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\lhead{\bf Math 126}      		%Course Number
\chead{\bf Final Examination}	      %Exam Name
\rhead{\bf Spring 2023}			%Term
\cfoot{}
\pagestyle{fancy}
%
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\begin{comment}
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\SetWatermarkScale{5}
\SetWatermarkText{DRAFT \#3}
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%

\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 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}
\setlength{\tabcolsep}{.2in}
\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&12 &\\
\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&12 &\\
\hline
7&14 &\\
\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, Spring 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 (4 points per part)} {\bf Parts (a), (b), and (c) are unrelated.}
\begin{enumerate}
\item The two lines $x=1+t$, $y=t$, $z=1-2t$ and $x = 1$, $y=4t$, $z=1-3t$ intersect at the point $(1,0,1)$.  Find the equation for the plane containing these two intersecting lines.
\vfill
\hfill equation for plane: \makebox[3in]{\hrulefill}
\vskip .2 in

\item Find parameteric equations for the line of intersection of the two planes $x+y+z=1$ and $2x+y-3z=-3$. 
\vfill
\hfill equations for line: \makebox[4in]{\hrulefill} 
\vskip .2 in

\item Find all points of intersection of the line through $(0,0,1)$ and $(3,4,1)$ and the paraboloid $100z = x^2+y^2$.
\vfill
\hfill intersection point(s): $(x,y,z)$ = \makebox[3in]{\hrulefill}
\vskip .3 in
\end{enumerate}

\pagebreak

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

\begin{enumerate}
\item Find a vector function for the curve of intersection between the surface $5x^2+y^2-z^2=4$ and the plane $z=x$.\\

\vfill

\hfill vector function: \makebox[3in]{\hrulefill}
\vskip .2 in

% Chapter 13 stuff
\item Find the curvature of $\mathbf{r}(t) = \langle t^3,t^2-1,3t+7\rangle$ at the point $(-8,3,1)$.
\vfill

\hfill curvature: \makebox[1.5in]{\hrulefill}
\vskip .3 in

\end{enumerate}

\newpage

% Chapter 14

\item
{\em (6 points per part)}  For parts (a) and (b), let $f(x,y)=x^2y^2-4x^2-y^2$.

\begin{enumerate}
\item
Find all the saddle points of $f(x,y)$.
\vfill

\hfill saddle points: \makebox[2.5in]{\hrulefill}
\vskip .2 in


\item
Find the equation of the tangent plane to the surface $z=f(x,y)$ at $x=2, y=1$ and use it to approximate $f(2.1, 0.8)$.
\vfill

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

\vskip .3 in

\end{enumerate}

\newpage

\item
{\em (12 points)}  Find the absolute maximum and absolute minimum values of the function 
$$f(x,y)=x^2-y^2+2y$$
on the region enclosed by $x=0$, $y=4$ and $y=x^2$.\\

\vfill

\hfill absolute extrema: \makebox[3 in]{\hrulefill}

\vskip .3 in

\newpage



% Chapter 15 stuff

\item {\em (12 points)} Reverse the order of integration and evaluate
$$\int_0^2\int_{x/2}^1 \frac{x}{y^3+1} \, dydx.$$
\vfill
\hfill answer = \makebox[1.5in]{\hrulefill}

\vskip .3 in

\newpage

\item {\em (12 points)} Let $R$ be the region inside the circle $x^2+y^2=4y$, outside the circle $x^2+y^2 = 8$, and in the first quadrant (shown below).  Evaluate
$$\iint\limits_{R} \frac{x}{x^2+y^2} \, dA$$.
\hfill \includegraphics[width=1.5in]{sp23finalpic.jpg} 
\vfill

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

\vskip .3 in

\newpage

% Taylor Poly

\item For parts (a)--(c), let $f(x)=\ln(2x-1)$.\\
\begin{enumerate}
\item {\em (5 points)}
 Find the second Taylor polynomial, $T_2(x)$, for $f(x)$ based at $b=1$.
\vspace{6cm}

\hfill $T_2(x) =$ \makebox[3in]{\hrulefill} 
\vskip .2 in

\item {\em (4 points)} Use your answer to part (a) to approximate $\ln(1.1)$.
\vspace{3 cm}

\hfill $\ln(1.1) \approx$ \makebox[1.5in]{\hrulefill} 
\vskip .2 in

\item {\em (5 points)} Find an upper bound (as sharp as possible) on the error for your answer from part (b).\\[.1 in]
\vfill

\hfill Error bound: \makebox[1.5in]{\hrulefill} 
\vskip .3 in

\end{enumerate}

\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)}
Find the Taylor series for $f(x)=\dsps\int_0^x e^{2t^2}\,dt$ based at $b=0$. Express your answer using $\sum$--notation.\\
\vspace{8cm}

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

\item {\em (3 points)}
Find the open interval of convergence for the series you found in (a).\\
\vspace{2.5cm}

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

\item {\em (5 points)}
Find $f^{(2023)}(0)$, i.e. the 2023$^{\text{rd}}$ derivative of $f$ at $0$.
\vfill

\hfill $f^{(2023)}(0) =$ \makebox[2in]{\hrulefill} 
\vskip .3 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}


