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Guia de Estudo de C\303\241lculo \302\251 Maplesoft, a division of Waterloo Maple Inc., 2005 Cap\303\255tulo 13: Derivadas de Fun\303\247\303\265es Trigonom\303\251tricas
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field> Agora temos todas as ferramentas necess\303\241rias para derivar fun\303\247\303\265es trigonom\303\251tricas (trigonometric functions). As fun\303\247\303\265es trigonom\303\251tricas fundamentais s\303\243o seno (sine) e cosseno (cosine). Uma vez conhecidas as derivadas (derivatives) destas, as derivadas da tangente, secante, cotangente e cossecante (cosecant) podem ser obtidas com as regras do Quociente, Pot\303\252ncia e da Cadeia. Portanto, nosso primeiro objetivo \303\251 encontrar a derivada do seno e do cosseno. Isto ser\303\241 feito atrav\303\251s da defini\303\247\303\243o e envolver\303\241 os dois limites limits especiais do Cap\303\255tulo 4 sobre limites ( limits) de fun\303\247\303\265es trigonom\303\251tricas. Exemplos adicionais permitir\303\243o praticar com todas as regras de deriva\303\247\303\243o. O DerivativePlot maplet \303\251 introduzido como uma ferramenta \303\272til para criar um gr\303\241fico de uma fun\303\247\303\243o e de uma ou mais de suas derivadas. O Differentiation maplet continua a ser uma ferramenta \303\272til para o uso das regras de deriva\303\247\303\243o (differentiation ). Mas , n\303\243o se torne dependente destes maplets --- certifique-se de que voc\303\252 pode encontrar as derivadas a m\303\243o!
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Cap\303\255tulo Gloss\303\241rio</Font></Text-field> Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio Matem\303\241tico Maple (Maple Math Dictionary). amplitude chainrule continuous cosecant cosine cotangent curve derivative differentiable differentiate differentiation domain double-angleformula explicit exponent factor function graph limit slope maximum minimum negative period powerrule product quotient radian rateofchange rationalfunction secant set sign sine slope tables tangent trigonometricfunction work zero
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Inicializa\303\247\303\243o</Font></Text-field> restart;
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Derivadas das Fun\303\247\303\265es de Seno e Cosseno </Font></Text-field> O Gr\303\241fico das fun\303\247\303\265es de seno e cosseno \303\251 cont\303\255nuo (continuous) e t\303\252m uma reta tangente (tangent) em todos os pontos. Isto significa que as fun\303\247\303\265es de seno e cosseno devem ser diferenci\303\241veis (differentiable) em todo lugar. Figura 13.1 mostra o resultado da aplica\303\247\303\243o de DerivativePlot maplet para a fun\303\247\303\243o seno(x) (sin(x)). Este maplet \303\251 equivalente a construir no Tutor de Derivada (Derivative Tutor) acess\303\255vel de Ferramentas/Tutores/C\303\241lculo (Tools/Tutors/Calculus) - menu Uma Vari\303\241vel (Single Variable) sob o nome Derivadas (Derivatives)... 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Figura 13.1 Aplica\303\247\303\243o do Tutor de Derivada (Derivative Tutor) para a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3
Para obter a Figura 13.1, inicie o DerivativePlot maplet, estabele\303\247a (set) para a Fun\303\247\303\243o (Function) ser (seno (x)) sin(x) , a =-2*Pi e b =2*P e ent\303\243o clique Display. O Gr\303\241fico (graph) da derivada da fun\303\247\303\243o seno parece familiar --- com um per\303\255odo (period ) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= , amplitude 1 e um maximum em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= --- ele se parece com a fun\303\247\303\243o cosseno. Isto nos leva a cojeturar 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 Agora, mude a Fun\303\247\303\243o (Function) para cos(x) e clique Display. O Tutor de Derivada (Derivative Tutor) aparecer\303\241 agora como como o da Figura 13.2. 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Figura 13.2 Aplica\303\247\303\243o do Tutor de Derivada (Derivative Tutor) para a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3
