



\documentclass[12pt]{article}
\input{rmb_local}
\input{rmb_macros}
\pagestyle{empty}

\topmargin -.5in
\begin{document}

\begin{flushleft}



\course \hfill Russell Brown \\
Exam 2  \hfill 18 November 1997

\bigskip

\vfill

Answer all of the following questions.
Use the backs of the question papers for scratch paper.
Additional sheets are available if necessary.  No books
or notes may be used.  When answering these questions,
please be sure to 1) check answers when possible, 2)
clearly indicate your answer and the reasoning used to
arrive at that answer {\bf (unsupported answers may receive
NO credit)}, and 3) label all variables and equations.

If you use your calculator to solve an equation or produce a graph,
please indicate this on your test paper. Otherwise the answer may
receive no credit.
\vfill

Name \rule {2in}{0.01in}

\smallskip

Section \rule{ 1in }{0.01in}

\vfill

\begin{center}
\begin{tabular}{|r|c|r|}
\hline
Question & Score & Total \\
\hline \hline
1 && 20 \\[0.1in]
\hline
2 && 15 \\[0.1in]
\hline
3 && 10 \\[0.1in]
\hline
4 && 10 \\[0.1in]
\hline
5 && 15 \\[0.1in]
\hline
6 && 15 \\[0.1in]
\hline
7 && 15 \\[0.1in]
\hline \hline
Total && 100 \\[0.1in]
\hline  
\end{tabular}
\end{center}


\vfill

\newpage 

\begin{enumerate}


\item  For each of the following functions, find the derivative. Do
not simplify your 
answer.  
\begin{enumerate}
\item $   x^ 2 + \ln x $

\item $\displaystyle{\frac {1-t^2}{ 1+t^2}} $

\item  $ x\cos (1/x)$

\item $ {(x^2+1) } e^{2x}$


\item $x^{\cos x}$

\end{enumerate}


\newpage
\item A mass is attached to a spring and slides  without friction
along the $x$-axis. Its
position is given by the function 
$x(t) = 2 \cos(t)$ where $x(t)$ denotes its displacement in
centimeters from the 
origin at time $t$ seconds. 

\begin{enumerate}
\item For which $t$ in $[0,2\pi]$ is $x(t)\geq 0$. 
\item What are the units of $x'(t)$?
\item For which $t$ in $[0,2\pi]$ is the mass moving to the right?
\item What is the total distance travelled for $0\leq t \leq 2\pi$?
(Hint: The answer is not zero.)
\end{enumerate}

\newpage

\item Use the quotient rule and the derivatives of $\sin$ and $\cos$
to find $$
\frac d { dx} \cot x .$$
Simplify your answer. 

\newpage

\item Let $f(x) = \tan x $ for $ \pi/2 < x < 3\pi /2$. 
\begin{enumerate}
\item Sketch the graph of $f(x)$ and $f^{-1} (x)$.
\item Find $$
\frac d { dx} f^{-1} (x).
$$
Simplify in order to express your answer as a rational function of
$x$. 
\end{enumerate}

\newpage

\item Consider the parametric curve
$$
x(t) = t - \sin(\pi t /2), \ y(t)= t^2-t.
$$
\begin{enumerate}
\item Find the two tangent lines at $(x,y) = (0,0)$. 
\item Do these tangent lines cross at a right angle?
\end{enumerate}


\newpage
\item  Let $y(x)$ satisfy the equation $ x^2 + y^3 = xy$. 

\begin{enumerate} \item 
 Find the linear approximation to $y(x)$ at
 $ (x,y) = (2, -2)$. 
\item Use your answer to part a) to estimate the values of $y(x)$ when
 $x= 2.2$ and $x= 1.95$. 

\end{enumerate}


\newpage


\item Find the limits, if they exist.  Indicate how you obtain each
limit. Answers obtained by guessing based on calculated values will
not receive full credit.

\begin{enumerate}
\item $\displaystyle{ \lim _{ x\rightarrow 0} \frac {x^3} { \sin (x)}}$
\item $\displaystyle {\lim _{ x\rightarrow 0} \frac  { \cos x} { x}}$
\item $\displaystyle{ \lim _{x\rightarrow 0^+} ( 1+x) ^{3/x}}$

\end{enumerate}

\end{enumerate}
\end{flushleft}

\end{document}

 
