C\303\241lculo via Web 2.0Cap\303\255tulo 20: Teorema do Valor M\303\251dio
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Introdu\303\247\303\243o</Font></Text-field>O Teorema do Valor M\303\251dio (TVM) (Mean Value Theorem) (MVT) \303\251 um dos marcos em c\303\241lculo. A natureza geom\303\251trica de TVM o torna mais f\303\241cil de acreditar e entender. Alguns dos mais importantes usos do TVM n\303\243o \303\251 t\303\243o geom\303\251trico e requer mais esfor\303\247o para compreender. Neste cap\303\255tulo o TVM ser\303\241 definido (estabelecido). No lugar de uma prova , exemplos ser\303\243o dados para mostrar como o TVM pode falhar se as hip\303\263teses (hypotheses) n\303\243o forem sarisfeitas (satisfied). Exemplos adicionais ilustram alguns resultados que s\303\243o baseados no TVM. O exemplo final \303\251 o fundamento para discuss\303\243o de antiderivadas na Unidade 4.O maplet Teorema do Valor M\303\251dio (MeanValueTheorem maplet) (ou o inserido no Tutor de Teorema do Valor M\303\251dio (Mean Value Theorem Tutor)) prov\303\252 uma \303\263tima ferramenta de visualiza\303\247\303\243o para a geometria b\303\241sica por tr\303\241s do TVM.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Cap\303\255tulo Gloss\303\241rio</Font></Text-field>Os seguintes termos, neste cap\303\255tulo, t\303\252m conex\303\243o com o Dicion\303\241rio de Matem\303\241tica Maple (Maple Math Dictionary).
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Inicializa\303\247\303\243o</Font></Text-field>restart;
with( plots ):
with( Student[Calculus1] ):
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Procedimentos Auxiliares</Text-field>InInterval := proc(x,S::list(numeric))
if has(x,I) then return false end if;
return evalb( evalf(x)>=op(1,S) and evalf(x)<=op(2,S) )
end proc:
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Teorema do Valor M\303\251dio (TVM)</Font></Text-field>Relembrando o Teorema 8.1, o Teorema do Valor Intermedi\303\241rio (TVI) (Intermediate Value Theorem) (IVT), ou seja, que uma fun\303\247\303\243o cont\303\255nua (continuousfunction) em um intervalo (interval) limitado e fechado (closed) atinge qualquer valor (value) entre os valores nos limites (do intervalo) para pelo menos um ponto no intervalo. O Teorema do Valor M\303\251dio (Mean Value Theorem) \303\251 similar ao Teorema do Valor Intermedi\303\241rio exceto que o TVM diz que existe pelo menos um ponto no interior do intervalo onde a primeira derivada (derivative) \303\251 igual a inclina\303\247\303\243o da reta secante (secant line) entre os limites do intervalo.
Teorema 20.1: Teorema do Valor M\303\251dio Se a fun\303\247\303\243o f \303\251 cont\303\255nua em 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 com derivada cont\303\255nua em 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, existe pelo menos um n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn em 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 para o quala inclina\303\247\303\243o da reta secante 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 = inclina\303\247\303\243o da reta tangente (tangent) em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImNGJ0Y0RjcvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZXRj4=A forma simbolicamente equivalente deste enunciado \303\251 expressa pela equa\303\247\303\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
Para a fun\303\247\303\243o 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 no intervalo 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, Figura 20.1, obtida do comando MeanValueTheorem (Teorema do Valor M\303\251dio) no pacote Student[Calculus1] (Estudante[C\303\241lculo]), d\303\241 uma interpreta\303\247\303\243o visual do TVM.
MeanValueTheorem(x^3 - x, x=-2..2, output = plot);Figura 20.1 Ilustra\303\247\303\243o do Teorema do Valor M\303\251dio
Neste exemplo existem dois pontos no intervalo 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 onde a inclina\303\247\303\243o da reta secante de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRKyZ1bWludXMwOzJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSsmdW1pbnVzMDs2RidGOC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEY4RitGWEY4 a ( 2, 6 ) \303\251 paralela \303\240 reta tangente ao gr\303\241fico (graph) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNihGKy1GIzYnRistSSVtc3VwR0YkNiUtRiw2JVEieEYnRjRGNy1JI21uR0YkNiRRIjNGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y+LUY7Ni1RKCZtaW51cztGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGZm9GZW5GX29GPkYrRl9vRj5GK0Zfb0Y+ . A reta secante \303\251 mostrada como uma reta azul e as duas retas tangentes s\303\243o mostradas em preto.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Exemplo 20.1: Um Enfoque nas Hip\303\263teses do TVM </Font></Text-field>No lugar de uma prova (proof) de TVM, esta cole\303\247\303\243o de exemplos ilustra como o TVM pode falhar quando as hip\303\263teses n\303\243o forem satisfeitas.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Caso 1: Hip\303\263tese n\303\243o satisfeita: Fun\303\247\303\243o cont\303\255nua mas n\303\243o diferenci\303\241vel</Font> </Text-field>Considere a fun\303\247\303\243o cuja regra \303\251f1 := abs(x):
a1 := -2:
b1 := 5:
f(x) = f1;no intervalo 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A reta secante entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRKyZ1bWludXMwOzJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSIyRidGOC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEY4RitGWEY4 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIjVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y0LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y4RjhGK0ZVRjg= tem inclina\303\247\303\243omsec1 := (eval(f1,x=b1)-eval(f1,x=a1))/(b1-a1):
m[secant] = msec1;Para ver que n\303\243o existe nenhum ponto onde a linha tangente ao gr\303\241fico de 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 tem inclina\303\247\303\243o 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 , considere a Figura 20.1, um gr\303\241fico de 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 e v\303\241rias retas com inclin\303\247\303\243o 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 .
