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\centerline{Mathematics 133 -- Intensive Calculus for Science}
\centerline{Final Examination}
\centerline{December 13, 2005}
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\noindent
{\it Directions}
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Do problems I - VIII in the blue exam booklet.  You may use an
approved calculator on any of the following.  There are 200 possible points.
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I.  (20) One of the functions tabulated below is approximately
linear and the other is approximately exponential.  Say which is which
and give possible formulas for each.
$$\vbox{
\halign{&\strut\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil &\vrule#&\hfil #\hfil\cr
$x$   &&$\ .3\ $&&$\ .6\ $&&$\ .9\ $&&$\ 1.2\ $\cr
\noalign{\hrule}
$f(x)\ $&&$\ 4.12\ $&&$\ 3.74\ $&&$\ 3.39\ $&&$\ 3.07$\cr
\noalign{\hrule}
$g(x)\ $&&$\ 4.31\ $&&$\ 4.07\ $&&$\ 3.83\ $&&$\ 3.59$\cr
}}$$
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II.  
\item{A)}  (10) Give a possible formula for the function plotted
here:
\vglue 2.0truein
\item{B)} (10)  How can you tell that the function from part A 
has an inverse function if it is restricted to the interval 
$-1 \le x \le 1$?  What is the domain of the inverse function?
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\noindent
III.
\item{A)} (10) Using the limit definition, compute $f'(x)$ for 
$f(x) = 5x^2 - x + 3$.

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Using appropriate derivative rules, compute the derivatives
of the following functions:
\item{B)} (10) $g(x) = 3^x - \ln(x^3+x)$
\item{C)} (10) $h(x) = 3 {\root 5\of x} + {7\over {\root 3\of x}} + 3x^8$
\item{D)} (10) $i(x) = {\cos(x)\over \sin^2(x) - 1}$ 
\item{E)} (10) $j(x) = \arctan(3x+2)$
\item{F)} (10) Find ${dy\over dx}$ if $y^2 \sqrt{x}  - e^y = 3$.
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\noindent
IV.  The following graph shows the derivative $y = f'(x)$ for
some function $f(x)$.
\vglue 1.5truein
\item{A)} (10)  Over which intervals is $f$ increasing? decreasing?
\item{B)} (10)  Over which intervals is $y = f(x)$ concave up?  concave 
down?
\item{C)} (10)  Give a qualitative sketch of the graph $y = f''(x)$.
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\noindent
V.  A frozen dinner is taken out of a grocery store freezer
case at time $t = 0$ and placed in a shopping cart.  Its temperature
in degrees Fahrenheit $t$ minutes later is given by $H(t) = 70-50e^{-t/10}$.
\item{A)} (5) What is the temperature of the dinner at $t = 0$?
\item{B)} (10) How fast is the temperature changing at $t = 10$?
Give your answer including correct units.
\item{C)} (5) How long does it take for the temperature to 
reach $60^\circ$F?
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VI. (15)  Find the (global) maximum and minimum values of 
$f(x) = e^{-x} \sin(x)$ on $[0,\pi]$.  
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VII. A town wants to build a water pumping station on a small
island 2 miles from the straight shoreline of the lake that
forms its water reservoir, as in this figure:
\vglue 1.5truein
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The town is 6 miles along the shore
from the point nearest to the island.  The town must
be connected to the pumping station by a pipeline made up of 
straight line segments.  It costs \$3 million  
per mile to lay pipe under the water and \$2 million per mile
to lay pipe along the shoreline. 
\item{A)} (10)  Give the cost of constructing the pipeline
as a function of $x$.  
\item{B)} (10) Where along the 
shoreline should the pipeline hit land to minimize the costs
of construction?
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VIII. (15)  A block of dry ice (solid $CO_2$) is 
evaporating and losing volume at the rate of $10$ cm${}^3$/min. 
It has the shape of a cube at all times.  How 
fast are the sides of cube shrinking when the block has volume 
$216$ cm${}^3$?  Give the units of your answer.
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\centerline{\it Have a peaceful and joyous holiday season!} 
\bye