A derivada do cosseno tem per\303\255odo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU=, amplitude 1, um zero em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=, um minimum pr\303\263ximo a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEkMS41RidGPkY+RitGPg== e outro zero pr\303\263ximo a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4=; isto parece ser o negativo (negative) da fun\303\247\303\243o seno. Baseados nisto, conjeturamos 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Para determinar se estas conjeturas est\303\243o corretas, retorne \303\240 defini\303\247\303\243o da derivada de 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, a saber, \303\240 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A seguir, olhe para a defini\303\247\303\243o da derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tRiM2Ji1GOzYtUSIoRidGNy9GP1EldHJ1ZUYnRkAvRkNGV0ZERkZGSEZKL0ZNUSwwLjE2NjY2NjdlbUYnL0ZQRlotRiM2JC1GLDYlUSJ4RicvRjVGVy9GOFEnaXRhbGljRidGNy1GOzYtUSIpRidGN0ZWRkBGWEZERkZGSEZKRlkvRlBRLDAuMTExMTExMWVtRidGN0YrRjdGK0Y3, a saber, para 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
Agora voc\303\252 j\303\241 deve entender porque gastamos tanto tempo no Cap\303\255tulo 4 olhando para os dois limites especiais 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNipGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LVEkbGltRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JVEnJiM5NTI7RicvJSdpdGFsaWNHRj9GOi1GNzYtUScmcmFycjtGJ0Y6Rj1GQEZCRkRGRi9GSUY/RktGTS9GUUZPLUkjbW5HRiQ2JFEiMEYnRjpGOi8lLGFjY2VudHVuZGVyR0Y/RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNGVtRicvJSZkZXB0aEdGZG8vJSpsaW5lYnJlYWtHUSVhdXRvRidGKy1GIzYmLUY3Ni1RIihGJ0Y6L0Y+RkpGQC9GQ0ZKRkRGRkZnbkZLL0ZORlJGUC1JJm1mcmFjR0YkNigtRiM2JkYrLUYjNictRmpuNiRRIjFGJ0Y6LUY3Ni1RKCZtaW51cztGJ0Y6Rj1GQEZCRkRGRkZnbkZLL0ZOUSwwLjIyMjIyMjJlbUYnL0ZRRmNxLUYjNictRiw2JVEkY29zRidGWEY6LUY3Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjpGPUZARkJGREZGRmduRktGTUZobi1GIzYmRl9wLUYjNiRGVUY6LUY3Ni1RIilGJ0Y6RmJwRkBGY3BGREZGRmduRktGZHAvRlFRLDAuMTExMTExMWVtRidGOkYrRjpGK0Y6RitGOkZfci8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGW3MvJSliZXZlbGxlZEdGP0ZhckY6RitGOi1GNzYtUSI9RidGOkY9RkBGQkZERkZGZ25GSy9GTlEsMC4yNzc3Nzc4ZW1GJy9GUUZkc0ZpbkY6RitGOg== Por causa da maneira como as derivadas de seno e cosseno s\303\243o relacionadas, seria interessante olhar para as derivadas de ordem superior destas fun\303\247\303\265es. Tabela 13.1, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JkYrLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSNkeEYnRjdGOi1JI21uR0YkNiRRIjBGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRj1GK0Y9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZVLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ== 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 = 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, isto \303\251, a derivada de ordem zero \303\251 a pr\303\263pria fun\303\247\303\243o j\303\241 que nenhuma derivada foi tomada, lista as derivadas das fun\303\247\303\265es de seno e cosseno at\303\251 a oitava ordem. <<k,seq(k,k=0..8)>|<d/dx^k,seq(diff(sin(x),[x$k]),k=0..8)>| <d/dx^k,seq(diff(cos(x),[x$k]),k=0..8)>>; 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 Tabela 13.1 Derivadas de senLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y1RjVGK0Y1RitGNQ== na Coluna 2, e de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 na Coluna 3
Da\303\255, concluimos que as derivadas de ordem superior de seno e cosseno s\303\243o c\303\255clicas (com periodo 4).