Figura 20.2 Prova visual de que TVM n\303\243o se aplica \303\240 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 em 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
Algebricamente, a derivada da fun\303\247\303\243o m\303\263dulo (ou valor absoluto) (absolute value) \303\251df1 := convert( diff(f1,x), piecewise, x ):
`f '`(x) = df1;Como os \303\272nicos valores desta derivada s\303\243o +1, n\303\243o existe nenhum ponto onde 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiUtSSNtbkdGJDYkUSIzRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y4LUYjNiUtRjU2JFEiN0YnRjhGO0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZILyUpYmV2ZWxsZWRHUSZmYWxzZUYnRitGO0Y4 .Isto n\303\243o contradiz o TVM porque a fun\303\247\303\243o m\303\263dulo(ou valor absoluto) n\303\243o \303\251 diferenci\303\241vel (differentiable) em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4= e assim, n\303\243o \303\251 diferenci\303\241vel em ( -2, 5 ).Quando o Teorema do Valor M\303\251dio (MeanValueTheorem) do Maple encontra este problema, ele responde com uma mensagem de erro apropriada , como se v\303\252 em MeanValueTheorem( f1, a1..b1, output=plot );
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Caso 2: <Hyperlink linktarget="Dictionary:Definition,hypothesis" hyperlink="true"><Font style="Heading 2" encoding="UTF-8">Hip\303\263tese n\303\243o satisfeita</Font></Hyperlink><Font encoding="UTF-8">: A fun\303\247\303\243o n\303\243o \303\251 cont\303\255nua</Font></Text-field>Considere a fun\303\247\303\243o cuja regra \303\251f2 := piecewise( x<1, x+1, x>=1, x-1 ):
a2 := 0:
b2 := 2:
f(x) = f2;no intervalo [ 0, 2 ].A reta secante entre ( 0, 1 ) e ( 2, 1 ) tem inclina\303\247\303\243omsec2 := (eval(f2,x=b2)-eval(f2,x=a2))/(b2-a2):
m[secant] = msec2;Para ver que n\303\243o existe nenhum ponto neste intervalo onde a reta tangente ao gr\303\241fico de 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 tem inclina\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=, considere a Figure 20.3, um gr\303\241fico de 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 e v\303\241rias retas com inclina\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=.
Figura 20.3 Prova visual de que o TVM n\303\243o n\303\243o se aplica \303\240 fun\303\247\303\243o descont\303\255nua e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y5 em 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
Algebricamente, a derivada desta fun\303\247\303\243o definida por partes (piecewise-defined) \303\251df2 := convert( diff(f2,x), piecewise, x ):
`f '`(x) = df2;Como esta derivada assume somente o valor +1, n\303\243o existe nenhum ponto onde 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 = 0.Isto n\303\243o contradiz o TVM porque esta fun\303\247\303\243o definida por partes (piecewise-defined) n\303\243o \303\251 cont\303\255nua em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4= .Quando o Teorema do Valor M\303\251dio (MeanValueTheorem) do Maple encontra este problema, ele responde com uma mensagem de erro apropriada , como se v\303\252 MeanValueTheorem( f2, a2..b2, output=plot );
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7">Caso 3: <Hyperlink linktarget="Dictionary:Definition,hypothesis" hyperlink="true"><Font style="Heading 2" encoding="UTF-8">Hip\303\263tese satisfeita</Font></Hyperlink></Text-field>Considere a fun\303\247\303\243o cuja regra \303\251f3 := sin(x)^2:
a3 := 0:
b3 := 3*Pi/2:
f(x) = f3;no intervalo 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 reta secante entre ( 0, 0 ) e ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KEYrLUkmbWZyYWNHRiQ2KC1GIzYnLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y6LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItRiw2JVEnJiM5NjA7RicvJSdpdGFsaWNHRkNGOi8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOi1GIzYlLUY3NiRRIjJGJ0Y6RlpGOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYW8vJSliZXZlbGxlZEdGQy1GPjYtUSIsRidGOkZBL0ZFUSV0cnVlRidGRkZIRkpGTEZORlAvRlRRLDAuMzMzMzMzM2VtRictRjc2JFEiMUYnRjpGWkY6RitGWkY6) tem inclina\303\247\303\243omsec3 := (eval(f3,x=b3)-eval(f3,x=a3))/(b3-a3):
m[secant] = msec3;Para ver que existe pelo menos um ponto neste intervalo onde a tangente ao gr\303\241fico de 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 tem inclina\303\247\303\243o 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 , considere a Figura 20.4, um gr\303\241fico de 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 e v\303\241rias retas com inclina\303\247\303\243o 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 .
