<Text-field style="Heading 1" layout="Heading 1">Introdu\303\247\303\243o</Text-field>

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Cap\303\255tulo 13

Derivadas de Fun\303\247\303\265es Trigonom\303\251tricas

Agora temos todas as ferramentas necess\303\241rias para derivar fun\303\247\303\265es trigonom\303\251tricas. As fun\303\247\303\265es trigonom\303\251tricas fundamentais s\303\243o seno e cosseno. Uma vez conhecidas as derivadas destas, as derivadas da tangente, secante, cotangente e cossecante 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 especiais do Cap\303\255tulo 4 sobre limites 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 maplet DerivativePlot \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 maplet Differentiation continua a ser uma ferramenta \303\272til para o uso das regras de diferencia\303\247\303\243o. 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">\303\215ndice de Termos</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.

amplitude

regra da cadeia

cont\303\255nuo

cossecante

cosseno

cotangente

curva

derivada

diferenci\303\241vel

diferenciada

diferencia\303\247\303\243o

dom\303\255nio

f\303\263rmula do arco duplo

expl\303\255cita

expoente

fator

fun\303\247\303\243o

gr\303\241fico

limite

inclina\303\247\303\243o

m\303\241ximo

m\303\255nimo

negativo

per\303\255odo

regra da pot\303\252ncia

produto

quociente

radiano

taxa de varia\303\247\303\243o

fun\303\247\303\243o racional

secante

conjunto

sinal

seno

inclina\303\247\303\243o

tabelas

tangente

fun\303\247\303\243o trigonom\303\251trica

trabalho

zero

<Text-field style="Heading 1" layout="Heading 1">Inicializa\303\247\303\243o</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Derivadas das Fun\303\247\303\265es de Seno e Cosseno </Text-field>

Os gr\303\241ficos das fun\303\247\303\265es seno e cosseno s\303\243o cont\303\255nuos e t\303\252m uma reta tangente em todos os pontos. Isto significa que as fun\303\247\303\265es de seno e cosseno devem ser diferenci\303\241veis em todo lugar.

A figura 13.1 mostra o resultado da aplica\303\247\303\243o do maplet DerivativePlot para a fun\303\247\303\243o sin(x). Este maplet \303\251 equivalente a construir no Tutor Derivadas acessado atrav\303\251s do menu Ferramentas/Tutores/C\303\241lculo - Uma Vari\303\241vel sob o nome Derivadas...

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Figura 13.1 Aplica\303\247\303\243o do Tutor Derivadas para a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3

Para obter a Figura 13.1, inicie o maplet DerivativePlot, ajuste para a op\303\247\303\243o Fun\303\247\303\243o para 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, a =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIyRidGLy1GLDYtUScmc2RvdDtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRJjAuMGVtRicvRkVGTi1JI21pR0YkNiVRI1BpRicvJSdpdGFsaWNHRjRGLy8lK2V4ZWN1dGFibGVHRjRGLw== e b =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1JI21pR0YkNiVRI1BpRicvJSdpdGFsaWNHRjhGLy1GMzYtUSJ+RidGL0Y2RjlGO0Y9Rj9GQUZDRkVGSC8lK2V4ZWN1dGFibGVHRjhGLw==e ent\303\243o clique Exibir. O gr\303\241fico da derivada da fun\303\247\303\243o seno parece familiar --- com um per\303\255odo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU= , amplitude 1 e um m\303\241ximo em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= --- ele se parece com a fun\303\247\303\243o cosseno. Isto nos leva a escrever

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

Agora, mude a op\303\247\303\243o Fun\303\247\303\243o para 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 e clique Exibir. O Tutor Derivadas aparecer\303\241 agora 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 m\303\255nimo pr\303\263ximo a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEkMS41RidGPkY+RitGPg== e outro zero pr\303\263ximo a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4=; isto parece ser o negativo da fun\303\247\303\243o seno. Baseados nisto, conjecturamos

