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\lhead{\bf Math 126}      		%Course Number
\chead{\bf Final Examination}	      %Exam Name
\rhead{\bf Autumn 2023}			%Term
\cfoot{}
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\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 8 problems on 6 double-sided pages, including this cover sheet. There is one blank page at the front and two 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

{
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\hspace{-5ex}\begin{tabular}{cc}
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\begin{tabular}{|c|c|c|}
\hline
Problem&Total Points&\parbox{.8in}{\hfil Score\hfil}\\
\hline
1&12 &\\
\hline
2&12 &\\
\hline
3&12 &\\
\hline
4&12 &\\
\hline
5&14 &\\
\hline
\end{tabular} 
&
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%\begin{minipage}[t]{3.5in}
\setlength{\tabcolsep}{.2in}
\begin{tabular}{|c|c|c|}
\hline
Problem&Total Points&\parbox{.8in}{\hfil Score\hfil}\\
\hline
6&12 &\\
\hline
7&12 &\\
\hline
8&14 &\\
\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 2023}
\chead{\bf Final Examination}
\rhead{\it Page \thepage~of 8}		%Number of pages
\setcounter{page}{1}

\begin{enumerate}

% Chapter 12.1-12.5 stuff
\item {\em (12 points)} Let ${\mathcal P}$ be the plane given by the equation
$$-x +y +z = 2.$$
The point $Q(2, 1, -1)$ is not on the plane ${\mathcal P}$. Let $L_1$ be the line through $Q$ and orthogonal to $\mathcal{P}$, and let $R$ be the point of intersection of $L_1$ and $\mathcal{P}$. Let $W$ be another point $(-1,3,1)$.\\[.1 in]
Find parametric equations of the line $L_2$ that passes through points
$W$ and $R$.

\vfill

\hfill Parametric equations: \makebox[3in]{\hrulefill}
\vskip .5 in

\newpage

% Chapter 12.6-13 stuff
\item {\em (6 points per part)} {\bf Parts (a) and (b) of this question are unrelated.}

\begin{enumerate}
\item Find the equation for the quadric surface consisting of all points that are equidistant to the plane 
$y=3$ and the point $(0, 1, 2)$. Then identify the quadric surface.

\vskip 2.8 in

\hfill Equation for surface: \makebox[3in]{\hrulefill}\\[.1 in]

\hfill Name of surface: \makebox[3in]{\hrulefill}\\[.05 in]
% Chapter 13 stuff
\item Let $P$ be the point where $\vr(t) = \left\langle t^2+t,t^3,\frac{-3}{2}t^2-2t\right\rangle$ intersects the plane $y=-8$.\\[.1 in]
Find the curvature of $\vr(t)$ at $P$.
\vfill

\hfill Curvature: \makebox[1.5in]{\hrulefill}
\vskip .5 in

\end{enumerate}

\newpage

% Chapter 14

\item {\em (12 points)} Find the equation of the plane tangent to the surface \[z=x^2y-\sqrt{3x-y}-\frac{x}{y}\]at the point $(6,2,65)$. {\bf Write your answer in the form $Ax+By+Cz=D$.}

\vfill

\hfill Tangent plane equation: \makebox[3in]{\hrulefill}

\vskip .5 in

\newpage

\item {\em (12 points)} Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and the vertex opposite the origin (i.e. the one marked $P$ in the picture below) on the surface $z = 20-x^2-y$.\\[.1 in]
{\em In order to receive full credit, you must show that your answer really is the maximum.}
\vspace{-.1 in}
\begin{flushright}
\begin{tikzpicture}[z={(0,1)},y={(1,0)},x={(-.5,0,-.4)}]
\draw [line width=3pt,->] (0,0) -- (3,0) node [below left] {$x$};
\draw [line width=3pt,->] (0,0) -- (0,3.5) node [right] {$y$};
\draw [line width=3pt,->] (0,0) -- (0,0,2) node [above] {$z$};
\draw [thick] (2,0,0) -- (2,3,0) -- (0,3,0) -- (0,3,1.5) -- (0,0,1.5) -- (2,0,1.5) -- cycle (2,3,0) -- (2,3,1.5) -- (0,3,1.5) (2,3,1.5) -- (2,0,1.5);
\node [circle,draw,fill,inner sep=2,label={$P$}] at (2,3,1.5) {};
\end{tikzpicture}
\end{flushright}

