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{\bf date:} the $17^{\rm th}$ of December, 1998 

{\bf place:} room 221 Hayes

{\bf time:} 8:00-10:00

{\bf 111 - Final}

This final contains 15 problems worth 10 points each. You may use only
a blank sheet of paper, a pencil, a rubber gum, a ruler and a small
calculator. You can use your calculator only to add, substract,
multiply or divide two numbers. This final is taken under the honor
code. \\

{\bf Name:}
\\

{\bf Recommendation}

{\it Never give a "solitary" answer without justifying it by previous
calculations or reasoning. } \\

$\ln.34=-1.0788$

$\ln1.6=0.47$

$\ln4=1.38$

$\ln4.5=1.50$

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{\bf Problems}\footnote{For some useful numerical values of $\ln$ see
the front page} \\

{\bf 1.} Find the equation and sketch the graph of the line through
$(2,0)$, with slope $5$.


{\bf 2.} Compute $\frac{d}{dP}(\root\of{1-3P})$. 

{\bf 3.} What is the slope of the graph of $f(x)=x^3-4x^2+6$ at $x=2$?
Write the equation of the tangent line to this curve at $x=2$.

{\bf 4.} A rectangular garden of area $75$ square feet is to be
surrounded on three sides by a brick wall costing \$$10$ per foot and
on one side by a fence costing \$$5$ per foot. Find the dimensions of
the garden such that the cost of materials is minimized.


{\bf 5.} Properties of a function $h$ are described next. Draw some
conclusions about the graph of the function. $$
h(-3)=4,\ h'(-3)=1,\ h''(-3)=0.  $$

{\bf 6.} Sketch the parabola $y=-2x^2+10x-10$.
Include its $x$- and $y$-intercepts.


{\bf 7.} Sketch the curve
$
y=x^3-3x^2-9x+7.
$

{\bf 8.} Differentiate the function
$
(4x-1)(3x+1)^4.
$

{\bf 9.} Find $\frac{dy}{dx}$ where $y$ is a function of $u$ such that
$\frac{dy}{du}=\frac{u}{u^2+1}$ and $u=\frac{5}{x}$.


{\bf 10.} Differentiate $y=e^{-2x}-2x$.

{\bf 11.} Differentiate the function $\ln(\frac{x}{x-3})$. 

{\bf 12.} The size of a certain insect population is given by
$P(t)=300e^{.01t}$, where $t$ is measured in days. At what time will
the population equal $1200$?

{\bf 13.} Sandals woven from strands of tree bark were found recently
in Fort Rock Creek Cave in Oregon. The bark contained $34$\% of the
level of $^{14}C$ found in living bark. Approximately how old are the
sandals? (Recall that the decay constant for $^{14}C$ is $.00012$.)

{\bf 14.} A parcel of land bought in 1985 for \$10,000 was worth
\$16,000 in 1990. If the land continues to appreciate at this rate, in
what year will it be worth \$45,000?

{\bf 15.} Determine $\int(x^2-x-1)dx$.\\ 

{\bf Good luck!}

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