\documentclass[10pt,twoside,addpoints]{exam}
\newcommand{\examnum}{Final Exam}
\newcommand{\examdate}{December 12, 2018}
\usepackage{multicol, tikz, pgfplots}
%\newcommand{\vect}[1]{\boldsymbol{#1}} % Shortcut definition for vectors with italic
%\renewcommand{\vec}[1]{\mathbf{#1}} % Shortcut for vectors, bold and no italic
%\renewcommand{\vec}[1]{\overrightarrow{ #1}} % Alternate vector notation (arrows)
%\renewcommand{\theenumi}{\bf{\arabic{enumi}}} %Change numbering style

\input{../PackageList.tex}
\input{../macros.tex}

\begin{document}
	%\NameAndSection
	\parindent 0ex
	%\pointname{ credit \underline{\hspace{0.25in}}}
	\hpword{Credit}
	\hsword{Credit Points}
	
	
	\pagestyle{headandfoot}
	%\pagestyle{head}
	\firstpageheader{}{}{}
	\runningheader{\class}{\examnum}{ Page  \thepage \, of  \numpages }
	\firstpagefooter{}{}{}
	\runningfooter{\bf\name }{}{\sectionnum}
	\NameAndSectionForExams
	%\makebox[\textwidth]{Name and section:\enspace\hrulefill}
	
	
	
	\addpoints
\begin{center}
 \begin{tabular}{|l|l|l|}
                  \hline
                  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
                  & & \\
                  $\ds \mathrm{d}A= r\, \mathrm{d}r\, \mathrm{d} \theta$ & $\ds \mathrm{d}V= r\, \mathrm{d}z\,\mathrm{d}r\,\mathrm{d}\theta $ & $\ds \mathrm{d}V = \rho^2 \, \sin \phi \,\mathrm{d} \rho \, \mathrm{d}\phi \, \mathrm{d} \theta$\\
                   & &\\
                  \hline
                \end{tabular}
\end{center}
\begin{questions}
    
    	\question Consider the integral  $\ds \iint \limits_{\mathcal{D}} \frac{1}{1-y^4} \, dA = \int_{0}^{1} \int_{0}^{\sqrt[4]{x}} \frac{1}{1-y^4}\,dy\,dx.$
    	\begin{parts}
    		\part[1] Sketch $\mathcal{D},$ the region of integration. \\
    		\includegraphics[width=2.5in]{GraphPaper}
    		\part[2] Calculate  $\ds \int_{0}^{1} \int_{0}^{\sqrt[4]{x}} \frac{1}{1-y^4}\,dy\,dx $ by switching the order of integration. \vfill \minisquare{}
    	\end{parts}
    	
\newpage  
\question[1] True/False. 
	\begin{parts}
		\part \TF{The two lines $\mathcal{L}_1: \mathbf{r}_1(t)= \langle 10-4t, 3+t, 4+t\rangle$ and $\mathcal{L}_2: \mathbf{r}_2(t)= \langle 6, mt, 9-t\rangle$
			intersect if $m=1.$	} \vfill
		\part \TF{The vectors $\mathbf{v}_1 = \langle 1, \lambda, -2\rangle$ and $ \mathbf{v}_2 = \la 4 \lambda, 3, 7\ra$   are orthogonal when $\lambda = 2.$}\vfill
		\part \TF{Consider the sphere $x^2 + y^2 + z^2 = 9.$  The plane $x=3$ is the tangent plane to the sphere at the point $(3,0,0).$}\vfill
		\part \TF{$\ds \int_1^2 \int _3^4 x^2e^y\, \mathrm{d}y \, \mathrm{d}x =  \left(\int_1^2 x^2\, \mathrm{d}x\right) \left( \int_3^4 e^y\, \mathrm{d}y \right)$ }\vfill
	\end{parts}
	

\question[1] The following equations are all results from theorems we learned this semester. \\

