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\lhead{\bf Math 126}      		%Course Number
\chead{\bf Final Examination}	      %Exam Name
\rhead{\bf Winter 2024}			%Term
\cfoot{}
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%
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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&12 &\\
\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&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 2024}
\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 (6 points per part)}
\begin{enumerate}
\item Find parametric equations for the line of intersection
of the planes $2x-y+3z+4=0$ and $-x+y-z=0$. %AND find the intersection point of this line with the plane $x+2y+z=12$. 
\vfill

\hfill Parametric equations: \makebox[4in]{\hrulefill}
\vskip .1 in

\item Find the intersection point of the line you found in part (a) with the plane $x+2y+z=12$.

\vfill

\hfill Intersection point: $(x,y,z)$ = \makebox[2in]{\hrulefill}
\vskip .5 in

%\item Find the equation of the plane that goes through the two points $P(2,-1,0)$ and $Q(4, 0, 3)$ and is parallel to the line $x=3t$, $y=1-t$, $z=4+t$.
%\vfill
%
%\hfill plane equation: \makebox[3in]{\hrulefill}
%\vskip .5 in
\end{enumerate}

\newpage

% Chapter 12.6-13 stuff
\item {\em (4 points per part)} Consider the curve given by the position function $\mathbf{r}_1(t) = \langle \ln(t), t^2+5, 3t \rangle$ for $t > 0$.
\begin{enumerate}
  \item Find the curvature at $t=1$.
  \vfill
	\hfill $\kappa(1)$ = \makebox[2in]{\hrulefill}
  \item Find the $(x,y,z)$ point on the curve at which the tangent line is orthogonal to the plane $x+8y+6z=7$.
  \vfill
	\hfill $(x,y,z)$ = \makebox[2in]{\hrulefill}
  \item Find the {\bf acute} angle of intersection of $\mathbf{r}_1(t)$ with the curve $\mathbf{r}_2(u) = \langle 0, u^{3}-u,\sqrt{u^2+5} \rangle$.\\[.1 in]
  (Give your answer rounded to the nearest degree.)
	\vfill
	\hfill Angle $\approx$ \makebox[2in]{\hrulefill} degrees
\vskip .5 in

\end{enumerate}


\newpage

% Chapter 14
\item
{\em (6 points per part)} Consider the function $f(x, y)=x^2+xy^2-y$.

\begin{enumerate}
\item
Find all the saddle point(s) of $f$ in $\mathbb{R}^2$. Justify your answer. 
\vspace{6cm}


\hfill Saddle point(s): $\rule[-0.2cm]{8cm}{0.15mm}$\\

\item
Let $D$ be the closed region bounded by $x=y^2$ and $x=1$. Find the absolute maximum and absolute minimum values of the function $f$ on the region $D$, and the points where these extrema occur. 
 
 \vfill
\hfill Absolute (global) maximum on $D$:  $f(\rule[-0.2cm]{1cm}{0.15mm},\rule[-0.2cm]{1cm}{0.15mm})= \rule[-0.2cm]{1.2cm}{0.15mm}$
\vskip .2 in

\hfill Absolute (global) minimum on $D$: $f(\rule[-0.2cm]{1cm}{0.15mm},\rule[-0.2cm]{1cm}{0.15mm})= \rule[-0.2cm]{1.2cm}{0.15mm}$
\vskip .5 in
 \end{enumerate}

\newpage

\item
({\it 12 points}) Consider the surface $x^2-(y+1)^2-3z^2=2$.\\[.1 in] 
%\begin{enumerate}
%\item 
%Find the equation of the tangent plane to the surface at $P(3, 1, 1)$.
%\vspace{6cm}
%\hfill tangent plane: $\rule[-0.2cm]{6cm}{0.15mm}$\\
%\item 
Find the point(s) on the surface that are closest to the point $A(0, 3, 1)$.\\[.1 in]
For full credit, you must show work to justify your answer.


