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Cap\303\255tulo 4 Limites Trigonom\303\251tricos
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field> Limites de express\303\265es envolvendo fun\303\247\303\265es trigonom\303\251tricas usualmente podem ser avaliados por substitui\303\247\303\243o direta--- a n\303\243o ser que uma divis\303\243o por zero seja encontrada ou seja encontrado um caso de oscila\303\247\303\243o infinita. Quando divis\303\243o por zero for encontrada, pode-se tentar simplifica\303\247\303\265es como as apresentadas no Cap\303\255tulo 1. Em muitos casos, entretanto, isto n\303\243o \303\251 suficiente para avaliar um limite . Os dois limites especiais 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 e 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 foram introduzidos -- graficamente -- no Cap\303\255tulo 1. Neste cap\303\255tulo, cada um destes limites ser\303\241 rigorosamente avaliado. Estas provas ser\303\243o nossa primeira aplica\303\247\303\243o real das leis de limites, particularmente o Teorema do Confronto.
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">\303\215ndice de Termos</Font></Text-field> Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio do Maple; para mais informa\303\247\303\265es v\303\241 em Dicion\303\241rio Matem\303\241tico. \303\241rea limite c\303\255rculo cont\303\255nuo cossecante cosseno cotangente dom\303\255mio avaliar expl\303\255cito fator fun\303\247\303\243o gr\303\241fico inequa\303\247\303\243o infinito teorema de bolzano limite limite inferior m\303\272ltiplo per\303\255odo prova real secante seno teorema do sandu\303\255che substitui\303\247\303\243o tabelas tangente fun\303\247\303\243o trigonom\303\251trica limite superior valor zero
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Inicializa\303\247\303\243o</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Procedimentos Auxiliares</Text-field> 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiVGKy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjNGNkYrRjNGNg==
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Leis de Limites para Fun\303\247\303\265es Trigonom\303\251tricas</Font></Text-field> As fun\303\247\303\265es seno e cosseno s\303\243o as bases de todas as fun\303\247\303\265es trigonom\303\251tricas. Estas fun\303\247\303\265es s\303\243o peri\303\263dicas, com per\303\255odoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbW5HRiQ2JFEiMkYnRi9GKy1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y0Ri9GLw== . Da Figura 4.1, os gr\303\241ficos destas duas fun\303\247\303\265es, 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 Figura 4.1 Tr\303\252s per\303\255odos das fun\303\247\303\265es seno e cosseno
\303\251 f\303\241cil ver na Tabela 4.1 que: Limite Restri\303\247\303\243o 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 nenhuma 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 nenhuma
Tab 4.1 Limites b\303\241sicos para fun\303\247\303\265es seno e cosseno
Os limites b\303\241sicos para outras quatro fun\303\247\303\265es trigonom\303\251tricas, tangente, cotangente, secante e cossecante, s\303\243o obtidos da Tabela 4.1 e do comportamento das fun\303\247\303\265es de seno e cosseno. Baseado nas Regras de Limite, particularmente aquela para quocientes, \303\251 f\303\241cil ver que muitos limites envolvendo estas fun\303\247\303\265es podem ser avaliados por substitui\303\247\303\243o. Por exemplo, Tabela 4.2 lista os mais \303\263bvios destes limites. Limite Restri\303\247\303\243o 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSUmbmU7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk8tSSZtZnJhY0dGJDYoLUYjNictRiw2I1EhRictRiM2Ji1GOTYtUSIoRidGPC9GP0YxRkEvRkRGMUZFRkdGSUZLRk1GUC1GIzYnRlctRiM2Ji1JI21uR0YkNiVRIjJGJ0YyRjwtRjk2LVExJkludmlzaWJsZVRpbWVzO0YnRjxGPkZBRkNGRUZHRklGS0ZNRlAtRiw2JlEibkYnRi9GMkY1RjwtRjk2LVEiK0YnRjxGPkZBRkNGRUZHRklGSy9GTlEsMC4yMjIyMjIyZW1GJy9GUUZdcC1GYG82JVEiMUYnRjJGPEY8LUY5Ni1RIilGJ0Y8RmluRkFGam5GRUZHRklGS0ZNL0ZRUSwwLjExMTExMTFlbUYnRjxGY28tRiw2JlEnJiM5NjA7RicvRjBGQEYyRjxGPC1GIzYkRl9vRjwvJS5saW5ldGhpY2tuZXNzR0ZhcC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZhcS8lKWJldmVsbGVkR0ZARldGPA== 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSUmbmU7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk8tRiM2Ji1GLDYmUSJuRidGL0YyRjUtRjk2LVExJkludmlzaWJsZVRpbWVzO0YnRjxGPkZBRkNGRUZHRklGS0ZNRlAtRiw2JlEnJiM5NjA7RicvRjBGQEYyRjxGPC1GLDYjUSFGJ0Y8 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSUmbmU7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk8tRiM2Ji1GLDYmUSJuRidGL0YyRjUtRjk2LVExJkludmlzaWJsZVRpbWVzO0YnRjxGPkZBRkNGRUZHRklGS0ZNRlAtRiw2JlEnJiM5NjA7RicvRjBGQEYyRjxGPC1GLDYjUSFGJ0Y8