\centerline{
	\begin{minipage}{.3\textwidth}
		\centering
		$\ds\oint\limits_{\partial D} \vec{F} \cdot \text{d}\vec{r} = \iint\limits_D \left(\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right) \,\text{d}A$\\
		\vspace{.15in}
		{\large  I}
	\end{minipage}
	~~
	\begin{minipage}{.3\textwidth}
		\centering
		$\ds\int\limits_a^b \int\limits_c^d f ~\text{d}y \text{d}x = \int\limits_c^d\int\limits_a^b f \,\text{d}x \text{d}y$\\
		\vspace{.15in}
		{\large  II}
	\end{minipage}
	~~\begin{minipage}{.3\textwidth}
		\centering
		$\ds \oint\limits_{\partial \mathcal{S}} \vec{F} \cdot \text{d}\vec{r}=\iint\limits_{\mathcal{S}} \text{curl}(\vec{F})\cdot \text{d}\vec{S}$\\
		\vspace{.15in}
		{\large  III}
	\end{minipage}
}
\vspace{40pt}
\centerline{
	\begin{minipage}{.3\textwidth}
		\centering
		$\ds\iint\limits_{\partial{\mathcal{W}}} \vec{F}\cdot \text{d}\vec{S} = \iiint\limits_\mathcal{W} \mathrm{div}(\vec{F})\, \text{d}V$\\
		\vspace{.15in}
		{\large  IV}
	\end{minipage}
	~~
	\begin{minipage}{.3\textwidth}
		\centering
		$\ds \frac{\partial^2f}{\partial x\partial y}  =  \frac{\partial^2f}{\partial y\partial x} $\\
		\vspace{.15in}
		{\large  V}
	\end{minipage}
	~~
	\begin{minipage}{.3\textwidth}
		\centering
		$\ds \int\limits_C \nabla f \cdot\text{d}\vec{r}=f(Q)-f(P)$\\
		\vspace{.15in}
		{\large  VI}
	\end{minipage}
}
\vspace{20pt}
Match each of the following theorems with its corresponding equation above. 
\begin{parts}
	\part Divergence Theorem\\
	\begin{oneparcheckboxes} \choice I \choice II \choice III \choice IV \choice V \choice VI\end{oneparcheckboxes}\\ \\
	\part Green's Theorem \\
	\begin{oneparcheckboxes} \choice I \choice II \choice III \choice IV \choice V \choice VI\end{oneparcheckboxes}\\ \\
	\part Fundamental Theorem for Conservative Vector Fields \\
	\begin{oneparcheckboxes} \choice I \choice II \choice III \choice IV \choice V \choice VI\end{oneparcheckboxes}\\ \\
	\part Stokes' Theorem \\
	\begin{oneparcheckboxes} \choice I \choice II \choice III \choice IV \choice V \choice VI\end{oneparcheckboxes}
\end{parts}

\newpage
\question
A radioactive substance with strength $$P(x,y,z) = e^{-x^2-y^2-(z+100)^2}$$ is suddenly discharged. A person standing at the point $(1,1,-100)$ must move away, in the direction of maximum \underline{decrease} of radiation. Note that the gradient of $P$ is given by \[\nabla P(x,y,z)=\langle -2x e^{-x^2-y^2-(z+100)^2},-2ye^{-x^2-y^2-(z+100)^2},-2(z+100)e^{-x^2-y^2-(z+100)^2}\rangle.\]


\begin{parts}
	\part[1]  A person standing at the point $(1,1,-100)$ must move away, in the direction of maximum \underline{decrease} of radiation. What direction should he/she choose to move?  \vspace{0.1in}\\
	
	\begin{oneparcheckboxes} \choice $\langle 1,1,0 \rangle$ \choice  $\langle 2,3,0 \rangle$ \choice  $\langle  -1,-1,0\rangle$ \choice  $\langle  -2,-3,0\rangle$ \choice $\la 1,-1\ra$ \end{oneparcheckboxes} \vfill
	
	\part[1]  A person standing at the point $(1,1,-100)$ must move away, in the direction of maximum \underline{decrease} of radiation.  What is the maximum rate of \underline{decrease} in radiation? \vspace{.1in} \\
	\bigskip
	\begin{oneparcheckboxes} \choice $-\sqrt{2}$ \choice  $-e^{-3}\sqrt{13}$ \choice  $-2\sqrt{2}e^{-2}$ \choice  $\sqrt{2}$ \choice $0$ \end{oneparcheckboxes}\vfill
	
	
	\part[1]  The person standing at the point $(1,1,-100)$ decided to move in the direction $\langle 0,1,0 \rangle$.   What is the rate of change in radiation in this direction?
	\vspace{.1in}\\
	\begin{oneparcheckboxes} \choice $2e^{3}$ \choice $-2e^{-3}$ \choice $0$ \choice $-2e^{-2}$ \end{oneparcheckboxes}\vfill
\end{parts}


        \newpage


\question[2] Find the equation of the plane containing the points $P=(1,2,1)$, $Q=(3,2,-1)$, $R=(1,1,1)$.
\vfill
	\longrec{}

\question[2] Find the tangent line to the curve parametrized by \[\vec{r}(t) = \la t + \cos(t), t \mathrm{e}^t, \ln(1+t) \ra ~~,\quad t>-1\]
at the point where $t=0$. 
\vfill
	\longrec{}