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

Para determinar se estas conjecturas est\303\243o corretas, retorne \303\240 defini\303\247\303\243o da derivada do senoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiOkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDRi8=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYjNihGKy1GIzYoRistRiM2Jy1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dGOkY1LUYyNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjVGOEY7Rj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS0Zlbi1GIzYmLUYyNi1RIihGJ0Y1L0Y5USV0cnVlRidGOy9GPkZdb0Y/RkFGQ0ZFL0ZIUSwwLjE2NjY2NjdlbUYnL0ZLRmBvLUYjNiQtRiw2JVEieEYnL0ZWRl1vL0Y2USdpdGFsaWNGJ0Y1LUYyNi1RIilGJ0Y1RlxvRjtGXm9GP0ZBRkNGRUZfby9GS1EsMC4xMTExMTExZW1GJ0Y1RitGNS1GMjYtUTEmSW52aXNpYmxlVGltZXM7RidGNUY4RjtGPUY/RkFGQ0ZFRlpGZm4tRiM2JkYrLUYjNipGKy1JJ211bmRlckdGJDYlLUYyNi1RJGxpbUYnRjVGOEY7Rj1GP0ZBL0ZERl1vRkVGWkZhby1GIzYmLUYsNiVRImhGJ0Znb0Zoby1GMjYtUScmcmFycjtGJ0Y1RjhGO0Y9Rj9GQUZDRkVGWkZmbi1JI21uR0YkNiRRIjBGJ0Y1RjUvJSxhY2NlbnR1bmRlckdGOkYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmByLyUqbGluZWJyZWFrR1ElYXV0b0YnRistRiM2JkYrLUkmbWZyYWNHRiQ2KC1GIzYmRistRiM2J0YrLUYjNictRiw2JVEkY29zRidGVUY1RlctRiM2JkZpbi1GIzYkRl9xRjVGam9GNUYrRjUtRjI2LVEoJm1pbnVzO0YnRjVGOEY7Rj1GP0ZBRkNGRS9GSFEsMC4yMjIyMjIyZW1GJy9GS0ZfdC1GZnE2JFEiMUYnRjVGNUYrRjVGaXMvJS5saW5ldGhpY2tuZXNzR0ZjdC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZodC8lKWJldmVsbGVkR0Y6RitGNUYrRjVGK0Y1RitGNS1GMjYtUSIrRidGNUY4RjtGPUY/RkFGQ0ZFRl50RmB0LUYjNihGKy1GIzYnRmRzRldGZ25GK0Y1Rl9wLUYjNiZGKy1GIzYqRitGZnBGK0ZbckYrLUYjNiZGKy1GXHM2KC1GIzYmRistRiM2J0ZSRldGZ3NGK0Y1RitGNUZpc0ZkdEZmdEZpdEZbdUYrRjVGK0Y1RitGNUYrRjVGK0Y1RitGNUYrRjU=

A seguir, olhe para a defini\303\247\303\243o da derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg==:

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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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 e 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

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. A Tabela 13.1, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JkYrLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSNkeEYnRjdGOi1JI21uR0YkNiRRIjBGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdGTEYrRj1GK0Y9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZULyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ== 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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.

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

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKV8kNDEm

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">Derivadas de Outras Fun\303\247\303\265es Trigonom\303\251tricas </Text-field>

As derivadas das fun\303\247\303\265es seno e cosseno juntamente com as Regras do Quociente, Pot\303\252ncia e da Cadeia s\303\243o as ferramentas necess\303\241rias 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.

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkmbWZyYWNHRiQ2KS1JI21vR0YkNi5RMCZEaWZmZXJlbnRpYWxEO0YnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GIzYnRi4tRi82LlEifkYnRjJGNS9GOVEmZmFsc2VGJy9GPEZSL0Y+RlIvRkBGUi9GQkZSL0ZERlIvRkZGUkZHRkotSSNtaUdGJDYmUSJ4RicvJSdpdGFsaWNHUSV0cnVlRidGMi9GNlEnaXRhbGljRidGMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zhby8lKWJldmVsbGVkR0ZSRjItRi82LUZQRjVGUUZTRlRGVUZWRldGWEZHRkotRlo2JVEkdGFuRicvRmhuRlJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZaNiVGZm5GZ25Gam5GNUY1LUYvNi1RIj1GJ0Y1RlFGU0ZURlVGVkZXRlgvRkhRLDAuMjc3Nzc3OGVtRicvRktGZ3AtRiw2KC1GIzYkLUYvNi1GMUY1RlFGU0ZURlVGVkZXRlhGR0ZKRjUtRiM2Ji1GWjYjUSFGJy1GIzYlRl1xRmFwRjVGYXFGNUZcb0Zfb0Zib0Zkby1GLzYtUTEmSW52aXNpYmxlVGltZXM7RidGNUZRRlNGVEZVRlZGV0ZYRkdGSi1GLDYoLUYjNiZGYXEtRiM2Jy1GWjYlUSRzaW5GJ0ZbcEY1LUYvNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjVGUUZTRlRGVUZWRldGWEZHRkotRiM2JC1GXXA2JC1GIzYmRmFxRl9wRmFxRjVGNUY1RmFxRjVGYXFGNS1GIzYmRmFxLUYjNictRlo2JVEkY29zRidGW3BGNUZickZlckZhcUY1RmFxRjVGXG9GX29GYm9GZG9GNQ==

