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\noindent
{\it Practice Final}
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\noindent
I.  
\item{A)}  (10) Give a possible formula for the function plotted
here:
\vglue1.3truein
\item{B)} (10) Does the function from part A have an inverse function
if $x$ is restricted to the interval $0 \le x \le 4$?  If
so, say why and give the domain of the inverse function.  If not,
say why not.
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\noindent
II.  (15) The function plotted here is a rational function 
with numerator and denominator both polynomials of degree 2 or 
less.  Find a possible formula.
\vglue1.3truein
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\noindent
III.  (20) 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).
$$\vbox{
\halign{&\strut\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil&\vrule #&\hfil#\hfil &\vrule#&\hfil #\hfil\cr
$x$   &&$\ .2\ $&&$\ .4\ $&&$\ .6\ $&&$\ .8\ $&&$\ 1.0\ $\cr
\noalign{\hrule}
$f(x)\ $&&$\ 2.30\ $&&$\ 2.51\ $&&$\ 2.72\ $&&$\ 2.93\ $&&$\ 3.14\ $\cr
\noalign{\hrule}
$g(x)\ $&&$\ 2.30\ $&&$\ 2.42\ $&&$\ 2.55\ $&&$\ 2.69\ $&&$\ 2.84\ $\cr
}}$$
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\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).
\item{A)} (5) What is the temperature of the chocolate at $t = 0$?
\item{B)} (10) How fast is the temperature changing at $t = 10$ (give
units).
\item{C)} (10) How long does it take for the temperature to 
reach $100^\circ$F?
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\noindent
V.
\item{A)} (10) Using the limit definition, compute $f'(x)$ for 
$f(x) = {1\over x+1}$.
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\noindent
Using appropriate derivative rules, compute the derivatives
of the following functions:
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\item{B)} (10) $g(x) = 3x^4 + {3\over \sqrt{x}} + 2 {\root 3\of x} + \pi^2$.
\medskip
\item{C)} (10) $g(x) = {\tan(x) + x\over \cos(2x)}$
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\item{D)} (10) $i(x) = 3 \ln(x^2+3^x)$
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\item{E)} (10) $j(x) = \arcsin(12x+2)$
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\noindent
VI.  (10) The following graph shows $y = f(x)$.
Make a qualitative sketch of the graph $y = f'(x)$.
\vglue 1.5truein
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\noindent
VII.  The following graph shows the {\it derivative}
$f'(x)$ for some function $f(x)$ defined on $-3/2\le x \le 3/2$.
Using the graph, {\it estimate}
\vglue 1.5truein
\item{A)} (5) The intervals on which $f$ is increasing.
\item{B)} (10) The critical points of $f(x)$.  Say what the type
(local max, local min, neither) of each critical point is.
\item{C)} (7.5) The intervals on which $y = f(x)$ is concave up. 
\item{D)} (7.5) The locations of the inflection points of $y = f(x)$.
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\noindent
VIII. (20) 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 60 ft.  What dimensions
will maximize the total area of the window?
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\noindent
IX.  (10)  Compute the LHS and RHS approximations 
to $\int_1^4 x^2 + x - 3\ dx$ using a partition
with $n = 6$ subdivisions.
\bye