\newpage


    
        \question[2]  Using the \emph{Divergence Theorem}, find the flux of the vector field $\vec{F}(x,y,z) = \la 3x, 4y, -2z\ra$ outwards across the surface of the box $\mathcal{W}=[0,1] \times[0,2] \times [-1,1]$.
        	\vfill
        	\shortrec{$\ds ~ \iint\limits_{\partial \mathcal{W}} \vec{F} \cdot \D \vec{S}=$ \hfill}


 
 \question[2] Given $\mathbf{u}=5\mathbf{i} + \mathbf{j} -3 \mathbf{k}$ and $\mathbf{v}= 4 \mathbf{i} +4 \mathbf{j}+2\mathbf{k},$ find the {\bf \underline{lengths}} $a$ and $b$ pictured below. \\
\includegraphics[width=2in]{Final_triangle} 

\vspace{30pt}
\framebox(60,45){~$a=$ \hfill} \hspace{1cm}
\framebox(60,45){~$b=$ \hfill} \\

 \newpage
	 \question[2] Let $\mathcal{S}$ be the cone given by $\{(x,y,z) \in \R^3 \mid z = \sqrt{x^2 + y^2} \text{ and } 0 \leq z \leq 1 \}$ and let the vector field $\vec{F}$ be given by $\vec{F}(x,y,z) = \la -y z, x z,  z^2 \ra.$
	 
	 	 Given that $\partial \mathcal{S}$ is parametrized by
	 		$\vec{r}(t) = \la \cos(t), \sin(t), 1 \ra, ~ 0 \leq t \leq 2 \pi,$ use \emph{Stokes' Theorem} to find the flux of $\mathrm{curl} (\vec{F})$ upwards across $\mathcal{S},$ that is, find $\ds \iint\limits_\mathcal{S} \mathrm{curl} (\vec{F}) \cdot \D \vec{S}.$
	 		\vfill
	 		\shortrec{$\ds ~\iint\limits_\mathcal{S} \mathrm{curl} (\vec{F}) \cdot \D \vec{S} =$ \hfill}




\question[2] Consider the surface $\mathcal{S}$ parametrized by $\vec{r}(u,v) = \la u+v, u-v, 2u + 3 v \ra$ where $0 \leq u \leq 1$ and $0 \leq v \leq 1$. Find the surface area of $\mathcal{S}$. 
	\vfill
	\minisquare{}
	



\newpage

\question[1] Consider the following quadratic surfaces in $\R^3$. 

	\centerline{
		\begin{minipage}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{Surfaces/y2equalsxplus1.png}\\
		{\large  I}
		\end{minipage}
		~~
		\begin{minipage}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{Surfaces/x2minusy2equalsz.png}\\
		{\large  II}
		\end{minipage}
		~~\begin{minipage}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{Surfaces/x2plusy2equalsz2plus1.png}\\
		{\large  III}
		\end{minipage}
		}
		\vspace{40pt}
	\centerline{
	\begin{minipage}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{Surfaces/x2plusy2equalsz2minus1.png}\\
		{\large  IV}
		\end{minipage}
		~~
		\begin{minipage}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{Surfaces/x2equalsy2plusz2.png}\\
		{\large  V}
		\end{minipage}
		~~
		\begin{minipage}{.3\textwidth}
		\centering
		\includegraphics[width=\textwidth]{Surfaces/ellipsoid.png}\\
		{\large  VI}
		\end{minipage}
		}
		\vspace{20pt}
Match each of the following equations with its corresponding graph above. 
\begin{parts}
	\part $z=x^2-y^2$\\
		\begin{oneparcheckboxes} \choice I \choice II \choice III \choice IV \choice V \choice VI\end{oneparcheckboxes}\\ \\
	\part $x^2 + y^2 = z^2 -1$ \\
		\begin{oneparcheckboxes} \choice I \choice II \choice III \choice IV \choice V \choice VI\end{oneparcheckboxes}\\ \\
	\part $z^2 = x^2+y^2$ \\
		\begin{oneparcheckboxes} \choice I \choice II \choice III \choice IV \choice V \choice VI\end{oneparcheckboxes}\\ \\
	\part $z = x^2-1$ \\
		\begin{oneparcheckboxes} \choice I \choice II \choice III \choice IV \choice V \choice VI\end{oneparcheckboxes}
\end{parts}

\newpage	
\question[2] Let $\mathcal{E}$ be the region in $\R^3$ where $1 \leq x^2+y^2 + z^2 \leq 4$ and $x,y,z \geq 0$ (i.e., in the first octant). Evaluate the integral \[ \iiint\limits_\mathcal{E} {(x^2+y^2+z^2)^{3/2}} \D V.\]
\vfill
\hfill \framebox(80,80){}
 \begin{center}
 	\gradetable[h]
 \end{center} 
\end{questions}
\end{document}
