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\section*{Quiz 4}
\section*{Name:\hspace{4cm}\;}
\begin{enumerate}[(a)]
\item Find a parameterization $\vec l(s)$ for the tangent line at the point $\vec f(0)$ for the vector valued function
\[\vec f(t)=\langle \sin t, \cos t, t\rangle\]
\solution We have that the velocity vector of a parameterized curve is given by taking derivatives in each of its components, giving us
\[\vec f'(t)=\langle \cos t, -\sin t , 1 \rangle\]
At $t=0$, this is the vector $\langle 1, 0, 1\rangle$. The line that travels in this direction that starts at the point $\vec f (0)$ is given by
\[\vec l(s)=\langle t+0, 1, t+0\rangle\]
\item Find the length of the curve between time $0$ and $\pi$.\\
\solution The length of the curve is given by $\int_0^2\pi \frac{ds}{dt} dt$, where
\[\frac{ds}{dt}=\sqrt{\frac{dx}{dt}+\frac{dy}{dt}+\frac{dz}{dt}}=\sqrt 2 \]
Therefore the length of the curve is $\pi\sqrt 2$.
\end{enumerate}
\noindent \textbf{Problem 2: The Saddle:}\\
The saddle is given by the function
\[z=x^2-y^2\]
Draw a contour plot of the saddle, for integer values of $z$ between $-3$ and $3$. \\
\solution Here is such a drawing. The level curves are not for the integer $z$ values between $-3$ and $3$, but the shape is generally correct.
\[\includegraphics[scale=.5]{saddle}\]
\newpage
\noindent \textbf{Problem 3: Continuity}\\
Consider the function
\[z=\left\{\begin{array}{cc}
\frac{xy}{x^2+y^2} & (x, y)\neq (0, 0)\\
0 & (x, y)=(0, 0)
\end{array}\right.\]
Determine range of the function, and the domain on which the function is continuous. \\
\solution Let's start by computing the domain of continuity. We know that the function must be continuous except possibly where the denominator $x^2+y^2=0$. Here, it is not continuous-- observe that
\[\lim_{n\to\infty} f(1/n, 1/n)=1/2\]
\[\lim_{n\to\infty} f(0, 1/n)=0\]
so the function is not continuous at the origin.\\
For the range, we have that $(x-y)^2=x^2-2xy+y^2\geq 0$, which tells us that $x^2+y^2\geq 2xy$. This tells us that the value of the function can never be bigger than 1/2. It indeed achieves this value along the line $x=y$. Likewise, we see that the minimum value of this function is $\frac{-1}{2}$, which it achieves along the line $x=-y$.
\subsection*{Interesting Puzzle, Will not be graded}
There is a prison run by a strange and mathematically inclined warden. He gathers 20 prisoners in a room, and gives them colorful hats. No prisoner can see their own hat, but can see all the other hats. The warden then says ``At least one of you is wearing a red hat. I will ask all of you together to raise your hand if you are wearing a red hat. If one of you raises your hand incorrectly, I will take away all of your meals for the week. If everybody who is wearing a red hat correctly identifies themselves, I will let you all go free. There is no penalty for not raising your hand. Also, I will continue to ask the group `Are you wearing a red hat' until either somebody is wrong, or all the red hats have been identified.''\\
So he begins: ``Are you wearing a red hat? '' \\
As nobody knows the color of their own hat, nobody raises their hands. And he asks the group again, ``Are you wearing a red hat? ''
Is there a way for a group of rational prisoners to save themselves ?
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