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

A derivada da fun\303\247\303\243o secante pode ser encontrada com as Regras da Pot\303\252ncia e da Cadeia como a seguir

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiZGKy1GIzYnLUYsNiVRJHNpbkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y8LyUmZmVuY2VHRjsvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1GIzYmLUZANi1RIihGJ0Y8L0ZEUSV0cnVlRidGRS9GSEZmbkZJRktGTUZPL0ZSUSwwLjE2NjY2NjdlbUYnL0ZVRmluLUYjNiQtRiw2JVEieEYnL0Y6RmZuL0Y9USdpdGFsaWNGJ0Y8LUZANi1RIilGJ0Y8RmVuRkVGZ25GSUZLRk1GT0Zobi9GVVEsMC4xMTExMTExZW1GJ0Y8RitGPEYrRjwtRiM2JkYrLUYjNictRiw2JVEkY29zRidGOUY8Rj9GVkYrRjxGK0Y8LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZkcC8lKWJldmVsbGVkR0Y7RitGPA==

Note o uso da Regra da Cadeia nesta deriva\303\247\303\243o.

As derivadas das fun\303\247\303\265es cotangente e cossecante podem ser obtidas de forma similar.

Todas as seis derivadas trigonom\303\251tricas est\303\243o resumidas na Tabela 13.2.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHhGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= 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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 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>

Seria muito interessante usar o maplet Differentiation 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="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tRiM2Ji1GOzYtUSIoRidGNy9GP1EldHJ1ZUYnRkAvRkNGV0ZERkZGSEZKL0ZNUSwwLjE2NjY2NjdlbUYnL0ZQRlotRiM2JkYrLUkmbWZyYWNHRiQ2KC1GIzYmLUYsNiVRJyYjOTYwO0YnRjRGNy1GOzYtUTEmSW52aXNpYmxlVGltZXM7RidGN0Y+RkBGQkZERkZGSEZKRkxGTy1GLDYlUSJ0RicvRjVGVy9GOFEnaXRhbGljRidGNy1GIzYkLUkjbW5HRiQ2JFEkMTgwRidGN0Y3LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZkcC8lKWJldmVsbGVkR0Y2RitGNy1GOzYtUSIpRidGN0ZWRkBGWEZERkZGSEZKRlkvRlBRLDAuMTExMTExMWVtRidGN0YrRjdGK0Y3">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</Equation> </Text-field>

A derivada de 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 \303\251 calculada usando a Regra da Cadeia, a qual d\303\241:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHRGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic= 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

O Maple confirma este resultado pela sintaxe:

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

Ly1JJGZ+J0c2IjYjSSJ0R0YlLCQqKCMiIiIiJCE9RistSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKihGKkYrSSNQaUdGMEYrRidGK0YrRitGNUYrISIi

O fator 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 taxa de varia\303\247\303\243o 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, temos que diferenciar um quociente. 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 arco duplo:

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 e 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

a qual fornece:

Simplifica\303\247\303\265es posteriores s\303\243o poss\303\255veis com outra aplica\303\247\303\243o da f\303\263mula do arco duplo para seno:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiQtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GO1Enbm9ybWFsRictRiM2JC1GLDYlUSNkdUYnRjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGPQ== 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3RGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ1RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjctRjs2LVExJkludmlzaWJsZVRpbWVzO0YnRjdGPkZARkJGREZGRkhGSkZMRk8tSSVtc3VwR0YkNiUtRiw2JVEkY3NjRidGNEY3LUkjbW5HRiQ2JFEiMkYnRjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGUUYrRjc=

Em retrospectiva, note que a fun\303\247\303\243o a ser derivada pode ser simplificada usando a f\303\263rmula do arco duplo para seno.

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Use o Differentiation para que a derivada desta express\303\243o concorde com os resultados anteriores deste exemplo.