Tab 4.2 O limites "\303\263bvios" de algumas fun\303\247\303\265es trigonom\303\251tricas
Estas f\303\263rmulas podem ser resumidas dizendo-se que limites envolvendo quaisquer destas seis fun\303\247\303\265es trigonom\303\251tricas podem ser avaliados por substitui\303\247\303\243o, desde que o ponto de limite esteja no dom\303\255nio da fun\303\247\303\243o. Para as fun\303\247\303\265es tangente e secante, pontos onde a fun\303\247\303\243o cosseno \303\251 zero (m\303\272ltiplos \303\255mpares de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiQtSSNtbkdGJDYkUSIyRidGOkY6LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZILyUpYmV2ZWxsZWRHRjlGK0Y6) devem ser exclu\303\255dos; para as fun\303\247\303\265es cotangente e cossecante, os "pontos ruins" s\303\243o os zeros da fun\303\247\303\243o seno, qualquer m\303\272ltiplo (inteiro) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy . Figuras 4.2 a 4.4, criadas com: 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 e apresentadas com: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUS1gRmlndXJlfjQuMmBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUS1gRmlndXJlfjQuM2BGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUS1gRmlndXJlfjQuNGBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=
Figura 4.2 Tr\303\252s per\303\255odos das fun\303\247\303\265es seno e cosseno
Figura 4.3 Seis per\303\255odos das fun\303\247\303\265es tangente e cotangente
 Figura 4.4 Tr\303\252s per\303\255odos das fun\303\247\303\265es secante e cossecante
 fornecendo os gr\303\241ficos destas seis fun\303\247\303\265es, pode ser \303\272til para relembrar os dom\303\255nios destas fun\303\247\303\265es. O maplet Limit pode ser usado para avaliar limites envolvendo fun\303\247\303\265es trigonom\303\251tricos. Para aplicar as regras, clique no bot\303\243o para a fun\303\247\303\243o ou pegue a fun\303\247\303\243o de uma das caixas que se abrem.
<Text-field style="Heading 1" layout="Heading 1">Dois Limites Especiais Calculados de Forma Rigorosa </Text-field> No cap\303\255tulo 1, argumentos gr\303\241ficos s\303\243o usados para justificar os enunciados: 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 and 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 Embora os argumentos baseados nos gr\303\241ficos de 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 e 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 sejam indicativos, eles n\303\243o s\303\243o provas. Para fornecer provas rigorosas de que estes limites existem e t\303\252m os valores sugeridos pelos gr\303\241ficos, as Regras de Limites, particularmente o Teorema do Confronto, ser\303\243o utilizadas.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova de que <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNipGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LVEkbGltRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JVEnJiM5NTI7RicvJSdpdGFsaWNHRj9GOi1GNzYtUScmcmFycjtGJ0Y6Rj1GQEZCRkRGRi9GSUY/RktGTS9GUUZPLUkjbW5HRiQ2JFEiMEYnRjpGOi8lLGFjY2VudHVuZGVyR0Y/RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNGVtRicvJSZkZXB0aEdGZG8vJSpsaW5lYnJlYWtHUSVhdXRvRidGKy1GIzYmLUY3Ni1RIihGJ0Y6L0Y+RkpGQC9GQ0ZKRkRGRkZnbkZLRk1GaG4tSSZtZnJhY0dGJDYoLUYjNiZGKy1GIzYnLUYsNiVRJHNpbkYnRlhGOi1GNzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y6Rj1GQEZCRkRGRkZnbkZLRk1GaG4tRiM2JkZfcC1GIzYkRlVGOi1GNzYtUSIpRidGOkZicEZARmNwRkRGRkZnbkZLRk0vRlFRLDAuMTExMTExMWVtRidGOkYrRjpGK0Y6RmNxLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zfci8lKWJldmVsbGVkR0Y/RmVxRjpGK0Y6LUY3Ni1RIj1GJ0Y6Rj1GQEZCRkRGRkZnbkZLL0ZOUSwwLjI3Nzc3NzhlbUYnL0ZRRmhyLUZqbjYkRlxyRjpGOkYrRjo=</Equation> </Text-field> Figuras 4.5 e 4.6 ilustram que a \303\241rea do tri\303\242ngulo POB \303\251 limitada acima pela \303\241rea do setor POA e limitada abaixo pela \303\241rea do setor COB. 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 Figura 4.5 Figura 4.6
Quando o \303\242ngulo feito com o eixo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= decresce at\303\251 zero, todas as tr\303\252s \303\241reas tamb\303\251m decrescem at\303\251 zero. Isto \303\251 claramente mostrado pela anima\303\247\303\243o na Figura 4.7. 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
Figura 4.7 Anima\303\247\303\243o suporte da prova de que: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
