativo de sinal aparece na Figura 18.30 SignChart( f4, x=0..4*Pi, [0,1,2] ); Figura 18.30 Exemplo 18.4: Visual informativo de sinal para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGOQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Rosa \342\207\222positivo, azul \342\207\222negativo Agora \303\251 um assunto bastante simples obter um gr\303\241fico preciso conectando estes pontos em sucess\303\243o. O gr\303\241fico resultante \303\251 dado na Figura 18.31. Pfn4 := plot( f4, x=0..4*Pi, color=green ): display( [Ppts4, Pfn4] ); Figura 18.31 Exemplo 18.4: O gr\303\241fico e pontos importantes Para concluir, \303\251 interessante comparar nosso gr\303\241fico com o da Figura 18.32, produzido pelo comando FunctionChart (Visual informativo da Fun\303\247\303\243o) do Maple no intervalo 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. FunctionChart( f4, 0..4*Pi, pointoptions=[symbolsize=20], slope=[thickness(2,2), color(red,blue)], concavity=[filled(pink,cyan)],title="" ); Figura 18.32 Exemplo 18.4: Comando FunctionChart (Visual Informativo da Fun\303\247\303\243o) aplicado em 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

Seja f uma fun\303\247\303\243o n\303\243o especificada com as propriedades: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEwJkFwcGx5RnVuY3Rpb247RicvJSVib2xkR1EmZmFsc2VGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdGQC8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUYjNiktRjs2LVEiKEYnRkEvRkRGNkZFL0ZIRjZGSUZLRk1GTy9GUlEsMC4xNjY2NjY3ZW1GJy9GVUZobi1GIzYkLUkjbW5HRiQ2JFEiMEYnRkFGQS1GOzYtUSIpRidGQUZlbkZFRmZuRklGS0ZNRk9GZ24vRlVRLDAuMTExMTExMWVtRictRjs2LVExJkludmlzaWJsZVRpbWVzO0YnRkFGQ0ZFRkdGSUZLRk1GT0ZRRlQtRjs2LVEiPUYnRkFGQ0ZFRkdGSUZLRk1GTy9GUlEsMC4yNzc3Nzc4ZW1GJy9GVUZccC1GXW82JFEsJnVtaW51czA7MTBGJ0ZBRkFGK0ZBRitGQQ==, e df5 := (x-4)*(x-2)/sqrt(x): dF5 := unapply( f5, x ): `f '`(x) = df5; Determine toda informa\303\247\303\243o necess\303\241ria para criar um gr\303\241fico da fun\303\247\303\243o f. Em particular, criar uma tabela de pontos importantes e o visual informativo de sinal (sign charts) para primeira e segunda derivada. E ent\303\243o, desenhar a fun\303\247\303\243o a m\303\243o. A an\303\241lise a seguir \303\251 enormemente motivada pelo gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== na Figura 18.33. plot(df5, x=0..7, -1..5); Figura 18.33 Exemplo 18.5: Gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ==

Como n\303\243o temos uma f\303\263rmula expl\303\255cita (explicit) para a fun\303\247\303\243o, n\303\243o se tem muito que fazer nos Passos 1-3.

A primeira derivada \303\251 `f '`(x) = df5; Os pontos estacion\303\241rios s\303\243o os zeros reais da primeira derivada, a saber, stat5 := solve( df5=0., x ): evalf[4]( stat5 ); Como a derivada existe somente para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=, a fun\303\247\303\243o f deve ser cont\303\255nua para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJGA+YEYnRjRGN0Y6LUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= (e, possivelmente, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 0). Entramos com o seguinte. sing5 := NULL: end5 := 0: Os pontos cr\303\255ticos desta fun\303\247\303\243o s\303\243o crit5 := [ stat5, sing5, end5 ]: evalf[4]( crit5 ); Os pontos cr\303\255ticos dividem o dom\303\255nio em intervalos em que a derivada n\303\243o muda de sinal. A Figura 18.34 cont\303\251m um visual informativo de sinal da primeira derivada, que \303\251 positivo nos intervalos rosa e negativo nos intervalos azuis. display(SignChart( df5, x=0..8, 0 ), view=[0..7,-1..0]); Figura 18.34 Exemplo 18.5: Visual informativo de sinal para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Rosa \342\207\222positivo, Azul \342\207\222negativo Deste visual informativo de sinal (sign chart) e do Teste da Primeira Derivada, est\303\241 claro que a fun\303\247\303\243o tem m\303\255nimo local em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 4 e tem um m\303\241ximo local em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 2. E tamb\303\251m, como a derivada \303\251 positiva imediatamente \303\240 direita de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 0, este limite (de intervalo) \303\251 um m\303\255nimo local. (N\303\243o existe nada que possa ser dito sobre extremo global.) Tudo que sabemos sobre os valores da fun\303\247\303\243o nos pontos cr\303\255ticos est\303\241 resumido abaixo: cp5 := [ 0, -10, `endpoint` ], [ 0, -10, `local minimum` ], [ 2, `>-10`, `local maximum` ], [ 4, `<f(2)`, `local minimum` ]: cp5;

