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%{\bf Name}: % \ \rule{5.0in}{.01in}
%Possible Final exam questions for Math 114 (Bob Strain)  \\
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%And sample title page (below)
%% \noindent {\bf Student~ID~number:} \ \rule{3.0in}{.01in}
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% %\noindent  {\bf TA (circle one):} \ \hfill
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% Zhe Liu \hfill  M 9-10(221)  \hfill
%M 10-11(222)  \hfill
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% Yin Xia \hfill  M 9-10(225)  \hfill
%M 10-11(226)  \hfill
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%{\bf Math 114-003  \ \ - \ \ Midterm Exam 3 \ \  - \ \  Prof. Robert Strain
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%Thursday, April 15, 2010 @ 12:00 - 1:30 PM
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%This is a 75 Minute Exam, 15 Minutes are allotted for attendance, etc.}
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%\noindent
%I certify that all of the work on this test is my own.\\
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%{\bf Signature:} \ \rule{3.0in}{.01in}
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%{\bf  INSTRUCTIONS}   \\
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%%{\bf  OFFICIAL USE ONLY}   \\
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\begin{center}
{\bf Math 114  \ \ - \ \ Final Exam  \ \  - \ \  University of Pennsylvania

Tuesday, May 4, 2010 @ 12:00 - 2:00 PM

%This is a 75 Minute Exam, 15 Minutes are allotted for attendance, etc.
}
\end{center}
\vspace{0.1in}
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\begin{itemize}
\item {\bf Name}:  \ \rule{5.0in}{.01in}

%\begin{tabular}{|c|c|} \hline
%Your Professor & Your TA  \\
%\hline
%\hline Dr. Powers & Dr. He & Dr. Wylie & Dr. Strain  \\
%\hline Dr. He & Dr. Wylie & Dr. Strain  \\
%\hline Dr. Wylie & Dr. Strain  \\
%\hline Dr. Strain  \\
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%
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%\hline
%\hline Total & 125 & \\
%\hline
%\end{tabular}


\item {\bf Your professor} {\bf (circle one)}:
Dr. Powers \hfill  Dr. He  \hfill Dr. Wylie  \hfill Dr. Strain  \hfill

\item {\bf Your TA}{\bf (circle one)} :
\begin{tabular}{p{0.6 cm}lccc}
& Yan Wang & Sohrab Shahshahani & \multicolumn{2}{c}{Zhe Liu} \\
&         & \multicolumn{2}{c}{Eugene So \quad \quad \quad Ryan Eberhart} & \quad Yin Xia \\
\end{tabular}

%\item {\bf Your TA}{\bf (circle one)} :
% \quad Zhe Liu \hfill  Yin Xia  \hfill
%Sohrab Shahshahani  \hfill   \\
% Eugene So  \hfill
%Ryan Eberhart  \hfill %{\bf (circle one)}
%Yan Wang  \hfill
%  \hfill
\item {\bf Recitation day and time}:    \ \rule{3.5in}{.01in}
\end{itemize}
\hrule

%\vspace {0.1in}

\noindent
I certify that all of the work on this test is my own.\\

%\vspace {.3in}


{\bf Signature:} \ \rule{3.0in}{.01in}

\vspace{0.1in}
\hrule
{\bf INSTRUCTIONS:}
\begin{enumerate}
\item Please complete the information requested above. There are 19 multiple choice problems.

\item Please show all your work on the exam itself. Correct answers with little or no supporting work will not be given credit.