Figura 20.4 Prova visual de que TVM se aplica \303\240 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 em 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
Neste gr\303\241fico parece existir (pelo menos) dois pontos onde a reta tangente tem a mesma inclina\303\247\303\243o que a da reta secante . Para encontrar estes pontos algebricamente, obtem-se a derivada, que \303\251df3 := diff(f3,x):
`f '`(x) = df3;O TVM garante que existe pelo menos uma solu\303\247\303\243o (solution) para a equa\303\247\303\243oMVTeq := df3 = msec3;Trabalhando a m\303\243o \303\251 necess\303\241rio avaliar 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 e colocar os resultados em seus pr\303\263prios quadrantes (quadrants). Isto n\303\243o \303\251 uma tarefa trivial, ent\303\243o chamamos o Maple, que forneceq1 := solve( MVTeq, x ):
q1;Existem (quatro?)solu\303\247\303\265es em [ LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y6RjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjVGK0ZRRjU= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGMg== ]; Pela periodicidade, existem mais quatro solu\303\247\303\265es em [ LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGMg== , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJyYjOTYwO0YnLyUnaXRhbGljR0Y+RjUvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjVGK0ZVRjU= ]. A representa\303\247\303\243o com ponto flutuante (Floating-point) de todas as oito solu\303\247\303\265es em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi1RIltGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJXRydWVGJy8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGSi1GIzYpRistRiM2Ji1GMjYtUSomdW1pbnVzMDtGJ0Y1L0Y5Rj1GOy9GP0Y9RkBGQkZERkYvRklRJjAuMGVtRicvRkxGVy1GLDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGPUY1LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y1LUYyNi1RIixGJ0Y1RlQvRjxGOkZVRkBGQkZERkZGVi9GTFEsMC4zMzMzMzMzZW1GJy1GIzYnLUkjbW5HRiQ2JFEiM0YnRjUtRjI2LVExJkludmlzaWJsZVRpbWVzO0YnRjVGVEY7RlVGQEZCRkRGRkZWRlhGWUZobkY1RitGaG5GNS1GMjYtUSJdRidGNUY4RjtGPkZARkJGREZGRkgvRkxRLDAuMTExMTExMWVtRidGaG5GNUYrRmhuRjU= s\303\243oq2 := evalf([q1, seq(z+2*Pi,z=q1)]):
q2;e os valores que est\303\243o em ( 0, 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 ) s\303\243oMVTc := select( InInterval, q2, [0,evalf(3*Pi/2)] ):
MVTc;Logo, existem quatro valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn satisfazendo 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 = 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 .Quando o Teorema do Valor M\303\251dio (MeanValueTheorem) do Maple encontra este problema, ele pode retornar ou ao gr\303\241fico na Figura 20.5
MeanValueTheorem( f3, a3..b3, output=plot );
Figura 20.5 Comando MeanValueTheorem (Teorema do Valor M\303\251dio)do Maple aplicado a uma fun\303\247\303\243o com m\303\272ltiplos valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn
ou os valores expl\303\255citos (explicit) de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn garantidos pelo TVM, a saber,q3 := MeanValueTheorem( f3, a3..b3, output=points ):
q3;A forma com ponto flutuante destas solu\303\247\303\265es seriam evalf( q3 );
<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):
with(Student[Calculus1]):
randomize():
nn1 := Generate(integer(range=1..4)):
nn2 := Generate(integer(range=2..5)):
while (nn1=nn2) do nn2 := Generate(integer(range=2..5)) od:
nn3 := Generate(integer(range=1..8)):
aa_ := Generate(integer(range=-4..-1)),0,-nn2,0,nn3+1:
bb_ := Generate(integer(range=1..4)),2^nn2,nn2^2-nn2,2,simplify((nn3)+64^(1/(nn1))):
fn_[1] := sort(Generate(polynom(integer(range=-5..5),x,degree=3)),x):
fn_[2]:= x^(nn1/nn2):
fn_[3] := x*sqrt(x+(nn2)):
fn_[4] := sin((nn3)/2*Pi*x):
fn_[5] := (x-(nn3))^(-(nn1)):
nn := Generate(integer(range=1..5)):
fn := fn_[nn];
aa := aa_[nn];
bb := bb_[nn];
ptsC := MeanValueTheorem( (fn), x=(aa)..(bb), output=points):
if (ptsC=[]) then ptsC := MeanValueTheorem( (fn), x=(aa)..(bb), output=points, numeric=true) fi:
ans := convert(ptsC,set);JSFH
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Quest\303\243o 20.1</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.Seja f(x) = `$fn`.Encontre todos os n\303\272meros c que satisfa\303\247am o Teorema do Valor M\303\251dio no intervalo [`$aa`,`$bb`].Entre com sua responta como c ou c1,c2,c3,etc., com precis\303\243o de at\303\251 5 d\303\255gitos significativos. Se o TVM n\303\243o se aplica, sua resposta deve ser a \303\272nica paravra: none(nenhuma)JSFH
<Text-field style="Heading 2" layout="Heading 2">Answer</Text-field>To evaluate a student's response, enter the correct answer. Complete each line of code with a semicolon. The last line must evaluate to an expression that will be stored as the variable `$ANSWER`.