A segunda derivada \303\251 ddf5 := simplify(diff( df5, x )): `f ''`(x) = ddf5; Os poss\303\255veis pontos de inflex\303\243o s\303\243o os zeros reais da segunda derivada, a saber, q1 := solve( ddf5=0., x ): evalf[4]( q1 ); O \303\272nico ponto de inflex\303\243o poss\303\255vel \303\251 possinfl5 := op(select( InInterval, [q1], [0,10] )): evalf[4]( possinfl5 ); a raiz (root) no dom\303\255nio da fun\303\247\303\243o. A concavidade nos dois intervalos definidos por este ponto \303\251 mais f\303\241cil exibir em termos do visual informativo de sinal (sign char) para a segunda derivada, dado na Figura 18.35. display(SignChart( df5, x=0..8, 1 ), view=[0..7,-1..0]); Figure 18.35 Example 18.5: Sign chart for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Pink \342\207\222positive, cyan \342\207\222negative Existe um ponto de inflex\303\243o j\303\241 que o sinal da segunda derivada muda no zero da segunda derivada. As coordenadas do ponto de inflex\303\243o no dom\303\255nio n\303\243o s\303\243o conhecidas, mas, porque a fun\303\247\303\243o \303\251 decrescente em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtSSNtb0dGJDYtUSIoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHUSZmYWxzZUYnLyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkwtRiM2Ji1JI21uR0YkNiRRIjJGJ0Y3LUY0Ni1RIixGJ0Y3L0Y7Rj8vRj5GPC9GQUY/RkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRlI2JFEiNEYnRjdGNy1GNDYtUSIpRidGN0Y6Rj1GQEZCRkRGRkZIRkovRk5RLDAuMTExMTExMWVtRidGN0YrRjdGK0Y3 sabemos que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2LkYrLUYjNictRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RMCZBcHBseUZ1bmN0aW9uO0YnLyUlYm9sZEdRJmZhbHNlRicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHRkIvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1GIzYmLUY9Ni1RIihGJ0ZDL0ZGRjhGRy9GSkY4RktGTUZPRlEvRlRRLDAuMTY2NjY2N2VtRicvRldGam4tRiM2JC1JI21uR0YkNiRRIjJGJ0ZDRkMtRj02LVEiKUYnRkNGZ25GR0ZobkZLRk1GT0ZRRmluL0ZXUSwwLjExMTExMTFlbUYnRkNGK0ZDLUY9Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZDRkVGR0ZJRktGTUZPRlFGU0ZWLUYsNiVRJGA+YEYnRjZGOUZnby1GIzYnRjNGPC1GIzYmRlotRiM2JC1GX282JFEmMi45MTVGJ0ZDRkNGYm9GQ0YrRkNGZ29Gam9GZ28tRiM2J0YzRjwtRiM2JkZaLUYjNiQtRl9vNiRRIjRGJ0ZDRkNGYm9GQ0YrRkNGK0ZDRitGQw==. Registramos isto como ip5 := seq( [ x, `<f(2) & >f(4)`, `inflection point` ], x=[possinfl5] ): evalf[4]( ip5 );

Os pontos importantes identificados nos Passos 4 & 5 incluem a interse\303\247\303\243o (com eixo-y), os zeros, os pontos cr\303\255ticos e o ponto de inflex\303\243o. Estes s\303\243o colecionados na Tabelea18.5, arranjados de acordo com os valores crescentes de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. important5 := sort( [ cp5, ip5 ], (a,b)->a[1]<b[1] ): tab5 := [ [`x`, `y`, `description`], [`-----`, `-----`,`---------------`], op(important5) ]: Matrix( evalf[4]( tab5 ) ); Tabela 18.5 Exemplo 18.5: interse\303\247\303\243o, zeros, pontos cr\303\255ticos e ponto de inflex\303\243o Esta \303\251 a tabela de valores importantes requisitados desde o princ\303\255pio do problema. N\303\243o \303\251 dif\303\255cil esbo\303\247ar a forma geral da fun\303\247\303\243o a partir desta informa\303\247\303\243o e da Figura 18.36, visual informativo de sinal para primeira e segunda derivada. display([SignChart( df5, x=0..10, [0,1]), textplot({[6,-.2,`y'`],[6,-.45,`y''`]})], view=[0..6,-1..0] ); Figura 18.36 Exemplo 18.5: Visual informativo de sinal (Sign charts) para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmcHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmUHJpbWU7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk1GOQ== Rosa \342\207\222positivo, Azul \342\207\222negativo

A tarefa de criar o gr\303\241fico de uma fun\303\247\303\243o pode ser subdividida em an\303\241lise da fun\303\247\303\243o, a primeira derivada e a segunda derivada. Algumas das conclus\303\265es, por exemplo, a classifica\303\247\303\243o de extremo local, pode ser dada em mais de um est\303\241gio do processo. De uma tabela de valores importantes compilados durante a an\303\241lise e o visual informativo de sinal (sign charts) da fun\303\247\303\243o e de sua duas primeiras derivadass, \303\251 usualmente bastante f\303\241cil completar o esbo\303\247o pela "conex\303\243o dos pontos." Use o Maple em qualque tempo enquanto estiver resolvendo (solving) estes problemas. Lembre-se, entretanto, de evitar usar o Maple para desenhar a fun\303\247\303\243o at\303\251 que voc\303\252 tenha descoberto todas as propriedades do gr\303\241fico. Use o comando FunctionChart (Visual Informativo da Fun\303\247\303\243o), no pacote Student[Calculus1] (Estudante[C\303\241lculo]), somente para compara\303\247\303\243o de seu trabalho.

O Cap\303\255tulo19 (Otimiza\303\247\303\243o), seguindo os cap\303\255tulos sobre An\303\241lise Global, An\303\241lise Local e Desenhar Gr\303\241fico, prov\303\252 uma discuss\303\243o completa do uso de c\303\241lculo para localiza\303\247\303\243o e classifica\303\247\303\243o dos pontos mais interessantes sobre o gr\303\241fico de uma fun\303\247\303\243o.

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