Usar o Maple para confirmar os c\303\241lculos manuais; al\303\251m disso, ser\303\241 necess\303\241rio dizer ao Maple explicitamente quais identidades trigonom\303\251tricas usar em v\303\241rios pontos ao longo do caminho. Come\303\247amos por anotar estas identidades:

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Ly1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiIyIiIkkidUdGKEYtRi0sJiokKS1GJDYjRi5GLEYtRi0qJCktSSRzaW5HRiVGM0YsRi0hIiI=

A derivada da fun\303\247\303\243o na sua forma original \303\251:

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

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComKS1JJGNvc0dGJTYjSSJ1R0YoIiIjIiIiKS1JJHNpbkdGJTYjLCQqJkYwRjFGL0YxRjFGMCEiIkYvLCYqKkYwRjFGLEYxRjJGOC1GNEYuRjFGOCoqIiIlRjFGK0YxKUYzIiIkRjgtRi1GNUYxRjg=

Combinando os dois termos no lado direito temos:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVElZGYyYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSVkZjJiRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComKS1JJGNvc0dGJTYjSSJ1R0YoIiIjIiIiKS1JJHNpbkdGJTYjLCQqJkYwRjFGL0YxRjFGMCEiIkYvLCQqKCNGMUYwRjFGLEYxKS1GNEYuIiIkRjhGOA==

o que concorda com o resultado obtido manualmente.

Quando a f\303\263mula do arco duplo para seno \303\251 aplicada \303\240 fun\303\247\303\243o antes da deriva\303\247\303\243o, o resultado \303\251:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEkZjJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYnLUYsNiVRKXNpbXBsaWZ5RidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRI2YyRidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRlxvNiYtRiM2J0YrLUYjNictRiw2JVEkaWQyRidGNkY5RmVuLUZcbzYkLUYjNiUtRiw2JVEidUYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmlwRkBGK0ZpcEZARkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGaXBGQEZARmlwRkBGK0ZpcEZARitGaXBGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZpcEZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEjZjJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiw2JVEkZjJhRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA

LyomKS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ1R0YqIiIjIiIiKS1JJHNpbkdGJzYjLCQqJkYtRi5GLEYuRi5GLSEiIiwkKiYjRi4iIiVGLiktRjFGK0YtRjVGLg==

e assim a derivada ser\303\241:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVElZGYyYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSVkZjJkRidGNkY5LyUrZXhlY3V0YWJsZUdGREZARitGWUZARitGWUZA

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComKS1JJGNvc0dGJTYjSSJ1R0YoIiIjIiIiKS1JJHNpbkdGJTYjLCQqJkYwRjFGL0YxRjFGMCEiIkYvLCQqKCNGMUYwRjFGLEYxKS1GNEYuIiIkRjhGOA==