Observe que, para \303\242ngulos pequenos, as tr\303\252s \303\241reas parecem ser iguais. F\303\263rmulas para as tr\303\252s \303\241reas podem ser encontradas da geometria b\303\241sica, e s\303\243o, respectivamente: \303\241rea(setor COB ) = 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSRjb3NGJ0Y0RjctSSNtbkdGJDYkUSIyRidGNy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y3RjotRiw2JVEnJiM5NTI7RidGNEY3RjdGK0Y3 = 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUklbXN1cEdGJDYlLUYsNiVRJGNvc0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtbkdGJDYkUSIyRidGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlgtRiw2JVEnJiM5NTI7RidGN0Y6RjpGK0Y6 \303\241rea(tri\303\242ngulo POB ) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjgtRiM2JC1GNTYkUSIyRidGOEY4LyUubGluZXRoaWNrbmVzc0dGNy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGOA== 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 \303\241rea(setor POA ) = 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy (1)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlRistSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjhGK0Y4 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiZGKy1GIzYmLUkjbW9HRiQ2LVEpJnZlcmJhcjtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dGP0Y6RjZGOkYrRjotRiM2JC1JI21uR0YkNiRRIjJGJ0Y6RjovJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlxvLyUpYmV2ZWxsZWRHRj9GK0Y6
que combinadas fornecem: 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 Supondo que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVElJm5lO0YnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFEiMEYnRjdGN0YrRjc= e 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, ent\303\243o \303\251 permitido multiplicar estas inequa\303\247\303\265es pela quantidade diferente de zero 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 e 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 Para qualquer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy em (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 ,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6LUYjNiQtSSNtbkdGJDYkUSIyRidGOkY6LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZILyUpYmV2ZWxsZWRHRjlGK0Y6 ) (excluindo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPUYnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFEiMEYnRjdGN0YrRjc= ), 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 > 0 e assim LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUkmbWZyYWNHRiQ2KC1GIzYmRistRiM2Ji1JI21vR0YkNi1RKSZ2ZXJiYXI7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiM2Jy1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dGQUY8LUY5Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjxGP0ZCL0ZFRkFGR0ZJRktGTUZPRlItRiM2Ji1GOTYtUSIoRidGPC9GQEZGRkJGREZHRklGS0ZNRk9GUi1GIzYkLUYsNiVRJyYjOTUyO0YnRllGPEY8LUY5Ni1RIilGJ0Y8Rl5vRkJGREZHRklGS0ZNRk8vRlNRLDAuMTExMTExMWVtRidGPEYrRjxGOEY8RitGPC1GIzYmRistRiM2JkY4RmFvRjhGPEYrRjwvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmJwLyUpYmV2ZWxsZWRHRkEtRjk2LVEiPUYnRjxGP0ZCRmhuRkdGSUZLRk0vRlBRLDAuMjc3Nzc3OGVtRicvRlNGW3EtRjI2KC1GIzYmRitGVEYrRjxGX29GXXBGYHBGY3BGZXBGK0Y8RitGPA== . Isto significa que: 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 Figura 4.8 confirma a comprova\303\247\303\243o -- e utilidade -- destas inequa\303\247\303\265es. 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 Figura 4.8 Teorema do Confronto para 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 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUkjbW5HRiQ2JFEiMEYnRjlGNS1GLDYlUSJjRidGL0YyRjk=
Conseq\303\274entemente, porque 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 = 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 = 1, o Teorema do Confronto \303\251 aplic\303\241vel e nos diz: 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
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Prova de que <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> </Text-field> Esta prova come\303\247a pela multiplica\303\247\303\243o de 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 por 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 e o uso de identidades trigonom\303\251tricas para simplificar: 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 = 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 = 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
Portanto, 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 = 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 = (1)(0) = 0
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><Font readonly="true">Maple T.A. Maple-Graded Question</Font></Text-field> Using the Maple-graded question type, you create a question that uses Maple functionality when creating the correct answer and determining if the student's response is correct. The Maple-graded question type has an option to plot the student's response. To include the plot option, insert the corresponding task template. For information on creating questions, making assignments, and exporting Maple T.A. course modules, see Overview of Maple T.A. Integration.
