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\course \hfill Russell Brown \\
Final Exam   \hfill  17 December 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.
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Name \rule {2in}{0.01in}

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Section \rule{ 1in }{0.01in}

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\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}


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\newpage 

\begin{enumerate}

\item \begin{enumerate}
\item State the definition of the derivative.
\item Use the definition of the derivative to find the $f'(2)$ for
$f(x) = 1/x$. 

      \end{enumerate}

\newpage

\item \begin{enumerate} \item Compute the limits.
\begin{enumerate}
\item $\displaystyle{\lim_{x\rightarrow \infty} x e^{-x}}$
\item $\displaystyle{\lim_{x\rightarrow 0} x^2 e^x}$
\item $\displaystyle{ \lim_{x\rightarrow 1} \frac {x^2-1}{x^2-2x+1}}$
\end{enumerate}

\vfill

\item Use that $ \sqrt{100} =10$ and a linear approximation to the
square root function to estimate $\sqrt{98}$.

\vfill


      \end{enumerate}

\newpage

\item A red balloon and an observer are 100 feet from each other on
level ground. The balloon is released and rises straight up at 8
feet/second. Find the rate of change of the balloon's angle of elevation
from the observer when the balloon is 120 feet above the ground. 
Is the angle of elevation increasing or decreasing?

\newpage

\item A function $f(x)$ is defined on $[0,10]$. Below are graphs of
the first and second derivatives of $f$.
\begin{enumerate}
\item Give all local maxima and minima
 for the function $f$. Explain
why each point is a local minimum or local maximum.

\item Give the intervals where $f$ is increasing and the intervals
where $f$ is decreasing. 

\item Give the intervals where the graph of $f$ is concave up and the
intervals where the graph  of $f$ is concave down.

\item Sketch a graph of $f$ which incorporates the information from
(a)-(c).

\end{enumerate}

\newpage

\item A box is to be built with a square base and no top. The volume
of the box will 
be 120 meter${}^3$. What are the side lengths for the box with the
smallest possible surface area?  Be sure to explain how you know you
have found the minimum. 

\newpage

\item \begin{enumerate} \item Below is the graph of a function
$f$. Beginning at the value $x_0$ marked on the $x$-axis, 
illustrate one step of Newton's method to  find $x_1$,  the next estimate to
the root of $f(x)=0$. 

\vfill

\item Show how to use Newton's method to find the negative root of the
equation $x^4-
\cos x = 0$. In your answer you should give an appropriate starting
point $x_0$, give the formula for finding $x_{n+1}$ in terms of $x_n$
and give approximate numerical values for $x_1, \ x_2, \
x_3\dots$ Your final approximate root should be correct to at least four
decimal places.


\vfill
      \end{enumerate}


\newpage


\item A bicycle is  travelling on a straight road. At time $ t=0$, the
brakes are applied to give a constant deceleration of 6
feet/second${}^2$.  One second after applying the brakes, the
bicycle's  velocity is 20 feet/second. 

\begin{enumerate}
\item Use antidifferentiation to find a function $h(t)$ which gives
the distance travelled from the time  the brakes were applied to time
$t$. 
\item  How many seconds after applying the brakes  does the bike come
to a stop? 
\item How far does the bike travel between the time the brakes are
applied and the time it stops?
\end{enumerate}


\end{enumerate}
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