Resumêra

Questão 1 – Calcule \(\displaystyle\int_{-1}^{2} 2\,dx\).

1.50 pontos Média

Resposta correta: C) 6

Usando o TFC: \(\int_{-1}^{2}2dx = 2x\big|_{-1}^{2}=2(2)-2(-1)=6\).

Questão 2 – Avalie \(\displaystyle\int_{0}^{\pi/4}\cos(2x)\,dx\).

2.50 pontos Difícil

A) \(\dfrac{1}{\sqrt{2}}\) B) \(\dfrac{1}{2}\) C) \(\dfrac{\sqrt{2}}{2}\) D) \(\dfrac{\pi}{8}\) E) 0

Resposta correta: B) \(\dfrac{1}{2}\)

Antiderivada: \(\displaystyle\int\cos(2x)dx = \frac{1}{2}\sin(2x)\). Avaliando: \(\frac{1}{2}\sin(\pi/2)-\frac{1}{2}\sin(0)=\frac{1}{2}\).

Questão 3 – Calcule \(\displaystyle\int_{1}^{2}\big(x^{3}+2x+1\big)\,dx\).

2.50 pontos Difícil

A) \(\dfrac{25}{4}\) B) \(\dfrac{27}{4}\) C) \(\dfrac{29}{4}\) D) \(\dfrac{31}{4}\) E) \(\dfrac{33}{4}\)

Resposta correta: D) \(\dfrac{31}{4}\)

Antiderivada: \(\displaystyle F(x)=\frac{x^{4}}{4}+x^{2}+x\). Avaliando: \(F(2)-F(1)=\big(\frac{16}{4}+4+2\big)-\big(\frac{1}{4}+1+1\big)=10-\frac{9}{4}= \frac{31}{4}\).

Questão 4 – Determine o limite \(\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{n}\sqrt{1+\Big(\frac{k}{n}\Big)^{2}}\).

3.50 pontos Extrema

A) \(\displaystyle\frac{\sqrt{2}+\ln(1+\sqrt{2})}{2}\) B) \(\displaystyle\frac{\sqrt{2}}{2}\) C) \(\displaystyle\ln(1+\sqrt{2})\) D) \(\displaystyle\frac{1}{2}\) E) \(\displaystyle\sqrt{2}\)

Resposta correta: A) \(\displaystyle\frac{\sqrt{2}+\ln(1+\sqrt{2})}{2}\)

O somatório é a soma de Riemann de \(\sqrt{1+x^{2}}\) em \([0,1]\). \(\displaystyle\int_{0}^{1}\sqrt{1+x^{2}}dx = \frac{1}{2}\Big[x\sqrt{1+x^{2}}+\sinh^{-1}x\Big]_{0}^{1}= \frac{\sqrt{2}+\ln(1+\sqrt{2})}{2}\).

Teorema Fundamental do Cálculo

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