<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field> Algorithmic variables can be used to create a question that contains randomly generated parameters. You can refer to algorithmic variables in the Question, Answer, Hints, and Feedback input regions, as well as in other variable definitions. To reference an algorithmic variable, include "`$variable`" as text or 2-D Math, for example, x := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique. with(RandomTools): randomize(): a1 := Generate(integer(range=-9..9)): while (a1 = 0) do a1 := Generate(integer(range=-9..9)) end do: b1 := Generate(integer(range=-9..9)): while (b1 = 0) do b1 := Generate(integer(range=-9..9)) end do: fn1 := Limit(sin(a1*theta)/(b1*theta), theta=0); ans11 := value(fn1); if member(ans11,{`infinity`,-`infinity`,`undefined`}) then ans1 := 'dne'; else ans1 := ans11; end if; LUkmTGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwkKiYtSSRzaW5HRiQ2IywkSSZ0aGV0YUdGJyIiIyIiIkYvISIiI0YxIiIpL0YvIiIh IyIiIiIiJQ== IyIiIiIiJQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 4.1</Font></Text-field> The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question. Calcule `$fn1` Se o limite \303\251 indefinido(ou infinito), entre com a sua resposta como dne. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans1; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field> To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$RESPONSE` is automatically initialized to the student's response. Using `$ANSWER` (the correct answer) and `$RESPONSE` (the student response), one example of evaluating to a Boolean value is, evalb(`$ANSWER`-`$RESPONSE`=0); Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region. if ((`$RESPONSE` = infinity or `$RESPONSE` = -infinity or `$RESPONSE` = undefined) and evalb(`$RESPONSE` = `$ans11`) = true) then rrr := 'dne': else rrr := `$RESPONSE`: end if: evalb(`$ANSWER` = rrr); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints. Be sure to look for singularities caused by attempts to divide by zero. A plot of the function near the limit point can help. Replace this text with hint 3.
with(StringTools): with(DocumentTools): response := GetProperty(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Be&MediumSpace;sure&MediumSpace;to&MediumSpace;look&MediumSpace;for&MediumSpace;singularities&MediumSpace;caused&MediumSpace;by&MediumSpace;attempts&MediumSpace;to&MediumSpace;divide&MediumSpace;by&MediumSpace;zero.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>A&MediumSpace;plot&MediumSpace;of&MediumSpace;the&MediumSpace;function&MediumSpace;near&MediumSpace;the&MediumSpace;limit&MediumSpace;point&MediumSpace;can&MediumSpace;help.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response. Replace this text with feedback.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := ans1; if ((eval(parse(StudentResponse)) = infinity or eval(parse(StudentResponse)) = -infinity or eval(parse(StudentResponse)) = undefined) and evalb(eval(parse(StudentResponse)) = ans11) = true) then rrr := 'dne':else rrr := eval(parse(StudentResponse)):end if: if evalb(ANSWER = rrr) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="&lt;NO RESPONSE&gt;"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect.&nbsp;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer:&nbsp;</mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><Font readonly="true">Maple T.A. Maple-Graded Question</Font></Text-field> Using the Maple-graded question type, you create a question that uses Maple functionality when creating the correct answer and determining if the student's response is correct. The Maple-graded question type has an option to plot the student's response. To include the plot option, insert the corresponding task template. For information on creating questions, making assignments, and exporting Maple T.A. course modules, see Overview of Maple T.A. Integration.