Enter Maple code in 1-D Math that evaluates to the correct answer. Add additional execution groups as needed.JSFHans;
<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`-`$RESPONSE`=0);
Enter Maple code to evaluate a student's response. To add additional Maple functionality, use the Answer input region.test_set_numeric:=proc()
local a, aa, dd, rr, r;
if nargs<2 then
return false
else
a:=args[1];
if not type(a,set) then
if type(a,list) then convert(a,set) else a := {a} end if;
end if:
if nargs=2 then
r:=args[2];
if not type(r,set) then
if type(r,list) then convert(r,set) else r := {r} end if;
end if:
else
r:={args[2..-1]}
end if:
end if:
if nops(a)<>nops(r) then return false end if;
aa := {seq( fnormal(evalf(1.0000*z),4,Float(5,-4)), z=a )};
rr := {seq( fnormal(evalf(1.0000*z),4,Float(5,-4)), z=r )};
aa := sort(convert(aa,list));
rr := sort(convert(rr,list));
dd := convert(zip( (x,y)->fnormal(evalf(x-y),4,Float(5,-4)), aa, rr ),set);
return evalb(nops(dd)=1 and dd[1]=0.);
end proc:
test_set_numeric(`$ANSWER`,`$RESPONSE`);JSFH
<Text-field style="Heading 2" layout="Heading 2">Hints</Text-field>A question can offer hints to students when requested. Hints can contain text, 2-D Math, and algorithmic variables. Adding hints to the question is optional.
Enter your hints in the following table. You can provide up to three hints.
Replace this text with hint 1.Replace this text with hint 2.Replace this text with hint 3.
<Text-field style="Heading 2" layout="Heading 2">Feedback</Text-field>A question can display feedback when a student provides an incorrect response. The feedback can contain text, 2-D Math, and algorithmic variables. The feedback can also refer to `$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.with(StringTools): with(DocumentTools): feedback:=""; StudentResponse := GetProperty(`TextArea0`, `value`); ANSWER := ans; test_set_numeric:=proc() local a, aa, dd, rr, r; if nargs<2 then return false else a:=args[1]; if not type(a,set) then if type(a,list) then convert(a,set) else a := {a} end if; end if: if nargs=2 then r:=args[2]; if not type(r,set) then if type(r,list) then convert(r,set) else r := {r} end if; end if: else r:={args[2..-1]} end if: end if: if nops(a)<>nops(r) then return false end if; aa := {seq( fnormal(evalf(1.0000*z),4,Float(5,-4)), z=a )}; rr := {seq( fnormal(evalf(1.0000*z),4,Float(5,-4)), z=r )}; aa := sort(convert(aa,list)); rr := sort(convert(rr,list)); dd := convert(zip( (x,y)->fnormal(evalf(x-y),4,Float(5,-4)), aa, rr ),set); return evalb(nops(dd)=1 and dd[1]=0.); end proc: if test_set_numeric(ANSWER,eval(parse(StudentResponse))) then feedback:=cat("<math><mrow><mtext mathvariant='bold'>Correct!