<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Exemplo 13.3: <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW9HRiQ2LVEwJkRpZmZlcmVudGlhbEQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk5GOi1GIzYmRistRiM2JUY2LUYsNiVRJyYjOTUyO0YnLyUnaXRhbGljR0Y/RjpGOkYrRjovJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmluLyUpYmV2ZWxsZWRHRj8tRjc2LVExJkludmlzaWJsZVRpbWVzO0YnRjpGPUZARkJGREZGRkhGSkZMRk8tRiM2Ji1GNzYtUSIoRidGOi9GPlEldHJ1ZUYnRkAvRkNGZ29GREZGRkhGSi9GTVEsMC4xNjY2NjY3ZW1GJy9GUEZqby1GIzYoRistRiM2Jy1GLDYlUSRzaW5GJ0ZYRjotRjc2LVEwJkFwcGx5RnVuY3Rpb247RidGOkY9RkBGQkZERkZGSEZKRkxGTy1GIzYmRmNvLUYjNiRGVUY6LUY3Ni1RIilGJ0Y6RmZvRkBGaG9GREZGRkhGSkZpby9GUFEsMC4xMTExMTExZW1GJ0Y6RitGOkZeby1GIzYnLUYsNiVRJHRhbkYnRlhGOkZjcC1GIzYmRmNvLUYjNiZGKy1GIzYmRistSSVtc3VwR0YkNiVGVS1JI21uR0YkNiRRIjJGJ0Y6LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjpGK0Y6RmpwRjpGK0Y6RitGOkZqcEY6RitGOkYrRjo=">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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 e a Regra da Cadeia. O resultado \303\251:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRidGKy1JI21vR0YkNi1RIj1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRy1GIzYoRistRiM2KEYrLUYjNiYtRjA2LVEiKEYnRjMvRjdRJXRydWVGJ0Y5L0Y8RlRGPUY/RkFGQy9GRlEsMC4xNjY2NjY3ZW1GJy9GSUZXLUYjNihGKy1JJm1mcmFjR0YkNigtRiM2JC1GMDYtUTAmRGlmZmVyZW50aWFsRDtGJ0YzRjZGOUY7Rj1GP0ZBRkMvRkZRJjAuMGVtRicvRklGXm9GMy1GIzYmRistRiM2JUZqbi1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dGOEYzRjNGK0YzLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZecC8lKWJldmVsbGVkR0Y4LUYwNi1RMSZJbnZpc2libGVUaW1lcztGJ0YzRjZGOUY7Rj1GP0ZBRkNGXW9GX28tRiM2Jy1GLDYlUSRzaW5GJ0Znb0YzLUYwNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjNGNkY5RjtGPUY/RkFGQ0Zdb0Zfby1GIzYmRlAtRiM2JEZkb0YzLUYwNi1RIilGJ0YzRlNGOUZVRj1GP0ZBRkNGVi9GSVEsMC4xMTExMTExZW1GJ0YzRitGM0YrRjNGYnFGM0ZjcC1GIzYnLUYsNiVRJHRhbkYnRmdvRjNGW3EtRiM2JkZQLUYjNiZGKy1GIzYmRistSSVtc3VwR0YkNiVGZG8tSSNtbkdGJDYkUSIyRidGMy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0YzRitGM0ZicUYzRitGM0YrRjMtRjA2LVEiK0YnRjNGNkY5RjtGPUY/RkFGQy9GRlEsMC4yMjIyMjIyZW1GJy9GSUZgcy1GIzYoRitGZnBGY3AtRiM2JkYrLUYjNihGK0ZlbkZjcEZncUYrRjNGK0YzRitGM0YrRjNGK0Yz

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

A computa\303\247\303\243o desta derivada no Maple poderia ser feita como se segue.

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

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComLUkkc2luR0YlNiNJJnRoZXRhR0YoIiIiLUkkdGFuR0YlNiMqJClGLiIiI0YvRi9GLiwmKiYtSSRjb3NHRiVGLUYvRjBGL0YvKipGNUYvRitGLywmRi9GLyokKUYwRjVGL0YvRi9GLkYvRi8=

Esta resposta difere da nossa somente na apar\303\252ncia. Note que o Maple informa:

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Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JJHRhbkdGJTYjSSJ4R0YoRi0sJiIiIkYvKiQpRioiIiNGL0Yv

Para fazer com que o Maple recoloque as pot\303\252ncias de tangente com secante, fornecemos uma quarta identidade trigonom\303\251trica, a saber,

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

LyokKS1JJHRhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YqIiIjIiIiLCYqJCktSSRzZWNHRidGK0YtRi5GLkYuISIi

Assim, fornecendo esta identidade para simplificar como uma rela\303\247\303\243o adicional:

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

Ly1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JJHRhbkdGJTYjSSJ4R0YoRi0qJCktSSRzZWNHRiVGLCIiIyIiIg==

Aplicando a mesma abordagem para este exemplo,

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Exceto pela separa\303\247\303\243o dos termos no fator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTUyO0YnLyUnaXRhbGljR0Y+RjVGNUYrRjU=, este \303\251 exatamente o nosso resultado.

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</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(): r1 := Generate(choose([-10..-1,1..10])): n1 := Generate(integer(range=r1)): n2 := Generate(integer(range=-10..10)): gg := n1*x+n2: fn8 := Generate(choose([sin(gg),cos(gg),tan(gg),csc(gg),sec(gg),cot(gg)])); ans1 := diff(fn8,x);

LCQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmSSJ4R0YoIiIpIiIiRi0hIiI=

LCQtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmSSJ4R0YoIiIpIiIiRi0hIik=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 13.1</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.

Usando as regras de derivada apropriadas, encontre a derivada de: f(x) = `$fn8` (Nota: Voc\303\252 deve inserir uma resposta antes de clicar em "Grade".)

