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\title{ESP Workshop, Worksheet \#15}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#15}

{\bf \large Thursday \today}

{\bf \large AI: Eric Katerman}
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\vspace{.15in}
\begin{enumerate}
\item Match the function with its graph (labeled A--F) and with its
  contour map (labeled I--IV). Give reasons for your choices.

\newpage
\item Consider the curve $C$ in $\mathbb{R}^3$ defined by the
  vector-valued function $\mathbf{r} = 2\sin t \mathbf{i} +2\cos t
  \mathbf{j} +\tan t \mathbf{k}$ on the interval $I = (-\pi/2,
  \pi/2)$.
\begin{enumerate}
\item Find the unit tangent vector \textbf{T}($t$) at the point where
  $t=\pi/4$.
\item Write $\mathbf{r}(t)$ parametrically, i.e. in terms of $x,y,$
  and $z$. Find parametric equations for the tangent line at the point from
  part (a).

\item A curve given by a vector function $\mathbf{r}(t)$ on an
  interval $I$ is called \textbf{smooth} if $\mathbf{r}'$ is
  continuous and $\mathbf{r}'(t) \neq 0$ (except possibly on the
  endpoints of $I$). Is $C$ smooth on $I$?



\end{enumerate}

\item Now instead of vector-valued functions $\mathbf{r}:\mathbb{R}
  \rightarrow \mathbb{R}^3$, we will work with a real-valued function
  of several variables, $f:\mathbb{R}^2 \rightarrow \mathbb{R}$. Let
  $f(x,y) = \sqrt{x+y}$.
\begin{enumerate}
\item What is the domain of $f$? Sketch it.
\item Draw a contour map for $f$. Remember, this is a graph of the
  level curves (i.e. $f(x,y) = c$ for some constant $c$) of $f$.
\item Use your picture from part (b) to sketch the graph of $f$ in
  $\mathbb{R}^3$.
\item Now suppose that $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ is
  defined by $f(x,y,z) = \sqrt{x+y}$. How do your answers to parts
  (a)--(c) change for this new function? In particular, can you draw
  the graph of $f$ in $\mathbb{R}^3$??

\end{enumerate}

\item (A Scorched Earth question!) A projectile is fired from the
  origin with angle of elevation $\alpha$ and initial speed $v_0$.
\begin{enumerate}
\item  Assuming that air resistance is negligible and that the only force
  acting on the projectile is gravity, $g$, show that the position
  vector of the projectile is $\mathbf{r}(t) = (v_0 \cos \alpha)t
  \mathbf{i} + [(v_0 \sin \alpha )t -\frac12 gt^2]\mathbf{j}$.

\item Show that the maximum horizontal distance of the projectile is achieved when $\alpha = 45^\circ$ and that in this case the range is $R = v_0^2/g$.

\item At what angle should the projectile be fired to achieve maximum height and what is the maximum height?

\item Fix the initial speed $v_0$ and consider the parabola $x^2 +2Ry -R^2 =0$, whose graph is shown in the figure.  Show that the projectile can hit any target inside or on the boundary of the region bounded by the parabola and the $x$-axis, and that it can't hit any target outside this region.

\item Suppose that the gun is elevated to an angle of inclination
  $\alpha$ in order to aim at a target that is suspended at a height
  $h$ directly over a point $D$ units downrange. The target is
  released at the instant the gun is fired. Show that the projectile
  always hits the target, regardless of the value $v_0$, provided the
  projectile does not hit the ground ``before'' D.

\end{enumerate}
\end{enumerate}

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