</mtext><mtext>
</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>
</mtext><mtext mathvariant='bold'>Your Answer: </mtext><mtext>", StudentResponse, "</mtext><mtext>
</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><mfenced open='{' close='}' separators=','><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mn>10</mn><mn>9</mn></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>4</mn><mn>9</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msqrt><mn>13</mn></msqrt></mrow></mrow></mrow></mrow></mfenced></mrow></mstyle></math><mtext>
</mtext>"); feedback:=cat(feedback, SubstituteAll("<mrow><mtext></mtext><mtext> </mtext><mtext></mtext><mtext>
</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><mfenced open='{' close='}' separators=','><mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>−</mo><mfrac><mn>10</mn><mn>9</mn></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>4</mn><mn>9</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>⁢</mo><mrow><msqrt><mn>13</mn></msqrt></mrow></mrow></mrow></mrow></mfenced></mrow></mstyle></math>"); feedback:=cat(feedback, "</mrow></math>"); fi; with(Maplets[Elements]): maplet := Maplet(Window('title' = "Maple TA Feedback", [[MathMLViewer('value' = feedback, 'fontsize'='12', 'width'='600', 'height'='200', 'breakwidth'='600')], [Button("OK", Shutdown())]])): if feedback <> "" then Maplets[Display](maplet); fi;
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Exemplo 20.2: Usando o Maplet Teorema do Valor M\303\251dio </Font></Text-field>O maplet Teorema do Valor M\303\251dio (MeanValueTheorem maplet) (ou equivalentemente, o inserido em Mean Value Theorem Tutor) \303\251 o mais apropriado para encontrar -- visualmente, simbolicamente, and numericamente -- todos os pontos em um intervalo onde a inclina\303\247\303\243o da reta \303\251 igual a inclina\303\247\303\243o da reta secante entre os dois pontos limites to intervalo. Para ilustrar, deixe f2 := x^2 * sqrt( 5-x ):
f(x) = f2;ser a regra para a fun\303\247\303\243o cujo dom\303\255nio (domain) \303\251 o intervalo 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. Note que o dom\303\255nio desta fun\303\247\303\243o poderia ser estendido 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, assim a fun\303\247\303\243o definida \303\251 cont\303\255nua em um intervalo 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. A derivada ,df2 := simplify( diff( f2, x ) ):
`f '`(x) = df2;\303\251 definida para x 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; assim f \303\251 claramente diferenci\303\241vel no intervalo abertoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRKyZ1bWludXMwOzVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSI1RidGOC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEY4RitGWEY4. Logo, as hip\303\263teses TVM s\303\243o satisfeitas e existir\303\241 pelo menos um n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn emLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRKyZ1bWludXMwOzVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSI1RidGOC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEY4RitGWEY4 para o 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 = 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\303\251 verdadeira. A manipula\303\247\303\243o alg\303\251brica que resolve (solve) esta equa\303\247\303\243o \303\251 bastante tediosa. Em lugar destes c\303\241lculos, estabelece-se o maplet Teorema do Valor M\303\251dio (MeanValueTheorem maplet). (Veja a Figura 20.6.)
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20.6 Mean Value Theorem Tutor (Tutor Teorema do Valor M\303\251dio)aplicado \303\240 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ4RidGNEY3LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y+Rj5GZm5GPi1GOzYtUSI9RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjI3Nzc3NzhlbUYnL0ZTRl1vLUklbXN1cEdGJDYlRlktSSNtbkdGJDYkUSIyRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKy1JJm1zcXJ0R0YkNiMtRiM2Jy1GY282JFEiNUYnRj4tRjs2LVEoJm1pbnVzO0YnRj5GQC9GREY2RkVGR0ZJRktGTUZPL0ZTUSwwLjMzMzMzMzNlbUYnRllGZm5GPkYrRmZuRj4= on 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
Na linha superior do maplet, entre com a Fun\303\247\303\243o (Function) ( x^2*sqrt(5-x) ) e os limites do intervalo ( a = -5 e b = 5 ). Clique no bot\303\243o Display (exibir). Observe que o gr\303\241fico da fun\303\247\303\243o tem uma reta tangente vertical em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNUYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4= . O mais relevante para este problema \303\251 a observa\303\247\303\243o de que existem duas localiza\303\247\303\265es no gr\303\241fico de 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 onde a reta tangente \303\251 paralela \303\240 reta secante 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 a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIjVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSIwRidGOC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEY4RitGWEY4. O maplet encontra estes pontos algebricamente e simbolicamente. A forma alg\303\251brica \303\251 bastante complicada (voc\303\252 deve se convencer de que estas ra\303\255zes v\303\252m da solu\303\247\303\243o de um polin\303\264mio de quarto grau); numericamente, as duas localiza\303\247\303\265es s\303\243o aproximadamente 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFElNC41MEYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=. Para terminar o , clique o bot\303\243o Close (fechar) .
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7">Exemplo 20.3: O TVM e Patrulha da Rodovia </Text-field>Um carro viaja a 203 milhas em 3 horas em uma rodovia com cuja velocidade limite \303\251 65 milhas/h. Supondo que a velocidade \303\251 cont\303\255nua em toda a viajem, o motorista acelerou alguma vez? Alguma vez a velocidade excedeu \303\240 velocidade limite de pelo menos 10 milhas/h?Note que a velocidade m\303\251dia, em milhas por hora, \303\251evalf(203/3);Esta \303\251 a inclina\303\247\303\243o da reta secante para o TVM, e o que a quest\303\243o realmente pergunta \303\251 se existe um tempo durante a viajem em que a velocidade instant\303\242nea, representada pela tangente, \303\251 sempre a mesma que a velocidade m\303\251dia (ou velocidade limite?).