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;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

To evaluate a student's response, you must evalute to a Boolean value (true or false). The variable `$RESPONSE` is automatically initialized to the student's response. Using `$ANSWER` (the correct answer) and `$RESPONSE` (the student response), one example of evaluating to a Boolean value is, evalb(`$ANSWER`-`$RESPONSE`=0); Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region.

evalb(simplify(`$ANSWER`-`$RESPONSE`)=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional. Enter your hints in the following table. You can provide up to three hints.

Use the derivatives of trigonometric functions.

Use the Chain Rule.

Try to simplify your answer after differentiating.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea2`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the derivatives of trigonometric functions.</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(`TextArea2`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the Chain Rule.</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(`TextArea2`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Try to simplify your answer after differentiating.</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(`TextArea2`, `value`); ANSWER := ans1; if evalb(simplify(ANSWER-eval(parse(StudentResponse)))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>8</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><mn>8</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mrow></mfenced></mrow></mrow></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>8</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><mn>8</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mrow></mfenced></mrow></mrow></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): n3 := Generate(integer(range=2..5)): fn6 := sort(Generate(polynom(integer(range=-10..10),x,degree=n3))): fn7 := Generate(choose([sin(fn6),cos(fn6),tan(fn6),sec(fn6),csc(fn6),cot(fn6)])); ans2 := diff(fn7,x);

LCQtSSR0YW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywsKiRJInhHRigiIiUiIicqJEYsIiIkISIoKiRGLCIiIyEiIkYsRjEiIioiIiJGNA==

LCQqJiwmIiIiRiUqJC1JJHRhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCwqJEkieEdGLCIiJSIiJyokRjAiIiQhIigqJEYwIiIjISIiRjBGNSIiKkYlRjdGJUYlLCpGMyIjQ0Y2ISNARjAhIiNGNUYlRiVGOA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 13.2</Text-field>

Usando as regras de derivada apropriadas, encontre a derivada de: f(x) = `$fn7` (Nota: Voc\303\252 deve inserir uma resposta antes de clicar em "Grade".)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans2;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(simplify(`$ANSWER`-`$RESPONSE`)=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

Use the derivatives of trigonometric functions.

Use the Chain Rule.

Try to simplify your answer after differentiating.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea1`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the derivatives of trigonometric functions.</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><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the Chain Rule.</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><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Try to simplify your answer after differentiating.</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>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea1`, `value`); ANSWER := ans2; if evalb(simplify(ANSWER-eval(parse(StudentResponse)))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msup><mrow><mi mathvariant='normal'>tan</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>9</mn></mrow></mrow></mfenced></mrow><mn>2</mn></msup></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mn>24</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>21</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>7</mn></mrow></mfenced></mrow></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msup><mrow><mi mathvariant='normal'>tan</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>9</mn></mrow></mrow></mfenced></mrow><mn>2</mn></msup></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mn>24</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>21</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>7</mn></mrow></mfenced></mrow></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): fn1 := Generate(choose([x,x^2,x^3])); fn2 := Generate(choose([x,x^2,x^3])); fn3 := Generate(choose([fn1,fn2,tan(fn1), sin(fn1), cos(fn1), sec(fn1), csc(fn1), cot(fn1)])); fn4 := Generate(choose([tan(fn2), sin(fn2), cos(fn2), sec(fn2), csc(fn2), cot(fn2)])); fn5 := Generate(choose([fn3*fn4,fn3/fn4,fn4/fn3]));

KiRJInhHNiIiIiQ=

LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJEkieEdGJyIiJA==

KiZJInhHNiIiIiQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyokRiNGJSEiIg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 13.3</Text-field>

Usando as regras de derivada apropriadas, encontre a derivada de:

f(x) = `$fn5` (Nota: Voc\303\252 deve inserir uma resposta antes de clicar em "Grade".)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=simplify(diff(fn5,x));

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(simplify(`$ANSWER`-`$RESPONSE`)=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

Remember to use the chain rule, product rule, and quotient rule, if necessary.

Try to simplify the function before differentiating.

