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\begin{document}

\noindent {\bf Math 107 Fall '12 \quad\quad\quad Exam 3 \hfill
\makebox[1.5 in]{Score: \hrulefill}\\

\noindent \makebox[3 in]{Name: \hrulefill} \hfill \makebox[3
in]{TA's Name: \hrulefill}}
\medskip \hrule\smallskip \noindent {\bf
Instructions:} You must show supporting work to receive full and
partial credits. No text book, notes, formula sheets allowed.
\smallskip\hrule
%\begin{center}
%\begin{tabular}[t]{|c|c|c|c|c|c|c|c|}
%\hline No.&\quad 1 \quad & \quad 2 \quad & \quad 3\quad &\quad
%4\quad & \quad 5 \quad &\quad 6\quad &
%\quad Total \quad\\ \hline score & & & & & & & \\
%\hline
%\end{tabular}
%\end{center}

\begin{enumerate}
\item[\bf 1(15pts)] (a) Find the first three nonzero terms for the
Taylor series of the function $f(x)=\sqrt{x}$ at the point $x=1$.
%\[
%\displaystyle{\sum_{k=0}^\infty\frac{(x+2)^k}{2n+1}}
%\displaystyle{\sum_{n=0}^\infty\frac{1}{2^n\sqrt{n+1}}(x+2)^n}
%\]

\vskip 3in

(b) Use the Taylor series $\ln (1+x)=\displaystyle{\sum_{n=1}^\infty
(-1)^{n-1}\frac{x^n}{n}}$ to find the Taylor series of $x\ln(1-2x)$
at $x=0$.

\vskip 2.5in

\item[\bf 2(10pts)] Which two of the three vectors are
perpendicular, $\vec u=\langle 1,2,3 \rangle$, $\vec v=\langle
1,0,-2 \rangle$, $\vec w=\langle 2,-1,0 \rangle$? Show work to
support your answer.


\vfill \hfill \makebox[1.5 in]{(\it Page \arabic{page}. Continue on
Next Page ... \rm )} \pagebreak

\item[\bf 3(15pts)] Use either the Lagrange Error Bound estimate or the Alternating
Series Error Bound estimate to determine the number of terms from
the Taylor series of $e^{x}$ that are needed to approximate the
value of $e^{-0.1}$ to 4 decimal places accuracy.


\vskip 5in

\item[\bf 4(10pts)] A freshly brewed cup of coffee of 200$^\circ$ F is
placed in a 70$^\circ$ F room. Use Newton's Law of Heating and
Cooling to write down a differential equation with initial condition
for the temperature $H$ of the coffee as a function of time in
minutes. {\bf Do Not Solve The Equations.}

\vskip 1.5in

%(b) If it takes 15 minutes for the coffee to cool to 150$^\circ$ F,
%then how much longer will it take for the coffee to cool to
%100$^\circ$ F? (You can use the fact that the solution to the
%equation is $H(t)=70+Ce^{-kt}$ for some parameter $k$ and constant
%$C$.)

\vfill \hfill \makebox[1.5 in]{(\it Page \arabic{page}. Continue on
Next Page ... \rm )} \pagebreak

\item[\bf 5(15pts)] Find the solution to the differential equation
$\displaystyle{\frac{dy}{dt}=\frac{2t+1}{e^y}}$ that also satisfies
the initial condition $y(0)=1$.

\vskip 6in

\item[\bf 6(10pts)] Which is traveling faster, a car whose velocity vector is $21\vec
i+35\vec j$, or a car whose velocity if $40\vec i$, assuming that
units are the same for both directions? Show work to support your
answer.


\vfill \hfill \makebox[1.5 in]{(\it Page \arabic{page}. Continue on
Next Page ... \rm )} \pagebreak

\item[\bf 7(15pts)] Let $\vec u=\langle 1,2,3 \rangle$ and
$\vec v=\langle -1,0,1 \rangle$.

(a) Find the angle $\theta$ between the two vectors.

\vskip 3in

(b) Find the projection, $\vec v_{\rm parallel}$, of vector $\vec v$
on vector $\vec u$.

\vskip 3in

\item[\bf 8(10pts)] Find an equation for the plane which contains
the point $(1,0,-2)$ and is perpendicular to the vector $\vec
n=\langle 3,2,1 \rangle$.


\vskip 2in


\end{enumerate}
\vfill \hrule
\medskip \noindent {\bf 2 Bonus Points}: Isaac Newton was born in:
(a) the 16th century, (b) the 17th century, (c) the 18th century,
(d) the 19th century. \hfill \makebox[1.5 in]{\hfill (\it Page
\arabic{page} ... The End\rm )}
\end{document}