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field>Designemos 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 a dist\303\242ncia percorrida(em milhas) como uma fun\303\247\303\243o do tempo, t, medidas em horas, iniciando com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEjMC5GJ0Y+LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y+RitGWEY+A velocidade \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVEjZHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y7USdub3JtYWxGJy1GIzYlLUYsNiVRI2R0RidGN0Y6Rj1GQC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGTC8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRj1GQA== = 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 . Em termos das hip\303\263teses do TVM, a continuidade da velocidade significa que a posi\303\247\303\243o \303\251, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, diferenci\303\241vel em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSIzRidGOC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEY4RitGWEY4 e cont\303\255nua em 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. O TVM garante a exist\303\252ncia de pelo menos um n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn emLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSIzRidGOC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEY4RitGWEY4 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 = 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 = 67.67 milhas/h. \303\251 verdadeiro. Portanto, existiu pelo menos um tempo --- de fato, pela continuidade de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, um intervalo de tempo contido emLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGOC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYkUSIzRidGOC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEY4RitGWEY4 --- quando a velocidade excedeu a velocidade limite de 65 milhas/h.O TVM n\303\243o tem nada a dizer sobre a velocidade ter excedido alguma vez a 75 milhas/h.
<Text-field style="Heading 3" layout="Heading 3"><Font opaque="true" background="[255,51,255]">Note</Font></Text-field>Este sistema \303\251 usado para dar bilhetes de velocidade speeding tickets nas cabines de ped\303\241gio toll booths em algumas das estradas com ped\303\241gio(toll roads) nos Estados Unidos toll roads in the United States.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Exemplo 20.4: M\303\251dias Geom\303\251tricas e Aritm\303\251ticas </Font></Text-field>Use e TVM para mostrar 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 < 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 para todos 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYnent\303\243o, mostre queLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNictRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSJiRidGN0Y6LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZBRitGWkZB < 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 para todos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn > 0
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field>Seja 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 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn > 0. Sejam LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn dois n\303\272meros positivos ( positive) com 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. A fun\303\247\303\243o f \303\251 cont\303\255nua 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 e diferenci\303\241vel em 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. (Relembre que 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 para todo 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.) sendo assim, o TVM garante que existe pelo menos um n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn em 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 para o 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\303\251 verdadeiro, isto \303\251, para o 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 verifica.Agora, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYsNiVRImNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGKy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjlRJ25vcm1hbEYn > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYsNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGKy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjlRJ25vcm1hbEYn de forma que 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 < 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 . Ap\303\263s a multiplica\303\247\303\243o por (a quantidade positiva) 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 ),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 < 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, e isto completa a prova da primeira desigualdade inequality.Para a segunda parte deste problema, multiplicar a primeira desigualdade por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYsNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGKy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjlRJ25vcm1hbEYn para 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 < 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 os termos da subtra\303\247\303\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 < 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 pequena \303\241lgebra a mais leva o lado esquerdo da desigualdade a ficar s\303\263 com o radical, e temos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNictRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSJiRidGN0Y6LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZBRitGWkZB < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNidGKy1GIzYqLUYsNiVRImJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSgmbWludXM7RicvRj1RJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRkcvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlYtRiw2JVEiYUYnRjlGPC1GQDYtUSIrRidGQ0ZFRkhGSkZMRk5GUEZSRlRGVy1GIzYnLUkjbW5HRiQ2JFEiMkYnRkMtRkA2LVExJkludmlzaWJsZVRpbWVzO0YnRkNGRUZIRkpGTEZORlBGUi9GVVEmMC4wZW1GJy9GWEZjb0ZZLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZDRitGZW9GQ0YrRmVvRkMtRiM2JUZbb0Zlb0ZDLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZfcC8lKWJldmVsbGVkR0ZHRitGZW9GQw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KEYrLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkmbWZyYWNHRiQ2KC1GIzYnRistRiM2Jy1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnLUYyNi1RIitGJ0Y1RjhGO0Y9Rj9GQUZDRkUvRkhRLDAuMjIyMjIyMmVtRicvRktGaW4tRiw2JVEiYkYnRlZGWS8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGNUYrRl5vRjUtRiM2JS1JI21uR0YkNiRRIjJGJ0Y1Rl5vRjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlxwLyUpYmV2ZWxsZWRHRjpGK0Zeb0Y1RitGXm9GNQ==A import\303\242ncia deste resultado \303\251 que a m\303\251dia geom\303\251trica de dois n\303\272meros, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNictRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSJiRidGN0Y6LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZBRitGWkZB , n\303\243o pode ser maior que a m\303\251dia geom\303\251trica dos mesmos dois n\303\272meros, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNidGKy1GIzYnLUYsNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIrRicvRj1RJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRkcvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlYtRiw2JVEiYkYnRjlGPC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQ0YrRmZuRkMtRiM2JS1JI21uR0YkNiRRIjJGJ0ZDRmZuRkMvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmRvLyUpYmV2ZWxsZWRHRkdGK0ZmbkZD . As m\303\251dias geom\303\251tricas e aritm\303\251ticas s\303\243o iguais somente quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImJGJ0Y0RjcvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZXRj4=.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Exemplo 20.5: Exist\303\252ncia de uma \303\232nica Raiz em um Intervalo </Font></Text-field>Mostre que 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 tem exatamente uma solu\303\247\303\243o real.
