\documentclass[12pt]{article}
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\lhead{\bf Math 126}      		%Course Number
\chead{\bf Final Examination}	      %Exam Name
\rhead{\bf Autumn 2022}			%Term
\cfoot{}
\pagestyle{fancy}
%
\newcommand{\diff}{\mathrm{d}}
\newcommand{\setR}{\mathbb{R}}

%vectors
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%\mathrm{Answer}&\mbox{:\hskip 1in}\\ & \\ &\\ & \\ &\end{array}$$}}

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\tikzstyle{printersafe}=[snake=snake,segment amplitude=0 pt]


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\newcommand{\ds}{\displaystyle\sum}
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\newcommand{\R}{\mathbf R}
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\newcommand{\totalpts}[1]{[{\bf #1 points total}] }
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\item[\bf \theproblemnumber.]}
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\begin{comment}
\usepackage{draftwatermark}
\SetWatermarkScale{5}
\SetWatermarkText{DRAFT \#3}
\SetWatermarkLightness{0.9}
\end{comment}



%

\begin{document}
%\thispagestyle{fancy}

\begin{tabular}{l@{\hspace{1in}}l}
Your Name & Your Signature\\
\blankbox{2.9in}{.45in} & \blankbox{2.9in}{.45in}
\\[.2in]
Student ID \# & \hfill Quiz Section\\
\blankbox{2in}{.33in}%
&
\hfill \blankbox{.33in}{.33in}\blankbox{.33in}{.33in}
\\[.2in]
Professor's Name & TA's Name\\
\blankbox{2.9in}{.45in} & \blankbox{2.9in}{.45in}
\end{tabular}

\begin{itemize}

\item CHECK that your exam contains 9 problems on 7 double-sided pages, including this cover sheet. There is one blank page at the front and three blank pages at the back reserved for scratch work or extra space.

\item
This exam is closed book. You may use one $8\frac12"\times11"$ sheet of notes and a TI-30X~IIS calculator.  Do not share notes or calculators. 

\item
Unless otherwise specified, you should give your answers in exact form. (For example, $\dsps \frac{\pi}4$ and $\sqrt{2}$ are in exact form and are preferable to their decimal approximations.)

\item
In order to receive full credit, you must show all of your work.

\item
\textbf{Place a box around \fbox{YOUR FINAL ANSWER} to each question.}

\item
If you need more room, use the back of the first page or either side of the last page {\bf and indicate that you have done so}. If you {\em still} need more room, ask for more scratch paper.

\item
\textbf{Do not write within 1 centimeter of the edge of the page.}

\item
Raise your hand if you have a question.

\end{itemize}
\vskip0.3in

{
\renewcommand{\baselinestretch}{1.8}
\setlength{\tabcolsep}{.2in}
\normalsize

\hspace{-5ex}\begin{tabular}{cc}
\setlength{\tabcolsep}{.2in}
\begin{tabular}{|c|c|c|}
\hline
Problem&Total Points&\parbox{.8in}{\hfil Score\hfil}\\
\hline
1&12 &\\
\hline
2&9 &\\
\hline
3&11&\\
\hline
4&12&\\
\hline
5&10&\\
\hline
\end{tabular} 
&
%\end{minipage}
%\begin{minipage}[t]{3.5in}
\setlength{\tabcolsep}{.2in}
\begin{tabular}{|c|c|c|}
\hline
Problem&Total Points&\parbox{.8in}{\hfil Score\hfil}\\
\hline
6&8&\\
\hline
7&14&\\
\hline
8&14&\\
\hline
9&10&\\
\hline
Total&100&\\
\hline
\end{tabular}
\end{tabular}

}
\newpage
\thispagestyle{empty}
You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see first page'' below a problem.