<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field> Algorithmic variables can be used to create a question that contains randomly generated parameters. You can refer to algorithmic variables in the Question, Answer, Hints, and Feedback input regions, as well as in other variable definitions. To reference an algorithmic variable, include "`$variable`" as text or 2-D Math, for example, x := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique. with(RandomTools): randomize(): a2 := Generate(integer(range=-4..4)): while (a2 = 0) do a2 := Generate(integer(range=-4..4)): end do: t2 := Generate(choose([1,2,3])): ff1 := (sin(x^a2),cos(x^a2),tan(x^a2))[t2]; fn2 := Limit(ff1,x=0); ans21 := value(fn2); if member(ans21,{-1..1,`infinity`,-`infinity`,`undefined`}) then ans2 := 'dne'; else ans2 := ans21; end if; LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJEkieEdGJyEiJA== LUkmTGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JJGNvc0dGJDYjKiRJInhHRichIiQvRi0iIiE= OyEiIiIiIg== SSRkbmVHNiI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o</Font> 4.2</Text-field> The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question. Calcule `$fn2` Se o limite \303\251 indefinido(ou infinito), entre com a sua resposta como dne. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans2;
<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field> To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$RESPONSE` is automatically initialized to the student's response. Using `$ANSWER` (the correct answer) and `$RESPONSE` (the student response), one example of evaluating to a Boolean value is, evalb(`$ANSWER`-`$RESPONSE`=0); Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region. if ((`$RESPONSE` = infinity or `$RESPONSE` = -infinity or `$RESPONSE` = undefined or `$RESPONSE`=-1..1) and evalb(`$RESPONSE` = `$ans21`) = true) then rrr2 := 'dne': else rrr2 := `$RESPONSE`: end if: evalb(`$ANSWER` = rrr2);
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints. Be sure to look for singularities caused by attempts to divide by zero. A plot of the function near the limit point can help. Replace this text with hint 3.
with(StringTools): with(DocumentTools): response := GetProperty(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Be&MediumSpace;sure&MediumSpace;to&MediumSpace;look&MediumSpace;for&MediumSpace;singularities&MediumSpace;caused&MediumSpace;by&MediumSpace;attempts&MediumSpace;to&MediumSpace;divide&MediumSpace;by&MediumSpace;zero.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet); with(StringTools): with(DocumentTools): response := GetProperty(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext>&lt;NO RESPONSE&gt;</mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>A&MediumSpace;plot&MediumSpace;of&MediumSpace;the&MediumSpace;function&MediumSpace;near&MediumSpace;the&MediumSpace;limit&MediumSpace;point&MediumSpace;can&MediumSpace;help.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response. Replace this text with feedback.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea1`, `value`); ANSWER := ans2; if ((eval(parse(StudentResponse)) = infinity or eval(parse(StudentResponse)) = -infinity or eval(parse(StudentResponse)) = undefined or eval(parse(StudentResponse))=-1..1) and evalb(eval(parse(StudentResponse)) = ans21) = true) then rrr2 := 'dne':else rrr2 := eval(parse(StudentResponse)):end if: if evalb(ANSWER = rrr2) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="&lt;NO RESPONSE&gt;"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect.&nbsp;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer:&nbsp;</mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dne</mi></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dne</mi></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" readonly="true" layout="Heading 1"><Font readonly="true">Maple T.A. Maple-Graded Question</Font></Text-field> Using the Maple-graded question type, you create a question that uses Maple functionality when creating the correct answer and determining if the student's response is correct. The Maple-graded question type has an option to plot the student's response. To include the plot option, insert the corresponding task template. For information on creating questions, making assignments, and exporting Maple T.A. course modules, see Overview of Maple T.A. Integration.