\vfill

\hfill Closest point(s): $\rule[-0.2cm]{6cm}{0.15mm}$\\
\vskip .2 in
%\end{enumerate}

\newpage

% Chapter 15 stuff

% Two volume problems
\item {\em (6 points per part)} The parts of this question are not related.
\begin{enumerate}

\item The region $D$ is bounded above by the curve $\displaystyle{y=\frac{1}{x}}$, below by the line $\displaystyle{y=\frac{1}{2}(x-1)}$ and to left by the line $x=1$.\\[.1 in]
Set up the integral $\displaystyle{\iint_D f(x,y)\,dA}$ using \textbf{two different orders of integration}.\\[.1 in]
You may have to split the region.

\vspace {3.2in}

\hfill One way: \makebox[3in]{\hrulefill}
\vskip .4 in
\hfill Another way: \makebox[3in]{\hrulefill}\\[.1 in]

\item Evaluate $\displaystyle{\int_0^1\,\int_{y^4}^{\sqrt{y}}}\,\left(\sqrt{x}-y^2\right)\,dx\,dy$.
 
  \end{enumerate}
\vfill

\hfill Answer: \makebox[3in]{\hrulefill}\\[.1 in]
\vskip .2 in

\newpage
	
% Volume
	\item {\em (12 points)} Find the volume of the region that is inside the ellipsoid $x^2+y^2+4z^2=25$ and above the cone $z=\sqrt{x^2+y^2}$.

\vfill

\hfill Volume = \makebox[3in]{\hrulefill}\\[.1 in]
\vskip .2 in

\newpage

% Taylor Poly

\item Let $f(x)=\sqrt{(2x+1)^3}$. 
\begin{enumerate}
\item {\em (6 points)} Find the second Taylor polynomial $T_2(x)$ for $f$ based at $b=4$.

\vfill

\hfill  $T_2(x) = $ \makebox[4in]{\hrulefill}\\[.1 in]

\item {\em (3 points)} Use your answer to part (a) to approximate $\sqrt{9.1^3}$.

\vskip 1.5 in

\hfill $\sqrt{9.1^3} \approx$ \makebox[3in]{\hrulefill}\\[.1 in]

\item {\em (5 points)} Use Taylor's inequality to find an upper bound (as sharp as possible) for the error in your approximation in part (b).\\[.1 in]

\vfill

\hfill Upper bound: \makebox[3in]{\hrulefill}\\[.1 in]

\end{enumerate}

\vskip .5 in

\newpage

\item
%For this problem, you may use the following basic Taylor series:
%$$\displaystyle\frac{1}{1-x}=\displaystyle\sum\limits_{k=0}^{\infty}x^k,\ \ \ e^x=\displaystyle\sum\limits_{k=0}^{\infty}\displaystyle\frac{x^k}{k!},\ \ \ \sin x=\displaystyle\sum\limits_{k=0}^{\infty}\displaystyle\frac{(-1)^k x^{2k+1}}{(2k+1)!},\ \ \ \cos x=\displaystyle\sum\limits_{k=0}^{\infty}\displaystyle\frac{(-1)^k x^{2k}}{(2k)!}$$

For this problem, let $f(x) = x\sin(x^4)+\dfrac{x^2}{4-x^3}$.

\begin{enumerate}

\item {\em (6 points)} Find $T_{10}(x)$, the 10th Taylor polynomial for $f$ based at $b=0$.

\vspace{8cm}

\hfill $T_{10}(x) =$ \makebox[3in]{\hrulefill} 

\vskip .2 in

\item {\em (3 points)} Find the largest open interval on which the Taylor series for $f$ based at $b=0$ converges.

\vspace{1 in}

\hfill Interval: \makebox[3in]{\hrulefill} 

\vskip .2 in

\item {\em (5 points)}
Find $f^{2024}(0)$. (That is, the 2024${\rm{th}}$ derivative 
of $f$ at 0.) Give an exact answer.
\vfill

\hfill $f^{(2024)}(0) =$ \makebox[3in]{\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}


