Use o Teorema da Estimativa do Resto para encontrar um intervalo contendo x = 0, no qual a função f(x) = ln(1+x) possa ser aproximada por p(x) = x - x^2 / 2 + x^3 / 3 com precisão de três casas decimais.
Represente em um mesmo plano cartesiano a função, p(x) e |f(x)-p(x)| sobre o intervalo obtido.

Passo 1

Ok, vamos lá! A gente tem que usar o Teorema da Estimativa do Resto para achar um intervalo em torno de onde a função pode ser aproximada por com uma precisão de três casas decimais. Para isso, vamos precisar entender um pouco sobre séries de Taylor e como calcular o resto dessa aproximação.

Passo 2

Primeiro, vamos lembrar que o Teorema da Estimativa do Resto nos ajuda a saber o quão boa é a nossa aproximação polinomial. Ele nos dá uma forma de calcular um limite para o erro entre a função real e o polinômio de aproximação. No caso, nosso polinômio é uma aproximação de terceiro grau de .

Passo 3

O próximo passo é encontrar a expressão para o termo de resto, que no caso da série de Taylor é dado por:
Onde está entre 0 e , e é o grau do polinômio. Como nosso polinômio é de grau 3, , então:

Passo 4

Vamos calcular a quarta derivada de :

Passo 5

Substituindo na expressão do resto:
Como está entre 0 e , para garantir a maior estimativa do resto, vamos considerar no extremo que maximiza . Isso ocorre quando é mínimo, ou seja, :

Passo 6

Queremos que esse erro seja menor que (já que queremos precisão de três casas decimais):
Multiplicando ambos os lados por 4:

Passo 7

Agora, vamos resolver para :
Então, o intervalo em torno de zero onde a aproximação é válida com a precisão desejada é aproximadamente .

Resposta

O intervalo onde pode ser aproxime por com precisão de três casas decimais é aproximadamente . Além disso, ao representar no mesmo plano cartesiano, você verá as curvas de e dentro desse intervalo, mostrando a boa aproximação da função real pelo polinômio.

Exercícios relacionados

<html><head></head><body><div>Determine a diferencial da função. <span class=“math-inline “>
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<html><head></head><body><div>Encontre a soma <span class=“math-inline “>
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<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.025ex;” xmlns=“http://www.w3.org/2000/svg” width=“1.357ex” height=“1.025ex” role=“img” focusable=“false” viewBox=“0 -442 600 453”><title>n</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D45B” d=“M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z”></path></g></g></g></svg></mjx-container></span> primeiros termos das seguintes séries:</div>
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Determine as derivadas parciais indicadas.