<Text-field style="Heading 2" layout="Heading 2" spaceabove="7"><Font encoding="UTF-8">Solu\303\247\303\243o</Font></Text-field>Este problema ser\303\241 enderessado (resolvido) em dois passos. Primeiro, o TVM ser\303\241 usado para mostrar que n\303\243o pode ter mais de uma raiz (root). E ent\303\243o, o Teorema do Valor Intermedi\303\241rio ser\303\241 usado para mostrar que existe pelo menos uma raiz. Combinados, estes resultados significam que existe exatamente uma solu\303\247\303\243o para esta equa\303\247\303\243o .Definir a fun\303\247\303\243of5 := x^3 + 7*x + 4*sin(x)^3 - 3*sin(2*x):
f(x) = f5;Supondo que a equa\303\247\303\243o 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 tenha mais que uma solu\303\247\303\243o. O objetivo \303\251 mostrar que esta suposi\303\247\303\243o (assumption) leva a uma conclus\303\243o contradit\303\263ria. Neste caso, a contradi\303\247\303\243o ser\303\241 que existe um ponto entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0ZB e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0ZB onde 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 .Sejam LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0ZB e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0ZB dois pontos distintos (i.e., 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 ) n\303\272meros com 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 = 1. Supondo 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. note que a fun\303\247\303\243o \303\251 cont\303\255nua em 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 e diferenci\303\241vel 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. (De fato, a fun\303\247\303\243o \303\251 cont\303\255nua e diferenci\303\241vel para todos os n\303\272meros reais (real numbers).) Isto significa que TVM pode ser aplicado. Como resultado, deve existir um n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0ZB e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0ZB 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 = 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 = 0 \303\251 verdadeiro. (Note que esta conclus\303\243o poderia ser feita se n\303\243o se supusesse que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0ZB e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y5USdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0ZB eram distintos.)Agora, vejamos a derivada, que \303\251df5 := diff( f5, x ):
`f '`(x) = df5;
`` = simplify( df5, trig );A manipula\303\247\303\243o manual ?? desta express\303\243o continua ao escrever 13 como 1 + 12, ent\303\243o fatorando ( factoring) os termos trigonom\303\251tricos (trigonometric) para produzir
Para qualquer valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, as quantidades 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 , (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 ), e (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 ) s\303\243o positivas ou zero. Quando combinadas como na f\303\263rmula para a derivada, a adi\303\247\303\243o de 1 \303\251 suficiente para concluir 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 > 1 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn. Em particular,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 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn em 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Esta \303\251 uma contradi\303\247\303\243o direta do resultado obtido pela aplica\303\247\303\243o do TVM. Como um enunciado verdadeiro n\303\243o pode levar a dois resultados inconsistentes, as hip\303\263teses devem ser falsas. N\303\243o pode existir duas solu\303\247\303\265es distintas. Portanto, existe no m\303\241ximo uma solu\303\247\303\243o real para 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.Para o segundo est\303\241gio do problema, a exist\303\252ncia de pelo menos uma solu\303\247\303\243o ser\303\241 confirmada. Agora, um c\303\241lculo simples mostra que \303\251 verdadeiro o seguinte .f(0) = eval( f5, x=0 );
f(Pi) = eval( f5, x=Pi );Observe mais ainda 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 = 0 <1 < 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, como a fun\303\247\303\243o f \303\251 cont\303\255nua em 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, O Teorema do Valor Intermedi\303\241rio implica que deve existir pelo menos um valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn 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 em que 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. Quando combinados com o resultado anterior, temos a prova completa de que existe exatamente uma raiz real da equa\303\247\303\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Al\303\251m disso, a raiz ocorre em algum lugar no 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. A Figura 20.7, um gr\303\241fico da fun\303\247\303\243o e a reta LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=, confirma estes resultados.
plot( [f5,1], x=-2..4 );
Figura 20.7 Exemplo 20.5: Gr\303\241fico de 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 e a reta LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y5
Deste gr\303\241fico a raiz real parece estar pr\303\263xima de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFElMC40MEYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4= e o Maple computa que esta raiz est\303\241 aproximadamente emfsolve( f5=1, x );
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Exemplo 20.6: Uma Pr\303\251via sobre Antiderivada (Integral indefinida)</Font></Text-field>Dada uma derivada, a fun\303\247\303\243o da qual esta derivada foi produzida \303\251 chamada de uma antiderivada. Uma derivada pode ter muitas antiderivadas, mas todas as antiderivadas deiferem uma da outra por uma constante aditiva. Esta obsserva\303\247\303\243o \303\251 expressa formalmente no Teorema 20.2.
