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

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

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

\begin{enumerate}
\item
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Let $\mathcal{A}$ be the astroid given by $f(t)=[\cos^3(t),\sin^3(t)]$ for $t\in[0,2\pi]$. Let $P(t)$ be a point on $\mathcal{A}$. Let $P_x$ and $P_y$ be the $x$- and $y$-intercepts of the line tangent to $\mathcal{A}$ at $P(t)$. Show that the line segment $\overline{P_xP_y}$ has constant length; i.e., the length of the segment is independent of $t.$}

{\vspace*{-0.9in} \hspace*{3.5in}\includegraphics[width=2.75in]{Astroid}}
\\[1ex]

\item Let $\vec{r}:\mathbb{R}\rightarrow\mathbb{R}^3$ be a vector-valued function that has $2$ continuous derivatives for all $t.$ Prove or disprove
	\[  \frac{d}{dt}[\vec{r}(t)\times\vec{r}^{\;\prime}(t)] = \vec{r}(t)\times\vec{r}^{\;\prime\prime}(t).  \]

\item Let $f(t) = \dfrac{2t^2}{1+t^2}$ and set $C_{6\pi}$ to be the curve given by $[f(t)\cos(2t), f(t)\sin(2t)]$ for $t\in[0,6\pi].$ Find the length of the curve $C.$ Can you make a conjecture concerning the ratio $\dfrac{\operatorname{length}(C_{2n\pi})}{4n}$ as $n\rightarrow\infty$ ?
\\[3ex]

\item Prove or disprove: \\
	Let $A_1=B(\langle1,0\rangle,1)$ and $A_{-1}=B(\langle-1,0\rangle,1)$ be open balls in $\mathbb{R}^2.$ Then $E=A_1\cup A_{-1}$ is not separated.
\\[3ex]

\item Problem 4, page 446. \\
	Add: (g) Show $f_x(0,0) = 0 = f_y(,0,0)$ even though $f$ is not continuous at $(0,0)$.\\[3ex]

\item A \emph{harmonic function} is one that satisfies \emph{Laplace's equation} $\nabla^2 f(x,y) = 0$ 
	where $\nabla^2 = \dfrac{\partial^2}{\partial x^2} + \dfrac{\partial ^2}{\partial y^2}.$ \begin{enumerate}
	\item Prove that the 
	functions
	\begin{enumerate}
		\item $f(x,y) = x^3-3x\, y^2$
		\item $g(x,y) = 3x^2\,y - y^3$ 
	\end{enumerate}
	are harmonic.\\
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	\item Find $\dfrac{d^2z}{dt^2}$ for $z=f(x,y)$ when $x(t)=\ln(t)$ and $y(t)=e^t$ without expanding $f$ in terms of $t.$
	\end{enumerate}
\end{enumerate}

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