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Derivadas de Outras Fun\303\247\303\265es Trigonom\303\251tricas </Font></Text-field> As derivadas das fun\303\247\303\265es seno e cosseno juntas com as Regras do Quociente, Pot\303\252ncia e da Cadeia s\303\243o tudo o que \303\251 necess\303\241rio para descobrir as derivadas das fun\303\247\303\265es tangente, secante, cotangente e cossecante . A derivada da fun\303\247\303\243o tangente pode ser obtida como se segue. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSR0YW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 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A derivada da fun\303\247\303\243o secante (secant) pode ser encontrada com as Regras de Pot\303\252ncia (Power) e da Cadeia como a seguir LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLDYlUSNkeEYnRjVGOC9GOVEnbm9ybWFsRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkcvJSliZXZlbGxlZEdRJmZhbHNlRidGK0ZA LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRzZWNGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 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Note o uso da Regra da Cadeia (Chain Rule) nesta deriva\303\247\303\243o. As derivadas das fun\303\247\303\265es cotangente (cotangent) e cossecante podem ser derivadas de forma similar. Todas as as seis derivadas trigonom\303\251tricas est\303\243o resumidas na Tabela 13.2. 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cscLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiVGKy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRistSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjdRJ2l0YWxpY0YnRjZGNkYrRjY= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= 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 Tabela 13.2 Derivadas das fun\303\247\303\265es trigonom\303\251tricas
Uma compara\303\247\303\243o das entradas na primeira coluna com a correspondente entrada na segunda coluna da Tabela 13.2 revela uma rela\303\247\303\243o entre as derivadas de uma fun\303\247\303\243o trigonom\303\251trica com sua "co'' fun\303\247\303\243o: mude o sinal (sign) e mude todas fun\303\247\303\265es para as suas co-fun\303\247\303\265es Por esta raz\303\243o, n\303\243o \303\251 essencial que voc\303\252 memorize as derivadas da cotangente e da cossecante.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplos</Text-field> Voc\303\252 deveria realmente usar o Differentiation maplet para seguir cada passo da deriva\303\247\303\243o em cada exemplo.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 13.1: <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1GLDYlUSNkdEYnRjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1GLDYlUSNkdEYnRjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ==</Equation> <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field> A derivada de 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\303\251 computada usando a Regra da Cadeia, a qual d\303\241 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHRGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYjNiYtRjI2LVEqJnVtaW51czA7RidGNUY4RjtGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnL0ZLRlItSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEnJiM5NjA7RicvJSdpdGFsaWNHRjpGNUY1LUYjNiQtSSNtbkdGJDYkUSQxODBGJ0Y1RjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNvLyUpYmV2ZWxsZWRHRjpGK0Y1RitGNUYrRjU= 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
O Maple confirma este resultado via a sintaxe f := cos( Pi*t/180 ): `f '`(t) = diff(f,t); O fator (factor) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiQtSSNtbkdGJDYkUSQxODBGJ0Y6RjovJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkgvJSliZXZlbGxlZEdGOUYrRjo= , nesta fun\303\247\303\243o, converte um \303\242ngulo de graus para radianos. Este resultado mostra que este fator tamb\303\251m relaciona a varia\303\247\303\243o( rate of change) do cosseno de um \303\242ngulo medido em radianos com a varia\303\247\303\243o do cosseno de um \303\242ngulo medido em graus.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 13.2: <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1GLDYlUSNkdUYnRjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1GLDYlUSNkdUYnRjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ==</Equation> <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field> Esta derivada requer o uso da regra da cadeia. Durante seu uso , um quociente (quotient) deve ser derivado (differentiated). Enquanto se deriva o quociente, a regra da cadeia deve ser novamente utilizada. O c\303\241lculo completo \303\251 como se segue. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1GLDYlUSNkdUYnRjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ== 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Para simplificar esta resposta, mais adiante use a f\303\263rmula do \303\242ngulo duplo (double angle) 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 and 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Em retrospectiva, note que a fun\303\247\303\243o a ser derivada pode ser simplificada usando a f\303\263rmula do \303\242ngulo duplo para seno. 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