\newpage
%
%			START OF PROBLEMS
%
\lhead{\bf Math 126, Autumn 2022}
\chead{\bf Final Examination}
\rhead{\it Page \thepage~of 9}		%Number of pages
\setcounter{page}{1}
\begin{enumerate}

% Plane stuff

\item ({\em 4 points per part}) For parts (a)--(c), let $\mathcal{P}$ be the plane $8x-4y+z=3$.
\begin{enumerate}
\item Find the acute angle between $\mathcal{P}$ and the plane $x=4$.
\vskip 2.4 in
\item Write parametric equations for the line of intersection between $\mathcal{P}$ and the $xz$-plane.
\vskip 2.4 in
\item Find the point on the plane $\mathcal{P}$ that is closest to $(26,-7,10)$.
\vfill
\end{enumerate}


\newpage

% Vector functions
\item {\em (9 points)} The acceleration of a particle at time $t$ is given by the vector function \[\mathbf{a}(t) = \langle 2t,4,45\sqrt{t} \rangle \,\, \text{m/s}^2\text{.}\]
The particle is in the same position at time $t=1$ as it is at time $t=4$.\\[.1 in]
What is the {\bf speed} of the particle at time $t=0$?\\

\newpage

% Partial derivative stuff

\item {\em (11 points)} Let $f(x,y)= 2x^2-2xy+y^2-3y$.
 
 \begin{enumerate}
 \Part Find and classify the critical points of $f$. 
 \vskip 2.2 in
 \Part Find the global minimum and global maximum of $f$ on the square $0 \leq x \leq 4$, $0 \leq y \leq 4$.
 \vfill
  \Part You do not need to explain your work on this part.\\[.3 in]
Does $f$ have a global minimum on $\mathbf{R}^2$? \qquad Yes \qquad No\\[.3 in]
Does $f$ have a global maximum on $\mathbf{R}^2$? \qquad Yes \qquad No\\[.3 in]
\vskip .5 in
 \end{enumerate} 
% 
%\newpage
%
%\item
% Let $S$ be the surface given by $x^2-y+\frac{z^2}{4}=0$. 
% \begin{enumerate}
% \item Find the tangent plane to $S$ at the point $(2, -1, 8)$. 
% \vfill
% \item Find the absolute absolute minimum and absolute maximum of $f(x,y)=xz+y^2$ on the surface $S$, if they exist. If not, explain why.
%\vfill
% \end{enumerate}
 
\newpage

\item {\em (6 points per part)} Evaluate the following double integrals:
\begin{enumerate}
\item $\displaystyle \int_0^1 \int_0^{\arctan(x)}\frac{1}{1-\tan(y)}\,dy\, dx$
\vskip 3.5 in
\item $\displaystyle \int\int_D \cos(\sqrt{x^2+y^2}) dA$, \par where $D$ is the region in the $xy$-plane given by $x^2+y^2\leq 4$, $x\geq 0$, and $y\geq 0$.
\vfill
\end{enumerate}

\newpage 

\item {\em (10 points)} In this problem, $a$ denotes a positive constant. A flat plate occupies the part of the disk $x^2+y^2\leq a^2$ that lies in the first quadrant. The plate has density $\rho(x,y)=xy^2$. The $y$-coordinate of the center of mass of the plate is $\bar{y} = 1$.\\[.1 in]
Find $a$.
\vfill
\newpage

%%%%%%%%%%%%%%%%%%%%%
\item {\em (2 points per part)} Match each of the following area integrals with one of the regions (A)-(F) below, corresponding to its domain of integration. You do not need to justify.  (Do NOT compute the integrals)
\vskip 0.2in


\begin{enumerate}
\item $\dsps \int_0^{10} \int_0^{\pi/4} r\,d\theta\, dr$ matches region: \minians 
\vskip 0.2in
\item $\dsps \int_0^{10} \int_0^y \,dx\, dy$ matches region: \minians 
\vskip 0.2in

\item $\dsps \int_0^{\pi/2} \int_{\left(\frac{20}{\pi}\right)\theta}^{10}\,\, r\,dr\, d\theta$ matches region:  \minians 
\vskip 0.2in

\item $\dsps \int_0^{10} \int_{x}^{10} \,dy\, dx$ matches region: \minians 
\vskip 0.7in