\vfill

\hfill Maximum volume: \makebox[1.5 in]{\hrulefill}

\vskip .5 in

\newpage

% Chapter 15 stuff

\item {\em (7 points per part)} {\bf Part (a) and (b) of this question are not related.}\\

\begin{enumerate}
\item
Compute $\displaystyle\int_{1}^{5}\displaystyle\int_{\sqrt{y-1}}^{2}\, \displaystyle\frac{x^2}{1+x^5}\,dx\,dy$ \ \ \ (Hint: reverse the order of integration)

\vfill

\hfill Answer = \makebox[1.5in]{\hrulefill}

\item
Set up (but {\bf do not evaluate}) an integral for the volume of the solid enclosed by the cylinder $y=x^2$ and the planes $y+z=4$ and $z=0$.
  
\vfill

\hfill Volume = {\begin{large}
$\displaystyle\int_{\ \ \frame{\rule{0pt}{0.8cm}\rule{1.5cm}{0pt}}}^{\ \frame{\rule{0pt}{0.8cm}\rule{1.5cm}{0pt}}} \displaystyle\int_{\ \ \frame{\rule{0pt}{0.8cm}\rule{1.5cm}{0pt}}}^{\ \frame{\rule{0pt}{0.8cm}\rule{1.5cm}{0pt}}}\ \frame{\rule{0pt}{1.2cm}\rule{4.5cm}{0pt}}\ d\,\frame{\rule{0pt}{0.8cm}\rule{1cm}{0pt}}\ d\,\frame{\rule{0pt}{0.8cm}\rule{1cm}{0pt}}$
\end{large}
}

 \end{enumerate}

\vskip .5 in

\newpage

\item {\em (6 points per part)} A lamina $D$ occupies the region {\bf in the first quadrant} that lies inside the circle $x^2+y^2=4$ and outside the circle $x^2+(y-2)^2=4$. 

The density at any point is $\rho(x, y)=\displaystyle\frac{x}{\sqrt{x^2+y^2}}$.

\begin{enumerate}
\item
Find the mass of $D$.\\[.1 in]
{\em (Hint: Set up and evaluate a double integral in polar coordinates.)}

\vfill

\hfill Mass = \makebox[1.5in]{\hrulefill}

\item
Find $\overline{y}$, which is the $y$-coordinate of the center of mass.

\vfill

\hfill $\overline{y}$ = \makebox[1.5in]{\hrulefill}

\end{enumerate}

\vskip .5 in

\newpage

% Taylor Poly

\item {\em (6 points per part)} Let $f(x)=\sin(x-1)+ x^2+ e^x$. 
\begin{enumerate}
\item Find the second Taylor polynomial $T_2(x)$ for $f$ based at $b=1$.

\vfill

\hfill $T_2(x)$ = \makebox[1.5in]{\hrulefill}\\[.1 in]

\item Find an upper bound for $|f(x)-T_2(x)|$ on the 
interval $\dsps\left[\frac{1}{2},\frac{3}{2}\right]$.\\[.1 in]
In order to receive credit, you must justify your answer.

\vfill

\hfill Upper bound = \makebox[1.5in]{\hrulefill}\\[.1 in]

%\item {\em (4 points)} Find a value of $a$ such that $0<a<\frac{1}{2}$ and 
%$|f(x)-T_2(x)|\leq 0.001$ for all $x$ on the interval
%$[1-a,1+a]$.
%
%\vfill
%
%\hfill $a$ = \makebox[1.5in]{\hrulefill}

\end{enumerate}

\vskip .5 in

\newpage

\item
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 {\em (6 points)}
Use the list of Basic Series to find the Taylor series 
for $f(x)=x^2\arctan(-3x)$ based at $b=0$. (Use $\Sigma$
notation.)

\vspace{8cm}

\hfill Taylor series: \makebox[3in]{\hrulefill} 
\vskip .2 in

\item {\em (3 points)}
Find the open interval on which the series in part (a) converges.\\
\vspace{2.5cm}

\hfill Interval of convergence: \makebox[2in]{\hrulefill} 
\vskip .2 in

\item {\em (5 points)}
Find $f^{2023}(0)$. (That is, the 2023${\rm{th}}$ derivative 
of $f$ at 0.) Give an exact answer.
\vfill

\hfill $f^{(2023)}(0) =$ \makebox[2in]{\hrulefill} 
\vskip .5 in


\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.

\end{document}


