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\myhead{Mat 5620}{Final Exam}{Fall '11}{ }

Work quickly and carefully, following directions closely. Answer all questions completely. \\[1ex]

%%%%%%%%
\S I. \textsc{Problems}. 

\begin{enumerate}
\item Prove or disprove: If $\vec{r}\!:\![a,b]\to\mathbb{R}^n$ is integrable, then
	\[ \left\|\, \int_a^b \vec{r}(t)\,dt\,\right\| \le \int_a^b \left\|\vec{r}(t)\right\|\,dt	\]
\vfill

\item Let $A = \left\{ \left(\frac1n\cos(\frac1n),\frac1{n^2}\sin(\frac1{n^2})\right): n\in\mathbb{N} \right\}\subset\mathbb{R}^2$
	\begin{enumerate}
	\item Is $A$ a connected set? If so, prove it; if not, why not?
	\item Is $A$ is separated from $B=\{\,\vec{0}\,\}$?
	\end{enumerate}
\vfill

\item Prove: Let $f$ and $g$ be differentiable functions at $\vec{a}$. Then $\nabla (f+g) = \nabla f + \nabla g$.
\vfill

\item Prove: Let $f$ be continuous and integrable on a closed and bounded region $R$. Then
	\[	\operatorname{area}(R)\cdot \min_{x\in R} f(x) \le \iint_R f\,dA \le \operatorname{area}(R)\cdot \max_{x\in R} f(x)	\]
\vfill

\item Let $C$ be a smooth rectifiable curve, and let $f$ be continuous everywhere. Then
	\[	\left| \int_C f\,ds \, \right| \le \operatorname{length}(C)\cdot \max_{\vec{x}\in C} \left|f(\vec{x})\right|	\]
\vfill

\item Show that any subset of set of measure zero has measure zero.
\vfill

\item Compute
	\begin{enumerate}
	\item $\displaystyle\int_{[0,1]} \chi_\mathbb{Q}\,d\mu$
	\item $\displaystyle\int_{[0,1]} T\,d\mu$ where $T(x)$ is Thomae's function.
	\end{enumerate}
\end{enumerate}
\vfill

\S II. \textsc{Extra Credit}.
\begin{enumerate}
\item Fix a real number $a\ge 0$. Define a sequence $f_n\!:\![0,\infty)\to\mathbb{R}$ by $f_n(x)=\dfrac{x^n}{1+x^n}$. Determine $\displaystyle\lim_{n\to\infty} \int_0^{a+1} f_n(x)\,dx$.
\end{enumerate}


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