\end{enumerate}

%[thick, domain=0:#1*pi, samples=200, smooth]
%  plot (xy polar cs:angle=\x r, radius={#2});

\begin{tikzpicture}
\foreach \xs/\ys/\l/\p in {
0/0/A/{plot [domain=0:90] (\x:\x/9)},
1/0/B/{(0,0) -- (10,0) arc (0:45:10) -- cycle},
2/0/C/{(0,10) arc (90:0:10) -- (0,0) plot [domain=0:90] (\x:\x/9)},
0/1/D/{(0,0) -- (10,0) -- (10,10) -- cycle},
1/1/E/{(10,0) -- (10,10) -- (0,10) -- cycle},
2/1/F/{(0,0) -- (10,10) -- (0,10) -- cycle}}
{
\begin{scope}[scale=.3,shift={(\xs*18,\ys*-23)}]
\draw [fill=black!50] \p;
\draw [line width=1.5pt,->] (0,-.1) -- (0,11) node[above] {$y$};
\draw [line width=1.5pt,->] (-.1,0) -- (11,0) node[right] {$x$};
\draw (10,.5) -- (10,-.5) node[below,scale=.8] {$10$};
\draw (.5,10) -- (-.5,10) node[left,scale=.8] {$10$};
\node [scale=2] at (5,12) {\textbf{\textsf{\l}}};
\end{scope}
}
\end{tikzpicture}

%
%\begin{center} \includegraphics[height=3.5in]{RegionsDoubleInt.JPG} \end{center}

\newpage

% Taylor Poly

\item
({\it 7 points per part}) For both part (a) and (b), let $f(x)=\sqrt{(x-1)^5}-\cos(x-2)$.\\
\begin{enumerate}
\item
 Find the second Taylor polynomial, $T_2(x)$, for $f(x)$ based at $b=2$.  %({\bf NOT at $b=0$!})
\vspace{10cm}

\item
 Use Taylor's inequality to find an upper bound (as sharp as possible) for $|f(x)-T_2(x)|$ on the interval $[1.8, 2.2]$.
\vspace{3cm}

\end{enumerate}
\pagebreak

\item {\em (14 points)}
For this problem, you may use the following basic Taylor series:
$$\displaystyle\frac{1}{1-x}=\displaystyle\sum\limits_{k=0}^{\infty}x^k,\ \ \ e^x=\displaystyle\sum\limits_{k=0}^{\infty}\displaystyle\frac{x^k}{k!},\ \ \ \sin x=\displaystyle\sum\limits_{k=0}^{\infty}\displaystyle\frac{(-1)^k x^{2k+1}}{(2k+1)!},\ \ \ \cos x=\displaystyle\sum\limits_{k=0}^{\infty}\displaystyle\frac{(-1)^k x^{2k}}{(2k)!}$$
\begin{enumerate}
\item
Find the Taylor series for $f(x)=\displaystyle\frac{x}{5+x^2}$ based at $b=0$. Express your answer using $\sum$--notation.\\
\vspace{6cm}


\item
Find the open interval of convergence for the series you found in (a).\\
\vspace{3cm}


\item
Use the Taylor series you found in (a) to find the Taylor series for $g(x)=\ln(5+x^2)$ based at $b=0$.\\


\end{enumerate}

\newpage

\item {\em (10 points)} Let $f(x,y)= 100- x^2-y^2+xy$.
\begin{enumerate}
\item Find a normal vector to the tangent plane to the surface $z = f(x,y)$ at the point $(x_0,y_0,f(x_0,y_0))$. (Your answer will depend on $x_0$ and $y_0$.)
\vskip 2 in
\item Find a point $P$ on the portion of the graph of $f$ above the region $-2 \leq x \leq 2$, $-2 \leq y \leq 2$ where the tangent plane at $P$ intersects the $xy$-plane at the largest possible (acute) angle.
\end{enumerate}

\end{enumerate}
\newpage
\thispagestyle{empty}
You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see last page'' below a problem.
\newpage
\thispagestyle{empty}
You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see last page'' below a problem.
\newpage
\thispagestyle{empty}
You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see last page'' below a problem.

\end{document}


