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\lhead{\bf Math 126}      		%Course Number
\chead{\bf Final Examination}	      %Exam Name
\rhead{\bf Spring 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} 
&
%\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 2024}
\chead{\bf Final Examination}
\rhead{\it Page \thepage~of 8}		%Number of pages
\setcounter{page}{1}

\begin{enumerate}

% Chapter 12.5 stuff
\item {\em (12 points)} Find the equation for the plane that passes through the point $(1,2,4)$ and contains the line given by $x=2t$, $y=3-t$, $z=2+5t$.  And give the $x$-intercept of this plane.

\vfill
\hfill Plane Equation: \makebox[3.5in]{\hrulefill} \\

\hfill $x$-intercept: $(x,y,z)$ = \makebox[3.5in]{\hrulefill}
\vskip .4 in

\newpage

% Chapter 13 stuff:
\item {\em (12 points)} Let a curve $\gamma$ be the intersection of the plane $y+z=1$ and the cylinder
$x^{2}+y^{2}=1$.\\[.1 in]
Find the points where $\gamma$ has maximum curvature.
\vfill
\hfill Points: $(x,y,z) = $ \makebox[3.5in]{\hrulefill}
\vskip .4 in

\newpage

% Chapter 14
\item
{\em (6 points per part)} Parts (a) and (b), consider the surface implicitly defined by the equation\[z(\cos(z)+2)  =x^{2}-y^{2}.\]

\begin{enumerate}
\item Find $\dsps\frac{\partial z}{\partial x}$ and $\dsps\frac{\partial z}{\partial y}$.

\vfill

\hfill $\dsps\frac{\partial z}{\partial x} = $ \makebox[3in]{\hrulefill}
\vskip .2 in
\hfill $\dsps\frac{\partial z}{\partial y} = $ \makebox[3in]{\hrulefill}
\vskip .2 in

\item Give the linear approximation, i.e. tangent plane, to the surface at the point where $(x,y) = (1,1)$.  And use this approximation to estimate the $z$-coordinate of a point on the surface where $(x,y) = (1.01,0.98)$.
%\item Find the linearization of the function $f(  x,y)  $ at $(1,1)$, and use it to approximate $f(  1.01,0.98)$.

\vfill

\hfill Linearization: \makebox[4in]{\hrulefill}
\vskip .2 in
\hfill $z \approx$ \makebox[2in]{\hrulefill}

\end{enumerate}
\vskip .3 in

\newpage

\item {\em (12 points)} Find the dimensions of a rectangular tent of maximum volume such that the sum
of the lengths of its $8$ posts (above the ground) is $36$ m.\\[.1 in]
\underline{To receive full credit, you must justify that your answer is indeed the maximum.}
\vspace{-.2 in}
\begin{flushright}
\begin{tikzpicture}[z={(0,1)},y={(1,0)},x={(0,-.7,-.8)}]
\foreach \x in {0,1} \foreach \y in {0,2} \draw [line width=1pt,double,double distance=2pt,cap=rect] (\x,\y,0) -- (\x,\y,1.2);
\draw [line width=1pt,double,double distance=2pt,cap=round,join=round,shift={(0,0,1.2)}] (0,0,0) -- (1,0,0) -- (1,2,0) -- (0,2,0) -- cycle;
\foreach \x in {0,1} \foreach \y in {0,2} \node [scale=1.5] at (\x,\y,1.2) {$\bullet$};
\draw (0,1,1.2) node[above] {$y$} (.5,0,1.2) node[above left] {$x$} (1,2,.6) node[left] {$z$};

\end{tikzpicture}
\end{flushright}

\vfill

\hfill Dimensions: \makebox[3in]{\hrulefill}
\vskip .4 in

\newpage

% Chapter 15 stuff

% Two volume problems
\item {\em (6 points per part)} For the following problems, set up an iterated integral as described.\\[.1 in]
\underline{\bf You do not have to evaluate the integrals. Just set them up.}
\begin{enumerate}

\item Reverse the order of integration of $\displaystyle \int_0^2 \int_{2^{(3-x)}}^{8-3x}\sin(x^2)\,dy\,dx$.

\vspace {3.2in}

\hfill Integral: \makebox[4in]{\hrulefill}
\vskip .2 in

\item Write a double integral for the volume below the cone $z=\sqrt{x^2+y^2}$ and above the paraboloid $z=x^2-x+y^2$.\\[.1 in]
{\em (Hint: convert to polar!)}
 
  \end{enumerate}
\vfill

\hfill Integral: \makebox[4in]{\hrulefill}\\[.1 in]
\vskip .2 in

\newpage
	
% Volume
	\item {\em (12 points)} Let $\mathcal{L}$ be the lamina whose shape is the triangular region with vertices $(0,0)$, $(0,2)$, and $(1,2)$, and whose density at any point is proportional to its distance to the $y$-axis.\\[.1 in]
	Find the center of mass of $\mathcal{L}$.

\vfill

\hfill $(\overline{x},\overline{y}) =$ \makebox[3in]{\hrulefill}\\[.1 in]
\vskip .2 in

\newpage

% Taylor Poly

\item For this problem, let $\displaystyle g(x) = (3+x)^{3/2}$ and consider the Taylor polynomials based at $b=1$.
\begin{enumerate}
\item {\em (5 points)} Find the second Taylor polynomial, $T_2(x)$, for the function $g$ based at $b=1$.
\vfill
\hfill $T_2(x) = $ \makebox[3.5in]{\hrulefill}
\vskip .2 in

\item {\em (4 points)} Use $T_2(x)$ to approximate the value of $\sqrt{4.4^3}$
\vfill
\hfill $\sqrt{4.4^3} \approx $ \makebox[3.5in]{\hrulefill}
\vskip .2 in

\item {\em (5 points)} Use Taylor's inequality to find an upper bound for the error in your approximation in part (b).
\vfill
\hfill Upper bound: \makebox[3.5in]{\hrulefill}
\vskip .4 in

\end{enumerate}

\newpage

\item For this problem, let $\displaystyle f(x) = \frac{16}{4-x^2}+6\cos(2x)$
\begin{enumerate}
\item {\em (6 points)} Give the Taylor series for $f$ based at $b=0$ using one sigma sign.
\vfill
\vfill
\vfill
\hfill Taylor series: \makebox[3.5in]{\hrulefill}
\vskip .2 in

\item {\em (3 points)} Find the largest open interval on which this Taylor series converges.
\vfill
\hfill interval: \makebox[3.5in]{\hrulefill}
\vskip .2 in

\item {\em (5 points)} Find $T_3(x)$, the third Taylor polynomial for $f$ based at $b=0$, then estimate the value of $\displaystyle \int_0^1 f(x) \, dx$ by replacing $f(x)$ with $T_3(x)$. 
\vfill
\vfill
\hfill $\displaystyle \int_0^1 f(x) \, dx \approx$ \makebox[3.5in]{\hrulefill}
\vskip .4 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}