Use o Differentiation maplet para que a derivada desta express\303\243o concorde com os resultados anteriores deste exemplo. Usar o Maple para confirmar as computa\303\247\303\265es manuais; al\303\251m disso, ser\303\241 necess\303\241rio dizer ao Maple explicitamente (explicitly) quais identidades trigonom\303\251tricas usar em v\303\241rios pontos ao longo do caminho. Come\303\247amos por anotar estas identidades: id1 := u -> cos(u)^2 = 1-sin(u)^2: id2 := u -> sin(2*u)=2*sin(u)*cos(u): id3 := u -> cos(2*u)=cos(u)^2-sin(u)^2: id1(u); id2(u); id3(u); A derivada da fun\303\247\303\243o na sua forma original \303\251 f2 := ( cos(u)/sin(2*u) )^2: df2a := Diff( f2, u ): df2a = value( df2a ); Combinando os dois termos no lado direito temos df2b := simplify( value(df2a) ): df2a = df2b; o que concorda com o resultado obtido manualmente. Quando a f\303\263mula do \303\242ngulo duplo para seno \303\251 aplicada \303\240 fun\303\247\303\243o antes da deriva\303\247\303\243o, o resultado \303\251 f2a := simplify( f2, [id2(u)] ): f2 = f2a; e assim a derivada ser\303\241 df2d := diff( f2a, u ): df2a = df2d;
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 13.3: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field> Note que o segundo termo \303\251 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 , e n\303\243o 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 . A requerida derivada pode ser encontrada usando a Regra do Produto (Product) e a Regra da Cadeia (Chain Rules). O resultado \303\251 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
A computa\303\247\303\243o desta derivada no Maple poderia ser feita como se segue. f3 := sin(theta)*tan(theta^2): df3 := Diff( f3, theta ): df3 = value( df3 ); Esta resposta difere da nossa somente na apar\303\252ncia. Note que o Maple informa q1 := Diff( tan(x), x ): q1 = value( q1 ); Para fazer com que o Maple recoloque as pot\303\252ncias de tangente com secante, fornecemos uma quarta identidade trigonom\303\251trica, a saber, id4 := u -> tan(u)^2=sec(u)^2-1: id4(x); Assim, fornecendo esta identidade para simplificar (simplify) como uma rela\303\247\303\243o a parte (side relation): q1 = simplify( value(q1), [id4(x)] ); Aplicando a mesma abordagem para este exemplo, df3 = simplify(value(df3), [id4(theta^2)]); Exceto pela separa\303\247\303\243o dos termos no fator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTUyO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= , este \303\251 exatamente o nosso resultado.
Quest\303\243o 13.1 Usando a regra apropriada da derivada, encontrar a derivada de: 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 (Nota: Voc\303\252 deve digitar uma resposta antes de clicar em "Grade".) Quest\303\243o 13.2 Usando a regra apropriada da derivada, encontrar a derivada de: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIj1GJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVi1GLDYlUSRjb3NGJy9GMEY/RkAtRjY2JC1GIzYtLUkjbW5HRiQ2JFEiNkYnRkAtRkM2LVEifkYnRkBGRkZIRkpGTEZORlBGUi9GVVEmMC4wZW1GJy9GWEZjby1JJW1zdXBHRiQ2JUY6LUYjNiUtRlxvNiRRIjJGJ0ZARi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRkM2LVEoJm1pbnVzO0YnRkBGRkZIRkpGTEZORlBGUi9GVVEsMC4yMjIyMjIyZW1GJy9GWEZkcC1GXG82JFEiM0YnRkBGX29GOkZgcC1GXG82JFEiNUYnRkBGPUZARkBGPUZA (Nota: Voc\303\252 deve digitar uma resposta antes de clicar em "Grade".) Quest\303\243o 13.3 Encontrar a derivada de: 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 (Nota: Voc\303\252 deve digitar uma resposta antes de clicar em "Grade".) Quest\303\243o 13.4 Usando a regra apropriada da derivada, encontrar a derivada de: 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 (Nota: Voc\303\252 deve digitar uma resposta antes de clicar em "Grade".)
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Cap\303\255tulo Resumo</Font></Text-field> As fun\303\247\303\265es trigonom\303\251tricas s\303\243o deferenci\303\241veis em todos os pontos em seus dom\303\255nios (domain). As derivadas das fun\303\247\303\265es fundamentais est\303\243o listadas na Tabela 13.2.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Qual \303\251 o Pr\303\263ximo?</Font></Text-field> Agora, voc\303\252 deveria estar bem confort\303\241vel usando o Differentiation maplet para checar seu trabalho. Embora esta ferramenta seja muito \303\272til e poderosa, lembre-se de que voc\303\252 deve tamb\303\251m desenvolver suas aptid\303\265es manuais. No pr\303\263ximo cap\303\255tulo, a Regra de Pot\303\252ncia (Power Rule) ser\303\241 verificada para todos os expoentes racionais (rational) e aprenderemos a encontrar a inclina\303\247\303\243o da reta tangente a qualquer curva (curve) - n\303\243o somente o gr\303\241fico de uma fun\303\247\303\243o. Isto concluir\303\241 a unidade sobre Deriva\303\247\303\243o.
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