<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field> Algorithmic variables can be used to create a question that contains randomly generated parameters. You can refer to algorithmic variables in the Question, Answer, Hints, and Feedback input regions, as well as in other variable definitions. To reference an algorithmic variable, include "`$variable`" as text or 2-D Math, for example, x := `$myvariable`. Adding algorithmic variables to the question is optional. In the following execution group, add an algorithmic variable. To include more variables, add more execution groups. If your assignment contains more than one algorithmic variable, ensure the variable names are unique. with(RandomTools): randomize(): a3 := Generate(choose([-1,0,1])): b3 := Generate(integer(range=1..9)): c3 := Generate(integer(range=1..9)): f3 := sin(b3*x): g3 := tan(c3*x): h3 := `if`(a3=0,f3/(c3*x),f3/g3^a3): fn3 := Limit(h3,x=0); ans31 := value(fn3); if member(ans31,{`infinity`,-`infinity`,`undefined`}) then ans3 := 'dne'; else ans3 := ans31; end if; LUkmTGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwkKiYtSSRzaW5HRiQ2IywkSSJ4R0YnIiIqIiIiRi8hIiIjRjEiIiYvRi8iIiE= IyIiKiIiJg== IyIiKiIiJg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 4.3</Font></Text-field> The question can contain text, 2-D Math, plots, images, and algorithmic variables. To insert a plot in the question, use the Plot Component. In the following text region, enter the full contents of the question. Calcule `$fn3` Se o limite \303\251 indefinido(ou infinito), entre com a sua resposta como dne. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field> To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`. Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans3;
<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field> To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$RESPONSE` is automatically initialized to the student's response. Using `$ANSWER` (the correct answer) and `$RESPONSE` (the student response), one example of evaluating to a Boolean value is, evalb(`$ANSWER`-`$RESPONSE`=0); Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region. if ((`$RESPONSE` = infinity or `$RESPONSE` = -infinity or `$RESPONSE` = undefined) and evalb(`$RESPONSE` = `$ans31`) = true) then rrr3 := 'dne': else rrr3 := `$RESPONSE`: end if: evalb(`$ANSWER` = rrr3);
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field> A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints. Replace this text with hint 1. Replace this text with hint 2. Replace this text with hint 3.
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field> A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$RESPONSE` (the student response) and `$ANSWER` (the correct answer). Adding feedback to the question is optional. In the following text region, enter feedback to an incorrect response. Replace this text with feedback.
with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea2`, `value`); ANSWER := ans3; if ((eval(parse(StudentResponse)) = infinity or eval(parse(StudentResponse)) = -infinity or eval(parse(StudentResponse)) = undefined) and evalb(eval(parse(StudentResponse)) = ans31) = true) then rrr3 := 'dne':else rrr3 := eval(parse(StudentResponse)):end if: if evalb(ANSWER = rrr3) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="&lt;NO RESPONSE&gt;"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect.&nbsp;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer:&nbsp;</mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer:&nbsp;</mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>9</mn><mn>5</mn></mfrac></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>9</mn><mn>5</mn></mfrac></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" layout="Heading 1">Resumo </Text-field> Limites envolvendo as fun\303\247\303\265es trigonom\303\251tricas fundamentais seno e cosseno s\303\243o f\303\241ceis de ser avaliadas se a divis\303\243o por zero for evitada e n\303\243o exista oscila\303\247\303\243o infinita no ponto de limite. (No Cap\303\255tulo 7 em continuidade vamos aprender que as fun\303\247\303\265es seno e cosseno s\303\243o cont\303\255nuas para todos os n\303\272meros reais.) Quando alguma divis\303\243o por zero \303\251 encontrada, simplifica\303\247\303\265es atrav\303\251s de fatora\303\247\303\265es podem ser tentadas. Em muitos casos, entretanto, isto n\303\243o ser\303\241 suficiente para avaliar um limite. Nestas situa\303\247\303\265es, deve-se lembrar dos dois limites especiais: 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 e 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 Mesmo que estas express\303\265es n\303\243o apare\303\247am explicitamente no limite original, lembre-se de explorar diferentes caminhos de que voc\303\252 j\303\241 deve ser capaz, para conduzir a express\303\243o a uma forma na qual um desses limites possa ser aplicado. Os m\303\251todos usados na avalia\303\247\303\243o rigorosa dos dois limites especiais --- pelos gr\303\241ficos e o Teorema do Confronto --- podem ser usados para outros problemas. Encontrar os limites superiores e inferiores apropriados para o Teorema do Confronto pode ser um desafio. Um gr\303\241fico pode \303\240s vezes ajudar, indicando escolhas razo\303\241veis para os limites de fun\303\247\303\265es.
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Pr\303\263ximo Cap\303\255tulo</Font></Text-field> Examinaremos limites no infinito, limites infinitos, e continuidade antes de concluir com uma das aplica\303\247\303\265es te\303\263ricas essenciais de continuidade, o Teorema do Valor Intermedi\303\241rio.
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