Try to simplify your expression after differentiating.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea0`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Remember to use the chain rule, product rule, and quotient rule, if necessary.</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><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Try to simplify the function before differentiating.</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><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Try to simplify your expression after differentiating.</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>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := simplify(diff(fn5,x)); if evalb(simplify(ANSWER-eval(parse(StudentResponse)))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow></mfenced></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow></mfenced></mrow></mrow></mrow></mfenced></mrow><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msup><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow></mfenced></mrow><mn>2</mn></msup></mrow></mrow></mrow></mfrac></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow></mfenced></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow></mfenced></mrow></mrow></mrow></mfenced></mrow><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msup><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mrow></mfenced></mrow><mn>2</mn></msup></mrow></mrow></mrow></mfrac></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" readonly="true" layout="Heading 1">Maple T.A. Maple-Graded Question</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Algorithmic Variables</Text-field>

with(RandomTools): randomize(): n4 := Generate(choose([2,3])): fn9 := sort(Generate(polynom(integer(range=-10..10),x,degree=n4))): fn10 := Generate(choose([sin(fn9),cos(fn9),tan(fn9),sec(fn9),csc(fn9),cot(fn9)])): fn11 := Generate(choose([sin(fn10),cos(fn10),tan(fn10),sec(fn10),csc(fn10),cot(fn10)])); ans4 := simplify(diff(fn11,x));

LUkkY3NjRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSSRjb3NHRiQ2IywoKiRJInhHRiciIiMiIiJGLiIiKCEiKkYw

LCQqKiwmSSJ4RzYiIiIjIiIoIiIiRiktSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2Iy1GKzYjLCgqJEYlRidGKUYlRighIipGKUYpLUkkc2luR0YsRjFGKSwmISIiRikqJEYqRidGKUY4Rjg=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Quest\303\243o 13.4</Text-field>

Usando as regras de derivada apropriadas, encontre a derivada de: f(x) = `$fn11` (Nota: Voc\303\252 deve inserir uma resposta antes de clicar em "Grade".)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ans4;

<Text-field style="Heading 2" layout="Heading 2">Evaluate Student's Response</Text-field>

evalb(simplify(`$ANSWER`-`$RESPONSE`)=0);

<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>

Use the derivatives of trigonometric functions.

Use the Chain Rule.

Try to simplify your answer after differentiating.

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the derivatives of trigonometric functions.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Use the Chain Rule.</mtext><mtext>&NewLine;</mtext></mrow></math>", "`\134$RESPONSE`", response); with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Hint", [[MathMLViewer('value' = hint, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): Maplets[Display](maplet);

with(StringTools): with(DocumentTools): response := GetProperty(`TextArea3`, `value`); if response = `` then response := "</mtext><mtext><NO RESPONSE></mtext><mtext>"; fi; hint := SubstituteAll("<math><mrow><mtext>Try to simplify your answer after differentiating.</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>

Replace this text with feedback.

with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea3`, `value`); ANSWER := ans4; if evalb(simplify(ANSWER-eval(parse(StudentResponse)))=0) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!&NewLine;</mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext></mrow></math>"); else if StudentResponse = "" then StudentResponse:="<NO RESPONSE>"; fi; feedback:=cat("<math><mrow><mtext mathvariant='bold'>Your answer is incorrect. </mtext><mtext>&NewLine;</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, "<mtext mathvariant='bold'>Correct Answer: </mtext><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>7</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>9</mn></mrow></mrow></mfenced></mrow></mrow></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>9</mn></mrow></mrow></mfenced></mrow></mrow><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msup><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>9</mn></mrow></mrow></mfenced></mrow></mrow></mfenced></mrow><mn>2</mn></msup></mrow></mrow></mrow></mfrac></mrow></mrow></mstyle></math><mtext>&NewLine;</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext>&NewLine;</mtext></mrow>", "`$RESPONSE`", StudentResponse)); feedback:=SubstituteAll(feedback, "`$ANSWER`", "<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>7</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>9</mn></mrow></mrow></mfenced></mrow></mrow></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>9</mn></mrow></mrow></mfenced></mrow></mrow><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msup><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>⁡</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mi>x</mi></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mn>9</mn></mrow></mrow></mfenced></mrow></mrow></mfenced></mrow><mn>2</mn></msup></mrow></mrow></mrow></mfrac></mrow></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;

<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Resumo</Text-field>

As fun\303\247\303\265es trigonom\303\251tricas s\303\243o diferenci\303\241veis em todos os pontos em seus dom\303\255nio. 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">Pr\303\263ximo Cap\303\255tulo</Text-field>

Agora, voc\303\252 deveria estar bem confort\303\241vel usando o maplet Differentiation 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 da Pot\303\252ncia ser\303\241 verificada para todos os expoentes racionais e aprenderemos a encontrar a inclina\303\247\303\243o da reta tangente a qualquer curva - 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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