\item You are allowed one hand-written sheet of paper with formulas. {\it No calculators, books or other aids are allowed. }  Please turn in your crib sheet together with your exam.
\end{enumerate}
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{\bf  OFFICIAL USE ONLY:} \hspace {1in}

\begin{footnotesize}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
Problem & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 \\
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Points & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10  \\
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\multirow{2}{*}{Your score} & & &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &   \\
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    {\bf Your total score}:  \ \rule{2.0in}{.01in}




%\begin{tabular}{| l | l | l |}
%\hline
%\multicolumn{3}{|c|}{Team sheet} \\
%\hline
%Goalkeeper & GK & Paul Robinson \\ \hline
%\multirow{4}{*}{Defenders} & LB & Lucus Radebe \\
% & DC & Michael Duberry \\
% & DC & Dominic Matteo \\
% & RB & Didier Domi \\ \hline
%\multirow{3}{*}{Midfielders} & MC & David Batty \\
% & MC & Eirik Bakke \\
% & MC & Jody Morris \\ \hline
%Forward & FW & Jamie McMaster \\ \hline
%\multirow{2}{*}{Strikers} & ST & Alan Smith \\
% & ST & Mark Viduka \\
%\hline
%\end{tabular}












\pagebreak


\begin{enumerate}








%\pagebreak










\item Let $P$ be the  plane that contains the points $(2,1, 3) $, $(2,2, 4)$, and $(1, 1, 6)$.  What is the distance from $P$ to the point  $(1, 1,1)$?
\begin{enumerate}
\item $\frac{5}{\sqrt{11}}$  {\bf **}
\item $\frac{5}{3}$
\item $\frac{5}{\sqrt{13}}$
\item $\frac{8}{\sqrt{11}}$
\item $\frac{8}{3}$
\item $\frac{8}{\sqrt{13}}$
\item $\frac{13}{\sqrt{11}}$
\item $\frac{11}{\sqrt{13}}$
\end{enumerate}


\vfill




\item Where does the the plane that contains both of  the lines
$$L_1 = <3-t, -4+t,  4+2t>
\quad  \& \quad
L_2 = <3+t, -4+t , 4-t>,
$$ intersect the $x$-axis?
\begin{enumerate}
\item $x=4$
\item $x=6$
\item $x=3$
\item $x=7$  {\bf **}
\item $x=-4$
\item $x=-6$
\item $x=-3$
\item $x=-7$
\end{enumerate}
\vfill









\pagebreak






\item Consider the surface $z = x^2+ 2y^2  - 2x + 4y$.  At one point
$\langle x_0,y _0, z_0\rangle$
the tangent plane to the surface is parallel to the $xy$-plane.  What is
the $z$-coordintate of that point?

\begin{enumerate}
\item $-3${\bf **}
\item  $-1$
\item  $0$
\item $4$
\item $6$
\item $7$
\item $9$
\item $10$
\end{enumerate}

\vfill












\item Find the maximum of the function $f(x,y,z) = 2x + 3y - 2z$ in the region
$$
4x^2 + 6y^2 + 2z^2 \leq 18.
$$
\begin{enumerate}
\item $0$
\item $1$
\item $3$
\item $4$
\item $6$
\item $9$ {\bf **}
\item $12$
\item $20$
\end{enumerate}
\vfill





\pagebreak


\item  The function $y(t)$ satisfies the equation $\frac{d y}{dt} + 2ty = y$ and $y(0) = 5$.
What is $y(1/2)$? \\

\begin{enumerate}
\item $5$

\item $0$

\item $e^{1/4}$

\item $5e^{1/4}$  {\bf **}

\item $-5e^{1/4}$

\item $-e^{1/4}$

\item $-5$

\item $e$
\end{enumerate}


\vspace{1 in}

\item  Jack and Jill plan to evaluate the line integral
$$
\int_C {\bf F} \cdot d{\bf r},
$$
along a path in the $xy$-plane from $(0,0)$ to $(1,1)$.
The force field is
$$
\mathbf{F} (x,y) = (x+2y,-x+y^2).
$$
Jack chooses the
path that runs along the $x$-axis from $(0,0)$ to $(1,0)$ and then runs along
the vertical line $x = 1$ from $(1,0)$ to the final point $(1,1)$.  Jill
chooses the direct path along the diagonal line $y = x$ from $(0,0)$ to
$(1,1)$.
Whose line integral is larger and by how much?


