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\lhead{\bf Math 126}      		%Course Number
\chead{\bf Final Examination}	      %Exam Name
\rhead{\bf Winter 2023}			%Term
\cfoot{}
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\begin{comment}
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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}
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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&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, Winter 2023}
\chead{\bf Final Examination}
\rhead{\it Page \thepage~of 8}		%Number of pages
\setcounter{page}{1}
\begin{enumerate}

%% Zhixu stuff
\item
({\it 4 points per part}) For parts (a)--(c), consider the plane $\mathcal{P}: 2x+y+3z=2$ and the line $L: \ x=1-4t,\  y=2t, \ z=5+t$.

\begin{enumerate}
\item
Find the acute angle between the line $L$ and the line $y=2, z=6$.


\vspace{5.5cm}

\item
Find the equation of the plane that contains the line $L$ and is perpendicular to the plane $\mathcal{P}$.  



\vspace{7cm}






\item
Find two points $A$ and $B$ on the plane $\mathcal{P}$ such that the vector $\overrightarrow{AB}$ is perpendicular to the line $L$.
\vspace{8cm}


\end{enumerate}

\pagebreak

% Chapter 13
\item
({\it 4 points per part}) For parts (a)--(c), a particle is moving along a space curve with  initial position vector ${\bf r}(0)=2{\bf i}+{\bf j}-{\bf k}$. The velocity vector of the particle is
$${\bf v}(t)=3\cos(t)\,{\bf i}+\,4t\,{\bf j}+\sin(t)\,{\bf k}\,$$


\begin{enumerate}
\item
Find the position vector function ${\bf r}(t)$.
\vspace{7.5cm}

\item
Find the equation of a surface that contains the space curve ${\bf r}(t)$. 

\vspace{4cm}

\item
Find the curvature of the space curve ${\bf r}(t)$ at $t=\pi$.
\vspace{7cm}


\end{enumerate}

\newpage

% Chapter 14

\item ({\it 12 points}) Let $f$ be the function given by:
\begin{equation*}
f(x,y) = x^2-y^2-2xy+4y,
\end{equation*}
and let $A$, $B$, and $C$ be the points $(0,0)$, $(0,4)$ and $(4,0)$. Find the global minimum and maximum of $f$ in the filled triangle $ABC$.\\
 
\vfill

\newpage

\item ({\it 12 points}) Find a point on the surface of equation $z = x^2 - y^2$ whose tangent plane is perpendicular to the line of equation
\begin{align*}
x &= t  \\ y &= 3t \\ z &= -2t.
\end{align*}

\newpage

% Chapter 15
\item ({\it 12 points}) Find the volume of the solid 
below the surface 
$$z=\frac{2x+y+1}{\sqrt{x^2+y^2}}$$ 
and above the region in the $xy$-plane
$$D=\{(x,y)\mid x\geq 0, y\geq 0, 1\leq x^2+y^2\leq 4\}.$$

\newpage

\item ({\it 12 points}) Evaluate the integral
$$\int_{0}^{1} \int_{\arcsin y}^{\pi/2} 
\cos x \sqrt{1+\cos^2 x} \; dx\; dy.$$

\newpage

% Taylor Poly

\item For parts (a)--(c), let $f(x)=\displaystyle\int_2^x \!\!\! \cos(\pi t^2)\,dt$.\\
\begin{enumerate}
\item {\em (5 points)}
 Find the second Taylor polynomial, $T_2(x)$, for $f(x)$ based at $b=2$.
\vspace{6cm}

\item {\em (3 points)} Use your answer to part (a) to approximate $\displaystyle\int_2^{2.1} \!\!\! \cos(\pi t^2)\,dt$.
\vspace{3 cm}

\item {\em (6 points)} Find an upper bound on the error for your answer from part (b).\\[.1 in]
{\footnotesize (Note: For full credit, you do not need to find a tight upper bound, but you must justify your answer with Taylor's inequality.)}

\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 $f(x)=\displaystyle\frac{x^3}{1+x^4}-3x\sin(x^2)$ based at $b=0$. Express your answer using $\sum$--notation.\\
\vspace{8cm}


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


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

\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}


