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\begin{document}

\centerline{Mathematics 131 -- Calculus for Physical and Life Sciences 1}
\centerline{Practice Final Examination}
\centerline{\it December 10, 2004}
\bigskip
\noindent
I.  
\begin{itemize}
\item{A)}  Give a possible formula for the function plotted
here:

\centerline{\psfig{figure=P1.ps,height=2.0in,width=2.0in,angle=-90}}

\item{B)} Does the function from part A have an inverse function
if $x$ is restricted to the interval $0 \le x \le 1$?  If
so, say why and give the domain of the inverse function.  If not,
say why not.
\end{itemize}
\bigskip
\noindent
II.  The function plotted here is a rational function 
with numerator and denominator both polynomials of degree 2 or 
less.  Find a possible formula.

\centerline{\psfig{figure=P2.ps,height=2.0in,width=3.0in,angle=-90}}

\bigskip
\noindent
III.  One of the functions tabulated below is approximately
linear and the other is approximately exponential.  Say which is which
and give a formula for either one (your choice).
$$\vbox{
\halign{&\strut\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil &\vrule#&\hfil #\hfil\cr
$x$   &&$\ .2\ $&&$\ .4\ $&&$\ .6\ $&&$\ .8\ $&&$\ 1.0\ $\cr
\noalign{\hrule}
$f(x)\ $&&$\ 2.30\ $&&$\ 2.51\ $&&$\ 2.72\ $&&$\ 2.93\ $&&$\ 3.14\ $\cr
\noalign{\hrule}
$g(x)\ $&&$\ 2.30\ $&&$\ 2.42\ $&&$\ 2.55\ $&&$\ 2.69\ $&&$\ 2.84\ $\cr
}}$$
\bigskip
\noindent
IV.  A cup of hot chocolate is set out on a counter at $t = 0$.  The 
temperature of the chocolate $t$ minutes later is
$70+80e^{-t/3}$ (in degrees F).
\begin{itemize}
\item{A)} What is the temperature of the chocolate at $t = 0$?
\item{B)} How fast is the temperature changing at $t = 10$ (give
units).
\item{C)} How long does it take for the temperature to 
reach $100^\circ$F?
\end{itemize}
\bigskip
\noindent
V.
\begin{itemize}
\item{A)} Using the limit definition, compute $f'(x)$ for 
$f(x) = {1\over x+1}$.
\medskip
\noindent
Using appropriate derivative rules, compute the derivatives
of the following functions:
\medskip
\item{B)} $g(x) = 3x^4 + {3\over \sqrt{x}} + 2 {\root 3\of x} + \pi^2$.
\medskip
\item{C)} $g(x) = {\tan(x) + x\over \cos(2x)}$
\medskip
\item{D)} $i(x) = 3 \ln(x^2+3^x)$
\medskip
\item{E)} $j(x) = \arcsin(12x+2)$
\end{itemize}
\bigskip
\noindent
VI.  
\begin{itemize}
\item{A)}  Give parametric equations for the circle
of radius 2 centered at $(0,0)$, traversed counterclockwise
with constant speed $1$.
\item{B)}  An object moves along the path given by your
parametric curve in part A.  Give a function of $t$ that
computes the object's distance from the point
$(1,3)$ at each time $t$.
\end{itemize}
\bigskip
\noindent
VII.  The following graph shows the {\it derivative}
$f'(x)$ for some function $f(x)$ defined on $-3/2\le x \le 3/2$.

\centerline{\psfig{figure=P3.ps,height=2.0in,width=3.0in,angle=-90}}

Using the graph, {\it estimate}
\begin{itemize}
\item{A)} The intervals on which $f$ is increasing.
\item{B)} The critical points of $f(x)$.  Say what the type
(local max, local min, neither) of each critical point is.
\item{C)} The intervals on which $y = f(x)$ is concave up. 
\item{D)} The locations of the inflection points of $y = f(x)$.
\end{itemize}
\bigskip
\noindent
VIII. {\it Note:  The actual exam will have only one of this
type of problem!} 
\begin{itemize}
\item{A)}  A window is to be made of a single
large piece of glass in the shape of a semicircle on top of a 
rectangle (the diameter of 
the semicircle is the same as the width of the rectangle).
The perimeter of the window is 30 ft.  What dimensions
will maximize the total area of the window (and the light
admitted)?
\item{B)}  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. 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.  Where along the 
shoreline should the pipeline hit land to minimize the costs
of construction?
\end{itemize}
\bigskip
\noindent
IX.  {\it Note:  The actual exam will have only one of this
type of problem!} 
\begin{itemize}
\item{A)}  A rocket rises vertically from a point on the ground
that is 100 m from an observer at ground level.  When the angle
of elevation of the rocket is $2\pi/3$, the angle of elevation
is increasing at a rate of $2\pi/15$ radians per second.  How
fast is the rocket moving at that time?
\item{B)} An oil spill from a tanker accident has happened in 
Pristine Bay.  The slick has the form of a (short) circular cylinder
floating on the water surface.
Oil-eating bacteria are being used to contain
the slick, and they are gobbling $5 {\rm ft}^3$ of oil per hour.
When the radius of the slick
is 500 ft, the thickness is .01 ft and the thickness is decreasing at a rate
of .001 ft/hour.  How is the radius of the slick changing at
that time?
\end{itemize}
\bigskip
\noindent
X.  This is the graph $y = f(x)$, made up of straight line segments
and a semicircle:

\centerline{\psfig{figure=P4.ps,height=2.0in,width=3.0in,angle=-90}}

\begin{itemize}
\item{A)} Use left- and right-hand sums with $\Delta x = 2$ to 
estimate $\int_{-5}^5 f(x)\ dx$.
\item{B)} Using the relation between the integral and area,
find the {\it exact} value of $\int_{-5}^5 f(x)\ dx$.
\end{itemize}
\end{document}
