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	{\noindent 21-120 Differential and Integral Calculus \hfill Summer Session I 2010\\}
	{\centering Quiz 10\medskip\\}
	\hrule
	{
		\begin{prob*}
			\begin{enumerate} \mbox{}
				\item $\ds\frac{d}{du} \int\limits_{-u}^u \arctan\left(\frac{1}{x}\right) \; dx$
				\item $\ds\frac{d}{dt} \int\limits_{-t^2}^1 \sqrt{4x^2+3} \; dx$
				\item $\ds\frac{d}{dv} \int\limits_4^v \sin(3x^2)\; dx$
			\end{enumerate}
		\end{prob*}
		\begin{soln*}
			\begin{enumerate}
				\item
				\begin{align*}
					\frac{d}{du} \int\limits_{-u}^u \arctan\left(\frac{1}{x}\right) dx &= \frac{d}{du} \left(\,\int\limits_{-u}^0 \arctan\left(\frac{1}{x}\right)  dx + \int\limits_{0}^u \arctan\left(\frac{1}{x}\right) dx \right)
					\\ &= \frac{d}{du} \left(-\int\limits_{0}^{-u} \arctan\left(\frac{1}{x}\right)  dx + \int\limits_{0}^u \arctan\left(\frac{1}{x}\right) dx \right)
					\\ &= -\arctan\left(-\frac{1}{u}\right)\cdot(-1) + \arctan\left(\frac{1}{u}\right)\\
					 &= \arctan\left(-\frac{1}{u}\right) + \arctan\left(\frac{1}{u}\right) \\
					 &= 0
				\end{align*}
				\item
				\begin{align*}
				\frac{d}{dt}\int\limits_{-t^2}^1 \sqrt{4x^2+3} \; dx	&= \frac{d}{dt}\left(-\int\limits^{-t^2}_1 \sqrt{4x^2+3} \; dx\right) \\
										&= -\sqrt{4(-t^2)^2+3}\cdot(\frac{d}{dt}(-t^2)) \\
										&= -\sqrt{4t^4+3}\cdot(-2t) \\
										&= 2t\sqrt{4t^4+3}
				\end{align*}
				\item
				\begin{align*}
					\frac{d}{dv} \int\limits_4^v \sin(3x^2)\; &= \sin(3v^2)
				\end{align*}


			\end{enumerate}

		\end{soln*}

	}
\end{document}