\begin{enumerate}
\item Jack's by  5/2
\item  Jack's by 2
\item Jack's by 1
\item They are equal.
\item Jill's by 1
\item Jill's by 3/2     {\bf **}
\item Jill's by 4
\item Jill's by 9/2
\end{enumerate}
\vfill
\pagebreak


\item  Find the integral of the function $F(x,y,z) = z$ in the region $E$
consisting of the top half of a sphere of radius 2 divided by the volume of that region which is $V
=16\pi/3$ (i.e. the region $E$ is $z \geq 0$ and $x^2 + y^2 + z^2 \leq 4)$:
$$
\text{Compute:} \qquad \overline {z} = \frac {1} {V} \iiint_E  z \,dV.
$$

\begin{enumerate}
\item  $1/3$
\item   $1/2$
\item   $1/\sqrt{2}$
\item   $3/4$   {\bf **}
\item   $\pi/4$
\item  $\sqrt{2}$
\item $\pi/2$
\item   $3/\pi$

\end{enumerate}
\vfill








\item Calculate the following integral
$$
\int_2^4 \int_{\frac{y-2}{2}}^1 e^{(x^2)}  dx dy.
$$
(You may need to change the order of integration.)

  \begin{enumerate}
\item $2e$
\item $e-1$  {\bf **}
\item $1 - \frac{1}{e}$
\item $\sqrt{e}-1$
\item $1 - \frac{1}{\sqrt{e}}$
\item $2e^2-2$
\item $2 \sqrt{2}$
\item $4e - 2$

\end{enumerate}



\vfill
\pagebreak


\item Use a double integral to find the area of the region enclosed by the curve
$$
r = \sqrt{\cos\left( \theta \right)},
$$
when $-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$.


\begin{enumerate}
\item $\frac{1}{2}$
\item $\frac{46\pi}{27}$
\item$\frac{\pi}{4}$
\item $\frac{\pi}{8}$
\item $\frac{\pi}{16}$
\item 1   {\bf **}
\item $46\pi$
\item$4\pi$
\end{enumerate}


\vfill







\item  The electrical potential $V$ is given by
$$
V(x,y,z) = x^2 - y^2 - z^2.
$$
In which direction does the potential increase most rapidly at $(1,1,0)$? \\

\begin{enumerate}

\item  $(1,1,0) $

\item  $(-2, 2, 0) $

\item  $(2,2,0) $

\item  $(-2, -2, 0)$

\item  $(1,0,0) $

\item  $(1, 1, 0) $

\item  $(1, -1,0)$   {\bf **}

\item  $(1,1,1)$

\end{enumerate}





\vfill


\pagebreak




\item Consider the function $f(x,y) = -2x^3 + 3 x^2 + 2 y^2 - 4y$.  Find the two critical points and determine their type.

\begin{enumerate}

\item local min at $(x,y) = (0,1)$, \\
local min at $(x,y) = (1,1)$.
\item  local min at $(0,1)$, \\
saddle at $(1,1)$.  {\bf **}
\item  local min at $(0,1)$, \\
local max at $(1,1)$.
\item saddle at $(0,1)$, \\
local min at $(1,1)$.
\item  saddle at $(0,1)$, \\
saddle at $(1,1)$.
\item  saddle at $(0,1)$, \\
local max at $(1,1)$.
\item  local max at $(0,1)$, \\
local min at $(1,1)$.
\item  local max at $(0,1)$, \\
saddle at $(1,1)$.
\end{enumerate}
\vfill

\pagebreak










\item Compute the work done by the force
\[ \mathbf{F}(x, y,z) = < 2x, 3y, -z> \]
along the path
\[ \mathbf{r}(t) = < t, t^2, t^3 > \qquad 0 \leq t \leq 1.  \]

Hint: This is given  by the line integral
\[ W = \int_C \mathbf{F} \cdot d\mathbf{r} \]

\begin{enumerate}
\item $-2$
\item $-1$
\item  $0$
\item $1$
\item $2$  {\bf **}
\item $3$
\item $4$
\item $5$
\end{enumerate}
\vfill

