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\begin{document}

\centerline{\bf MAT 0146 - C\'alculo 1 para Economia} 

\centerline{\bf Prova Substitutiva - 27 de junho de 2016}

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\noindent {\bf Quest\ao\ 1} (2,5 pts) Considere ${\displaystyle f(x)=\sqrt{1+x}-\left(1+\frac{x}{2}-\frac{x^2}{8}\right)}$, $x>-1$. 
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(a) Calcule $f'(x)$ e $f''(x)$.
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(b) Mostre que $f''(x)>0$, para todo $x>0$.
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(c) Explique por que o resultado do item (b) implica que $f'(x)>0$, para todo $x>0$.
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(d) Mostre que $1+\frac{x}{2}-\frac{x^2}{8}<\sqrt{1+x}<1+\frac{x}{2}$, para todo $x>0$.


\newpage

\noindent {\bf Quest\ao\ 2} (2,5 pts) Considere $f(x)=x^4-6x^2$, $x\in\R$.
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(a) Determine os intervalos em que $f'(x)$ \'e positiva ou negativa.
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(b) Determine os intervalos em que $f''(x)$ \'e positiva ou negativa.
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(c) Esboce o gr\'afico de $f$, assinalando os pontos cr\'iticos e os pontos de inflex\ao.
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(d) Qual \'e o menor valor assumido por $f(x)$?

\newpage

\noindent {\bf Quest\ao\ 3} (2 pts) Mostre que:\ \ \ \  (a) ${\displaystyle \int_1^\infty\frac{1}{\sqrt{x^5+1}}\,dx<1}$,\ \ \ \ 
(b) ${\displaystyle \int_0^\infty\frac{1}{\sqrt{x^5+1}}\,dx<2}$.

\noindent
{\tt Dica}: N\ao\ tente calcular exatamente as integrais.

\newpage

\noindent {\bf Quest\ao\ 4} (3 pts) Calcule ${\displaystyle \int_0^\infty x^3e^{-x}\,dx}$.


\end{document}