Teorema 20.2: Antiderivadas se diferem por Constante Aditiva Se 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 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn em um intervalo, ent\303\243o 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 neste intervalo. Portanto, fun\303\247\303\265es com a mesma derivada se diferem no m\303\241ximo por uma constante (constant) aditiva.
A prova deste teorema envolve o Teorema do Valor M\303\251dio. Em lugar de dar esta prova, considere a seguinte ilustra\303\247\303\243o deste resultado.Considere as fun\303\247\303\265esf6a := sin(x)^2:
g6a := -cos(x)^2:
f(x) = f6a;
g(x) = g6a;As primeiras derivadas destas fun\303\247\303\265es s\303\243odf6a := diff( f6a, x ):
dg6a := diff( g6a, x ):
`f '`(x) = df6a;
`g '`(x) = dg6a;Como estas derivadas s\303\243o iguais para todos os n\303\272meros reais LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, deve existir uma constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn para cada 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 para todos os n\303\272meros reais LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn. A constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn pode ser determinada selecionando qualquer valor para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn onde seja f\303\241cil avaliar (evaluate)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 e 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. Por exemplo, com 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 ,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 = 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 = 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tal queLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNilGKy1GIzYoLUYsNiVRImZGJ0Y0RjctRjs2LVEwJkFwcGx5RnVuY3Rpb247RidGPkZARkNGRUZHRklGS0ZNL0ZQUSYwLjBlbUYnL0ZTRmluLUYjNictRjs2LVEiKEYnRj4vRkFGNkZDL0ZGRjZGR0ZJRktGTS9GUFEsMC4xNjY2NjY3ZW1GJy9GU0Zjby1GIzYnRistSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVEnJiM5NjA7RicvRjVGQkY+LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y+LUYjNiUtSSNtbkdGJDYkUSI2RidGPkZgcEY+LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZecS8lKWJldmVsbGVkR0ZCRitGYHBGPi1GOzYtUSIpRidGPkZgb0ZDRmFvRkdGSUZLRk1GYm8vRlNRLDAuMTExMTExMWVtRidGYHBGPkYrRmBwRj4tRjs2LVEoJm1pbnVzO0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0Zcci1GIzYoLUYsNiVRImdGJ0Y0RjdGZW5GW29GK0ZgcEY+RitGYHBGPkYrRmBwRj5GK0ZgcEY+ = 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 = 1 Isto seria uma prova baseada no c\303\241lculo da identidade trigonom\303\251trica 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 que esta mesma identidade \303\251 usada para provar que 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, e este limite \303\251 necess\303\241rio para obter a derivada da fun\303\247\303\243o cosseno (cosine) .Uma apresenta\303\247\303\243o gr\303\241fica deste resultado \303\251 mostrada na Figura 20.8.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Cap\303\255tulo Sum\303\241rio</Font></Text-field>O Teorema do Valor M\303\251dio (TVM) faz sentido intuitivamente. (Se voc\303\252 n\303\243o concorda com esta argumenta\303\247\303\243o, isto quer dizer que voc\303\252 n\303\243o entendeu a geometria do TVM. Por favor reveja este cap\303\255tulo e o seu livro texto; se necess\303\241rio consiga mais ajuda (extra)fora.) O Exemplo 1 illustra porque \303\251 necess\303\241rio que a fun\303\247\303\243o seja cont\303\255nua em um intervalo fechado e diferenci\303\241vel no mesmo intervalo, exceto possivelmente nos pontos limites do intervalo. Encontrar a localiza\303\247\303\243o onde a reta tangente ao gr\303\241fico \303\251 paralela \303\240 reta secante que conecta os pontos limites do gr\303\241fico no dado intervalo, como no Exemplo 2, fica melhor se feito com a ajuda do maplet Teorema do Valor M\303\251dio ( MeanValueTheorem maplet). As aplica\303\247\303\265es mais interessantes do TVM (Exemplos 3 - 5) usam o TVM sem encontar a localiza\303\247\303\243o dos pontos garantida pelo TVM.O exemplo final, Exemplo 20.6, prov\303\252 uma primeira olhada first look em antiderivadas (ou integral indefinida). Isto ser\303\241 discutido em maiores detalhes no Cap\303\255tulo 21.
<Text-field style="Heading 1" layout="Heading 1" spaceabove="7"><Font encoding="UTF-8">Qual \303\251 o Pr\303\263ximo?</Font></Text-field>Este conclui a unidade sobre aplica\303\247\303\265es de derivadas. Porquanto existam muito mais aplica\303\247\303\265es de derivadas, aquelas apresentadas nesta unidade prov\303\252em um fundamento em que voc\303\252 pode se tornar apto a entender muitas outras aplica\303\247\303\265es.Chapter Navigation<< Prev12345678910111213141516171819 20 2122232425262728293031Next >>
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