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\title{ESP Workshop, Worksheet \#16}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#16}

{\bf \large Tuesday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in} 

OMG!  In fact, ZOMG!!!  You've got \textit{another} exam coming up!
Don't be a N00b!  In order to avoid getting PWNED by the l33t math
wizard Doctor Durbin, let's practice with some of his old exam
problems... w00t!

\begin{enumerate}
\item Consider the curve represented parametrically by the equations
\begin{displaymath}
x= t^2 -4t, \qquad y=t^2
\end{displaymath}

\begin{enumerate}
\item Sketch the graph for $0\leq t \leq 2$. (Suggestion: use the
  graph paper provided.)
\item Find $\frac{dy}{dx}$ and $\frac{d^2 y}{dx^2}$.
\item Find all points (if there are any) of horizontal and vertical
  tangency. [Do not restrict $t$ as in part (a).]
\end{enumerate}

\item 
\begin{enumerate}
\item Draw the graph of $r = 2+\sin \theta$.
\item Write but do not evaluate an integral that will give the length
  of the graph in part (a) over $0\leq \theta \leq 2\pi$.
\item Write but do not evaluate an integral that will give the area
  inside the graph in part (a).
\end{enumerate}

\item The acceleration of an object at each time $t$ is given by
  $\mathbf{a}(t) = \mathbf{i} -2\mathbf{j} +\mathbf{k}$. Also,
  $\mathbf{v}(0) = 3\mathbf{k}$ and $\mathbf{r}(0) = \mathbf{j}$
  ($\mathbf{v}$ for velocity, $\mathbf{r}$ for position). Find
  $\mathbf{r}(t)$ and the position at time $t = 2$.

\item Find a set of parametric equations for the line tangent to the
  curve $\mathbf{r}(t) = t^2 \mathbf{i} +t\mathbf{j} +\mathbf{k}$ at
  (4,2,1).

\item Sketch a contour map for $f(x,y) = 2x^2 +y$ using the level
  curves corresponding to $c=0,-1,$ and $2$.

\item Use the limit definition of partial derivative to find $f_x
  (x,y)$ given $f(x,y) = x^2 y^3$. (Write your solution very
  carefully. For example, do not be sloppy with limit notation.)

\item Given $f(x,y) = 2xy^3$, find $f_y (x,y)$ by forming the
  appropriate difference quotient and taking the limit as $h$ tends to
  zero.

\item Find the equation of the tangent plane to the surface $x^2 - y^2
  -z^2 = 4$ at (3,1,-2).

\newpage

\item (A Chemistry problem!)  The gas law for a fixed mass $m$ of an
  ideal gas at absolute temperature $T$, pressure $P$, and volume $V$
  is $PV = mRT$, where $R$ is the gas constant. 

\begin{enumerate}

\item Show that
\begin{displaymath}
\frac{\partial P}{\partial V} \frac{\partial V}{\partial T} \frac{\partial T}{\partial P} = -1
\end{displaymath}

\item For the ideal gas of part (a), show that
\begin{displaymath}
T \frac{\partial P}{\partial T} \frac{\partial V}{\partial T} = mR
\end{displaymath}

\item Now assume that the pressure, volume, and temperature of a mole
  of an ideal gas are related by the equation $PV = 8.31 T$, where $P$
  is measured in kilopascals, $V$ in liters, and $T$ in kelvins. Use
  differentials to find the approximate change in the pressure if the
  volume increases from 12 L to 12.3 L and the temperature decreases
  from 310 K to 305 K.
\end{enumerate}

\item (An Electrical Engineering problem!) If $R$ is the total
  resistance of the three resistors, connected in parallel, with
  resistances $R_1, R_2, R_3$, then
\begin{displaymath}
\frac{1}{R} = \frac{1}{R_1} +\frac{1}{R_2} +\frac{1}{R_3}
\end{displaymath}

\begin{enumerate}
\item Rewrite this equation to make $R$ a function of $R_1,R_2,$ and
  $R_3$. (All I mean is that you should invert both sides and possibly
  simplify to get $R = f(R_1,R_2,R_3)$.)

\item Use your solution to part (a) to find $\partial R / \partial
  R_1$, i.e. the partial derivative of the function $R$ with respect
  to the variable $R_1$.

\item If the resistances are measured in ohms as $R_1 = 25\ \Omega$,
  $R_2 = 40\ \Omega$, and $R_3 = 50\ \Omega$, with a possible error of
  $0.5 \%$ in each case, estimate the maximum error in the calculated
  value of $R$.
\end{enumerate}

\item (A Marine Biology problem!) Marine biologists have determined
  that when a shark detects the presence of blood in the water, it
  will swim in the direction in which the concentration of blood (in
  parts per million) at a point $P(x,y)$ on the surface of the
  seawater is approximated by
\begin{displaymath}
C(x,y) = e^{-(x^2 +2y^2)/10^4}
\end{displaymath}
where $x$ and $y$ are measured in meters in a rectangular coordinate
system with the blood source at the origin.
\begin{enumerate}
\item Identify the level curves of the concentration function and
  sketch several members of this family together with a path that the
  shark will follow to the source.

\item (Challenge.) Suppose a shark is at the point $(x_0, y_0)$ when
  it first detects the presence of blood in the water. Find and
  equation of the shark's path by setting up and solving a
  differential equation.
\end{enumerate}

\end{enumerate}

\end{document}