\item Compute the integral
\[ \iint_R \cos \left( \frac{x-y}{x+y} \right) dA \]
where $R$ is the region inside the triangle with vertices $(0,0)$, $(0,1)$, and $(1,0)$.

\begin{enumerate}
\item  $-\sqrt{2} $
\item $\frac{\pi}{4} $
\item $1 - \cos(1) $
\item $ 1 - \frac{1}{\sqrt{2}}$
\item $ \frac{\pi}{2}$
\item $\frac{\sin(1)}{2}$  {\bf **}
\item $0$
\item $\sin(1) - \cos(1)$
\end{enumerate}
\vfill

\pagebreak

\item Suppose a tennis canon located at ground level fires a ball at a $45^{\circ}$ angle to the ground with a speed of 80 feet per second.  In calculating the trajectory of the ball you may exclude the effect of air resistance and assume the only force acting on the ball is gravity which accelerates the ball downward at a rate of 32 feet per second. How high  is the ball at its highest point?

\begin{enumerate}
\item $20$ feet
\item $20 \sqrt{2}$ feet
\item $32$ feet
\item $50$ feet  {\bf **}
\item $50 \sqrt{2}$ feet
\item $64$ feet
\item $64 \sqrt{2}$ feet
\item $100$ feet
\end{enumerate}
\vfill

\item Let $\kappa(x)$ be the curvature of the parabola $y = x^2$ at $(x,x^2)$. What is maximal value
of $\kappa(x)$? \\

\begin{enumerate}
\item $1/2 $
\item $2  $  {\bf **}
\item $1 $
\item $4 $
\item $1/4 $
\item $3 $
\item$1/3 $
\item $0$
\end{enumerate}
\vfill



\pagebreak

\item
A helix has the vector equation $\mathbf{r}(t)=(\cos t, \sin t, 3 t)$. Find the arc length from
$\mathbf{r}(0)$ to $\mathbf{r}(2\pi)$. \\

\begin{enumerate}
\item $2\pi$
\item$4 \pi$
\item$2\sqrt{10}\pi $ {\bf **}
\item $2\sqrt{2}\pi $
\item $\sqrt{10} \pi $
\item $\sqrt{2} \pi$
\item $4\sqrt{2}\pi $
\item$20 \pi$
\end{enumerate}
\vfill



\item What is  the volume of  the region above the plane $z=2$ and inside the sphere $x^2+y^2+z^2=9$. (i.e.  $z\geq 2$ and $x^2+y^2=z^2 \leq 9$) ?

\begin{enumerate}
\item $2 \pi$
\item $\frac{8 \pi}{3}$  {\bf **}
\item $\frac{5 \pi}{2}$
\item $4 \pi$
\item $\frac{9 \pi}{4}$
\item $\frac{11 \pi}{3}$
\item $7 \pi$
\end{enumerate}
\vfill


\pagebreak

\item Evaluate
\[ \iint_D x ~dA \]
where $D$ is the region in the $xy$-plane bounded by the curves $y=2x$ and $y=x^2$.
\begin{enumerate}
\item $-1$
\item $-\frac{1}{2}$
\item $0$
\item $\frac{3}{8}$
\item $\frac{3}{4}$
\item $1$
\item $\frac{4}{3}$  {\bf **}
\item $\frac{5}{2}$
\end{enumerate}
\vfill




\item
Evaluate the following line integral.
\begin{equation*}
\int_C (6y+x) dx + (y+2x) dy
\end{equation*}
$C$ is the circle $(x-2)^2 + (y-3)^2 = 4$ oriented counterclockwise. \\

\begin{enumerate}
\item $16 \pi $
\item $8 \pi $
\item $32\pi $
\item$4\pi $
\item $-16 \pi $  {\bf **}
\item $-8 \pi $
\item $-32 \pi$
\item $-4 \pi$
\end{enumerate}
\vfill







\end{enumerate}

\end{document}
